In this article, we revisit the initial data rigidity theorem of Eichmair, Galloway and Mendes (arxiv:2009.09527). The goal is to strengthen their result by showing that the initial data sets concerned carry a vector field that is lightlike and parallel in an ambient sense. This will be used in a second step to show that among the spacetimes satisfying the dominant energy condition there exists locally essentially one spacetime extending these initial data sets. This local uniqueness theorem also applies in the context of other initial data rigidity theorems. Notably, the one in the spin case due the author (arxiv:2304.02331) and a recent study of the mass zero case in the positive energy theorem due to Hirsch and Zhang (arxiv:2403.15984).

We show that the variety of residuated lattices does not have the amalgamation property.

Let $\mathcal{L}$ be the infinitesimal generator of an analytic semigroup $\big\{e^{-t\mathcal L}\big\}_{t>0}$ satisfying the Gaussian upper bounds. For given $0<\alpha<n$, let $\mathcal L^{-\alpha/2}$ be the generalized fractional integral associated with $\mathcal{L}$, which is defined as \begin{equation*} \mathcal L^{-\alpha/2}(f)(x):=\frac{1}{\Gamma(\alpha/2)}\int_0^{+\infty} e^{-t\mathcal L}(f)(x)t^{\alpha/2-1}dt, \end{equation*} where $\Gamma(\cdot)$ is the usual gamma function. For a locally integrable function $b(x)$ defined on $\mathbb R^n$, the related commutator operator $\big[b,\mathcal L^{-\alpha/2}\big]$ generated by $b$ and $\mathcal{L}^{-\alpha/2}$ is defined by \begin{equation*} \big[b,\mathcal L^{-\alpha/2}\big](f)(x):=b(x)\cdot\mathcal{L}^{-\alpha/2}(f)(x)-\mathcal{L}^{-\alpha/2}(bf)(x). \end{equation*} A new class of Morrey--Campanato spaces associated with $\mathcal{L}$ is introduced in this paper. The authors establish some new estimates for the commutators $\big[b,\mathcal L^{-\alpha/2}\big]$ on Morrey--Campanato spaces. The corresponding results for higher-order commutators$\big[b,\mathcal L^{-\alpha/2}\big]^m$($m\in \mathbb{N}$) are also discussed.

In this paper the cosheaf homology is investigated from different viewpoints: the behavior under site morphisms, connections with Cech homology via spectral sequences, and description of cosheaf homology using hypercoverings. It is proved that in the case of Hausdorff paracompact spaces, the cosheaf homology in general is isomorphic to the Cech homology, and for a constant cosheaf is isomorphic to the shape pro-homology. In the case of Alexandroff spaces, including finite and locally finite spaces, the cosheaf homology is isomorphic to the singular homology.

We consider the perturbed Mann's iterative process \begin{equation} x_{n+1}=(1-\theta_n)x_n+\theta_n f(x_n)+r_n, \end{equation} where $f:[0,1]\rightarrow[0,1]$ is a continuous function, $\{\theta_n\}\in [0,1]$ is a given sequence, and $\{r_n\}$ is the error term. We establish that if the sequence $\{\theta_n\}$ converges relatively slowly to $0$ and the error term $r_n$ becomes enough small at infinity, any sequences $\{x_n\}\in [0,1]$ satisfying the process converges to a fixed point of the function $f$. We also study the asymptotic behavior of the trajectories $x(t)$ as $t\rightarrow\infty$ of a continuous version of the the considered. We investigate the similarities between the asymptotic behaviours of the sequences generated by the considered discrete process and the trajectories $x(t)$ of its corresponding continuous version.

In this article we find locally an eigenfunctions for a particular nonlinear hyperbolic differential operator $\Delta_H u^{n}$, where $\Delta_H$ is the hyperbolic Laplacian in the half-plane of Poincair\'e. We have proved that these eigenfunctions are an analytic and non-exact whose coefficients satisfy a specific algebraic recursive rule. The existence of these eigenfunctions allows us to find non-exact solutions respecting the spatial coordinate of nonlinear diffusive PDEs on the Poincair\'e half-plane, which could describe a possible one-dimensional physical model.

We present a new approach to the problem of proving global stability, based on symplectic geometry and with a focus on systems with several conserved quantities. We also provide a proof of instability for integrable systems whose momentum map is everywhere regular. Our results take root in the recently proposed notion of a confining function and are motivated by ghost-ridden systems, for whom we put forward the first geometric definition.

High-dimensional partial differential equations (PDEs) pose significant computational challenges across fields ranging from quantum chemistry to economics and finance. Although scientific machine learning (SciML) techniques offer approximate solutions, they often suffer from bias and neglect crucial physical insights. Inspired by inference-time scaling strategies in language models, we propose Simulation-Calibrated Scientific Machine Learning (SCaSML), a physics-informed framework that dynamically refines and debiases the SCiML predictions during inference by enforcing the physical laws. SCaSML leverages derived new physical laws that quantifies systematic errors and employs Monte Carlo solvers based on the Feynman-Kac and Elworthy-Bismut-Li formulas to dynamically correct the prediction. Both numerical and theoretical analysis confirms enhanced convergence rates via compute-optimal inference methods. Our numerical experiments demonstrate that SCaSML reduces errors by 20-50% compared to the base surrogate model, establishing it as the first algorithm to refine approximated solutions to high-dimensional PDE during inference. Code of SCaSML is available at https://github.com/Francis-Fan-create/SCaSML.

