In this paper, we present a general framework for the derivation of interesting finite combinatorial sums starting with certain classes of polynomial identities. The sums that can be derived involve products of binomial coefficients and also harmonic numbers and squared harmonic numbers. We apply the framework to discuss combinatorial sums associated with some prominent polynomial identities from the recent past.

Two kinds of novel generalizations of Nesbitt's inequality are explored in various cases regarding dimensions and parameters in this article. Some other cases are also discussed elaborately by using the semiconcave-semiconvex theorem. The general inequalities are then employed to deduce some alternate inequalities and mathematical competition questions. At last, a relation about Hurwitz-Lerch zeta functions is obtained.

Can you find an xy-equation that, when graphed, writes itself on the plane? This idea became internet-famous when a Wikipedia article on Tupper's self-referential formula went viral in 2012. Under scrutiny, the question has two flaws: it is meaningless (it depends on typography) and it is trivial (for reasons we will explain). We fix these flaws by formalizing the problem, and we give a very general solution using techniques from computability theory.

In this paper, we give a description of the self-injective dimension of string algebras and obtain a necessary and sufficient condition for a string algebra to be Gorenstein.

The OEIS sequence A051221 consists of nonnegative integers of the form 10^x - y^2. The known values are those less than or equal to 2000 with x <= 7, and it is conjectured that no new values in this range appear for x >= 8. In this paper, we give an elementary proof that this is indeed the case by using Pell-type equations.

We investigate forms of filter extension properties in the two-cardinal setting involving filters on $P_\kappa(\lambda)$. We generalize the filter games introduced by Holy and Schlicht in \cite{HolySchlicht:HierarchyRamseyLikeCardinals} to filters on $P_\kappa(\lambda)$ and show that the existence of a winning strategy for Player II in a game of a certain length can be used to characterize several large cardinal notions such as: $\lambda$-super/strongly compact cardinals, $\lambda$-completely ineffable cardinals, nearly $\lambda$-super/strongly compact cardinals, and various notions of generic super and strong compactness. We generalize a result of Nielson from \cite{NielsenWelch:games_and_Ramsey-like_cardinals} connecting the existence of a winning strategy for Player II in a game of finite length and two-cardinal indescribability. We generalize the result of \cite{ForMagZem} to construct a fine $\kappa$-complete precipitous ideal on $P_\kappa(\lambda)$ from a winning strategy for Player II in a game of length $\omega$. Finally, we improve Theorems 1.2 and 1.4 from \cite{ForMagZem} and partially answer questions Q.1 and Q.2 from \cite{ForMagZem}.

We introduce the notion of the weak Brehmer's condition and prove that the Cauchy transform for a representation of a right-angled Artin monoid is bounded under such conditions. As a result, we obtain the Poisson transform and $*$-regular dilation for a family of operators that satisfies the weak Brehmer's condition and the property (P). This generalizes Popescu's notion of Cauchy and Poisson transforms for commuting families of row contractions.

We look at the question of which distance-regular graphs are core-complete, meaning they are isomorphic to their own core or have a complete core. We build on Roberson's homomorphism matrix approach by which method he proved the Cameron-Kazanidis conjecture that strongly regular graphs are core-complete. We develop the theory of the homomorphism matrix for distance-regular graphs of diameter $d$. We derive necessary conditions on the cosines of a distance-regular graph for it to admit an endomorphism into a subgraph of smaller diameter $e

In this paper, we show a physics-free derivation of a Landau-Zener type integral introduced by Kholodenko and Silagadze.

Given $k$ graphs $G_{1}, \ldots, G_{k}$, their intersection is the graph $(\cap_{i\in [k]}V(G_{i}), \cap_{i\in [k]}E(G_{i}))$. Given $k$ graph classes $\mathcal{G}_{1}, \ldots , \mathcal{G}_{k}$, we call the class $\{G: \forall i \in[k], \exists G_{i} \in \mathcal{G}_{i} \text{ such that } G=G_{1}\cap \ldots \cap G_{k}\}$ the graph-intersection of $\mathcal{G}_{1}, \ldots , \mathcal{G}_{k}$. The main motivation for the work presented in this paper is to try to understand under which conditions graph-intersection preserves $\chi$-boundedness. We consider the following two questions: Which graph classes have the property that their graph-intersection with every $\chi$-bounded class of graphs is $\chi$-bounded? We call such a class intersectionwise $\chi$-guarding. We prove that classes of graphs which admit a certain kind of decomposition are intersectionwise $\chi$-guarding. We provide necessary conditions that a finite set of graphs $\mathcal{H}$ should satisfy if the class of $\mathcal{H}$-free graphs is intersectionwise $\chi$-guarding, and we characterize the intersectionwise $\chi$-guarding classes which are defined by a single forbidden induced subgraph. Which graph classes have the property that, for every positive integer $k$, their $k$-fold graph-intersection is $\chi$-bounded? We call such a class intersectionwise self-$\chi$-guarding. We study intersectionwise self-$\chi$-guarding classes which are defined by a single forbidden induced subgraph, and we prove a result which allows us construct intersectionwise self-$\chi$-guarding classes from known intersectionwise $\chi$-guarding classes.

It is proved that the Minkowski question-mark function comes from the K-theory of Cuntz-Pimsner algebras. We apply this result to calculate the action of Frobenius endomorphism at the infinite prime. Such a problem was raised by Serre and Deninger in the theory of local factors of zeta functions of projective varieties.

This paper introduces an effective framework for designing memoryless dissipative full-state feedbacks for general linear delay systems via the Krasovski\u{i} functional (KF) approach, where an unlimited number of pointwise and general distributed delays (DDs) exists in the state, input and output. To handle the infinite dimensionality of DDs, we employ the Kronecker-Seuret Decomposition (KSD) which we recently proposed for analyzing matrix-valued functions in the context of delay systems. The KSD enables factorization or least-squares approximation of any number of $\fL^2$ DD kernel from any number of DDs without introducing conservatism. This also facilitates the construction of a complete-type KF with flexible integral kernels, following from an application of a novel integral inequalities derived from the least-squares principle. Our solution includes two theorems and an iterative algorithm to compute controller gains without relying on nonlinear solvers. A challenging numerical example, intractable for existing methods, underscores the efficacy of this approach.

In this article we study the dynamics of one-dimensional relativistic billiards containing particles with positive and negative energy. We study configurations with two identical positive masses and symmetric positions with two massless particles between them of negative energy and symmetric positions. We show that such systems have finitely many collisions in any finite time interval. This is due to a phenomenon we call \textit{tachyonic collision}, which occur at small scales and produce changes in the sign of the energy of individual particles. We also show that depending on the initial parameters the solutions can be bounded with certain periodicity or unbounded while obeying an inverse square law at large distances.

We show that for a compact surface without boundary $M$ the set of cw-expansive homeomorphisms is dense in the set of all the homeomorphisms of $M$ with respect to the $C^0$ topology. After this we show that for a generic homeomorphism $f$ of $M$ it holds that: for all $\epsilon>0$ there is a cw-expansive homeomorphism $g$ of $M$ which is $\epsilon$-close to $f$ and is semiconjugate to $f$; moreover, if $\pi\colon M\to M$ is this semiconjugacy then $\pi^{-1}(x)$ is connected, does not separate $M$ and has diameter less than $\epsilon$ for all $x\in M$.

We generalize the problem of reconstructing strings from their substring compositions first introduced by Acharya et al. in 2015 motivated by polymer-based advanced data storage systems utilizing mass spectrometry. Namely, we see strings as labeled path graphs, and as such try to reconstruct labeled graphs. For a given integer t, the subgraph compositions contain either vectors of labels for each connected subgraph of order t (t-multiset-compositions) or the sum of all labels of all connected subgraphs of order t (t-sum-composition). We ask whether, given a graph of which we know the structure and an oracle whom you can query for compositions, we can reconstruct the labeling of the graph. If it is possible, then the graph is reconstructable; otherwise, it is confusable, and two labeled graphs with the same compositions are called equicomposable. We prove that reconstructing through a brute-force algorithm is wildly inefficient, before giving methods for reconstructing several graph classes using as few compositions as possible. We also give negative results, finding the smallest confusable graphs and trees, as well as families with a large number of equicomposable non-isomorphic graphs. An interesting result occurs when twinning one leaf of a path: some paths are confusable, creating a twin out of a leaf sees the graph alternating between reconstructable and confusable depending on the parity of the path, and creating a false twin out of a leaf makes the graph reconstructable using only sum-compositions in all cases.

This article is about the shadowing property of homeomorphisms on compact metric spaces and the map associating a point of the space to each pseudo-orbit, called 'shadowing map'. Based on some particular dynamical properties, as expansivity, we develop a brief theory and a hierarchy of such maps. We consider examples as odometers, shifts on infinite spaces, topologically hyperbolic homeomorphisms and north-shouth dynamics. We revisit a well-known technique for proving shadowing of expansive homeomorphisms with canonical coordinates due to R. Bowen, to obtain a shadowing map with the property we call 'self-tuning' from a hyperbolic bracket. This notion is introduced as part of the hierarchy of shadowing maps studied in this paper.

We show that Baumslag-Solitar groups are virtually 2-avoidable, that is, they admit finite index subgroup whose first homology is devoid of $\mathbb{Z}_2$ summand. We also prove virtual 2-avoidability for some other classes of one-relator groups, which generalizes non-orientable surface groups. This along with result from a previous paper confirms Gromov's Conjecture about macroscopic dimension of universal cover of PSC manifolds, for all closed oriented spin manifolds whose fundamental group is product of Baumslag-Solitar groups, the one-relator groups under consideration.

An efficient beamforming design is proposed for continuous aperture array (CAPA)-based point-to-point multiple-input multiple-output (MIMO) systems. In contrast to conventional spatially discrete array (SPDA)-MIMO systems, whose optimal beamforming can be obtained using singular-value decomposition, CAPA-MIMO systems require solving the eigendecomposition of a Hermitian kernel operator, which is computationally prohibitive. To address this challenge, an explicit closed-form expression for the achievable rate of CAPA-MIMO systems is first derived as a function of the continuous transmit beamformer. Subsequently, an iterative weighted minimum mean-squared error (WMMSE) algorithm is proposed, directly addressing the CAPA-MIMO beamforming optimization without discretization approximation. Closed-form updates for each iteration of the WMMSE algorithm are derived via the calculus of variations (CoV) method. For low-complexity implementation, an equivalent matrix-based iterative solution is introduced using Gauss-Legendre quadrature. Our numerical results demonstrate that 1) CAPA-MIMO achieves substantial performance gain over the SPDA-MIMO, 2) the proposed WMMSE algorithm enhances performance while significantly reducing computational complexity compared to state-of-the-art Fourier-based approaches, and 3) the proposed WMMSE algorithm enables practical realization of parallel, non-interfering transmissions.

In this survey paper we review classical results and recent progress about a certain topic in the spectral theory of two-dimensional canonical systems. Namely, we consider the questions whether the spectrum $\sigma$ is discrete, and if it is, what is its density. Here we measure density by the growth of the counting function of $\sigma$ in the sense of integrability or $\limsup$-conditions relative to suitable comparison functions. These questions have been around for many years. However, full answers have been obtained only very recently -- in partly still unpublished work. The way we measure density of eigenvalues must be clearly distinguished from spectral asymptotics, where one asks for an asymptotic expansion of eigenvalues. We explicitly and on purpose do not go into this direction and do not present any results about spectral asymptotics. We understand this survey as a focussed presentation of results revolving around the initially stated questions, and not as an exhaustive account on the literature which is in the one or other way related to the area. It does not contain any proofs, but we do comment on proof methods. The theorems we present are very diverse. They rely on different methods from operator theory, complex analysis, and classical analysis, and beautifully invoke the interplay between these areas. Besides the fundamental theorems solving the problem and several selected additions, we decided to include a number of results devoted to the spectral theory of Jacobi matrices, in other words, to the Hamburger moment problem. The connection between the theories of canonical systems and moment problems gives rise to important new insights, yet seems to be less widely known than other connections, e.g., with Krein-Feller or Schr\"odinger operators. We compile all necessary prerequisites from the general theory to make the exposition as self-contained as possible. It is our aim that this survey can be read and enjoyed by a broad community, including non-specialist readers.

Infinite sequences are of tremendous theoretical and practical importance, and in the Information Age sequences of 0s and 1s are of particular interest. Over the past century, the field of symbolic dynamics has developed to study sequences with entries from a finite set. In this paper we will show you an interesting method for constructing sequences, and then we'll show you a fun variant of the method. The we'll discuss what theoretical computer scientists and mathematicians call the "complexity" of a sequence. We conclude by showing a technique you can use to compute the complexity of a sequence, illustrating it on our concrete examples.

This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^2$. Assuming that the principal coefficients satisfy the Dini mean oscillation condition, we establish the equivalence between regular points for $L$ and those for the Laplace operator. This result bridges a gap in the authors' recent work on higher-dimensional cases. Furthermore, we construct the Green's function for $L$ in regular two-dimensional domains, extending a result by Dong and Kim.

In this paper, we introduce the notion of strong geometry, a structure composed by both the chirotope of the oriented matroid associated to a set of points X in the d-dimensional space and a new chirotope induced by the hyperplanes spanned by X. With this on hand, we answer positively a strong geometry version of a question due to M. Las Vergnas about reconstructing polygonal knots via oriented matroids.

For a family of log points with constant log structure and for a proper SNCL scheme with an SNCD over the family, we construct a fundamental l-adic bifiltered complex as a geometric application of the theory of the derived category of (bi)filtered complexes in our papers. By using this bifiltered complex, we give the formulation of the log l-adic relative monodromy-weight conjecture with respect to the filtration arising from the SNCD. That is, we state that the relative l-adic monodromy filtration should exist for the Kummer log etale cohomological sheaf of the proper SNCL scheme with an SNCD and it should be equal to the l-adic weight filtration.

Let $K$ be a field with a discrete valuation, and let $p$ and $\ell$ be (possibly equal) primes which are not necessarily different from the residue characteristic. Given a superelliptic curve $C : y^p = f(x)$ which has split degenerate reduction over $K$, with Jacobian denoted by $J / K$, we describe the action of an element of the inertia group $I_K$ on the $\ell$-adic Tate module $T_\ell(J)$ as a product of powers of certain transvections with respect to the $\ell$-adic Weil pairing and the canonical principal polarization of $J$. The powers to which the transvections are taken are given by a formula depending entirely on the cluster data of the roots of the defining polynomial $f$. This result is demonstrated using Mumford's non-archimedean uniformization of the curve $C$.

We consider Fredholm integral equation of the first kind, present an efficient new iterated Tikhonov method to solve it. The new Tikhonov iteration method has been proved which can achieve the optimal order under a-priori assumption. In numerical experiments, the new iterated Tikhonov regularization method is compared with the classical iterated Tikhonov method, Landweber iteration method to solve the corresponding discrete problem, which indicates the validity and efficiency of the proposed method.

We calculate the second rational homology group of the Torelli group for $g \geq 6$.

This paper is devoted to the analysis of the incompressible Euler equation in a time-dependent fluid domain, whose interface evolution is governed by the law of linear elasticity. Our main result asserts that the Cauchy problem is globally well-posed in time for any irrotational initial data in the energy space, without any smallness assumption. We also prove continuity with respect to the initial data and the propagation of regularity. The main novelty is that no dissipative effect is assumed in the system. In the absence of parabolic regularization, the key observation is that the system can be transformed into a nonlinear Schr\"odinger-type equation, to which dispersive estimates are applied. This allows us to construct solutions that are very rough from the point of view of fluid dynamics-the initial fluid velocity has merely one-half derivative in $L^2$. The main difficulty is that the problem is critical in the energy space with respect to several key inequalities from harmonic analysis. The proof incorporates new estimates for the Dirichlet-to-Neumann operator in the low-regularity regime, including refinements of paralinearization formulas and shape derivative formulas, which played a key role in the analysis of water waves.

We study the first homology group of the mapping class group and Torelli group with coefficients in the first rational homology group of the universal abelian cover of the surface. We prove two contrasting results: for surfaces with one boundary component these twisted homology groups are finite-dimensional, but for surfaces with one puncture they are infinite-dimensional. These results play an important role in a recent paper of the authors calculating the second rational homology group of the Torelli group.

Spectral estimation is an important tool in time series analysis, with applications including economics, astronomy, and climatology. The asymptotic theory for non-parametric estimation is well-known but the development of non-asymptotic theory is still ongoing. Our recent work obtained the first non-asymptotic error bounds on the Bartlett and Welch methods for $L$-mixing stochastic processes. The class of $L$-mixing processes contains common models in time series analysis, including autoregressive processes and measurements of geometrically ergodic Markov chains. Our prior analysis assumes that the process has zero mean. While zero-mean assumptions are common, real-world time-series data often has unknown, non-zero mean. In this work, we derive non-asymptotic error bounds for both Bartlett and Welch estimators for $L$-mixing time-series data with unknown means. The obtained error bounds are of $O(\frac{1}{\sqrt{k}})$, where $k$ is the number of data segments used in the algorithm, which are tighter than our previous results under the zero-mean assumption.

We give a new technique for constructing presentations by generators and relations for representations of groups like $SL_n(\mathbb{Z})$ and $Sp_{2g}(\mathbb{Z})$. Our results play an important role in recent work of the authors calculating the second homology group of the Torelli group.

We introduce stable reflection length in Coxeter groups, as a way to study the asymptotic behaviour of reflection length. This creates connections to other well-studied stable length functions in groups, namely stable commutator length and stable torsion length. As an application, we give a complete characterisation of elements whose reflection length is unbounded on powers.

In this second chapter, we analyse transmission problems between a dielectric and a dispersive negative material. In the first part, we consider a transmission problem between two half-spaces, filled respectively by the vacuum and a Drude material, and separated by a planar interface. In this setting, we answer to the following question: does this medium satisfy a limiting amplitude principle? This principle defines the stationary regime as the large time asymptotic behavior of a system subject to a periodic excitation. In the second part, we consider the transmission problem of an infinite strip of Drude material embedded in the vacuum and analyse the existence and dispersive properties of guided waves. In both problems, our spectral analysis enlighten new and unusual physical phenomena for the considered transmission problems due to the presence of the dispersive negative material. In particular, we prove the existence of an interface resonance in the first part and the existence of slow light phenomena for guiding waves in the second part.