We construct a rank-$2$ indecomposable vector bundle on $\mathbb P^2\times\mathbb P^2$ in characteristic $2$ that does not come from a bundle on $\mathbb P^2$ by factor projection nor from a bundle on $\mathbb P^{m} $ by central projection. We show that the zero-sets of a suitable twist of $E$ form a family of nonclassical smooth Enriques surfaces of bidegree (4, 4) whose general member is 'singular' in the sense that Frobenius acts isomorphically on $H^1$, and there is a smooth divisor consisting of smooth supersingular surfaces (Frobenius acts as zero). Every nonclassical Enriques surface of bidegree (4, 4) in $\mathbb P^2\times\mathbb P^2$ that is bilinearly normal arises as a zero-set in this way.

Line-search methods are commonly used to solve optimization problems. A simplest line search method is steepest descent where one always moves in the direction of the negative gradient. Newton's method on the other hand is a second-order method that uses the curvature information in the Hessian to pick the descent direction. In this work, we propose a new line-search method called Constrained Gradient Descent (CGD) that implicitly changes the landscape of the objective function for efficient optimization. CGD is formulated as a solution to the constrained version of the original problem where the constraint is on a function of the gradient. We optimize the corresponding Lagrangian function thereby favourably changing the landscape of the objective function. This results in a line search procedure where the Lagrangian penalty acts as a control over the descent direction and can therefore be used to iterate over points that have smaller gradient values, compared to iterates of vanilla steepest descent. We establish global linear convergence rates for CGD and provide numerical experiments on synthetic test functions to illustrate the performance of CGD. We also provide two practical variants of CGD, CGD-FD which is a Hessian free variant and CGD-QN, a quasi-Newton variant and demonstrate their effectiveness.

The aim of this monograph is twofold: to explain various nonautonomous integrable systems (discrete Painlev\'e all the way up to the elliptic level, as well as generalizations \`a la Garnier) using an interpretation of difference and differential equations as sheaves on noncommutative projective surfaces, and to develop the theory of such surfaces enough to allow one to apply the usual GIT construction of moduli spaces of sheaves. This requires a fairly extensive development of the theory of birationally ruled noncommutative projective surfaces, both showing that the analogues of Cremona transformations work and understanding effective, nef, and ample divisor classes. This combines arXiv:1307.4032, arXiv:1307.4033, arXiv:1907.11301, as well as those portions of arXiv:1607.08876 needed to make things self-contained. Some additional results appear, most notably a proof that the resulting discrete actions on moduli spaces of equations are algebraically integrable.

The Mandelbrot set is a fractal which classifies the behaviour of complex quadratic polynomials. Although its remarkably simple definition: $\mathcal{M}:=\{c \in \mathbb{C}\,|\,Q_c(0)^n \nrightarrow \infty \mbox{ as } n\rightarrow \infty, \mbox{ where } Q_c(z)=z^2+c\}$, it is a central object in complex dynamics, and it has been charming and intriguing since it has first been defined and drawn. A fascinating fact is the presence of little copies of the Mandelbrot set in the Mandelbrot set itself (and in many other parameter planes). There exist two different kinds of little copies of the Mandelbrot set within the Mandelbrot set: the primitive copies, visually similar to the Mandelbrot set with a cusp at the root of the principal hyperbolic component, and satellite copies, whose principal hyperbolic component has no cusp, i.e. has smooth boundary across the root point. Lyubich proved that the primitive copies of $\mathcal{M}$ satisfy a stronger regularity condition: they are quasiconformally homeomorphic to $\mathcal{M}$. The satellite copies are homeomorphic to $\mathcal{M}$, but the homeomorphism is only quasiconformal outside neighbourhoods of the root. The question that remained open was: are the satellite copies mutually quasiconformally homeomorphic? In a previous work, we showed that, satellite copies with rotation numbers with different denominators are not quasiconformally homeomorphic. Here we complete the picture, by showing that for any $q$ the satellite copies $\mathcal{M}_{p/q}$ and $\mathcal{M}_{p'/q}$ of the Mandelbrot set $\mathcal{M}$ for rotation numbers with the same denominator $q$ are quasiconformal homeomorphic.

A bicirculant is a regular graph that admits an automorphism having two orbits of the same size. A bicirculant can be described as follows. Given an integer $m \ge 1$ and sets $R,S,T \subseteq \mathbb{Z}_m$ such that $R=-R$, $T=-T$, $0 \not\in R \cup T$ and $0 \in S$, the graph $B(m;R,S,T)$ has vertex set $V=\{u_0,\dots,u_{m-1},v_0,\dots,v_m-1\}$ and edge set $E=\{u_iu_{i+j}| \ i \in \mathbb{Z}_m, j \in R\} \cup \{v_iv_{i+j}| \ i \in \mathbb{Z}_m, j \in T\} \cup \{u_iv_{i+j}| \ i \in \mathbb{Z}_m, j \in S\}.$ Let $m \ge 3$ be a positive integer and $a,b,c \in \mathbb{Z}_m \setminus\{0\}$ with $a,b \ne m/2$. If we take $R = \{a,-a\}$, $S = \{0,c\}$ and $T = \{ b, -b\}$, the graph $B(m;R,S,T)$ is a generalized rose window graph. A rose window graph has the additional property that at least one of $a,b$ is relatively prime to $m$. In this paper we show that all generalized rose window graphs are hamiltonian. As a consequence we obtain that every connected bicirculant $B(m;R,S,T)$ with $|S| \ge 3$ is hamiltonian if $m$ is a product of at most three prime powers. In particular, every connected bicirculant $B(m;R,S,T)$ with$|S| \ge 3$ is hamiltonian for even $m<210$ and odd $m < 1155$.