This paper presents a novel approach for steering the state of a stochastic control-affine system to a desired target within a finite time horizon. Our method leverages the time-reversal of diffusion processes to construct the required feedback control law. Specifically, the control law is the so-called score function associated with the time-reversal of random state trajectories that are initialized at the target state and are simulated backwards in time. A neural network is trained to approximate the score function, enabling applicability to both linear and nonlinear stochastic systems. Numerical experiments demonstrate the effectiveness of the proposed method across several benchmark examples.

We investigate in this chapter the mathematical models for electromagnetic wave propagation in dispersive isotropic passive linear media for which the dielectric permittivity $\varepsilon$ and magnetic permeability $\mu$ depend on the frequency. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notions of causality and passivity and its connection to the existence of Herglotz functions that determine the dispersion of the material. We consider successively the cases of the general passive media and the so-called local media for which $\varepsilon$ and $\mu$ are rational functions of the frequency. This leads us to analyse the important class of non dissipative and dissipative generalized Lorentz models. In particular, we discuss the connection between mathematical and physical properties of models through the notions of stability, energy conservation, dispersion and modal analyses, group and phase velocities and energy decay in dissipative systems.

This article considers the problem of reconstructing unknown driving forces based on incomplete knowledge of the system and its state. This is studied in both a linear and nonlinear setting that is paradigmatic in geophysical fluid dynamics and various applications. Two algorithms are proposed to address this problem: one that iteratively reconstructs forcing and another that provides a continuous-time reconstruction. Convergence is shown to be guaranteed provided that observational resolution is sufficiently high and algorithmic parameters are properly tuned according to the prior information; these conditions are quantified precisely. The class of reconstructable forces identified here include those which are time-dependent and potentially inject energy at all length scales. This significantly expands upon the class of forces in previous studies, which could only accommodate those with band-limited spectra. The second algorithm moreover provides a conceptually streamlined approach that allows for a more straightforward analysis and simplified practical implementation.

In survival analysis, the estimation of the proportion of subjects who will never experience the event of interest, termed the cure rate, has received considerable attention recently. Its estimation can be a particularly difficult task when follow-up is not sufficient, that is when the censoring mechanism has a smaller support than the distribution of the target data. In the latter case, non-parametric estimators were recently proposed using extreme value methodology, assuming that the distribution of the susceptible population is in the Fr\'echet or Gumbel max-domains of attraction. In this paper, we take the extreme value techniques one step further, to jointly estimate the cure rate and the extreme value index, using probability plotting methodology, and in particular using the full information contained in the top order statistics. In other words, under sufficient or insufficient follow-up, we reconstruct the immune proportion. To this end, a Peaks-over-Threshold approach is proposed under the Gumbel max-domain assumption. Next, the approach is also transferred to more specific models such as Pareto, log-normal and Weibull tail models, allowing to recognize the most important tail characteristics of the susceptible population. We establish the asymptotic behavior of our estimators under regularization. Though simulation studies, our estimators are show to rival and often outperform established models, even when purely considering cure rate estimation. Finally, we provide an application of our method to Norwegian birth registry data.

This paper is concerned with the partially observed linear system identification, where the goal is to obtain reasonably accurate estimation of the balanced truncation of the true system up to the order $k$ from output measurements. We consider the challenging case of system identification under adversarial attacks, where the probability of having an attack at each time is $\Theta(1/k)$ while the value of the attack is arbitrary. We first show that the $l_1$-norm estimator exactly identifies the true Markov parameter matrix for nilpotent systems under any type of attack. We then build on this result to extend it to general systems and show that the estimation error exponentially decays as $k$ grows. The estimated balanced truncation model accordingly shows an exponentially decaying error for the identification of the true system up to the similarity transformation. This work is the first to provide the input-output analysis of the system with partial observations under arbitrary attacks.

This note is an addendum to the work initiated by Eberlein, Kabanov, and Schmidt and developed further by Kabanov and Promyslov on the asymptotics of the ruin probabilities in the Sparre Andersen model with investments in a risky asset. Using more advanced methods of the implicit renewal theory, we provide complements to some results of the mentioned works.

We study the optimization problem over the weakly Pareto set of a convex multiobjective optimization problem given by polynomial functions. Using Lagrange multiplier expressions and the weight vector, we give three types of representations for the weakly Pareto set. Using these representations, we reformulate the optimization problem over the weakly Pareto set as a polynomial optimization problem. We then apply the Moment--SOS hierarchy to solve it and analyze its convergence properties under certain conditions. Numerical experiments are provided to demonstrate the effectiveness of our methods. Applications in multi-task learning are also presented.

For every family of orthogonal polynomials, we define a new realization of the Yangian of ${\mathfrak{gl}}_n$. Except in the case of Chebyshev polynomials, the new realizations do not satisfy the RTT relation. We obtain an analogue of the Christoffel-Darboux formula. Similar construction can be made for any family of functions satisfying certain recurrence relations, for example, $q$-Pochhhammer symbols and Bessel functions. Furthermore, using an analogue of the Jordan-Schwinger map, we define the ternary Yangian for an arbitrary semisimple Lie algebra as a flat deformation of the current algebra of certain ternary extension of the given Lie algebra.

The character of every irreducible finite-dimensional representation of a simple Lie algebra has the highest weight property. The invariance of the character under the action of the Weyl group W implies that there is a similar "extremal weight property" for every weight obtained by applying an element of W to the highest weight. In this paper we conjecture an analogous "extremal monomial property" of the q-characters of simple finite-dimensional modules over the quantum affine algebras, using the braid group action on q-characters defined by Chari. In the case of the identity element of W, this is the highest monomial property of q-characters proved in arXiv:math/9911112. Here we prove it for simple reflections. Somewhat surprisingly, the extremal monomial property for each w in W turns out to be equivalent to polynomiality of the "X-series" corresponding to w, which we introduce in this paper. We show that these X-series are equal to certain limits of the generalized Baxter operators for all w in W. Thus, we find a new bridge between q-characters and the spectra of XXZ-type quantum integrable models associated to quantum affine algebras. This leads us to conjecture polynomiality of all generalized Baxter operators, extending the results of arXiv:1308.3444.

We provide a new axiom system for flag matroids, characterize representability of uniform flag matroids, and give forbidden minor characterizations of full flag matroids that are representable over $\mathbb{F}_2$ and $\mathbb{F}_3$.

We describe the transformation of a polynomial planar dynamical system into a second order differential equation by means of a polynomial change of variables. We then, by means of the Krylov-Bogoliubov-Mitropolsky averaging method, identify sufficient conditions involving said change of variables so that a limit cycle exists.

We prove the existence of full replica symmetry breaking (FRSB) for the Sherrington-Kirkpatrick (SK) model at low temperature. More specifically, we verify that the support of the Parisi measure of the SK model contains an interval slightly beyond the high temperature regime.

We define and study higher analogues of Rankin-Selberg integrals for GL_n*GL_n-1 over function fields and relate the integral values to higher derivatives of unramified Rankin-Selberg L-functions. Our method reveals some relation between special cycle classes over function fields and the geometric conjectures of Ben-Zvi-Sakellaridis-Venkatesh and can be regarded as a first step towards a generalization of the higher Gross-Zagier formula of Yun-Zhang for higher dimensional spherical varieties.

Using Real Seiberg--Witten theory, Miyazawa introduced an invariant of certain 4-manifolds with involution and used this invariant to construct infinitely many exotic involutions on $\mathbb{CP}^2$ and infinitely many exotic smooth embeddings of $\mathbb{RP}^2$ in $S^4$. In this paper we extend Miyazawa's construction to a large class of 4-manifolds, giving many infinite families of involutions on 4-manifolds which are conjugate by homeomorphisms but not by diffeomorphisms and many infinite families of exotic embeddings of non-orientable surfaces in 4-manifolds, where exotic means continuously isotopic but not smoothly isotopic. Exoticness of our construction is detected using Real Seiberg--Witten theory. We study Miyazawa's invariant, relate it to the Real Seiberg--Witten invariants of Tian--Wang and prove various fundamental results concerning the Real Seiberg--Witten invariants such as: relation to positive scalar curvature, wall-crossing, a mod 2 formula for spin structures, a localisation formula relating ordinary and Real Seiberg--Witten invariants, a connected sum formula and a fibre sum formula.

The generator $\mathbf{L}$ of the linearized evolution equation of adiabatic oscillations of a gaseous star, ELASO, is a second order integro-differential operator and is realized as a self-adjoint operator in the Hilbert space of square integrable unknown functions with weight, which is the density distribution of the compactly supported background. Eigenvalues and eigenfunctions of the operator $\mathbf{L}$ have been investigated in practical point of view of eigenmode expansion of oscillations. But it should be examined whether continuous spectra are absent in the spectrum of $\mathbf{L}$ or not. In order to discuss this question, the existence of essential spectra in a closely related evolution problem is established.

The anti-van der Waerden number of a graph $G$ is the fewest number of colors needed to guarantee a rainbow $3$-term arithmetic progression in $G$, denoted $\operatorname{aw}(G,3)$. It is known that the anti-van der Waerden number of graph products is $3 \le \operatorname{aw}(G\square H,3)\le 4$. Previous work has been done on classifying families of graph products into $\operatorname{aw}(G\square H,3) = 3$ and $\operatorname{aw}(G\square H,3) = 4$. Some of these families include the product of two paths, the product of paths and cycles, the product of two cycles, and the product of odd cycles with any graph. Recently, a partial characterization of the product of two trees was established. This paper completes the characterization for $\operatorname{aw}(T\square T',3)$ where $T$ and $T'$ are trees. Moreover, this result extends to a full classification of products of forests.

We introduce the liquid bin model as a continuous-time deterministic dynamics, arising as the hydrodynamic limit of a discrete-time stochastic interacting particle system called the infinite bin model. For the liquid bin model, we prove the existence and uniqueness of a stationary evolution, to which the dynamics converges exponentially fast. The speed of the front of the system is explicitly computed as a continuous piecewise rational function of the parameters of the model, revealing an underlying wall-crossing phenomenon. We show that the regions on which the speed is rational are of non-empty interior and are naturally indexed by Dyck paths. We provide a complete description of the adjacency structure of these regions, which generalizes the Stanley lattice for Dyck paths. Finally we point out an intriguing connection to the topic of extensions of partial cyclic orders to total cyclic orders.

Let $(G, X)$ be a Shimura datum. In previous work with Christian Klevdal, we showed that the canonical $G(\mathbb{Q}_{\ell})$-valued local systems on Shimura varieties for $G$ form compatible systems after projection to the adjoint group of $G$. In this note, we strengthen this result to prove compatibility for the $G(\mathbb{Q}_{\ell})$-local systems themselves. We also include the crystalline compatibility, extending the adjoint case established in our joint work Jake Huryn, Kiran Kedlaya, and Christian Klevdal.

We compute the stringy E-functions of determinantal varieties and establish that the stringy E-function of a determinantal variety coincides with the E-function of the product of a Grassmannian and an affine space. Furthermore, a similar result holds for the projectivization of determinantal varieties. As an application, we verify the non-negativity conjecture for the stringy Hodge numbers of these varieties.

The canonical polyadic tensor decomposition has a long history. But it becomes challenging to find a tensor decomposition when the rank is between the largest and the second-largest dimension. In such cases, traditional optimization methods, such as nonlinear least squares or alternative least squares methods, often fail to find a tensor decomposition. There are also direct methods, such as the normal form algorithm and the method by Domanov and De Lathauwer, that solve tensor decompositions algebraically. However, these methods can be computationally expensive and require significant memory, especially when the rank is high. This paper proposes a novel two-stage algorithm for the order-3 nonsymmetric tensor decomposition problem when the rank is not greater than the largest dimension. It transforms the tensor decomposition problem into two optimization problems. When the first-stage optimization is not fully solved, the partial solution will also be leveraged in the second-stage optimization problem. We prove the equivalence between tensor decompositions and the global minimizers of the two-stage optimization problems. Our numerical experiments demonstrate the proposed two-stage optimization method is very efficient and robust, capable of finding tensor decompositions where other commonly used state-of-the-art methods fail.

Chung and Graham introduced a method to uniquely represent each positive integer using even-indexed Fibonacci terms. We generalize this result to represent each positive integer using other Fibonacci terms with equally-spaced indices.

A new Chebyshev-type family of stabilized explicit methods for solving mildly stiff ODEs is presented. Besides conventional conditions of order and stability we impose an additional restriction on the methods: their stability function must be monotonically increasing and positive along the largest possible interval of negative real axis. Although stability intervals of the proposed methods are smaller than those of classic Chebyshev-type methods, their stability functions are more consistent with the exponent, they have more convex stability regions and smaller error constants. These properties allow the monotonic methods to be competitive with contemporary stabilized second-order methods, as the presented results of numerical experiments demonstrate.

A cross matrix $X$ can have nonzero elements located only on the main diagonal and the anti-diagonal, so that the sparsity pattern has the shape of a cross. It is shown that $X$ can be factorized into products of matrices that are at most rank-two perturbations to the identity matrix and can be symmetrically permuted to block diagonal form with $2\times 2$ diagonal blocks and, if $n$ is odd, a $1\times 1$ diagonal block. The permutation similarity implies that any well-defined analytic function of $X$ remains a cross matrix. By exploiting these properties, explicit formulae for the determinant, inverse, and characteristic polynomial are derived. It is also shown that the structure of cross matrix can be preserved under matrix factorizations, including the LU, QR, and SVD decompositions.

This paper is devoted to examining the stability of Runge-Kutta methods for solving stiff nonlinear Volterra delay-integro-differential-algebraic equations (DIDAEs) with constant delay. Hybrid numerical schemes combining Runge-Kutta methods and compound quadrature rules are analyzed for nonlinear stiff DIDAEs. Criteria for ensuring the global and asymptotic stability of the proposed schemes are established. Several numerical examples are provided to validate the theoretical findings.

Let ${\Bbb Z}_{m}$ be the additive group of residue classes modulo $m$ and $s(m_{1},m_{2})$ denote the number of subgroups of the group ${\Bbb Z}_{m_{1}}\times {\Bbb Z}_{m_{2}}$, where $m_{1}$ and $m_{2}$ are arbitrary positive integers. We consider sums of type $\sum\limits_{\substack{m_{1}m_{2}\leq x \\ m_{1}m_{2}\in N_{k}}}s(m_{1},m_{2})$, where $N_{k}$ is the set of $k$-th power of natural numbers. In particular, we deduce asymptotic formulas with $k=2$ and $k=3$.

In this paper, we present an algorithm to computea fiberwise Lagrangian torus in quasi-periodic (QP) Hamiltonian systems, whose convergence is proved in the [CHP25]. We exhibit the algorithm with two models. The first is a Tokamak model [CVC+05, VL21], which proposes a control method to create barriers to the diffusion of magnetic field lines through a small modification in the magnetic perturbation. The second model [dCN00], known as the vorticity defect model, describes the nonlinear evolution of localized vorticity perturbations in a constant vorticity flow. This model was originally derived in the context of plasma physics and fluid dynamics.

The linear Boltzmann model for proton beams is a six-dimensional partial differential equation (PDE). We propose a deterministic solver for the linear Boltzmann model based on scattering decomposition and depth-splitting methods. The main idea is to first divide the protons into primary protons and scattering protons, whose equations are derived using the source iteration method. We then treat depth as the time variable in classical time-evolutionary problems and apply the depth-splitting method. In the depth-splitting method, the full operator is decomposed into three parts, with each subsystem being easily parallelizable, which is crucial for efficient simulations. The resulting discretization exhibits second-order convergence in both the depth and energy variables. The dose distributions obtained from our solver are compared with those from Monte Carlo simulations for various materials and heterogeneous cases.

This paper investigates the dynamical behaviors of a Holling type I Leslie-Gower predator-prey model where the predator exhibits an Allee effect and is subjected to constant harvesting. The model demonstrates three types of equilibrium points under different parameter conditions, which could be either stable or unstable nodes (foci), saddle nodes, weak centers, or cusps. The system exhibits a saddle-node bifurcation near the saddle-node point and a Hopf bifurcation near the weak center. By calculating the first Lyapunov coefficient, the conditions for the occurrence of both supercritical and subcritical Hopf bifurcations are derived. Finally, it is proven that when the predator growth rate and the prey capture coefficient vary within a specific small neighborhood, the system undergoes a codimension-2 Bogdanov-Takens bifurcation near the cusp point.

Accurate communication performance prediction is crucial for wireless applications such as network deployment and resource management. Unlike conventional systems with a single transmit and receive antenna, throughput (Tput) estimation in antenna array-based multiple-output multiple-input (MIMO) systems is computationally intensive, i.e., requiring analysis of channel matrices, rank conditions, and spatial channel quality. These calculations impose significant computational and time burdens. This paper introduces Geo2ComMap, a deep learning-based framework that leverages geographic databases to efficiently estimate multiple communication metrics across an entire area in MIMO systems using only sparse measurements. To mitigate extreme prediction errors, we propose a sparse sampling strategy. Extensive evaluations demonstrate that Geo2ComMap accurately predicts full-area communication metrics, achieving a median absolute error of 27.35 Mbps for Tput values ranging from 0 to 1900 Mbps.

We formulate a form of square-root cancellation for the operator which sums a mean-zero function over a hyperplane in $R^d$ for $R$ a possibly noncommutative finite ring. Using an argument of Hart, Iosevich, Koh, and Rudnev, we show that this square-root cancellation occurs when $R$ is a finite field. We then show that this square-root cancellation does not occur over finite rings which are not finite fields. This extends an earlier result of the author to an operator which is not translation-invariant.

Let $X$ and $Y$ be two graphs of order $n$. The friends-and-strangers graph $\textup{FS}(X,Y)$ of $X$ and $Y$ is a graph whose vertex set consists of all bijections $\sigma: V(X)\rightarrow V(Y)$, in which two bijections $\sigma$ and $ \sigma'$ are adjacent if and only if they agree on all but two adjacent vertices of $X$ such that the corresponding images are adjacent in $Y$. The most fundamental question about these friends-and-strangers graphs is whether they are connected. In this paper, we provide a sufficient condition regarding the maximum degree $\Delta(X)$ and vertex connectivity $\kappa(Y)$ that ensures the graph $\textup{FS}(X,Y)$ is $s$-connected. As a corollary, we improve upon a result by Bangachev and partially confirm a conjecture he proposed. Furthermore, we completely characterize the connectedness of $\textup{FS}(X,Y)$, where $X\in\textup{DL}_{n-k,k}$.

Different from conventional passive reconfigurable intelligent surfaces (RISs), incident signals and thermal noise can be amplified at active RISs. By exploiting the amplifying capability of active RISs, noticeable performance improvement can be expected when precise channel state information (CSI) is available. Since obtaining perfect CSI related to an RIS is difficult in practice, a robust transmission design is proposed in this paper to tackle the channel uncertainty issue, which will be more severe for active RIS-aided systems. To account for the worst-case scenario, the minimum achievable rate of each user is derived under a statistical CSI error model. Subsequently, an optimization problem is formulated to maximize the sum of the minimum achievable rate. Since the objective function is non-concave, the formulated problem is transformed into a tractable lower bound maximization problem, which is solved using an alternating optimization method. Numerical results show that the proposed robust design outperforms a baseline scheme that only exploits estimated CSI.

We study the K-theory of the Cuntz-Nica-Pimsner C*-algebra of a rank-two product system that is an extension determined by an invariant ideal of the coefficient algebra. We use a construction of Deaconu and Fletcher that describes the Cuntz-Nica-Pimsner C*-algebra of the product system in terms of two iterations of Pimsner's original construction of a C*-algebra from a right-Hilbert bimodule. We apply our results to the product system built from two commuting surjective local homeomorphisms of a totally disconnected space, where the Cuntz-Nica-Pimsner C*-algebra is isomorphic to the C*-algebra of the associated rank-two Deaconu--Renault groupoid. We then apply a theorem of Spielberg about stable finiteness of an extension to obtain sufficient conditions for stable finiteness of the C*-algebra of the Deaconu-Renault groupoid.

In this paper, we study numerical approximations for optimal control of a class of stochastic partial differential equations with partial observations. The system state evolves in a Hilbert space, whereas observations are given in finite-dimensional space $\rr^d$. We begin by establishing stochastic maximum principles (SMPs) for such problems, where the system state is driven by a cylindrical Wiener process. The corresponding adjoint equations are characterized by backward stochastic partial differential equations. We then develop numerical algorithms to solve the partially observed optimal control. Our approach combines the stochastic gradient descent method, guided by the SMP, with a particle filtering algorithm to estimate the conditional distributions of the state of the system. Finally, we demonstrate the effectiveness of our proposed algorithm through numerical experiments.

The sine-Gordon equation is a fundamental nonlinear partial differential equation that governs soliton dynamics and phase evolution in a variety of physical systems, including Josephson junctions and superconducting circuits. In this study, we analyze the effects of external perturbations such as damping and driving forces on the stability of soliton solutions. Using a rigorous Hilbert space framework, we establish well-posedness and derive regularity results for the perturbed equation. In particular, we provide sufficient conditions for the boundedness of the perturbation function, which plays a crucial role in determining the persistence of the soliton structure. Furthermore, we propose a neural network-based approach for solving the inverse problem of estimating unknown initial conditions. By training a data-driven model on simulated PDE solutions, we demonstrate that the network can accurately recover the initial states from limited or noisy observations. Numerical simulations validate the theoretical results and highlight the potential of combining mathematical analysis with machine learning techniques to study nonlinear wave phenomena. This approach offers valuable insights into soliton behavior and has potential applications in the design of quantum computing systems based on Josephson junctions.

Most existing generation scheduling models for power systems under demand uncertainty rely on energy-based formulations with a finite number of time periods, which may fail to ensure that power supply and demand are balanced continuously over time. To address this issue, we propose a robust generation scheduling model in a continuous-time framework, employing a decision rule approach. First, for a given set of demand trajectories, we formulate a general robust generation scheduling problem to determine a decision rule that maps these demand trajectories and time points to the power outputs of generators. Subsequently, we derive a surrogate of it as our model by carefully designing a class of decision rules that are affine in the current demand, with coefficients invariant over time and constant terms that are continuous piecewise affine functions of time. As a result, our model can be recast as a finite-dimensional linear program to determine the coefficients and the function values of the constant terms at each breakpoint, solvable via the cutting-plane method. Our model is non-anticipative unlike most existing continuous-time models, which use Bernstein polynomials, making it more practical. We also provide illustrative numerical examples.

This paper studies the rank-1 tensor completion problem for cubic tensors when there are noises for observed tensor entries. First, we propose a robust biquadratic optimization model for obtaining rank-1 completing tensors. When the observed tensor is sufficiently close to be rank-1, we show that this biquadratic optimization produces an accurate rank-1 tensor completion. Second, we give an efficient convex relaxation for solving the biquadratic optimization. When the optimizer matrix is separable, we show how to get optimizers for the biquadratic optimization and how to compute the rank-1 completing tensor. When that matrix is not separable, we apply its spectral decomposition to obtain an approximate rank-1 completing tensor. Numerical experiments are given to explore the efficiency of this biquadratic optimization model and the proposed convex relaxation.

This paper investigates the stability and stabilization of diffusively coupled network dynamical systems. We leverage Lyapunov methods to analyze the role of coupling in stabilizing or destabilizing network systems. We derive critical coupling parameter values for stability and provide sufficient conditions for asymptotic stability under arbitrary switching scenarios, thus highlighting the impact of both coupling strength and network topology on the stability analysis of such systems. Our theoretical results are supported by numerical simulations.

The existence of perfect state transfer (PST) on quantum spin networks is a fundamental problem in mathematics and physics. Various works in the literature have explored PST in graphs with arithmetic origins, such as gcd-graphs over $\mathbb{Z}$ and cubelike graphs. In this article, building on our recent work on gcd-graphs over an arbitrary finite Frobenius ring, we investigate the existence of PST on these graphs. Our approach is algebraic in nature, enabling us to unify various existing results in the literature.

Pre-filtering and post-filtering steps can be added to many of the traditional numerical methods to generate new, higher order methods with strong stability properties. Presented in this paper are a variable step pre-filter and post-filter that allow adaptive time stepping for a filtered method based on Implicit Euler.

A dormant oper is a specific type of principal bundle with a flat connection, defined on an algebraic curve in positive characteristic. The moduli spaces of dormant opers and their variants, known as dormant Miura opers, have been studied in various contexts. This paper focuses on the case where the underlying spaces are (possibly nodal) elliptic curves and provides a detailed examination of the geometric structures of their moduli spaces. In particular, we explicitly describe dormant (generic Miura) opers in terms of regular elements in an associated Lie algebra and establish the connectedness of these moduli spaces. We also explore generalizations to higher level and prime-power characteristic.

We characterize Martin-L\"of randomness and Schnorr randomness in terms of the merging of opinions, along the lines of the Blackwell-Dubins Theorem. After setting up a general framework for defining notions of merging randomness, we focus on finite horizon events, that is, on weak merging in the sense of Kalai-Lehrer. In contrast to Blackwell-Dubins and Kalai-Lehrer, we consider not only the total variational distance but also the Hellinger distance and the Kullback-Leibler divergence. Our main result is a characterization of Martin-L\"of randomness and Schnorr randomness in terms of weak merging and the summable Kullback-Leibler divergence. The main proof idea is that the Kullback-Leibler divergence between $\mu$ and $\nu$, at a given stage of the learning process, is exactly the incremental growth, at that stage, of the predictable process of the Doob decomposition of the $\nu$-submartingale $L(\sigma)=-\ln \frac{\mu(\sigma)}{\nu(\sigma)}$. These characterizations of algorithmic randomness notions in terms of the Kullback-Leibler divergence can be viewed as global analogues of Vovk's theorem on what transpires locally with individual Martin-L\"of $\mu$- and $\nu$-random points and the Hellinger distance between $\mu,\nu$.

We show the existence of local and global in time weak martingale solutions for a stochastic version of the Othmer-Dunbar-Alt kinetic model of chemotaxis under suitable assumptions on the turning kernel and stochastic drift coefficients, using dispersion and stochastic Strichartz estimates. The analysis is based on new Strichartz estimates for stochastic kinetic transport. The derivation of these estimates involves a local in time dispersion analysis using properties of stochastic flows, and a time-splitting argument to extend the local in time results to arbitrary time intervals.

We consider the free boundary problems of degenerate elliptic equations that describe the level set formulation of the interface motion evolved by anisotropic forced mean curvature flows. The type of free boundary problems in this paper was initially studied as the first-order Hamilton-Jacobi-Isaacs equations arising in pursuit-evasion differential games and applied to the models of first-order front propagation in Soravia (1994). In this paper, we consider an extension of these free boundary problems to the second-order equations and give a deterministic game representation based on a discrete approximation scheme in Kohn and Serfaty (2006). Furthermore, we prove the comparison principle for our free boundary problems by using the framework of time-discrete games.

When solving partial differential equations using classical schemes such as finite difference or finite volume methods, sufficiently fine meshes and carefully designed schemes are required to achieve high-order accuracy of numerical solutions, leading to a significant increase in computational costs, especially for three-dimensional (3D) time-dependent problems. Recently, machine learning-assisted numerical methods have been proposed to enhance accuracy or efficiency. In this paper, we propose a data-driven finite difference numerical method to solve the hyperbolic equations with smooth solutions on coarse grids, which can achieve higher accuracy than classical numerical schemes based on the same mesh size. In addition, the data-driven schemes have better spectrum properties than the classical schemes, although the spectrum properties are not explicitly optimized during the training process. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method, as well as its good performance on dispersion and dissipation.

In this paper we obtain a new equivalent of the Fermat-Wiles theorem based on a kind of cross-bred of Ingham and D. R. Heath-Brown formula. Further, we prove the existence of infinite set of finite chains of a kind of equivalent expressions of mathematical analysis.

Flow Matching (FM) is a recent generative modelling technique: we aim to learn how to sample from distribution $\mathfrak{X}_1$ by flowing samples from some distribution $\mathfrak{X}_0$ that is easy to sample from. The key trick is that this flow field can be trained while conditioning on the end point in $\mathfrak{X}_1$: given an end point, simply move along a straight line segment to the end point (Lipman et al. 2022). However, straight line segments are only well-defined on Euclidean space. Consequently, Chen and Lipman (2023) generalised the method to FM on Riemannian manifolds, replacing line segments with geodesics or their spectral approximations. We take an alternative point of view: we generalise to FM on Lie groups by instead substituting exponential curves for line segments. This leads to a simple, intrinsic, and fast implementation for many matrix Lie groups, since the required Lie group operations (products, inverses, exponentials, logarithms) are simply given by the corresponding matrix operations. FM on Lie groups could then be used for generative modelling with data consisting of sets of features (in $\mathbb{R}^n$) and poses (in some Lie group), e.g. the latent codes of Equivariant Neural Fields (Wessels et al. 2025).

Many theories of physical interest, which admit a Hamiltonian description, exhibit symmetries under a particular class of non -strictly conformal transformation, known as dynamical similarities. The presence of such symmetries allows a reduction process to be carried out, eliminating a single degree of freedom from the system, which we associate with an overall scale. This process of `contact reduction' leads to theories of a frictional nature, in which the physically-observable quantities form an autonomous subsystem, that evolves in a predictable manner. We demonstrate that this procedure has a natural generalisation to theories of higher order; detailed examples are provided, and physical implications discussed.

We consider the Aubry set for the XY model, symbolic dynamics $([0,1]^{\mathbb{N}_0},\sigma)$ with the uncountable symbol $[0,1]$, and study its action-optimizing properties. Moreover, for a potential function that depends on the first two coordinates we obtain an explicit expression of the set of optimal periodic measures and a detailed description of the Aubry set. We also show the typicality of periodic optimization for 2-locally constant potentials with the twist condition. Our approach combines the weak KAM method for symbolic dynamics and variational techniques for twist maps.

Higher form symmetry, one of the generalized symmetries, considers mainly the action of abelian groups. This is due to the topological nature of symmetry defect operators. In this study, we extend the vector space (or vector bundle) in which the charged operator takes values in order to describe the action of non-abelian groups while preserving this topological property.

In this paper, we study the parabolic equations of the form $$ \left\{ \begin{array}{rcll} Lu(y,t) &=& f, \qquad &(y,t)\in Q,\\ u(y,t)&=& 0, \qquad &(y,t)\in \partial Q, \\ u(y,t)&& \hspace{-8mm}\mbox{is uniformly bounded from below}, \qquad &(y,t)\in Q, \end{array} \right. $$ where $Q=\Omega\times\mathbb{R}\subset\mathbb{R}^{n+1}$ and $\Omega\subset\mathbb{R}^{n}$ is a bounded Lipschitz domain with $0\in\Omega$. Here $L$ is a general second order uniformly parabolic differential operator in non-divergence form or divergence form. For $f=0$, we establish the structure of the solution space, which is one dimensional and the solutions in this space grow exponentially at one end and decay exponentially at the other. For $f\neq0$, we show that all solutions can be presented by the solutions corresponding to the homogenous equations($f=0$) and a bounded special solution of the inhomogeneous equations. Our method is based on maximum principle in $Q$ and the Harnack type inequalities.

Networks provide a popular representation of complex data. Often, different types of relational measurements are taken on the same subjects. Such data can be represented as a \textit{multislice network}, a collection of networks on the same set of nodes, with connections between the different layers to be determined. For the analysis of multislice networks, we take inspiration from the analysis of simple networks, for which small subgraphs (motifs) have proven to be useful; motifs are even seen as building blocks of complex networks. A particular instance of a motif is a triangle, and while triangle counts are well understood for simple network models such as Erd\H{o}s-R\'enyi random graphs, with i.i.d. distributed edges, even for simple multislice network models little is known about triangle counts. Here we address this issue by extending the analysis of triadic structures to multislice Erd\H{o}s-R\'enyi networks. Again taking inspiration from the analysis of sparse Erd\H{o}s-R\'enyi random graphs, we show that the distribution of triangles across multiple layers in a multislice Erd\H{o}s-R\'enyi network can be well approximated by an appropriate Poisson distribution. This theoretical result opens the door to statistical goodness of fit tests for multislice networks.

Consider a random block matrix model consisting of $D$ random systems arranged along a circle, where each system is modeled by an independent $N\times N$ complex Hermitian Wigner matrix. The neighboring systems interact through an arbitrary deterministic $N\times N$ matrix $A$. In this paper, we extend the localization-delocalization transition of this model, established in arxiv:2312.07297 for the bulk eigenvalue spectrum, to the entire spectrum, including spectral edges. More precisely, let $[E^-,E^+]$ denote the support of the limiting spectrum, and define $\kappa_E:=|E-E^+|\wedge |E-E^-|$ as the distance of an energy $E\in[E^-,E^+]$ from the spectral edges. We prove that for eigenvalues near $E$, a localization-delocalization transition of the corresponding eigenvectors occurs when $\|A\|_{\mathrm{HS}}$ crosses the critical threshold $(\kappa_E+N^{-2/3})^{-1/2}$. Moreover, in the delocalized phase, we show that the extreme eigenvalues asymptotically follow the Tracy-Widom law, while in the localized phase, the edge eigenvalue statistics asymptotically behave like $D$ independent copies of GUE statistics, up to a deterministic shift. Our result recovers the findings of arxiv:2312.07297 in the bulk with $\kappa_E\asymp 1$, and also implies the existence of mobility edges at $E^{\pm}$ when $1\ll \|A\|_{\mathrm{HS}} \ll N^{1/3}$: bulk eigenvectors corresponding to eigenvalues within $[E^{-}+\varepsilon,E^{+}-\varepsilon]$ are delocalized, whereas edge eigenvectors near $E^{\pm}$ are localized.

Factor Analysis is an effective way of dimensionality reduction achieved by revealing the low-rank plus sparse structure of the data covariance matrix. The corresponding model identification task is often formulated as an optimization problem with suitable regularizations. In particular, we use the nonconvex discontinuous $\ell_0$ norm in order to induce the sparsity of the covariance matrix of the idiosyncratic noise. This paper shows that such a challenging optimization problem can be approached via an interior-point method with inner-loop Newton iterations. To this end, we first characterize the solutions to the unconstrained $\ell_0$ regularized optimization problem through the $\ell_0$ proximal operator, and demonstrate that local optimality is equivalent to the solution of a stationary-point equation. The latter equation can then be solved using standard Newton's method, and the procedure is integrated into an interior-point algorithm so that inequality constraints of positive semidefiniteness can be handled. Finally, numerical examples validate the effectiveness of our algorithm.

We characterise the isomorphisms of Tits--Kantor--Koecher Lie algebras of JB*-triples as a class of surjective linear isometries and show how these algebras form a category equivalent to that of JB*-triples. We introduce the concepts of tripotent, and orthogonality and order amongst tripotents for Tits--Kantor--Koecher Lie algebras. This leads to showing that a graded or negatively graded order isomorphism between certain subsets of tripotents of two Tits--Kantor--Koecher Lie algebras of atomic JB*-triples, which commutes with involutions, preserves orthogonality and is continuous at a non-zero tripotent of a specific type, can be extended as a real-linear isomorphism between the algebras.

We show that the maximal Fock space $F^\infty_\alpha$ on $C^n$ is a Lipschitz space, that is, there exists a distance $d_\alpha$ on $C^n$ such that an entire function $f$ on $C^n$ belongs to $F^\infty_\alpha$ if and only if $$|f(z)-f(w)|\le Cd_\alpha(z,w)$$ for some constant $C$ and all $z,w\in C^n$. This can be considered the Fock space version of the following classical result in complex analysis: a holomorphic function $f$ on the unit ball $B_n$ in $C^n$ belongs to the Bloch space if and only if there exists a positive constant $C$ such that $|f(z)-f(w)|\le C\beta(z,w)$ for all $z,w\in B_n$, where $\beta(z,w)$ is the distance on $B_n$ in the Bergman metric. We also present a new approach to Hardy-Littlewood type characterizations for $F^p_\alpha$.