We study a particular family of elements in the cohomology of the $\mathbb{C}$-motivic Steenrod algebra, also known as the $\mathbb{C}$-motivic Adams $E_2$-page. This family exhibits unusual periodicity properties, and it is related both to $h_1$-localization and to the algebraic Hurewicz image of the motivic modular forms spectrum $\mathrm{mmf}$.

Value Decomposition and Eckart-Young-like optimality results. Underlying the tubal tensor framework is a view of a tensor as a matrix of finite sized tubes. In this work, we lay the mathematical and computational foundations for working with tensors with infinite size tubes: matrices whose elements are elements from a separable Hilbert space. A key challenge is that existence of important desired matrix-mimetic features of tubal tensors rely on the existence of a unit element in the ring of tubes. Such unit element cannot exist for tubes which are elements of an infinite-dimensional Hilbert space. We sidestep this issue by embedding the tubal space in a commutative unital C*-algebra of bounded operators. The resulting quasitubal algebra recovers the structural properties needed for decomposition and low-rank approximation. In addition to laying the theoretical groundwork for working with tubal tensors with infinite dimensional tubes, we discuss computational aspects of our construction, and provide a numerical illustration where we compute a finite dimensional approximation to a infinitely-sized synthetic tensor using our theory. We believe our theory opens new exciting avenues for applying matrix mimetic tensor framework in the context of inherently infinite dimensional problems.

We study generalized solutions of an evolutionary equation related to some densely defined skew-symmetric operator in a real Hilbert space. We establish existence of a contractive semigroup, which provides generalized solutions, and suggest a criteria of uniqueness of this semigroup. We also find a stronger criteria of uniqueness of generalized solutions. Applications to transport equations with solenoidal (and generally discontinuous) coefficients are given.

A Tauberian theorem deduces an asymptotic for the partial sums of a sequence of non-negative real numbers from analytic properties of an associated Dirichlet series. Tauberian theorems appear in a tremendous variety of applications, ranging from well-known classical applications in analytic number theory, to new applications in arithmetic statistics and at the intersection of number theory and algebraic geometry. The goal of this article is to provide a useful reference for practitioners who wish to apply a Tauberian theorem. We explain the hypotheses and proofs of two types of Tauberian theorems: one with and one without an explicit remainder term. We furthermore provide counterexamples that illuminate that neither theorem can reach an essentially stronger conclusion unless its hypothesis is strengthened.

Deligne and Mostow investigated period maps on the configuration spaces $M_{0,n}$ of $n$ ordered points on $\mathbb{P}^1$. The images of these maps are open subsets of certain ball quotients. Moreover, they extend to isomorphisms between GIT-quotients and the Baily-Borel compactifications. Building on a theorem of Gallardo, Kerr and Schaffler, the period maps lift to isomorphisms between two natural compactifications, namely the Kirwan blow-up and the toroidal compactification. In this paper, we look at the more general situation where we also allow unordered or partially ordered $n$-tuples. Our main result is an easily verifiable criterion that, in this broader setting, determines when the Deligne-Mostow period maps still lift to isomorphisms between the Kirwan blow-up and the toroidal compactification. We further investigate a partial ordering among Deligne-Mostow varieties, which reduces this problem to considering minimal or maximal Deligne-Mostow varieties with respect to this partial ordering. As a byproduct, we prove that, in general, Kirwan's resolution pair is not a log canonical log minimal model and not log $K$-equivalent to the unique toroidal compactification.

In this paper, we describe the moduli space of rank-2 free algebras, along with several of its natural subspaces, as presheaf quotients: specifically, as the quotient of the functor of points of a scheme by the functor of points of a group scheme acting on it.

We extend the classical Heisenberg uncertainty principle to a fractional $L^p$ setting by investigating a novel class of uncertainty inequalities derived from the fractional Schr\"odinger equation. In this work, we establish the existence of extremal functions for these inequalities, characterize their structure as fractional analogues of Gaussian functions, and determine the sharp constants involved. Moreover, we prove a quantitative stability result showing that functions nearly attaining the equality in the uncertainty inequality must be close -- in an appropriate norm -- to the set of extremizers. Our results provide new insights into the fractional analytic framework and have potential applications in the analysis of fractional partial differential equations.

Let $f(z)$ be a holomorphic function, and let $\langle,\; rangle $ denote the inner product defined over an analytic Hilbert space with Gaussian measure. In this work, we demonstrate that the numerical values of the derivatives $f^{(n)}(z)$ at a point $z_{0}$ can be computed by evaluating an inner product of the form $\langle z^{n},f(z)\rangle$, divided by a constant. Specifically, if the inner product is taken over the Bargmann space (the analytic Hilbert space with Gaussian weight and orthogonal monomials), the constant is $\pi$. This result assumes that $f(z)$ is a holomorphic function of a single complex variable. If the function $f(z)$ is square-integrable, then the accuracy of the computed derivative values depends on the precision and reliability of the numerical routine used to evaluate the inner products. We introduce the projection coefficients algorithm , which determines the leading terms of the Taylor series expansion for a given holomorphic function from a graph perspective, and analyze the associated truncation errors. Furthermore, the projection coefficients provide clear insights into certain properties of functions, such as whether they are odd or even, and whether the $n$-th derivatives exist. This study lays the groundwork for further applications in numerical analysis and approximation theory within Hilbert spaces equipped with Gaussian measures. Additionally, it might contribute to advancements in reproducing kernel Hilbert space (RKHS) methods, which are widely used in support vector machines (SVM) and other areas of machine learning. Also, it might have impact in probabilistic numerics.

For $\mathbb Z_3$-orbifold limits of K3, we provide a counterpart to the extensive studies by Nikulin and others of the geometry and symmetries of classical Kummer surfaces. In particular, we determine the group of holomorphic symplectic automorphisms of $\mathbb Z_3$-orbifold limits of K3. We moreover track this group within two of the Mathieu groups, which involves a variation of Kondo's lattice techniques that Taormina and Wendland introduced earlier in their study of the symmetries of Kummer surfaces and the genesis of their symmetry surfing programme. Specifically, we realise the finite group of symplectic automorphisms of this class of K3 surfaces as a subgroup of the sporadic groups Mathieu 12 and Mathieu 24 in terms of permutations of 12, resp. 24 elements. As a proof of concept, we construct an embedding that yields the largest Mathieu group when the symmetry group of $\mathbb Z_3$-orbifold K3s is combined with all symmetries of Kummer surfaces.

Modularity is a parameter indicating the presence of community structure in the graph. Nowadays it lies at the core of widely used clustering algorithms. We study the modularity of the most classical random graph, binomial $G(n,p)$. In 2020 McDiarmid and Skerman proved, taking advantage of the spectral graph theory and a specific subgraph construction by Coja-Oghlan from 2007, that there exists a constant $b$ such that with high probability the modularity of $G(n,p)$ is at most $b/\sqrt{np}$. The obtained constant $b$ is very big and not easily computable. We improve upon this result showing that a constant under $3$ may be derived here. Interesting is the fact that it might be obtained by basic probabilistic tools. We also address the lower bound on the modularity of $G(n,p)$ and improve the results of McDiarmid and Skerman from 2020 using estimates of bisections of random graphs derived by Dembo, Montanari, and Sen in 2017.

A proper Euler's magic matrix is an integer $n\times n$ matrix $M\in\mathbb Z^{n\times n}$ such that $M\cdot M^t=\gamma\cdot I$ for some nonzero constant $\gamma$, the sum of the squares of the entries along each of the two main diagonals equals $\gamma$, and the squares of all entries in $M$ are pairwise distinct. Euler constructed such matrices for $n=4$. In this work, we construct examples for $n=8$ and prove that no such matrix exists for $n=3$.

We introduce \emph{Term Coding}, a novel framework for analysing extremal problems in discrete mathematics by encoding them as finite systems of \emph{term equations} (and, optionally, \emph{non-equality constraints}). In its basic form, all variables range over a single domain, and we seek an interpretation of the function symbols that \emph{maximises} the number of solutions to these constraints. This perspective unifies classical questions in extremal combinatorics, network/index coding, and finite model theory. We further develop \emph{multi-sorted Term Coding}, a more general approach in which variables may be of different sorts (e.g., points, lines, blocks, colours, labels), possibly supplemented by variable-inequality constraints to enforce distinctness. This extension captures sophisticated structures such as block designs, finite geometries, and mixed coding scenarios within a single logical formalism. Our main result shows how to determine (up to a constant) the maximum number of solutions \(\max_{\mathcal{I}}(\Gamma,n)\) for any system of term equations (possibly including non-equality constraints) by relating it to \emph{graph guessing numbers} and \emph{entropy measures}. Finally, we focus on \emph{dispersion problems}, an expressive subclass of these constraints. We discover a striking complexity dichotomy: deciding whether, for a given integer \(r\), the maximum code size that reaches \(n^{r}\) is \emph{undecidable}, while deciding whether it exceeds \(n^{r}\) is \emph{polynomial-time decidable}.

We give new results for problems in computational and statistical machine learning using tools from high-dimensional geometry and probability. We break up our treatment into two parts. In Part I, we focus on computational considerations in optimization. Specifically, we give new algorithms for approximating convex polytopes in a stream, sparsification and robust least squares regression, and dueling optimization. In Part II, we give new statistical guarantees for data science problems. In particular, we formulate a new model in which we analyze statistical properties of backdoor data poisoning attacks, and we study the robustness of graph clustering algorithms to ``helpful'' misspecification.

This paper studies the hypothesis testing problem to determine whether m > 2 unlabeled graphs with Gaussian edge weights are correlated under a latent permutation. Previously, a sharp detection threshold for the correlation parameter \rho was established by Wu, Xu and Yu for this problem when m = 2. Presently, their result is leveraged to derive necessary and sufficient conditions for general m. In doing so, an interval for \rho is uncovered for which detection is impossible using 2 graphs alone but becomes possible with m > 2 graphs.

We consider the elliptic equation with boundary singularities \begin{equation} \begin{cases} -\Delta u=-\lambda |x|^{-s_{1}}|u|^{p-2}u+|x|^{-s_{2}}|u|^{q-2}u &\text { in } \varOmega , u(x)=0 &\text { on } \partial \varOmega , \end{cases} \end{equation} where $0\leq s_1 < s_2 < 2$, $2<p< 2^{*}(s_1)$, $q< 2^{*}(s_2)$. Which is the subcritical approximations of the Li-Lin's open problem proposed by Li and Lin (Arch Ration Mech Anal 203(3): 943-968, 2012). We find a positive solution which is a local minimum point of the energy functional on the Nehari manifold when $p>q>\frac{2-s_2}{2-s_1}p+\frac{2s_2-2s_1}{2-s_1}$. We also discuss the asymptotic behavior of the positive solution and find a new class of blow-up points by blowing up analysis. These blow-up points are on the boundary of the domain, which are not similar with the usual.