$D_4$ triality invariants are modular forms as well as polynomial invariants for a fiber product of the modular group and the Weyl group of type $F_4$. We show that the ring of $D_4$ triality invariants satisfying a certain cusp condition is isomorphic to the ring of joint covariants of a binary cubic and a binary quadratic form.

Envelopes of parameterized families of plane curves is an important topic, both for the mathematics involved and for its applications. Nowadays, it is generally studied in a technology-rich environment, and automated methods are developed and implemented in software. The exploration involves a dialog between a Dynamic Geometry System (used mostly for interactive exploration and conjectures) and a Computer Algebra System (for algebraic manipulations). We study envelopes of families of circles centered on the so-called kiss curve and offsets of this curve, observing the differences between constructs. Both parametric presentations and implicit equations are used, switching from parametric to polynomial representation being based on packages for Gr\"obner bases and Elimination. Singular points, both cusps and points of self-intersection (crunodes), are analyzed.

A star coloring of a graph $G$ is a proper coloring where vertices of every two color classes induce a forest of stars. A strict partial order is defined on the set of all star colorings of $G$. We introduce the star b-chromatic number $S_b(G)$, analogous to the b-chromatic number, as the maximum number of colors in a minimum element of the mentioned order. We present several combinatorial properties of $S_b(G)$, compute the exact value for $S_b(G)$ for several known families and compare $S_b(G)$ with several invariants naturally connected to $S_b(G)$.

We present a variational integrator based on the Lobatto quadrature for the time integration of dynamical systems issued from the least action principle. This numerical method uses a cubic interpolation of the states and the action is approximated at each time step by Lobatto's formula. Numerical analysis is performed on a harmonic oscillator. The scheme is conditionally stable, sixth-order accurate, and symplectic. It preserves an approximate energy quantity. Simulation results illustrate the performance of the proposed method.

We continue our study of the unidirectional elephant random walk (uERW) initiated in {\it {Electron. Commun. Probab.}} ({\bf 29} 2024, article no. 78). In this paper we obtain definitive results when the memory exponent $\beta\in (-1, p/(1-p))$. In particular using a coupling argument we obtain the exact asymptotic rate of growth of $S_n$, the location of the uERW at time $n$, for the case $\beta\in (-1, 0] $. Also, for the case $\beta\in (0, p/(1-p))$ we show that $P(S_n \to \infty) \in (0,1)$ and conditional on $\{S_n \to \infty\}$ we obtain the exact asymptotic rate of growth of $S_n$. In addition we obtain the central limit theorem for $S_n$ when $\beta \in (-1, p/(1-p))$.

We prove optimal H\"older boundary regularity for a non-local operator with a singular, symmetric kernel that depends on the distance to the boundary of the underlying domain. Additionally, we prove higher boundary regularity of solutions.

We study quasi-cyclic codes of index 2 over finite fields. We give a classification of such codes. Their duals with respect to the Euclidean, symplectic and Hermitian inner products are investigated. We describe self-orthogonal and dual-containing codes. Lower bounds for minimum distances of quasi-cyclic codes are given. A quasi-cyclic code of index 2 is generated by at most two elements. We describe conditions when such a code (or its dual) is generated by one element.

We consider a special family of 2-dimensional timelike surfaces in the Minkowski 4-space $\mathbb{R}^4_1$ which lie on rotational hypersurfaces with timelike axis and call them meridian surfaces of elliptic type. We study the following basic classes of timelike meridian surfaces of elliptic type: with constant Gauss curvature, with constant mean curvature, with parallel mean curvature vector field, with parallel normalized mean curvature vector field. The results obtained for the last class are used to give explicit solutions to the background systems of natural PDEs describing the timelike surfaces with parallel normalized mean curvature vector field in $\mathbb{R}^4_1$.

Consider a graph $\Gamma$. A set $ S $ of vertices in $\Gamma$ is called a {cyclic vertex cutset} of $\Gamma$ if $\Gamma - S$ is disconnected and has at least two components containing cycles. If $\Gamma$ has a cyclic vertex cutset, then it is said to be {cyclically separable}. The {cyclic vertex connectivity} is the minimum cardinality of a cyclic vertex cutset of $\Gamma$. The power graph $\mathcal{P}(G)$ of a group $G$ is the undirected simple graph with vertex set $G$ and two distinct vertices are adjacent if one of them is a positive power of the other. If $G$ is a cyclic, dihedral, or dicyclic group, we determine the order of $G$ such that $\mathcal{P}(G)$ is cyclically separable. Then we characterize the equality of vertex connectivity and cyclic vertex connectivity of $\mathcal{P}(G)$ in terms of the order of $G$.

Based on a class of moderately interacting particle systems, we establish a quantitative approximation for density-dependent McKean-Vlasov SDEs and the corresponding nonlinear, nonlocal PDEs. The SDE is driven by both Brownian motion and pure-jump L\'evy processes. By employing Duhamel's formula, density estimates, and appropriate martingale functional inequalities, we derive precise convergence rates for the empirical measure of particle systems toward the law of the McKean-Vlasov SDE solution. Additionally, we quantify both weak and pathwise convergence between the one-marginal particle and the solution to the McKean-Vlasov SDE. Notably, all convergence rates remain independent of the noise type.

In this paper, we derive power guarantees of some sequential tests for bounded mean under general alternatives. We focus on testing procedures using nonnegative supermartingales which are anytime valid and consider alternatives which coincide asymptotically with the null (e.g. vanishing mean) while still allowing to reject in finite time. Introducing variance constraints, we show that the alternative can be broaden while keeping power guarantees for certain second-order testing procedures. We also compare different test procedures in multidimensional setting using characteristics of the rejection times. Finally, we extend our analysis to other functionals as well as testing and comparing forecasters. Our results are illustrated with numerical simulations including bounded mean testing and comparison of forecasters.

The main concern of this paper is to investigate the problem whether two Elephant Random Walks (ERWs) on $\mathbb{Z}$ with different memory parameters can meet each other infinitely often, extending the result by Roy, Takei, and Tanemura. We also study the asymptotic behavior of the distance between them by providing an elementary and accessible proof of the classical Law of Iterated Logarithm (LIL) for centered continuous self-similar Gaussian processes under a certain decay condition on the covariance kernel.

We develop a new approach on free wreath products, generalizing the constructions of Bichon and of Fima-Pittau. We show stability properties for certain approximation properties such as exactness, Haagerup property, hyperlinearity and K-amenability. We study qualitative properties of the associated von Neumann algebra: factoriality, primeness and absence of Cartan subalgebra and we give a formula for Connes' T-invariant. Finally, we give some explicit computations of K-theory groups for C*-algebras of generalized free wreath products.

This paper proposes an adaptive hyper-reduction method to reduce the computational cost associated with the simulation of parametric particle-based kinetic plasma models, specifically focusing on the Vlasov-Poisson equation. Conventional model order reduction and hyper-reduction techniques are often ineffective for such models due to the non-sparse nature of the nonlinear operators arising from the interactions between particles. To tackle this issue, we propose an adaptive, structure-preserving hyper-reduction method that leverages a decomposition of the discrete reduced Hamiltonian into a linear combination of terms, each depending on a few components of the state. The proposed approximation strategy allows to: (i) preserve the Hamiltonian structure of the problem; (ii) evaluate nonlinear non-sparse operators in a computationally efficient way; (iii) overcome the Kolmogorov barrier of transport-dominated problems via evolution of the approximation space and adaptivity of the rank of the solution. The proposed method is validated on numerical benchmark simulations, demonstrating stable and accurate performance with substantial runtime reductions compared to the full order model.

We present a nonlinear data-driven Model Predictive Control (MPC) algorithm for deep brain stimulation (DBS) for the treatment of Parkinson's disease (PD). Although DBS is typically implemented in open-loop, closed-loop DBS (CLDBS) uses the amplitude of neural oscillations in specific frequency bands (e.g. beta 13-30 Hz) as a feedback signal, resulting in improved treatment outcomes with reduced side effects and slower rates of patient habituation to stimulation. To date, CLDBS has only been implemented in vivo with simple control algorithms, such as proportional or proportional-integral control. Our approach employs a multi-step predictor based on differences of input-convex neural networks to model the future evolution of beta oscillations. The use of a multi-step predictor enhances prediction accuracy over the optimization horizon and simplifies online computation. In tests using a simulated model of beta-band activity response and data from PD patients, we achieve reductions of more than 20% in both tracking error and control activity in comparison with existing CLDBS algorithms. The proposed control strategy provides a generalizable data-driven technique that can be applied to the treatment of PD and other diseases targeted by CLDBS, as well as to other neuromodulation techniques.

Efficiently retrieving relevant data from massive Internet of Things (IoT) networks is essential for downstream tasks such as machine learning. This paper addresses this challenge by proposing a novel data sourcing protocol that combines semantic queries and random access. The key idea is that the destination node broadcasts a semantic query describing the desired information, and the sensors that have data matching the query then respond by transmitting their observations over a shared random access channel, for example to perform joint inference at the destination. However, this approach introduces a tradeoff between maximizing the retrieval of relevant data and minimizing data loss due to collisions on the shared channel. We analyze this tradeoff under a tractable Gaussian mixture model and optimize the semantic matching threshold to maximize the number of relevant retrieved observations. The protocol and the analysis are then extended to handle a more realistic neural network-based model for complex sensing. Under both models, experimental results in classification scenarios demonstrate that the proposed protocol is superior to traditional random access, and achieves a near-optimal balance between inference accuracy and the probability of missed detection, highlighting its effectiveness for semantic query-based data sourcing in massive IoT networks.

We introduce a model of infinite horizon linear dynamic optimization with linear constraints and obtain results concerning feasibility of trajectories and optimal solutions necessarily satisfying conditions that resemble the Euler condition and transversality condition. We prove results about optimal trajectories of strictly alternating problems, eventually conclusive problems, strongly eventually conclusive problems and two-phase problems.

In this paper, we show that Gibbs measures on self-conformal sets generated by a $C^{1+\alpha}$ conformal IFS on $\mathbb{R}^d$ satisfying the OSC are exponentially mixing. We exploit this to obtain essentially sharp asymptotic counting statements for the recurrent and the shrinking target subsets associated with any such set. In particular, we provide explicit examples of dynamical systems for which the recurrent sets exhibit (unexpected) behaviour that is not present in the shrinking target setup. In the process of establishing our exponential mixing result we extend Mattila's rigidity theorem for self-similar sets to self-conformal sets without any separation condition and for arbitrary Gibbs measures.

We classify all subterminal schemes by characterizing their point structure, stalks, and topologies. This extends our previous classification of subterminal affine schemes, which correspond to spectra of solid rings.

In phylogenetics, a key problem is to construct evolutionary trees from collections of characters where, for a set X of species, a character is simply a function from X onto a set of states. In this context, a key concept is convexity, where a character is convex on a tree with leaf set X if the collection of subtrees spanned by the leaves of the tree that have the same state are pairwise disjoint. Although collections of convex characters on a single tree have been extensively studied over the past few decades, very little is known about coconvex characters, that is, characters that are simultaneously convex on a collection of trees. As a starting point to better understand coconvexity, in this paper we prove a number of extremal results for the following question: What is the minimal number of coconvex characters on a collection of n-leaved trees taken over all collections of size t >= 2, also if we restrict to coconvex characters which map to k states? As an application of coconvexity, we introduce a new one-parameter family of tree metrics, which range between the coarse Robinson-Foulds distance and the much finer quartet distance. We show that bounds on the quantities in the above question translate into bounds for the diameter of the tree space for the new distances. Our results open up several new interesting directions and questions which have potential applications to, for example, tree spaces and phylogenomics.

This paper aims to develop a theory for linear-quadratic Nash systems and Master equations in possibly infinite-dimensional Hilbert spaces. As a first step and motivated by the recent results in [31], we study a more general model in the linear quadratic case where the dependence on the distribution enters just in the objective functional through the mean. This property enables the Nash systems and the Master equation to be reduced to two systems of coupled Riccati equations and backward abstract evolution equations. We show that solutions for such systems exist and are unique for all time horizons, a result that is completely new in the literature in our setting. Finally, we apply the results to a vintage capital model, where capital depends on time and age, and the production function depends on the mean of the vintage capital.

We develop and analyse residual-based a posteriori error estimates for the virtual element discretisation of a nonlinear stress-assisted diffusion problem in two and three dimensions. The model problem involves a two-way coupling between elasticity and diffusion equations in perturbed saddle-point form. A robust global inf-sup condition and Helmholtz decomposition for $\mathbf{H}(\mathrm{div}, \Omega)$ lead to a reliable and efficient error estimator based on appropriately weighted norms that ensure parameter robustness. The a posteriori error analysis uses quasi-interpolation operators for Stokes and edge virtual element spaces, and we include the proofs of such operators with estimates in 3D for completeness. Finally, we present numerical experiments in both 2D and 3D to demonstrate the optimal performance of the proposed error estimator.

The first-order approach to boundary value problems for second-order elliptic equations in divergence form with transversally independent complex coefficients in the upper half-space rewrites the equation algebraically as a first-order system, much like how harmonic functions in the plane relate to the Cauchy-Riemann system in complex analysis. It hinges on global Lp -bounds for some p > 2 for the resolvent of a perturbed Dirac-type operator acting on the boundary. At the same time, gradients of local weak solutions to such equations exhibit higher integrability for some p > 2, expressed in terms of weak reverse H{\"o}lder estimates. We show that the optimal exponents for both properties coincide. Our proof relies on a simple but seemingly overlooked connection with operator-valued Fourier multipliers in the tangential direction.

We consider the space of Siegel modular forms of genus $g$ of weight two relative to the main congruence subgroup of level 2 and to Igusa's group $\Gamma_g(4, 8) and $\Gamma_g(2,4)$.One of the main results of this paper is that in the case $g\ge 8$ the space $[\Gamma_g[4,8],2]$ is generated by the products of 4 theta nullwerte. Thus this note can be considered as a completion of the example at the end of [Fr].

We consider the convected Helmholtz equation with a generalized Myers boundary condition (a boundary condition of the second-order) and characterize the set of physical parameters for which the problem is weakly well-posed. The model comes from industrial applications to absorb acoustic noise in jet engines filled with absorbing liners (porous material). The problem is set on a 3D cylinder filled with a d-upper regular boundary measure, with a real 1 < d $\le$ 2. This setup leads to a parametric shape optimization problem, for which we prove the existence of at least one optimal distribution for any fixed volume fraction of the absorbing liner on the boundary that minimizes the total acoustic energy on any bounded wavenumber range.

We construct a new compact semi-explicit three-level in time fourth-order finite-difference scheme for numerical solving the general multidimensional acoustic wave equation, where both the speed of sound and density of a medium are variable. The scheme is three-point in each spatial direction, has the truncation order $\mathcal{O}(|h|^4+h_t^4)$ and is easily implementable. It seems to be the first compact scheme with such properties for the equation under consideration. It generalizes a semi-explicit compact scheme developed and studied recently in the much simpler case of the variable speed of sound only. Numerical experiments confirm the high precision of the scheme and its fourth error order not only in the mesh $C$ norm but in the mesh $C^1$ norm as well.

The purpose of this note is to investigate the coupling of Dirichlet and Neumann numerical boundary conditions for the transport equation set on an interval. When one starts with a stable finite difference scheme on the lattice $\mathbb{Z}$ and each numerical boundary condition is taken separately with the Neumann extrapolation condition at the outflow boundary, the corresponding numerical semigroup on a half-line is known to be bounded. It is also known that the coupling of such numerical boundary conditions on a compact interval yields a stable approximation, even though large time exponentially growing modes may occur. We review the different stability estimates associated with these numerical boundary conditions and give explicit examples of such exponential growth phenomena for finite difference schemes with ''small'' stencils. This provides numerical evidence for the optimality of some stability estimates on the interval.

In this article, we determine all inequivalent irreducible rational matrix representations of groups of order $p^5$, where $p$ is an odd prime. We also derive combinatorial formulations for the Wedderburn decomposition of rational group algebras of these $p$-groups, using results from their rational representations.

On normed vector spaces there is a well-known connection between the Tikhonov well-posedness of a minimisation problem and the differentiability of an associated convex conjugate function. We show how this duality naturally generalises to the setting of asymmetrically normed spaces.

We consider the existence of stationary wave solutions with prescribed mass to a supercritical nonlinear Schr{\"o}dinger equation on a noncompact connected metric graph without a small mass assumption.

We introduce the so-called weak Pinsker dynamical filtrations, whose existence in any ergodic system follows from the universality of the weak Pinsker property, recently proved by Austin. These dynamical filtrations appear as a potential tool to describe and classify positive entropy systems. We explore the links between the asymptotic structure of weak Pinsker filtrations and the properties of the underlying dynamical system. A central question is whether, on a given system, the structure of weak Pinsker filtrations is unique up to isomorphism. We give a partial answer, in the case where the underlying system is Bernoulli. We conclude our work by giving two explicit examples of weak Pinsker filtrations.

In this paper we continue the study of the higher-rank graphs associated to finite-dimensional complex semisimple Lie algebras, introduced by the author and R. Yuncken, whose construction relies on Kashiwara's theory of crystals. First we prove that the Bruhat graphs of the corresponding Weyl groups, both weak and strong, can be embedded into the higher-rank graphs as colored graphs. Next, specializing to Lie algebras of type $A$, we connect some aspects of the construction of the higher-rank graphs with some well-known notions in combinatorics, most notably the keys of Lascoux and Sch\"utzenberger.

Calegari and Venkatesh did construct, modulo small torsion, a surjection from the degree 2 homology of the rank 2 projective general linear group over a ring of algebraic integers (of odd class number, and with enough embeddings) to the 2nd algebraic K-group of that ring. They asked whether this surjection becomes an isomorphism when passing to the quotient modulo the Eisenstein ideal on the left hand side. We provide a partial answer to this question, by lifting elements against the direction of the surjection, enabled by a theorem we establish in a more general setting which passes via Steinberg homology.

The two-phase Navier-Stokes Cahn-Hilliard (NSCH) mixture model is a key framework for simulating multiphase flows with non-matching densities. Developing fully discrete, energy-stable schemes for this model remains challenging, due to the possible presence of negative densities. While various methods have been proposed, ensuring provable energy stability under phase-field modifications, like positive extensions of the density, remains an open problem. We propose a simple, fully discrete, energy-stable method for the NSCH mixture model that ensures stability with respect to the energy functional, where the density in the kinetic energy is positively extended. The method is based on an alternative but equivalent formulation using mass-averaged velocity and volume-fraction-based order parameters, simplifying implementation while preserving theoretical consistency. Numerical results demonstrate that the proposed scheme is robust, accurate, and stable for large density ratios, addressing key challenges in the discretization of NSCH models.

We consider the problem of optimally steering the state covariance matrix of a discrete-time linear stochastic system to a desired terminal covariance matrix, while inducing the control input to be zero over many time intervals. We propose to induce sparsity in the feedback gain matrices by using a sum-of-norms version of the iteratively reweighted $\ell_1$-norm minimization. We show that the lossless convexification property holds even with the regularization term. Numerical simulations show that the proposed method produces a Pareto front of transient cost and sparsity that is not achievable by a simple $\ell_1$-norm minimization and closely approximates the $\ell_0$-norm minimization obtained from brute-force search.

We present a new sumcheck protocol called Fold-DCS (Fold-Divide-and-Conquer-Sumcheck) for multivariate polynomials based on a divide-and-conquer strategy. Its round complexity and soundness error are logarithmic in the number of variables, whereas they are linear in the classical sumcheck protocol. This drastic improvement in number of rounds and soundness comes at the expense of exchanging multivariate polynomials, which can be alleviated using polynomial commitment schemes. We first present Fold-DCS in the PIOP model, where the prover provides oracle access to a multivariate polynomial at each round. We then replace this oracle access in practice with a multivariate polynomial commitment scheme; we illustrate this with an adapted version of the recent commitment scheme Zeromorph [KT24], which allows us to replace most of the queries made by the verifier with a single batched evaluation check.

This paper concerns the eigenvalues of the Neumann-Poincar\'e operator, a boundary integral operator associated with the harmonic double-layer potential. Specifically, we examine how the eigenvalues depend on the support of integration and prove that the map associating the support's shape to the eigenvalues is real-analytic. We then compute its first derivative and present applications of the resulting formula. The proposed method allows for handling infinite-dimensional perturbation parameters for multiple eigenvalues and perturbations that are not necessarily in the normal direction.

We show that a positive proportion of linear forms in four variables admit a solution in the primes that is as small as one would heuristically expect. Out of the linear forms that satisfy certain local solvability conditions, almost all admit small prime solutions.

In this paper, we present performance estimates for stochastic economic MPC schemes with risk-averse cost formulations. For MPC algorithms with costs given by the expectation of stage cost evaluated in random variables, it was recently shown that the guaranteed near-optimal performance of abstract MPC in random variables coincides with its implementable variant coincide using measure path-wise feedback. In general, this property does not extend to costs formulated in terms of risk measures. However, through a turnpike-based analysis, this paper demonstrates that for a particular class of risk measures, this result can still be leveraged to formulate an implementable risk-averse MPC scheme, resulting in near-optimal averaged performance.

We axiomatically derive a family of wavelets for an orientation score, lifting from position space $\mathbb{R}^2$ to position and orientation space $\mathbb{R}^2\times S^1$, with fast reconstruction property, that minimise position-orientation uncertainty. We subsequently show that these minimum uncertainty states are well-approximated by cake wavelets: for standard parameters, the uncertainty gap of cake wavelets is less than 1.1, and in the limit, we prove the uncertainty gap tends to the minimum of 1. Next, we complete a previous theoretical argument that one does not have to train the lifting layer in (PDE-)G-CNNs, but can instead use cake wavelets. Finally, we show experimentally that in this way we can reduce the network complexity and improve the interpretability of (PDE-)G-CNNs, with only a slight impact on the model's performance.

We develop a structural approach to simultaneous embeddability in temporal sequences of graphs, inspired by graph minor theory. Our main result is a classification theorem for 2-connected temporal sequences: we identify five obstruction classes and show that every 2-connected temporal sequence is either simultaneously embeddable or admits a sequence of improvements leading to an obstruction. This structural insight leads to a polynomial-time algorithm for deciding the simultaneous embeddability of 2-connected temporal sequences. The restriction to 2-connected sequences is necessary, as the problem is NP-hard for connected graphs, while trivial for 3-connected graphs. As a consequence, our framework also resolves the rooted-tree SEFE problem, a natural extension of the well-studied sunflower SEFE. More broadly, our results demonstrate the applicability of graph minor techniques to evolving graph structures and provide a foundation for future algorithmic and structural investigations in temporal graph theory.

Let $(x_n)_{n=1}^\infty$ be a sequence of integers. We study the number variance of dilations $(\alpha x_n)_{n=1}^\infty$ modulo 1 in intervals of length $S$, and establish pseudorandom (Poissonian) behavior for Lebesgue-almost all $\alpha$ throughout a large range of $S$, subject to certain regularity assumptions imposed upon $(x_n)_{n=1}^\infty$. For the important special case $x_n = p(n)$, where $p$ is a polynomial with integer coefficients of degree at least 2, we prove that the number variance is Poissonian for almost all $\alpha$ throughout the range $0 \leq S \leq (\log N)^{-c}$, for a suitable absolute constant $c>0$. For more general sequences $(x_n)_{n=1}^\infty$, we give a criterion for Poissonian behavior for generic $\alpha$ which is formulated in terms of the additive energy of the finite truncations $(x_n)_{n=1}^N$.

We elaborate on the intimate connection between the largest volume of an empty axis-parallel box in a set of $n$ points from $[0,1]^d$ and cover-free families from the extremal set theory. This connection was discovered in a recent paper of the authors. In this work, we apply a very recent result of Michel and Scott to obtain a whole range of new lower bounds on the number of points needed so that the largest volume of such a box is bounded by a given $\varepsilon$. Surprisingly, it turns out that for each of the new bounds, there is a choice of the parameters $d$ and $\varepsilon$ such that the bound outperforms the others.

We propose a fast fourth-order cut cell method for solving constant-coefficient elliptic equations in two-dimensional irregular domains. In our methodology, the key to dealing with irregular domains is the poised lattice generation (PLG) algorithm that generates finite-volume interpolation stencils near the irregular boundary. We are able to derive high-order discretization of the elliptic operators by least squares fitting over the generated stencils. We then design a new geometric multigrid scheme to efficiently solve the resulting linear system. Finally, we demonstrate the accuracy and efficiency of our method through various numerical tests in irregular domains.

Throughout physics Brownian dynamics are used to describe the behaviour of molecular systems. When the Brownian particle is confined to a bounded domain, a particularly important question arises around determining how long it takes the particle to encounter certain regions of the boundary from which it can escape. Termed the first passage time, it sets the natural timescale of the chemical, biological, and physical processes that are described by the stochastic differential equation. Probabilistic information about the first passage time can be studied using spectral properties of the deterministic generator of the stochastic process. In this work we introduce a framework for bounding the leading eigenvalue of the generator which determines the exponential rate of escape of the particle from the domain. The method employs sum-of-squares programming to produce nearly sharp numerical upper and lower bounds on the leading eigenvalue, while also giving good numerical approximations of the associated leading eigenfunction, the quasi-stationary distribution of the process. To demonstrate utility, the method is applied to prototypical low-dimensional problems from the literature.

In this paper we provide sufficient conditions for sequences of stochastic processes of the form $\int_{[0,t]} f_n(u) \theta_n(u) du$, to weakly converge, in the space of continuous functions over a closed interval, to integrals with respect to the Brownian motion, $\int_{[0,t]} f(u)W(du)$, where $\{f_n\}_n$ is a sequence satisfying some integrability conditions converging to $f$ and $\{\theta_n\}_n$ is a sequence of stochastic processes whose integrals $\int_{[0,t]}\theta_n(u)du$ converge in law to the Brownian motion (in the sense of the finite dimensional distribution convergence), in the multidimensional parameter set case.

The Eberlein diagonalization method is an iterative Jacobi-type method for solving the eigenvalue problem of a general complex matrix. In this paper we develop the block version of the Eberlein method. We prove the global convergence of our block method and present several numerical examples.

We provide a unique decomposition of every 4-connected graph into parts that are either quasi-5-connected, cycles of triangle-torsos and 3-connected torsos on $\leq 5$ vertices, generalised double-wheels, or thickened $K_{4,m}$'s. The decomposition can be described in terms of a tree-decomposition but with edges allowed in the adhesion-sets. Our construction is explicit, canonical, and exhibits a defining property of the Tutte-decomposition. As a corollary, we obtain a new Tutte-type canonical decomposition of 3-connected graphs into parts that are either quasi-4-connected, generalised wheels or thickened $K_{3,m}$'s. This decomposition is similar yet different from the tri-separation decomposition. As an application of the decomposition for 4-connectivity, we obtain a new theorem characterising all 4-connected vertex-transitive finite graphs as quasi-5-connected, the $K_4$-expansion of a quasi-5-connected graph, or on a short explicit list of graphs.

We consider the two-species totally asymmetric simple exclusion process on $\mathbb{Z}$ with a translation-invariant stationary measure as the initial condition. We establish the asymptotic decoupling of the marginal height profiles along characteristic lines and prove the decay of the two-point functions in the large-time limit, thus confirming predictions of the nonlinear fluctuating hydrodynamics theory. Our approach builds on the queueing construction of the stationary measure introduced in [Angel'06, Ferrari-Martin'07] and extends the theory of backwards paths for height functions developed in [Bufetov-Ferrari'22, Ferrari-Nejjar'24]. The arguments for asymptotic decoupling also apply to further homogeneous initial data.

We design a recursive algorithm to compute the partition function of the Ising model, summed over cubic maps with fixed size and genus. The algorithm runs in polynomial time, which is much faster than methods based on a Tutte-like, or topological, recursion. We construct this algorithm out of a partial differential equation that we derive from the first equation of the KP hierarchy satisfied by the generating function of bipartite maps. This series is indeed related to the Ising partition function by a change of variables. We also obtain inequalities on the coefficients of this partition function, which should be useful for a probabilistic study of cubic Ising maps whose genus grows linearly with their size.

It is a challenging problem that solving the \textit{multivariate linear model} (MLM) $\mathbf{A}\mathbf{x}=\mathbf{b}$ with the $\ell_1 $-norm approximation method such that $||\mathbf{A}\mathbf{x}-\mathbf{b}||_1$, the $\ell_1$-norm of the \textit{residual error vector} (REV), is minimized. In this work, our contributions lie in two aspects: firstly, the equivalence theorem for the structure of the $\ell_1$-norm optimal solution to the MLM is proposed and proved; secondly, a unified algorithmic framework for solving the MLM with $\ell_1$-norm optimization is proposed and six novel algorithms (L1-GPRS, L1-TNIPM, L1-HP, L1-IST, L1-ADM, L1-POB) are designed. There are three significant characteristics in the algorithms discussed: they are implemented with simple matrix operations which do not depend on specific optimization solvers; they are described with algorithmic pseudo-codes and implemented with Python and Octave/MATLAB which means easy usage; and the high accuracy and efficiency of our six new algorithms can be achieved successfully in the scenarios with different levels of data redundancy. We hope that the unified theoretic and algorithmic framework with source code released on GitHub could motivate the applications of the $\ell_1$-norm optimization for parameter estimation of MLM arising in science, technology, engineering, mathematics, economics, and so on.

In this paper we consider large-scale composite nonconvex optimization problems having the objective function formed as a sum of three terms, first has block coordinate-wise Lipschitz continuous gradient, second is twice differentiable but nonseparable and third is the indicator function of some separable closed convex set. Under these general settings we derive and analyze a new cyclic coordinate descent method, which uses the partial gradient of the differentiable part of the objective, yielding a coordinate gradient descent scheme with a novel adaptive stepsize rule. We prove that this stepsize rule makes the coordinate gradient scheme a descent method, provided that additional assumptions hold for the second term in the objective function. We also present a worst-case complexity analysis for this new method in the nonconvex settings. Numerical results on orthogonal nonnegative matrix factorization problem also confirm the efficiency of our algorithm.

We show that Miyaoka's bound for the number of conics on a degree-2h K3 surface is attained for high h, and analogously for higher even degree (smooth) rational curves.

We prove, with no claim to originality, a relative version of the Fujita-Zariski theorem. When the base is a field, this result is due to Fujita (1983) and states that if an invertible sheaf on a proper variety is ample on its base locus, its sufficiently high powers are globally generated. The special case where the base locus is finite was proved by Zariski (1962), whence the name.

In this paper, we consider a modified projected Gauss-Newton method for solving constrained nonlinear least-squares problems. We assume that the functional constraints are smooth and the the other constraints are represented by a simple closed convex set. We formulate the nonlinear least-squares problem as an optimization problem using the Euclidean norm as a merit function. In our method, at each iteration we linearize the functional constraints inside the merit function at the current point and add a quadratic regularization, yielding a strongly convex subproblem that is easy to solve, whose solution is the next iterate. We present global convergence guarantees for the proposed method under mild assumptions. In particular, we prove stationary point convergence guarantees and under Kurdyka-Lojasiewicz (KL) property for the objective function we derive convergence rates depending on the KL parameter. Finally, we show the efficiency of this method on the power flow analysis problem using several IEEE bus test cases.

In this paper, we investigate reconfigurable pixel antenna (RPA)-based electronic movable antennas (REMAs) for multiuser communications. First, we model each REMA as an antenna characterized by a set of predefined and discrete selectable radiation positions within the radiating region. Considering the trade-off between performance and cost, we propose two types of REMA-based arrays: the partially-connected RPA-based electronic movable-antenna array (PC-REMAA) and fully-connected REMAA (FC-REMAA). Then, we formulate a multiuser sum-rate maximization problem subject to the power constraint and hardware constraints of the PC-REMAA or FC-REMAA. To solve this problem, we propose a two-step multiuser beamforming and antenna selection scheme. In the first step, we develop a two-loop joint beamforming and antenna selection (TL-JBAS) algorithm. In the second step, we apply the coordinate descent method to further enhance the solution of the TL-JBAS algorithm. In addition, we revisit mechanical movable antennas (MMAs) to establish a benchmark for evaluating the performance of REMA-enabled multiuser communications, where MMAs can continuously adjust the positions within the transmission region. We also formulate a sum-rate maximization problem for MMA-enabled multiuser communications and propose an alternating beamforming and antenna position optimization scheme to solve it. Finally, we analyze the performance gap between REMAs and MMAs. Based on Fourier analysis, we derive the maximum power loss of REMAs compared to MMAs for any given position interval. Specifically, we show that the REMA incurs a maximum power loss of only 3.25\% compared to the MMA when the position interval is set to one-tenth of the wavelength. Simulation results demonstrate the effectiveness of the proposed methods.

In this paper we prove the unirationality of the locus of bielliptic curves in the Hilbert scheme of canonical curves of genus $g \geq 11$. As a consequence, we obtain another proof for the unirationality of the bielliptic locus in the moduli space of curves of genus $g \geq 11$.

Korn-Maxwell-Sobolev (KMS) inequalities represent a tool for estimating differential expressions and have gained particular importance in recent years, especially concerning elliptic operators. In my Master's thesis, together with Peter Lewintan (University of Duisburg-Essen), we extended this concept to also apply to operators of constant rank. This makes it possible to cover more complex structures such as the curl or divergence of vector fields. A key difference from the elliptic theory is that in the constant rank case, a correction term $\Pi_\mathbb{B}$ is necessary on the left-hand side of the inequality. Results were also obtained for the limiting case $p=1$, although additional assumptions are required here. This article provides an illustrative introduction to KMS inequalities and demonstrates their application in both the elliptic and constant rank cases.

We deal with the asymptotic analysis for Laplace's integral. For this problem, the so-called Laplace's method by P.S. Laplace (1812) is well-known and it has been developed in various forms over many years of studies. In this paper, we derive some formulas of the higher-order asymptotic expansions for that integral, with error estimates, which generalize previous results. Moreover, we discuss a comparison of these asymptotic formulas with approximations using numerical integral.

In this paper, we consider the initial value problem for some nonlinear second-order ODEs of Duffing type. We study the large time behavior of the solutions to this problem, from both the perspectives of mathematical and numerical analysis. First, we derive the decay estimate of the solutions, by using the energy method. Moreover, we numerically investigate the large time behavior of the energy function related to this problem, by using a structure-preserving difference method.

In 2022, Bergelson and Richter established a new dynamical generalization of the prime number theorem (PNT). Let $f(x)\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\ge 1$. Let $k\ge2$ be an integer. The number of integers $n$ such that $f(n)$ is $k$-free is widely studied in the literature. In principle, one expects that $f(n)$ is $k$-free infinitely often, if $f$ has no fixed $k$-th power divisor. Inspired by the work of Bergelson and Richter, we expect that their generalization of the PNT also holds over such integers of power-free polynomial values. In this note, we establish such variant of Bergelson and Richter's theorem for several polynomials studied by Estermann, Hooley, Heath-Brown, Booker and Browning.

The goal of this paper is to prove that the index of intersection of two complex curves in a two-dimensional complex manifold touching each other at a common boundary point is positive. This is achieved via the construction of a totally real surface such that the curves in question are attached to it by some parts of their boundaries and then defining a certain ``boundary intersection index'' of two complex ``half-disks'' with their edges on a totally real surface. We prove that this index is always positive. This second result holds true, more generally, for pseudoholomorphic curves with cusps in a two dimensional almost complex manifold, but to our best knowledge is new even for integrable structures, unless the totally real surface in question is supposed to be real analytic.

In this paper, we introduce and study the notion of the Ces\'aro operator $C_{\mathscr T}$ on a rooted directed tree $\mathscr T.$ If $\mathscr T$ is the rooted tree with no branching vertex, then $C_{\mathscr T}$ is unitarily equivalent to the classical Ces\'aro operator $C_{0}$ on the sequence space $\ell^2(\mathbb N).$ Besides some basic properties related to boundedness and spectral behavior, we show that $C_{\mathscr T}$ is subnormal if and only if $\mathscr T$ is isomorphic to the rooted directed tree $\mathbb N$ provided $\mathscr T$ is of finite branching index. In particular, the verbatim analogue of Kriete-Trutt theorem fails for Ces\'aro operators on rooted directed trees. However, if, in addition, $\mathscr T$ is locally finite, then $C_{\mathscr T}$ is a compact perturbation of a subnormal operator.