This paper demonstrates the impact of a phase field method on shape registration to align shapes of possibly different topology. It yields new insights into the building of discrepancy measures between shapes regardless of topology, which would have applications in fields of image data analysis such as computational anatomy. A soft end-point optimal control problem is introduced whose minimum measures the minimal control norm required to align an initial shape to a final shape, up to a small error term. The initial data is spatially integrable, the paths in control spaces are integrable and the evolution equation is a generalized convective Allen-Cahn. Binary images are used to represent shapes for the initial data. Inspired by level-set methods and large diffeomorphic deformation metric mapping, the controls spaces are integrable scalar functions to serve as a normal velocity and smooth reproducing kernel Hilbert spaces to serve as velocity vector fields. The existence of mild solutions to the evolution equation is proved, the minimums of the time discretized optimal control problem are characterized, and numerical simulations of minimums to the fully discretized optimal control problem are displayed. The numerical implementation enforces the maximum-bounded principle, although it is not proved for these mild solutions. This research offers a novel discrepancy measure that provides valuable ways to analyze diverse image data sets. Future work involves proving the existence of minimums, existence and uniqueness of strong solutions and the maximum bounded principle.

For a smooth formal scheme $\mathfrak{X}$ over the Witt vectors $W$ of a perfect field $k$, we construct a functor $\mathbb{D}_\mathrm{crys}$ from the category of prismatic $F$-crystals $(\mathcal{E},\varphi_\mathcal{E})$ (or prismatic $F$-gauges) on $\mathfrak{X}$ to the category of filtered $F$-crystals on $\mathfrak{X}$. We show that $\mathbb{D}_\mathrm{crys}(\mathcal{E},\varphi_\mathcal{E})$ enjoys strong properties (e.g., strong divisibility in the sense of Faltings) when $(\mathcal{E},\varphi_\mathcal{E})$ is what we call locally filtered free (lff). Most significantly, we show that $\mathbb{D}_\mathrm{crys}$ actually induces an equivalence between the category of prismatic $F$-gauges on $\mathfrak{X}$ with Hodge--Tate weights in $[0,p-2]$ and the category of Fontaine--Laffaille modules on $\mathfrak{X}$. Finally, we use our functor $\mathbb{D}_\mathrm{crys}$ to enhance the study of prismatic Dieduonn\'e theory of $p$-divisible groups (as initiated by Ansch\"{u}tz--Le Bras) allowing one to recover the filtered crystalline Dieudonn\'e crystal from the prismatic Dieudonn\'e crystal. This in turn allows us to clarify the relationship between prismatic Dieudonn\'e theory and the work of Kim on classifying $p$-divisible groups using Breuil--Kisin modules.

Let $p\geq 5$ be a prime number. In this paper, we construct Galois representations associated with modular forms for which the dimension of the $p$-torsion in the Bloch-Kato Selmer group can be made arbitrarily large. Our result extends similar results known for small primes, such as Matsuno's work on Tate-Shafarevich groups of elliptic curves. Extending the technique of Hamblen and Ramakrishna, we lift residually reducible Galois representations to modular representations for which the associated Greenberg Selmer groups are minimally generated by a large number of elements over the Iwasawa algebra. We deduce that there is an isogenous lattice for which the Bloch-Kato Selmer group has large $p$-rank.

Nudging is a data assimilation method amenable to both analysis and implementation. It also has the (reported) advantage of being insensitive to model errors compared to other assimilation methods. However, nudging behavior in the presence of model errors is little analyzed. This report gives an analysis of nudging to correct model errors. The analysis indicates that the error contribution due to the model error decays as the nudging parameter $\chi \to \infty$ like $\mathcal{O}(\chi^{-\frac{1}{2}})$, Theorem 3.2. Numerical tests verify the predicted convergence rates and validate the nudging correction to model errors.

We study the traveling wave solutions of the Burgers-Huxley equation from a geometric point of view via the qualitative theory of ordinary differential equations. By using the Poincar\'e compactification we study the global phase portraits of a family of polynomial ordinary differential equations in the plane related to the Burgers-Huxley equation. We obtain the traveling wave solutions and their asymptotic behaviors from the orbits that connect equilibrium points taking into account the restrictions of the studied equation.

Hoeffding's formulation and solution to the universal hypothesis testing (UHT) problem had a profound impact on many subsequent works dealing with asymmetric hypotheses. In this work, we introduce a quantum universal hypothesis testing framework that serves as a quantum analog to Hoeffding's UHT. Motivated by Hoeffding's approach, which estimates the empirical distribution and uses it to construct the test statistic, we employ quantum state tomography to reconstruct the unknown state prior to forming the test statistic. Leveraging the concentration properties of quantum state tomography, we establish the exponential consistency of the proposed test: the type II error probability decays exponentially quickly, with the exponent determined by the trace distance between the true state and the nominal state.

In mathematical phylogenetics, the time-consistent galled trees provide a simple class of rooted binary network structures that can be used to represent a variety of different biological phenomena. We study the enumerative combinatorics of unlabeled and labeled time-consistent galled trees. We present a new derivation via the symbolic method of the number of unlabeled time-consistent galled trees with a fixed number of leaves and a fixed number of galls. We also derive new generating functions and asymptotics for labeled time-consistent galled trees.

For an $(\infty,n)$-category $\mathscr E$ we define an $(\infty,1)$ category $\mathrm{TwAr}(\mathscr E)$ and provide an isomorphism between the stabilization of the overcategory of $\mathscr E$ in $\mathrm{Cat}_{(\infty,n)}$ and the $\infty$-category of spectrum-valued functors on $\mathrm{TwAr}(\mathscr E)$. We use this to develop the deformation theory of $(\infty,n)$-categories and apply it to given an $\infty$-categorical characterization of lax-idempotent monads.

Modern control algorithms require tuning of square weight/penalty matrices appearing in quadratic functions/costs to improve performance and/or stability output. Due to simplicity in gain-tuning and enforcing positive-definiteness, diagonal penalty matrices are used extensively in control methods such as linear quadratic regulator (LQR), model predictive control, and Lyapunov-based control. In this paper, we propose an eigendecomposition approach to parameterize penalty matrices, allowing positive-definiteness with non-zero off-diagonal entries to be implicitly satisfied, which not only offers notable computational and implementation advantages, but broadens the class of achievable controls. We solve three control problems: 1) a variation of Zermelo's navigation problem, 2) minimum-energy spacecraft attitude control using both LQR and Lyapunov-based methods, and 3) minimum-fuel and minimum-time Lyapunov-based low-thrust trajectory design. Particle swarm optimization is used to optimize the decision variables, which will parameterize the penalty matrices. The results demonstrate improvements of up to 65% in the performance objective in the example problems utilizing the proposed method.

We investigate convexification for convex quadratic optimization with step function penalties. Such problems can be cast as mixed-integer quadratic optimization problems, where binary variables are used to encode the non-convex step function. First, we derive the convex hull for the epigraph of a quadratic function defined by a rank-one matrix and step function penalties. Using this rank-one convexification, we develop copositive and semi-definite relaxations for general convex quadratic functions. Leveraging these findings, we construct convex formulations to the support vector machine problem with 0--1 loss and show that they yield robust estimators in settings with anomalies and outliers.

Thurston proposed, in part of an unfinished manuscript, to study surface group actions on $S^1$ by using an $S^1$-connection on the suspension bundle obtained from a harmonic measure. Following the approach and previous work of the authors, we study the actions of general lattices of $\mathrm{PSU}(1,1)$ on $S^1$. We prove the Gauss--Bonnet formula for the $S^1$-connection associated with a harmonic measure, and show that a harmonic measure on the suspension bundle of the action with maximal Euler number has rigidity, having a form closely related to the Poisson kernel. As an application, we prove a semiconjugacy rigidity for foliations with maximal Euler number, which is analogous to theorems due to Matsumoto, Minakawa and Burger--Iozzi--Wienhard.

This paper investigates an incompressible steady free boundary problem of Euler equations with helical symmetry in $3$ dimensions and with nontrivial vorticity. The velocity field of the fluid arises from the spiral of its velocity within a cross-section, whose global existence, uniqueness and well-posedness with fixed boundary were established by a series of brilliant works. A perplexing issue, untouched in the literature, concerns the free boundary problem with (partial) unknown domain boundary in this helically symmetric configuration. We address this gap through the analysis of the optimal regularity property of the scalar stream function as a minimizer in a semilinear minimal problem, establishing the $C^{0,1}$-regularity of the minimizer, and the $C^{1,\alpha}$-regularity of its free boundary. More specifically, the regularity results are obtained in arbitrary cross-sections through smooth helical transformation by virtue of variational method and the rule of "flatness implies $C^{1,\alpha}$".

Given a class $\mathcal{P}$ of groups we say that a group $G$ is fully residually $\mathcal{P}$ if for any finite subset $F$ of $G$, there exists an epimorphism from $G$ to a group in $\mathcal{P}$ which is injective on $F$. It is known that any non-trivial knot group is fully residually finite. For hyperbolic knots, its knot group is fully residually closed hyperbolic $3$--manifold group, and fully residually simple. In this article, we show that every non-trivial knot group is fully residually perfect, closed $3$--manifold group.

We provide an extension of the unbiased simulation method for SDEs developed in Henry-Labordere et al. [Ann Appl Probab. 27:6 (2017) 1-37] to a class of path-dependent dynamics, pertaining for Asian options. In our setting, both the payoff and the SDE's coefficients depend on the (weighted) average of the process or, more precisely, on the integral of the solution to the SDE against a continuous function with bounded variations. In particular, this applies to the numerical resolution of the class of path-dependent PDEs whose regularity, in the sens of Dupire, is studied in Bouchard and Tan [Ann. I.H.P., to appear].

We develop a theory of nearby and vanishing cycles in the context of finite-coefficient Zariski-constructible sheaves over a non-archimedean field which is non-trivially valued, complete, algebraically closed, and of mixed characteristic or equal characteristic zero. Apart from basic properties, we show that they preserve Zariski-constructibility, have a Milnor fibre interpretation, satisfy Beilinson's gluing construction, are perverse t-exact, and commute with Verdier duality.

We consider a collection of linear stochastic bandit problems, each modeling the random response of different agents to proposed interventions, coupled together by a global safety constraint. We assume a central coordinator must choose actions to play on each bandit with the objective of regret minimization, while also ensuring that the expected response of all agents satisfies the global safety constraints at each round, in spite of uncertainty about the bandits' parameters. The agents consider their observed responses to be private and in order to protect their sensitive information, the data sharing with the central coordinator is performed under local differential privacy (LDP). However, providing higher level of privacy to different agents would have consequences in terms of safety and regret. We formalize these tradeoffs by building on the notion of the sharpness of the safety set - a measure of how the geometric properties of the safe set affects the growth of regret - and propose a unilaterally unimprovable vector of privacy levels for different agents given a maximum regret budget.