In this paper, we compare the performance of different numerical schemes in approximating Pontryagin's Maximum Principle's necessary conditions for the optimal control of nonholonomic systems. Retraction maps are used as a seed to construct geometric integrators for the corresponding Hamilton equations. First, we obtain an intrinsic formulation of a discretization map on a distribution $\mathcal{D}$. Then, we illustrate this construction on a particular example for which the performance of different symplectic integrators is examined and compared with that of non-symplectic integrators.

Recently, there has been a surge of research on a class of methods called feedback optimization. These are methods to steer the state of a control system to an equilibrium that arises as the solution of an optimization problem. Despite the growing literature on the topic, the important problem of enforcing state constraints at all times remains unaddressed. In this work, we present the first feedback-optimization method that enforces state constraints. The method combines a class of dynamics called safe gradient flows with high-order control barrier functions. We provide a number of results on our proposed controller, including well-posedness guarantees, anytime constraint-satisfaction guarantees, equivalence between the closed-loop's equilibria and the optimization problem's critical points, and local asymptotic stability of optima.

Considering the $B$-branes over a complex manifold as the objects of the bounded derived category of coherent sheaves on that manifold, we extend the definition of holomorphic gauge fields on vector bundles to $B$-branes. We construct a family of coherent sheaves on the complex projective space, which generates the corresponding bounded derived category and such that the supports of the elements of this family are two by two disjoint. Using that family, we prove that the cardinal of the set of holomorphic gauge fields on any $B$-brane over the projective space is less than $2.$

We prove level raising results for $p$-adic automorphic forms on definite unitary groups $U(3)/\mathbb{Q}$ and deduce some intersection points on the eigenvariety. Let $l$ be an inert prime where the level subgroups varies, if there is a non-very-Eisenstein point $\phi$ on the old component (generically parametrizing forms old at $l$) satisfying $T_{l}(\phi)=l(l^3+1)$, then this point also lies in the new component (generically parametrizing forms new at $l$). This provides a $p$-adic analogue of Bella\"iche and Graftieaux's mod $p$ level raising for classical automorphic forms on $U(3)$, and also generalizes James Newton's $p$-adic level raising results for definite quaternion algebras. Key ingredients include abelian Ihara lemma (proved for any definite unitary group $U(n)$) and some duality arguments about certain Hecke modules. Finally we also discuss some methods to construct such points explicitly and further development.

We propose a data-driven approach for large-scale cellular network optimization, using a production cellular network in London as a case study and employing Sionna ray tracing for site-specific channel propagation modeling. We optimize base station antenna tilts and half-power beamwidths, resulting in more than double the 10\%-worst user rates compared to a 3GPP baseline. In scenarios involving aerial users, we identify configurations that increase their median rates fivefold without compromising ground user performance. We further demonstrate the efficacy of model generalization through transfer learning, leveraging available data from a scenario source to predict the optimal solution for a scenario target within a similar number of iterations, without requiring a new initial dataset, and with a negligible performance loss.

In this article, we introduce a novel geometric constant $L_X(t)$, which provides an equivalent definition of the von Neumann-Jordan constant from an orthogonal perspective. First, we present some fundamental properties of the constant $L_X(t)$ in Banach spaces, including its upper and lower bounds, as well as its convexity, non-increasing continuity. Next, we establish the identities of $L_X(t)$ and the function $\gamma_X(t)$, the von Neumann-Jordan constant, respectively. We also delve into the relationship between this novel constant and several renowned geometric constants (such as the James constant and the modulus of convexity). Furthermore, by utilizing the lower bound of this new constant, we characterize Hilbert spaces. Finally, based on these findings, we further investigate the connection between this novel constant and the geometric properties of Banach spaces, including uniformly non-square, uniformly normal structure, uniformly smooth, etc.

In the past, Takahashi has introduced the James type constants $\mathcal{J}_{\mathcal{X} ,t}(\tau)$. Building upon this foundation, we introduce an innovative skew James type constant, denoted as $\mathcal{J}_t[\tau,\mathcal{X}]$, which is perceived as a skewed counterpart to the traditional James type constants. We delineate a novel constant, and proceed to ascertain its equivalent representations along with certain attributes within the context of Banach spaces, and then an investigation into the interrelation between the skewness parameter and the modulus of convexity is conducted, after that we define another new constant $\mathcal{G}_t(\mathcal{X})$, and some conclusions were drawn.

In this work, we prove that the product of a function belonging to a Hardy-Orlicz space $H^{\Phi_{1}}$ and a function from another Hardy-Orlicz space $H^{\Phi_{2}}$ belongs to a third Hardy-Orlicz space $H^{\Phi_{3}}$. Moreover, we establish the converse: any holomorphic function in the space $H^{\Phi_{3}}$ can be expressed as the product of two functions, one from $H^{\Phi_{1}}$ and the other from $H^{\Phi_{2}}$. Subsequently, we use this factorization result in Hardy-Orlicz spaces to study the continuity of the Hankel operator in these spaces. More specifically, we provide gain and loss estimates for the norms of the Hankel operator in the context of analyzing its continuity in Hardy-Orlicz spaces.

After the discovery of Khovanov homology, which categorifies the Jones polynomial, an analogous categorification of the chromatic polynomial, known as chromatic homology, was introduced. Its graded Euler characteristic recovers the chromatic polynomial. In this paper, we present a spanning tree model for the chromatic complex, i.e., we describe a chain complex generated by certain spanning trees of the graph that is chain homotopy equivalent to the chromatic complex. We employ the spanning tree model over $\mathcal{A}_m:= \frac{\mathbb{Z}[x]}{}$ algebra to answer two open questions. First, we establish the conjecture posed by Sazdanovic and Scofield regarding the homological span of chromatic homology over $\\mathcal{A}_m$ algebra, demonstrating that for any graph $G$ with $v$ vertices and $b$ blocks, the homological span is $v - b$. Additionally, we prove a conjecture of Helme-Guizon, Przytycki, and Rong concerning the existence of torsion of order dividing $m$ in chromatic homology over $\mathcal{A}_m$ algebra.

Motzkin chain is a model of nearest-neighbor interacting quantum $s=1$ spins with open boundary conditions. It is known that it has a unique ground state which can be viewed as a sum of Motzkin paths. We consider the case of periodic boundary conditions and provide several conjectures about structure of the ground state space and symmetries of the Hamiltonian. We conjecture that the ground state is degenerate and independent states distinguished by the eigenvalue of the third component of the total spin operator. Each of these states can be described as a sum of paths, similar to the Motzkin paths. Moreover, there exist two operators commuting with the Hamiltonian, which play the roles of lowering and raising operators when acting at these states. We conjecture also that these operators generate the Lie algebra of $C$-type of the rank equal to the number of sites. The symmetry algebra of the Hamiltonian is actually wider, and extended, besides the cyclic shift operator, by a central element contained in the third component of the total spin operator.

Spingarn's method of partial inverses and the progressive decoupling algorithm address inclusion problems involving the sum of an operator and the normal cone of a linear subspace, known as linkage problems. Despite their success, existing convergence results are limited to the so-called elicitable monotone setting, where nonmonotonicity is allowed only on the orthogonal complement of the linkage subspace. In this paper, we introduce progressive decoupling+, a generalized version of standard progressive decoupling that incorporates separate relaxation parameters for the linkage subspace and its orthogonal complement. We prove convergence under conditions that link the relaxation parameters to the nonmonotonicity of their respective subspaces and show that the special cases of Spingarn's method and standard progressive decoupling also extend beyond the elicitable monotone setting. Our analysis hinges upon an equivalence between progressive decoupling+ and the preconditioned proximal point algorithm, for which we develop a general local convergence analysis in a certain nonmonotone setting.

We construct a family of finitely generated infinite periodic groups. The basic example is a 2-group, called the tetrahedron group. We generalize the construction by suggesting a family of infinite finitely generated dice groups. We provide weak conditions under which dice groups are periodic, where orders of elements are products involving finitely many given primes.

For curves singularities the dimension of smoothing components in the deformation space is an invariant of the singularity, but in general the deformation space has components of different dimensions. We are interested in the question what the generic singularities are above these components. To this end we revisit the known examples of non-smoothable singularities and study their deformations. There are two general methods available to show that a curve is not smoothable. In the first method one exhibits a family of singularities of a certain type and then uses a dimension count to prove that the family cannot lie in the closure of the space of smooth curves. The other method is specific for curves and uses the semicontinuity of a certain invariant, related to the Dedekind different. This invariant vanishes for Gorenstein, so in particular for smooth curves. With these methods and computations with computer algebra systems we study monomial curves and cones over point sets in projective space. We also give new explicit examples of non-smoothable singularities. In particular, we find non-smoothable Gorenstein curve singularities. The cone over a general self-associated point set is not smoothable, as the point set cannot be a hyperplane section of a canonical curve, if the genus is at least 11.

We construct a time-independent, incompressible, and Lipschitz-continuous velocity field in $\mathbb{R}^3$ that generates a fast kinematic dynamo - an instability characterized by exponential growth of magnetic energy, independent of diffusivity. Specifically, we show that the associated vector transport-diffusion equation admits solutions that grow exponentially fast, uniformly in the vanishing diffusivity limit $\varepsilon\to 0$. Our construction is based on a periodic velocity field $U$ on $\mathbb{T}^3$, such as an Arnold-Beltrami-Childress flow, which satisfies a generic spectral instability property called alpha-instability, established via perturbation theory. This provides a rigorous mathematical framework for the alpha-effect, a mechanism conjectured in the late 1960s to drive large-scale magnetic field generation. By rescaling with respect to $\varepsilon$ and employing a Bloch-type theorem, we extend the solution to the whole space. Finally, through a gluing procedure that spatially localizes the instability, we construct a globally defined velocity field $u$ in $\mathbb{R}^3$ that drives the dynamo instability.

An alternating torus knot or link may be constructed from a repeating double helix after connecting its two ends. A structure with additional helices may be closed to form a non-alternating torus knot or link. Previous work has optimized the dimensions and pitch of double helices to derive upper bounds on the ropelength of alternating torus knots, but non-alternating knots have not been studied extensively and are known to be tighter. Here, we examine concentric helices as units of non-alternating torus knots and discuss considerations for minimizing their contour length. By optimizing both the geometry and combinatorics of the helices, we find efficient configurations for systems with between 3 and 39 helices. Using insights from those cases, we develop an efficient construction for larger systems and show that concentric helices distributed between many shells have an optimized ropelength of approximately 7.83Q^(3/2) where Q is the total number of helices or the minor index of the torus knot, and the prefactor is exact and a 75 percent reduction from previous work. Links formed by extending these helices and bending them into a T(3Q,Q) torus link have a ropelength that is approximately 12 times the three-quarter power of the crossing number. These results reduce the ratio between the upper and lower bounds of the ropelength of non-alternating torus knots from 29 to between 1.4 and 3.8.

In this note, we prove the Zilber--Pink conjecture for subvarieties of mixed Shimura varieties, which are not defined over~$\overline{\mathbb Q}$ in a strong sense. We prove similar results for general variations of mixed Hodge structure of geometric origin, assuming furthermore that they are absolute.

In this paper, we provide a geometric characterization of virtual nonlinear nonholonomic constraints from a symplectic perspective. Under a transversality assumption, there is a unique control law making the trajectories of the associated closed-loop system satisfy the virtual nonlinear nonholonomic constraints. We characterize them in terms of the symplectic structure on $TQ$ induced by a Lagrangian function and the almost-tangent structure. In particular, we show that the closed-loop vector field satisfies a geometric equation of Chetaev type. Moreover, the closed-loop dynamics is obtained as the projection of the uncontrolled dynamics to the tangent bundle of the constraint submanifold defined by the virtual constraints.

In the article we study the simple unital communitative three-dimensional algebras over an algebraically closed field of characteristic not equal to 2. It is proved that every simple unital communitative three-dimensional algebra of nil-rank 2 is isotopic to Jordan algebra of a symmetric bilinear nondegenerate form on a two-dimensional vector space.

We show that mean field optimal controls satisfy a first order optimality condition (at a.e. time) without any a priori requirement on their spatial regularity. This principle is obtained by a careful limit procedure of the Pontryagin maximum principle for finite particle systems. In particular, our result applies to the case of mean field selective optimal control problems for multipopulation and replicator dynamics.

In this work, we address the efficient computation of parameterized systems of linear equations, with possible nonlinear parameter dependence. When the matrix is highly sensitive to the parameters, mean-based preconditioning might not be enough. For this scenario, we explore an approach in which several preconditioners are placed in the parameter space during a precomputation step. To determine the optimal placement of a limited number of preconditioners, we estimate the expected number of iterations with respect to a given preconditioner a priori and use a location-allocation strategy to optimize the placement of the preconditioners. We elaborate on our methodology for the Helmholtz problem with exterior Dirichlet scattering at high frequencies, and we estimate the expected number of GMRES iterations via a gray-box Gaussian process regression approach. We illustrate our approach in two practical applications: scattering in a domain with a parametric refractive index and scattering from a scatterer with parameterized shape. Using these numerical examples, we show how our methods leads to runtime savings of about an order of magnitude. Moreover, we investigate the effect of the parameter dimension and the importance of dimension anisotropy on their efficacy.

We introduce the Brackets package for the computer algebra system Macaulay2, which provides convenient syntax for computations involving the classical invariants of the special linear group. We describe our implementation of bracket rings and Grassmann-Cayley algebras, and illustrate basic functionality such as the straightening algorithm on examples from projective and enumerative geometry.

Robust optimization(RO) is an important tool for handling optimization problem with uncertainty. The main objective of RO is to solve optimization problems due to uncertainty associated with constraints satisfying all realizations of uncertain values within a given uncertainty set. The challenge of RO is to reformulate the constraints so that the uncertain optimization problem is transformed into a tractable deterministic form. In this paper, we have given more emphasis to study the robust counterpart(RC) of the RO problems and have developed a mathematical model on the solution strategy for robust linear optimization problems, where the constraints only are associated with uncertainties. The box and ellipsoidal uncertainty sets are considered and some illustrative numerical examples have been solved in each corresponding case for validating our proposed method.

We show a stability result for the recently established Brunn--Minkowski inequality in compact simple Lie groups. Namely, we prove that if two compact subsets $A, B$ of a compact simple Lie group $G$ satisfy $$ \mu(AB)^{1/d'} \leq (1 + \epsilon)\left(\mu(A)^{1/d'} + \mu(B)^{1/d'}\right)$$ where $AB$ is the Minkowski product $\{ab : a \in A, b \in B\}$, $d'$ denotes the minimal codimension of a proper closed subgroup and $\mu$ is a Haar measure, then $A$ and $B$ must approximately look like neighbourhoods of a proper subgroup $H$ of codimension $d'$, with an error that depends quantitatively on $d', \epsilon$ and the ratio $\frac{\mu(A)}{\mu(B)}$. This result implies an improved error rate in the Brunn--Minkowski inequality in compact simple Lie groups $$\mu(AB)^{\frac{1}{d'}} \geq (1-C\mu(A)^{\frac{2}{d'}})\left(\mu(A)^{\frac{1}{d'}} + \mu(B)^{\frac{1}{d'}}\right) $$ sharp, up to the constant $C$ which depends on $d'$ and $\frac{\mu(A)}{\mu(B)}$ alone. Our approach builds upon an earlier paper of the author proving the Brunn--Minkowski inequality, and stability in the case $A=B$. We employ a combinatorial multi-scale analysis and study so-called density functions. Additionally, the asymmetry between $A$ and $B$ introduces new challenges, requiring the use of non-abelian Fourier theory and stability results for the Pr\'ekopa--Leindler inequality.

A toric amplitude is a rational function associated to a simplicial polyhedral fan. The definition is inspired by scattering amplitudes in particle physics. We prove algebraic properties of such amplitudes and study the geometry of their zero loci. These hypersurfaces play the role of Warren's adjoint via a dual volume interpretation. We investigate their Fano schemes and singular loci via the nef cone and toric irrelevant ideal of the fan.

We prove that, as $m$ grows, any family of $m$ homotopically distinct closed curves on a surface induces a number of crossings that grows at least like $(m \log m)^2$. We use this to answer two questions of Pach, Tardos and Toth related to crossing numbers of drawings of multigraphs where edges are required to be non-homotopic. Furthermore, we generalize these results, obtaining effective bounds with optimal growth rates on every orientable surface.

We address the problem of nonparametric estimation of the spectral density for a centered stationary Gaussian time series under local differential privacy constraints. Specifically, we propose new interactive privacy mechanisms for three tasks: estimating a single covariance coefficient, estimating the spectral density at a fixed frequency, and estimating the entire spectral density function. Our approach achieves faster rates through a two-stage process: we apply first the Laplace mechanism to the truncated value and then use the former privatized sample to gain knowledge on the dependence mechanism in the time series. For spectral densities belonging to H\"older and Sobolev smoothness classes, we demonstrate that our estimators improve upon the non-interactive mechanism of Kroll (2024) for small privacy parameter $\alpha$, since the pointwise rates depend on $n\alpha^2$ instead of $n\alpha^4$. Moreover, we show that the rate $(n\alpha^4)^{-1}$ is optimal for estimating a covariance coefficient with non-interactive mechanisms. However, the $L_2$ rate of our interactive estimator is slower than the pointwise rate. We show how to use these estimators to provide a bona-fide locally differentially private covariance matrix estimator.

For a jointly integrable partially hyperbolic diffeomorphism $f$ on a 3-manifold $M$ with virtually solvable fundamental group which satisfies Diophantine condition along the center foliation, we show that the cohomological equation $\varphi = u\circ f - u + c$ has a continuous solution $u$ if and only if $\varphi$ has trivial periodic cycle functional.