Let $(m_1, m_2)$ be a pair of positive integers. Denote by $\mathbb{P}^1$ the complex projective line, and by $\mathbb{P}^1_{m_1,m_2}$ the orbifold complex projective line obtained from $\mathbb{P}^1$ by adding $\mathbb{Z}_{m_1}$ and $\mathbb{Z}_{m_2}$ orbifold points. In this paper we introduce a matrix linear difference equation, prove existence and uniqueness of its formal Puiseux-series solutions, and use them to give conjectural formulas for $k$-point ($k\ge2$) functions of Gromov--Witten invariants of $\mathbb{P}^1_{m_1,m_2}$. Explicit expressions of the unique solutions are also obtained. We carry out concrete computations of the first few invariants by using the conjectural formulas. For the case when one of $m_1,m_2$ equals 1, we prove validity of the conjectural formulas with $k\ge3$.

The exponential strong converse for a coding problem states that, if a coding rate is beyond the theoretical limit, the correct probability converges to zero exponentially. For the lossy source coding with side-information, also known as the Wyner-Ziv (WZ) problem, a lower bound on the strong converse exponent was derived by Oohama. In this paper, we derive the tight strong converse exponent for the WZ problem; as a special case, we also derive the tight strong converse exponent for the distributed function computation problem. For the converse part, we use the change-of-measure argument developed in the literature and the soft Markov constraint introduced by Oohama; the matching achievability is proved via the Poisson matching approach recently introduced by Li and Anantharam. Our result is build upon the recently derived tight strong converse exponent for the Wyner-Ahlswede-Korner (WAK) problem; however, compared to the WAK problem, more sophisticated argument is needed. As an illustration of the necessity of the soft Markov constraint, we present an example such that the soft Markov constraint is strictly positive.

Nonlocal models and their associated theories have been extensively investigated in recent years. Among these, nonlocal versions of the classical Laplace operator have attracted the most attention, while higher-order nonlocal operators have been studied far less. In this work, we focus on the nonlocal counterpart of the classical biharmonic operator together with clamped boundary condition ($u$ and $\frac{\partial u}{\partial n}$ are given on boundary). We develop the variational formulation of a nonlocal biharmonic model, establish the existence and uniqueness of its solution, and analyze its convergence as the nonlocal horizon approaches zero. In addition, numerical experiments are presented to further illustrate the analytical properties of the model and its solution.

In anabelian geometry, we consider to what extent the \'{e}tale or tame fundamental groups of schemes reflect geometric properties of the schemes. Although there are many known results (mainly for smooth curves) in this area, general singular curves have rarely been treated. One reason is that we cannot determine the isomorphism classes of singular curves themselves from their \'{e}tale or tame fundamental groups in general. On the other hand, Das proved that the structure of the tame fundamental group of a singular curve over an algebraically closed field is determined by its normalization and an invariant relating to its singularities. In the present paper, by using this and known anabelian results, we determine the isomorphism classes of the normalizations of singular curves over various fields under certain conditions.

In this paper, we investigate the nonlinear stability and transition threshold for the 3D Boussinesq system in Sobolev space under the high Reynolds number and small thermal diffusion in $\mathbb{T}\times\mathbb{R}\times\mathbb{T} $. It is proved that if the initial velocity $v_{\rm in}$ and the initial temperature $ \theta_{\rm in} $ satisfy $ \|v_{\rm in}-(y,0,0)\|_{H^{2}}\leq \varepsilon\nu, \|\theta_{\rm in}\|_{H^{2}}\leq \varepsilon\nu^{2} $, respectively for some $ \varepsilon>0 $ independent of the Reynolds number or thermal diffusion, then the solutions of 3D Boussinesq system are global in time.

In this paper, as a continuation of [Fernandez-Guasti, \textit{Celest Mech Dyn Astron} 137, 4 (2025)], we demonstrate the maximal superintegrability of the reduced Hamiltonian, which governs the four-body choreographic planar motion along the lima\c{c}on trisectrix (resembling a folded figure eight), in the six-dimensional space of relative motion. The corresponding eleven integrals of motion in the Liouville-Arnold sense are presented explicitly. Specifically, it is shown that the reduced Hamiltonian admits complete separation of variables in Jacobi-like variables. The emergence of this choreography is not a direct consequence of maximal superintegrability. Rather, it originates from the existence of \textit{particular integrals} and the phenomenon of \textit{particular involution}. The fragmentation of a more general four-body choreographic motion into two isomorphic two-body choreographies is discussed in detail. This model combines choreographic motion with maximal superintegrability, a seldom-studied interplay in classical mechanics.

We introduce the operation of even periodization on nonconnective spectral stacks. We show how to recover from it the even filtration of Hahn-Raksit-Wilson, and (a Nygaard-completion of) the filtered prismatization stack of Bhatt-Lurie and Drinfeld. We prove that the moduli stack of oriented elliptic curves of Goerss-Hopkins-Miller and Lurie is the even periodization of the spectrum of topological modular forms. Likewise, the chromatic moduli stack, which we previously studied under the name of the moduli stack of oriented formal groups, arises via even periodization from the sphere spectrum. Over the complex bordism spectrum, we relate even periodization with the symmetric monoidal shearing, in the sense of Devalapurkar, of free gradings.