We construct moduli spaces of framed logarithmic connections and also moduli spaces of framed parabolic connections. It is shown that these moduli spaces possess a natural algebraic symplectic structure. We also give an upper bound of the transcendence degree of the algebra of regular functions on the moduli space of parabolic connections.

In this paper, we consider the class $\mathcal{C}^d$ of sphere intersection graphs in $\mathbb{R}^d$ for $d \geq 2$. We show that for each integer $t$, the class of all graphs in $\mathcal{C}^d$ that exclude $K_{t,t}$ as a subgraph has strongly sublinear separators. We also prove that $\mathcal{C}^d$ has asymptotic dimension at most $2d+2$.

In 1980, Thomassen stated his Weak Linkage conjecture: for odd positive integers $k$, if a graph $G$ is $k$-edge-connected, then, for any collection of $k$ pairs of vertices $\{s_1,t_1\}$, ..., $\{s_k,t_k\}$ in $G$, not necessarily distinct, there are pairwise edge-disjoint paths $P_1,...,P_k$ in $G$, with $P_i$ joining $s_i$ and $t_i$. In 1991, Huck proved that the conclusion holds if $G$ is finite and $(k+1)$-edge-connected. We prove that Huck's theorem holds also for all infinite graphs, extending and improving a result of Ok, Richter and Thomassen for 1-ended, locally finite graphs. A novel key tool in the proof is the Linking Fan Proposition proved in Section 3. To show the potential and usefulness of this proposition in other contexts, we present in the last section a new result, similar to a result of Thomassen, on the existence of $2k$-edge-connected finite immersions in $(2k+2)$-edge-connected infinite graphs, and as a corollary we also reprove a statement on $k$-arc-connected orientations.

We consider finite normal covers of an infinite translation surface with a large Veech group - the Chamanara surface. We give a characterization for these covers to have Veech groups which are large themselves, that is, which are of finite index in the Veech group of the Chamanara surface. For covers of degree 2, we study these large Veech groups in detail. As an application, we show that every free group in finitely many generators can be realized as the projective Veech group of a translation surface of finite area.

In this work, we propose an exponentially convergent numerical method for the Caputo fractional propagator $S_\alpha(t)$ and the associated mild solution of the Cauchy problem with time-independent sectorial operator coefficient $A$ and Caputo fractional derivative of order $\alpha \in (0,2)$ in time. The proposed methods are constructed by generalizing the earlier developed approximation of $S_\alpha(t)$ with help of the subordination principle. Such technique permits us to eliminate the dependence of the main part of error estimate on $\alpha$, while preserving other computationally relevant properties of the original approximation: native support for multilevel parallelism, the ability to handle initial data with minimal spatial smoothness, and stable exponential convergence for all $t \in [0, T]$. Ultimately, the use of subordination leads to a significant improvement of the method's convergence behavior, particularly for small $\alpha < 0.5$, and opens up further opportunities for efficient data reuse. To validate theoretical results, we consider applications of the developed methods to the direct problem of solution approximation, as well as to the inverse problem of fractional order identification.

For every countable infinite group that admits $\mathbb{Z}$ as a homomorphic image, we show that for each $m\in\mathbb{N}$, there exists a minimal action whose topological sequence entropy is $\log(m)$. Furthermore, for every countable infinite group $G$ that contains a finite index normal subgroup $G'$ isomorphic to $\mathbb{Z}^r$, and for every $m\in \mathbb{N}$, we found a free minimal action with topological sequence entropy $\log(n)$, where $m\leq n\leq m^{2^r[G:G']}$. In both cases, we also show that the aforementioned minimal actions admit non-trivial independence tuples of size $n$ but do not admit non-trivial independence tuples of size $n+1$ for some $n\geq m$.

We study the distribution of the set of copies of some given graph $H$ in the random graph $G(n,p)$, focusing on the case when $H = K_r$. Our main results capture the 'leading term' in the difference between this distribution and the 'independent hypergraph model', where (in the case $H = K_r$) each copy is present independently with probability $\pi = p^{\binom{r}{2}}$. As a concrete application, we derive a new upper bound on the number of $K_r$-factors in $G(n,p)$ above the threshold for such factors to appear. We will prove our main results in a much more general setting, so that they also apply to random hypergraphs, and also (for example) to the case when $p$ is constant and $r = r(n) \sim 2\log_{1/p}(n)$.

We describe the eigenvalues and eigenvectors of real-analytic, non-self-adjoint Berezin--Toeplitz operators, up to exponentially small error, on complex one-dimensional compact manifolds, under the hypothesis of regularity of the energy levels. These results form a complex version of the Bohr-Sommerfeld quantization conditions; they hold under a hypothesis that the skew-adjoint part is small but can be of principal order with respect to the semiclassical parameter. To this end, we develop a calculus of Fourier Integral Operators and Lagrangian states associated with complex Lagrangians; these tools can be of independent interest.

We construct symmetric square type $L$-series for vector-valued modular forms transforming under the Weil representation associated to a discriminant form. We study Hecke operators and integral representations to investigate their properties, deriving functional equations and infinite product expansions.

Let $\mathcal{G}$ be a quantum graph without quantum sources and $E_\mathcal{G}$ be the quantum edge correspondence for $\mathcal{G}.$ Our main results include sufficient conditions for simplicity of the Cuntz-Pimsner algebra $\mathcal{O}_{E_\mathcal{G}}$ in terms of $\mathcal{G}$ and for defining a surjection from the quantum Cuntz-Krieger algebra $\mathcal{O}(\mathcal{G})$ onto a particular relative Cuntz-Pimsner algebra for $E_\mathcal{G}$. As an application of these two results, we give the first example of a quantum graph with distinct quantum Cuntz-Krieger and local quantum Cuntz-Krieger algebras. We also characterize simplicity of $\mathcal{O}_{E_\mathcal{G}}$ for some fundamental examples of quantum graphs, including rank-one quantum graphs on a single full matrix algebra, complete quantum graphs, and trivial quantum graphs. Along the way, we provide an equivalent condition for minimality of $E_\mathcal{G}$ and sufficient conditions for aperiodicity of $E_\mathcal{G}$ in terms of the underlying quantum graph $\mathcal{G}$.

In this paper we discuss nonholonomic contact Lagrangian and Hamiltonian systems, that is, systems with a kind of dissipation that are also subject to nonholonomic constraints. We introduce the so-called contact Eden bracket that allows us to obtain the evolution of any observable. Finally, we present a particular vector subspace of observables where the dynamics remain unconstrained.

Let $\mathcal G$ be a Hilbert space and $\mathfrak B(\mathcal G)$ the algebra of bounded operators, $\mathcal H=L_2([0,\infty);\mathcal G)$. An operator-valued function $Q\in L_{\infty,\rm loc}\left([0,\infty);\mathfrak B(\mathcal G)\right)$ determines a multiplication operator in $\mathcal H$ by $(Qy)(x)=Q(x)y(x)$, $x\geqslant0$. We say that an operator $L_0$ in a Hilbert space is a Schr\"odinger type operator, if it is unitarily equivalent to $-d^2/dx^2+Q(x)$ on a relevant domain. The paper provides a characterization of a class of such operators. The characterization is given in terms of properties of an evolutionary dynamical system associated with $L_0$. It provides a way to construct a functional Schr\"odinger model of $L_0$.

We introduce ninfty, a header-only C++ library distributed under an MIT Open Source License designed for the study of enumeration problems arising in homotopical combinatorics. The ninfty repository moreover contains a folder with data files for many common finite groups. This is in addition to Sage code which can be used to generate input data for further finite groups, and Sage code for generating input data for abstract lattices which may not arise at the subgroup lattice of a group.

In an abelian category, the (bi)fibration of subobjects is isomorphic to the (bi)fibration of quotients. This property captures a substantial information about the exactness structure of a category. Indeed, as it was shown by the second author and T.~Weighill, categories equipped with a proper factorization system such that the opfibration of subobjects relative to the factorization system is isomorphic to the fibration of relative quotients are precisely the Grandis exact categories. Motivated by on-going work of the second author with \"Ulo Reimaa, indicating that the 2-category of abelian categories (with suitably chosen morphisms) satisfies some 2-dimensional exactness conditions, in this paper we propose a 2-dimensional notion of a Grandis exact category. We reach such definition as an outcome of characterizing those $(1,1)$-proper factorization systems on a 2-category in the sense of M.~Dupont and E.~Vitale, for which the weak 2-opfibration of relative 2-subobjects is biequivalent to the weak 2-fibration of relative 2-quotients.

We study MaxCut on 3-regular graphs of minimum girth $g$ for various $g$'s. We obtain new lower bounds on the maximum cut achievable in such graphs by analyzing the Quantum Approximate Optimization Algorithm (QAOA). For $g \geq 16$, at depth $p \geq 7$, the QAOA improves on previously known lower bounds. Our bounds are established through classical numerical analysis of the QAOA's expected performance. This analysis does not produce the actual cuts but establishes their existence. When implemented on a quantum computer, the QAOA provides an efficient algorithm for finding such cuts, using a constant-depth quantum circuit. To our knowledge, this gives an exponential speedup over the best known classical algorithm guaranteed to achieve cuts of this size on graphs of this girth. We also apply the QAOA to the Maximum Independent Set problem on the same class of graphs.

We propose a tensor generalized approximate message passing (TeG-AMP) algorithm for low-rank tensor inference, which can be used to solve tensor completion and decomposition problems. We derive TeG-AMP algorithm as an approximation of the sum-product belief propagation algorithm in high dimensions where the central limit theorem and Taylor series approximations are applicable. As TeG-AMP is developed based on a general TR decomposition model, it can be directly applied to many low-rank tensor types. Moreover, our TeG-AMP can be simplified based on the CP decomposition model and a tensor simplified AMP is proposed for low CP-rank tensor inference problems. Experimental results demonstrate that the proposed methods significantly improve recovery performances since it takes full advantage of tensor structures.

This present results lay the foundations for the study of the optimal allocation of vaccine in the simple epidemiological SIS model where one consider a very general heterogeneous population. In the present setting each individual has a type x belonging to a general space, and a vaccination strategy is a function $\eta$ where $\eta$(x) $\in$ [0, 1] represents the proportion of non-vaccinated among individuals of type x. We shall consider two loss functions associated to a vaccination strategy $\eta$: either the effective reproduction number, a classical quantity appearing in many models in epidemiology, and which is given here by the spectral radius of a compact operator that depends on $\eta$; or the overall proportion of infected individuals after vaccination in the maximal endemic state.By considering the weak-* topology on the set $\Delta$ of vaccination strategies, so that it is a compact set, we can prove that those two loss functions are continuous using the notion of collective compactness for a family of operators. We also prove their stability with respect to the parameters of the SIS model. Eventually, we consider their monotonicity and related properties in particular when the model is ``almost'' irreducible.

The Axiom-Based Atlas is a novel framework that structurally represents mathematical theorems as proof vectors over foundational axiom systems. By mapping the logical dependencies of theorems onto vectors indexed by axioms - such as those from Hilbert geometry, Peano arithmetic, or ZFC - we offer a new way to visualize, compare, and analyze mathematical knowledge. This vector-based formalism not only captures the logical foundation of theorems but also enables quantitative similarity metrics - such as cosine distance - between mathematical results, offering a new analytic layer for structural comparison. Using heatmaps, vector clustering, and AI-assisted modeling, this atlas enables the grouping of theorems by logical structure, not just by mathematical domain. We also introduce a prototype assistant (Atlas-GPT) that interprets natural language theorems and suggests likely proof vectors, supporting future applications in automated reasoning, mathematical education, and formal verification. This direction is partially inspired by Terence Tao's recent reflections on the convergence of symbolic and structural mathematics. The Axiom-Based Atlas aims to provide a scalable, interpretable model of mathematical reasoning that is both human-readable and AI-compatible, contributing to the future landscape of formal mathematical systems.

We initiate a study of local operator algebras at the boundary of infinite tensor networks, using the mathematical theory of inductive limits. In particular, we consider tensor networks in which each layer acts as a quantum code with complementary recovery, a property that features prominently in the bulk-to-boundary maps intrinsic to holographic quantum error-correcting codes. In this case, we decompose the limiting Hilbert space and the algebras of observables in a way that keeps track of the entanglement in the network. As a specific example, we describe this inductive limit for the holographic HaPPY code model and relate its algebraic and error-correction features. We find that the local algebras in this model are given by the hyperfinite type II$_\infty$ factor. Next, we discuss other networks that build upon this framework and comment on a connection between type II factors and stabilizer circuits. We conclude with a discussion of MERA networks in which complementary recovery is broken. We argue that this breaking possibly permits a limiting type III von Neumann algebra, making them more suitable ans\"atze for approximating subregions of quantum field theories.

This paper presents a new algorithm for set-based state estimation of nonlinear discrete-time systems with bounded uncertainties. The novel method builds upon essential properties and computational advantages of constrained zonotopes (CZs) and polyhedral relaxations of factorable representations of nonlinear functions to propagate CZs through nonlinear functions, which is usually done using conservative linearization in the literature. The new method also refines the propagated enclosure using nonlinear measurements. To achieve this, a lifted polyhedral relaxation is computed for the composite nonlinear function of the system dynamics and measurement equations, in addition to incorporating the measured output through equality constraints. Polyhedral relaxations of trigonometric functions are enabled for the first time, allowing to address a broader class of nonlinear systems than our previous works. Additionally, an approach to obtain an equivalent enclosure with fewer generators and constraints is developed. Thanks to the advantages of the polyhedral enclosures based on factorable representations, the new state estimation method provides better approximations than those resulting from linearization procedures. This led to significant improvements in the computation of convex sets enclosing the system states consistent with measured outputs. Numerical examples highlight the advantages of the novel algorithm in comparison to existing CZ methods based on the Mean Value Theorem and DC programming principles.

Drawing from set theory, this article contributes to a deeper understanding of the no-arbitrage principle in multiple-priors settings and its application in mathematical finance. In the quasi-sure discrete-time frictionless market framework of Bouchard and Nutz, the equivalence between the quasi-sure no-arbitrage condition and the existence of a probability measure for which the local one-prior no-arbitrage condition holds and the affine hull of the support is equal to the quasi-sure support, all of this in a quasi-sure sense, was established by Blanchard and Carassus. We aim to extend this result to the projective setup introduced by Carassus and Ferhoune. This setup allows for standardised measurability assumptions, in contrast to the framework of Bouchard and Nutz, where prices are assumed to be Borel measurable, strategies and stochastic kernels universally measurable, and the graphs of one-step priors analytic sets. To achieve this, we assume the classical axioms of Zermelo-Fraenkel set theory, including the axiom of choice (ZFC), supplemented by the Projective Determinacy (PD) axiom. In ZFC+PD the existence of such probability measures was assumed by Carassus and Ferhoune to prove the existence of solutions in a quasi-sure nonconcave utility maximisation problem. The equivalence with the quasi-sure no-arbitrage was only conjectured.

Fermions in two-dimensional space, commonly called $(1+2)$-dimensional fermions, exhibit intriguing and distinctive characteristics that distinguish them from their higher-dimensional counterparts. This paper offers a comprehensive theoretical examination of planar fermionic systems, presenting novel findings by incorporating nonminimal coupling. Our analysis includes the computation of the non-relativistic limit up to second-order corrections in the Dirac equation. We also explore the Schr\"odinger equation under the influence of a harmonic potential and an electric field. Furthermore, we investigate how the coupling parameter affects physical properties relevant to condensed matter systems. Our results demonstrate that this parameter significantly impacts electronic properties and Hall conductivity. The interplay between an external electric field and the coupling parameter also influences energy levels and the system's polarizability. These findings underscore the novel effects of including nonminimal coupling in wave equations, offering new insights into the physics of coupled systems.

In this work, we propose an output-feedback tube-based model predictive control (MPC) scheme for linear systems under dynamic uncertainties that are described via integral quadratic constraints (IQC). By leveraging IQCs, a large class of nonlinear and dynamic uncertainties can be addressed. We leverage recent IQC synthesis tools to design a dynamic controller and an observer that are robust to these uncertainties and minimize the size of the resulting constraint tightening in the MPC. Thereby, we show that the robust estimation problem using IQCs with peak-to-peak performance can be convexified. We guarantee recursive feasibility, robust constraint satisfaction, and input-to-state stability of the resulting MPC scheme.

This paper addresses the problem of generating dynamically admissible trajectories for control tasks using diffusion models, particularly in scenarios where the environment is complex and system dynamics are crucial for practical application. We propose a novel framework that integrates system dynamics directly into the diffusion model's denoising process through a sequential prediction and projection mechanism. This mechanism, aligned with the diffusion model's noising schedule, ensures generated trajectories are both consistent with expert demonstrations and adhere to underlying physical constraints. Notably, our approach can generate maximum likelihood trajectories and accurately recover trajectories generated by linear feedback controllers, even when explicit dynamics knowledge is unavailable. We validate the effectiveness of our method through experiments on standard control tasks and a complex non-convex optimal control problem involving waypoint tracking and collision avoidance, demonstrating its potential for efficient trajectory generation in practical applications.

The theory of diffusion seeks to describe the motion of particles in a chaotic environment. Classical theory models individual particles as independent random walkers, effectively forgetting that particles evolve together in the same environment. Random Walks in a Random Environment (RWRE) models treat the environment as a random space-time field that biases the motion of particles based on where they are in the environment. We provide a universality result for the moderate deviations of the transition probability of this model over a wide class of choices of random environments. In particular, we show the convergence of moments to those of the multiplicative noise stochastic heat equation (SHE), whose logarithm is the Kardar-Parisi-Zhang (KPZ) equation. The environment only filters into the scaling limit through one parameter, which depends explicitly on the statistical description of the environment. This forms the basis for our introduction, in arXiv:2406.17733, of the extreme diffusion coefficient.

As rapidly growing AI computational demands accelerate the need for new hardware installation and maintenance, this work explores optimal data center resource management by balancing operational efficiency with fault tolerance through strategic rack positioning considering diverse resources and locations. Traditional mixed-integer programming (MIP) approaches often struggle with scalability, while heuristic methods may result in significant sub-optimality. To address these issues, this paper presents a novel two-tier optimization framework using a high-level deep reinforcement learning (DRL) model to guide a low-level gradient-based heuristic for local search. The high-level DRL agent employs Leader Reward for optimal rack type ordering, and the low-level heuristic efficiently maps racks to positions, minimizing movement counts and ensuring fault-tolerant resource distribution. This approach allows scalability to over 100,000 positions and 100 rack types. Our method outperformed the gradient-based heuristic by 7\% on average and the MIP solver by over 30\% in objective value. It achieved a 100\% success rate versus MIP's 97.5\% (within a 20-minute limit), completing in just 2 minutes compared to MIP's 1630 minutes (i.e., almost 4 orders of magnitude improvement). Unlike the MIP solver, which showed performance variability under time constraints and high penalties, our algorithm consistently delivered stable, efficient results - an essential feature for large-scale data center management.

Measurement based quantum computing is preformed by adding non-Clifford measurements to a prepared stabilizer states. Entangling gates like CZ are likely to have lower fidelities due to the nature of interacting qubits, so when preparing a stabilizer state, we wish to minimize the number of required entangling states. This naturally introduces the notion of CZ-distance. Every stabilizer state is local-Clifford equivalent to a graph state, so we may focus on graph states $\left\vert G \right\rangle$. As a lower bound for general graphs, there exist $n$-vertex graphs $G$ such that the CZ-distance of $\left\vert G \right\rangle$ is $\Omega(n^2 / \log n)$. We obtain significantly improved bounds when $G$ is contained within certain proper classes of graphs. For instance, we prove that if $G$ is a $n$-vertex circle graph with clique number $\omega$, then $\left\vert G \right\rangle$ has CZ-distance at most $4n \log \omega + 7n$. We prove that if $G$ is an $n$-vertex graph of rank-width at most $k$, then $\left\vert G \right\rangle$ has CZ-distance at most $(2^{2^{k+1}} + 1) n$. More generally, this is obtained via a bound of $(k+2)n$ that we prove for graphs of twin-width at most $k$. We also study how bounded-rank perturbations and low-rank cuts affect the CZ-distance. As a consequence, we prove that Geelen's Weak Structural Conjecture for vertex-minors implies that if $G$ is an $n$-vertex graph contained in some fixed proper vertex-minor-closed class of graphs, then $\left\vert G \right\rangle$ has CZ-distance at most $O(n\log n)$. Since graph states of locally equivalent graphs are local Clifford equivalent, proper vertex-minor-closed classes of graphs are natural and very general in this setting.

We investigate strong gravitational lensing by a marginally unstable photon sphere in a Reissner-Nordstr\"{o}m naked singularity spacetime. Using the Picard-Fuchs equation, we derive full-order power series expressions for the deflection angle in various regimes, including the strong deflection limits from both outside and inside the photon sphere. We show that the deflection angle diverges non-logarithmically in both cases, refining existing asymptotic formulae. Comparing truncated approximations with numerical results, we find that higher-order corrections are essential to achieve comparable accuracy to logarithmic divergence cases. Using these improved formulae, we also derive precise approximations for image positions that are not restricted to the almost perfectly aligned cases.

We present a lower bound for Pauli Manipulation Detection (PMD) codes, which enables the detection of every Pauli error with high probability and can be used to construct quantum erasure and tamper-detection codes. Our lower bound reveals the first trade-off between the error and the redundancy parameters in PMD codes.

Safely navigating around obstacles while respecting the dynamics, control, and geometry of the underlying system is a key challenge in robotics. Control Barrier Functions (CBFs) generate safe control policies by considering system dynamics and geometry when calculating safe forward-invariant sets. Existing CBF-based methods often rely on conservative shape approximations, like spheres or ellipsoids, which have explicit and differentiable distance functions. In this paper, we propose an optimization-defined CBF that directly considers the exact Signed Distance Function (SDF) between a polytopic robot and polytopic obstacles. Inspired by the Gilbert-Johnson-Keerthi (GJK) algorithm, we formulate both (i) minimum distance and (ii) penetration depth between polytopic sets as convex optimization problems in the space of Minkowski difference operations (the MD-space). Convenient geometric properties of the MD-space enable the derivatives of implicit SDF between two polytopes to be computed via differentiable optimization. We demonstrate the proposed framework in three scenarios including pure translation, initialization inside an unsafe set, and multi-obstacle avoidance. These three scenarios highlight the generation of a non-conservative maneuver, a recovery after starting in collision, and the consideration of multiple obstacles via pairwise CBF constraint, respectively.

Deep neural networks (DNNs) trained through end-to-end learning have achieved remarkable success across diverse machine learning tasks, yet they are not explicitly designed to adhere to the Minimum Description Length (MDL) principle, which posits that the best model provides the shortest description of the data. In this paper, we argue that MDL is essential to deep learning and propose a further generalized principle: Understanding is the use of a small amount of information to represent a large amount of information. To this end, we introduce a novel theoretical framework for designing and evaluating deep Variational Autoencoders (VAEs) based on MDL. In our theory, we designed the Spectrum VAE, a specific VAE architecture whose MDL can be rigorously evaluated under given conditions. Additionally, we introduce the concept of latent dimension combination, or pattern of spectrum, and provide the first theoretical analysis of their role in achieving MDL. We claim that a Spectrum VAE understands the data distribution in the most appropriate way when the MDL is achieved. This work is entirely theoretical and lays the foundation for future research on designing deep learning systems that explicitly adhere to information-theoretic principles.

Many robotics tasks, such as path planning or trajectory optimization, are formulated as optimal control problems (OCPs). The key to obtaining high performance lies in the design of the OCP's objective function. In practice, the objective function consists of a set of individual components that must be carefully modeled and traded off such that the OCP has the desired solution. It is often challenging to balance multiple components to achieve the desired solution and to understand, when the solution is undesired, the impact of individual cost components. In this paper, we present a framework addressing these challenges based on the concept of directional corrections. Specifically, given the solution to an OCP that is deemed undesirable, and access to an expert providing the direction of change that would increase the desirability of the solution, our method analyzes the individual cost components for their "consistency" with the provided directional correction. This information can be used to improve the OCP formulation, e.g., by increasing the weight of consistent cost components, or reducing the weight of - or even redesigning - inconsistent cost components. We also show that our framework can automatically tune parameters of the OCP to achieve consistency with a set of corrections.

Multivariate phase relationships are important to characterize and understand numerous physical, biological, and chemical systems, from electromagnetic waves to neural oscillations. These systems exhibit complex spatiotemporal dynamics and intricate interdependencies among their constituent elements. While classical models of multivariate phase relationships, such as the wave equation and Kuramoto model, give theoretical models to describe phenomena, the development of statistical tools for hypothesis testing and inference for multivariate phase relationships in complex systems remains limited. This paper introduces a novel probabilistic modeling framework to characterize multivariate phase relationships, with wave-like phenomena serving as a key example. This approach describes spatial patterns and interactions between oscillators through a pairwise exponential family distribution. Building upon the literature of graphical model inference, including methods like Ising models, graphical lasso, and interaction screening, this work bridges the gap between classical wave dynamics and modern statistical approaches. Efficient inference methods are introduced, leveraging the Chow-Liu algorithm for directed tree approximations and interaction screening for general graphical models. Simulated experiments demonstrate the utility of these methods for uncovering wave properties and sparse interaction structures, highlighting their applicability to diverse scientific domains. This framework establishes a new paradigm for statistical modeling of multivariate phase relationships, providing a powerful toolset for exploring the complexity of these systems.

Gaussian processes (GPs) are crucial in machine learning for quantifying uncertainty in predictions. However, their associated covariance matrices, defined by kernel functions, are typically dense and large-scale, posing significant computational challenges. This paper introduces a matrix-free method that utilizes the Non-equispaced Fast Fourier Transform (NFFT) to achieve nearly linear complexity in the multiplication of kernel matrices and their derivatives with vectors for a predetermined accuracy level. To address high-dimensional problems, we propose an additive kernel approach. Each sub-kernel in this approach captures lower-order feature interactions, allowing for the efficient application of the NFFT method and potentially increasing accuracy across various real-world datasets. Additionally, we implement a preconditioning strategy that accelerates hyperparameter tuning, further improving the efficiency and effectiveness of GPs.

Computational methods in drug repositioning can help to conserve resources. In particular, methods based on biological networks are showing promise. Considering only the network topology and knowledge on drug target genes is not sufficient for quantitative predictions or predictions involving drug combinations. We propose an iterative procedure alternating between system identification and drug response experiments. Data from experiments are used to improve the model and drug effect knowledge, which is then used to select drugs for the next experiments. Using simulated data, we show that the procedure can identify nearly optimal drug combinations.

We present a theoretical framework for analyzing linear attention models through matrix-valued state space models (SSMs). Our approach, Parallel Flows, provides a perspective that systematically decouples temporal dynamics from implementation constraints, enabling independent analysis of critical algorithmic components: chunking, parallelization, and information aggregation. Central to this framework is the reinterpretation of chunking procedures as computations of the flows governing system dynamics. This connection establishes a bridge to mathematical tools from rough path theory, opening the door to new insights into sequence modeling architectures. As a concrete application, we analyze DeltaNet in a generalized low-rank setting motivated by recent theoretical advances. Our methods allow us to design simple, streamlined generalizations of hardware-efficient algorithms present in the literature, and to provide completely different ones, inspired by rough paths techniques, with provably lower complexity. This dual contribution demonstrates how principled theoretical analysis can both explain existing practical methods and inspire fundamentally new computational approaches.

Quantum groups have been widely explored as a tool to encode possible nontrivial generalisations of reference frame transformations, relevant in quantum gravity. In quantum information, it was found that the reference frames can be associated to quantum particles, leading to quantum reference frames transformations. The connection between these two frameworks is still unexplored, but if clarified it will lead to a more profound understanding of symmetries in quantum mechanics and quantum gravity. Here, we establish a correspondence between quantum reference frame transformations and transformations generated by a quantum deformation of the Galilei group with commutative time, taken at the first order in the quantum deformation parameter. This is found once the quantum group noncommutative transformation parameters are represented on the phase space of a quantum particle, and upon setting the quantum deformation parameter to be proportional to the inverse of the mass of the particle serving as the quantum reference frame. These results allow us to show that quantum reference frame transformations are physically relevant when the state of the quantum reference frame is in a quantum superposition of semiclassical states. We conjecture that the all-order quantum Galilei group describes quantum reference frame transformations between more general quantum states of the quantum reference frame.

The spread of infectious diseases often concentrates within specific subgroups of a broader population. For instance, during recent mpox outbreaks in non-endemic countries, transmission primarily affected men who have sex with men (MSM). However, the internal structure of these subpopulations plays a crucial role in disease dynamics and should be accurately represented in mathematical models. In this study, we highlight the importance of modeling interactions between distinct subgroups and their impact on transmission patterns. We consider a stochastic SEIR-based model with two core groups embedded into the general population, and investigate the outcome of the outbreak with different levels of symmetry between these groups and assortativity in their contacts. Our results indicate that the efficiency of commonly used non-pharmaceutical interventions is greatly influenced by these factors, hence they should be considered in the design of intervention strategies.

Finding deletion-correcting codes of maximum size has been an open problem for over 70 years, even for a single deletion. In this paper, we propose a novel approach for constructing deletion-correcting codes. A code is a set of sequences satisfying certain constraints, and we construct it by greedily adding the highest-priority sequence according to a priority function. To find good priority functions, we leverage FunSearch, a large language model (LLM)-guided evolutionary search proposed by Romera et al., 2024. FunSearch iteratively generates, evaluates, and refines priority functions to construct large deletion-correcting codes. For a single deletion, our evolutionary search finds functions that construct codes which match known maximum sizes, reach the size of the largest (conjectured optimal) Varshamov-Tenengolts codes where the maximum is unknown, and independently rediscover them in equivalent form. For two deletions, we find functions that construct codes with new best-known sizes for code lengths \( n = 12, 13 \), and \( 16 \), establishing improved lower bounds. These results demonstrate the potential of LLM-guided search for information theory and code design and represent the first application of such methods for constructing error-correcting codes.

We discuss the notion of generating a cosmic inflation without any big bang singularity. It has been proved recently by Good and Linder (arXiv : 2503.02380v1) that such an expansion of the universe can be driven by quantum fluctuations embedded in vacuum. The rate of expansion is guided by a cosmological sum rule defined through the Schwarzian derivative. We explore the thermodynamic roots of Schwarzian and connect it with the surface gravity associated with an apparent horizon. In General Relativity the cosmological sum rule can be enforced only if the early universe is a Milne vacuum. We show that this restriction can be removed by considering an entropic source term in the Einstein-Hilbert action.

We study entangled resource state recycling after one round of probabilistic port-based teleportation. We analytically characterize its degradation and, for the case of the resource state consisting of $N$ EPR pairs, we demonstrate the possibility of reusing it for a subsequent round of teleportation in the $N \to \infty$ limit. For the case of an optimized resource state, we compare the protocol's performance to multi-port-based teleportation, indicating that the resource state reuse is possible.

Generative networks have shown remarkable success in learning complex data distributions, particularly in generating high-dimensional data from lower-dimensional inputs. While this capability is well-documented empirically, its theoretical underpinning remains unclear. One common theoretical explanation appeals to the widely accepted manifold hypothesis, which suggests that many real-world datasets, such as images and signals, often possess intrinsic low-dimensional geometric structures. Under this manifold hypothesis, it is widely believed that to approximate a distribution on a $d$-dimensional Riemannian manifold, the latent dimension needs to be at least $d$ or $d+1$. In this work, we show that this requirement on the latent dimension is not necessary by demonstrating that generative networks can approximate distributions on $d$-dimensional Riemannian manifolds from inputs of any arbitrary dimension, even lower than $d$, taking inspiration from the concept of space-filling curves. This approach, in turn, leads to a super-exponential complexity bound of the deep neural networks through expanded neurons. Our findings thus challenge the conventional belief on the relationship between input dimensionality and the ability of generative networks to model data distributions. This novel insight not only corroborates the practical effectiveness of generative networks in handling complex data structures, but also underscores a critical trade-off between approximation error, dimensionality, and model complexity.

Market participants regularly send bid and ask quotes to exchange-operated limit order books. This creates an optimization challenge where their potential profit is determined by their quoted price and how often their orders are successfully executed. The expected profit from successful execution at a favorable limit price needs to be balanced against two key risks: (1) the possibility that orders will remain unfilled, which hinders the trading agenda and leads to greater price uncertainty, and (2) the danger that limit orders will be executed as market orders, particularly in the presence of order submission latency, which in turn results in higher transaction costs. In this paper, we consider a stochastic optimal control problem where a risk-averse trader attempts to maximize profit while balancing risk. The market is modeled using Brownian motion to represent the price uncertainty. We analyze the relationship between fill probability, limit price, and order submission latency. We derive closed-form approximations of these quantities that perform well in the practical regime of interest. Then, we utilize a mean-variance method where our total reward function features a risk-tolerance parameter to quantify the combined risk and profit.

We consider the relationship between learnability of a ``base class'' of functions on a set X and learnability of a class of statistical functions derived from the base class. For example, we refine results showing that learnability of a family of functions implies learnability of the family of functions mapping a function in the class to its expectation under a distribution. We will look at both Probably Approximately Correct (PAC) learning, where example inputs and outputs are chosen at random, and online learning, where the examples are chosen adversarially. We establish improved bounds on the sample complexity of learning for statistical classes, stated in terms of combinatorial dimensions of the base class. We do this by adapting techniques introduced in model theory for ``randomizing a structure''. We give particular attention to classes derived from logical formulas, and relate learnability of the statistical classes to properties of the formula. Finally, we provide bounds on the complexity of learning the statistical classes built on top of a logic-based hypothesis class.

The Haldane model is a standard tight-binding model describing electrons hopping on a hexagonal lattice subject to a transverse, dipolar magnetic field. We consider its interacting version for values of the interaction strength that are small compared to the bandwidth. We study the critical case at the transition between the trivial and the `topological' insulating phases, and we rigorously establish that the transverse conductivity on the dressed critical line is quantized at a half-integer multiple of $e^2/h$: this is the average of the integer values of the Hall conductivity in the insulating phases on either side of the dressed critical line. Together with previous results, this fully characterizes the nature of the phase transition between different Hall plateaus and proves its universality with respect to many-body interactions. The proof is based on a combination of constructive renormalization group methods and exact lattice Ward identities.

We study an axion soliton, which weakly interacts with background matter and magnetic fields. A mirror-symmetric soliton, for which the magnetic flow is due to secondary magnetic helicity invariant, is described by the Iroshnikov-Kreichnan spectrum. For a large-scale magnetic field dynamo is not observed. In a mirror axionic soliton, a phase transition, which produces a magnetic helical flow, is possible. Using this transition, the soliton becomes mirror-asymmetric. When the mirror symmetry is broken, the axion soliton allows the magnetic energy, which is the result of the transformation of the axionic energy. In the main result, for an initial stage of the process, we calculate a scale for which the generation of large scale magnetic fields is the most intense. By making numerical simulations, we received that lower lateral harmonics of the magnetic field have greater amplitudes compared to higher ones. A simplest statistical ensemble, which is defined by the projection of all harmonics onto principal harmonics is constructed. We put forward an assumption that it was the indication to some instability in axionic MHD. Now, we can provide a possible explanation of this feature. When the mirror symmetry of the axion soliton is broken, the $\gamma$-term in the axionic mean field equation interacts with principal harmonics. As the result, the axion soliton acquires the magnetic energy and becomes helical.

Real-world networks grow over time; statistical models based on node exchangeability are not appropriate. Instead of constraining the structure of the \textit{distribution} of edges, we propose that the relevant symmetries refer to the \textit{causal structure} between them. We first enumerate the 96 causal directed acyclic graph (DAG) models over pairs of nodes (dyad variables) in a growing network with finite ancestral sets that are invariant to node deletion. We then partition them into 21 classes with ancestral sets that are closed under node marginalization. Several of these classes are remarkably amenable to distributed and asynchronous evaluation. As an example, we highlight a simple model that exhibits flexible power-law degree distributions and emergent phase transitions in sparsity, which we characterize analytically. With few parameters and much conditional independence, our proposed framework provides natural baseline models for causal inference in relational data.