Quantum resonances described by non-Hermitian tridiagonal-matrix Hamiltonians $H$ with complex energy eigenvalues $E_n \in {\mathbb C}$ are considered. The method of evaluation of quantities $\sigma_n=\sqrt{E_n^*E_n}$ known as the singular values of $H$ is proposed. Its basic idea is that the quantities $\sigma_n$ can be treated as square roots of eigenvalues of a certain auxiliary self-adjoint operator $\mathbb{H}$. As long as such an operator can be given a block-tridiagonal matrix form, we construct its resolvent as a matrix continued fraction. In an illustrative application of the formalism, a discrete version of conventional Hamiltonian $H=-d^2/dx^2+V(x)$ with complex local $V(x) \neq V^*(x)$ is considered. The numerical convergence of the recipe is found quick, supported also by a fixed-point-based formal proof.

We study the spectral stability of the one-dimensional small-amplitude periodic traveling wave solutions of the (1+1)-dimensional Caudrey-Dodd-Gibbon-Sawada-Kotera equation. We show that these waves are spectrally stable with respect to co-periodic as well as square integrable perturbations.

This article explores the connection between Chebyshev polynomials and knot theory, specifically in relation to Gram determinants. We reveal intriguing formulae involving the Chebyshev polynomial of the first and second kind. In particular we show that for Mersenne numbers, $M_k=2^k-1$ where $k\geq 2$, the $M_k$-th Chebyshev polynomial of the second kind is the product of Chebyshev polynomials of the first kind. We then discuss the Gram determinant of type $(Mb)_1$, restate the conjecture of its closed formula in terms of mostly products of Chebyshev polynomials of the second kind, and prove a factor of the determinant that supports the conjecture. We also showcase an algorithm for calculating the Gram determinant's corresponding matrix. Furthermore, we restate Qi Chen's conjectured closed formula for the Gram determinant of type Mb and discuss future directions.

Compliant mechanisms achieve motion through elastic deformation. In this work, we address the synthesis of a compliant cross-hinge mechanism capable of large angular strokes while approximating the behavior of an ideal revolute joint. To capture the competing demands of kinematic fidelity, rotational stiffness, and resistance to parasitic motion, we formulate a multi-objective optimization problem based on kinetostatic performance measures. A hybrid design strategy is employed: an efficient beam-based structural model enables extensive exploration of a high-dimensional design space using evolutionary algorithms, followed by fine-tuning with high-fidelity three-dimensional finite element analysis. The resulting Pareto-optimal designs reveal diverse geometric configurations and performance trade-offs.

In this paper, we prove that there exists a residual subset of contact forms $\lambda$ (if any) on a compact orientable manifold $M$ for which the foliation de Rham cohomology of the associated Reeb foliation $F_\lambda$ is trivial in that both $H^0(F_\lambda,{\mathbb R})$ and $H^1(F_\lambda,{\mathbb R})$ are isomorphic to $\mathbb R$. For any choice of $\lambda$ from the aforementioned residual subset, this cohomological result can be restated as any of the following two equivalent statements: (1) The functional equation $R_{\lambda}[f] = u$ is \emph{uniquely} solvable (modulo the addition by constant) for any $u$ satisfying $\int_M u\, d\mu_\lambda =0$, or (2) The Lie algebra of the group of strict contactomorphisms is isomorphic to the span of Reeb vector fields, and so isomorphic to the 1 dimensional abelian Lie algebra $\mathbb R$. This result is also a key ingredient for the proof of the generic scarcity result of strict contactomorphisms by Savelyev and the author.

The expansion exponent (or expansion constant) for maps was introduced by Schreiber in \cite{s}. In this paper, we introduce the analogous exponent for measures. We shall prove the following results: The expansion exponent of a measurable maps is equal to the minimum of the expansion exponent taken over the Borel probability measures. In particular, a map expands small distances (in the sense of Reddy \cite{r}) if and only if every Borel probability has positive expansion exponent. Any nonatomic invariant measure with positive expansion exponent is positively expansive in the sense of \cite{m}. For ergodic invariant measures, the Kolmogorov-Sinai entropy is bounded below by the product of the expansion exponent and the measure upper capacity. As a consequence, any ergodic invariant measure with both positive upper capacity and positive expansion exponent must have positive entropy.

The notion of non-projectible contact forms on a given compact manifold $M$ is introduced by the first-named author in [Ohb], the set of which he also shows is a residual subset of the set of (coorientable) contact forms. In this paper, we prove that for any non-projectible contact form $\lambda$ the set, denoted by $\text{\rm Cont}^{\text{\rm st}}(M,\lambda)$, consisting of strict contactomorphisms of $\lambda$ is a a countable disjoint union of real lines $\mathbb R$, one for each connected component.

We investigate the ground states of neutron stars and white dwarfs in the Hartree-Fock (HF) and Hartree-Fock-Bogoliubov (HFB) theories. It is known that the system is stable below a critical mass, which depends on the gravitational constant, while it becomes unstable if the total mass exceeds the critical mass. We prove that if the total mass is at the critical mass, then the HFB minimizers do not exist for any gravitational constant, while the HF minimizers exist for every gravitational constant except for a countable set, which is fully characterized by the Gagliardo-Nirenberg inequality for orthonormal systems. Our results complement the existence results in the sub-critical mass case established in [E. Lenzmann and M. Lewin, Duke Math. J., 2010].

We collect some results and notions concerning generalizations for block Jacobi matrices of several concepts, which have been important for spectral studies of the simpler and better known scalar Jacobi case. We focus here on some issues related to the matrix Titchmarsh-Weyl function, but we also consider generalizations of some other tools used by subordinacy theory, including the matrix orthogonal polynomials, the notion of finite cyclicity, a variant of a notion of nonsubordinacy, as well as Jitomirskaya-Last type semi-norms. The article brings together some issues already known, our new concepts, and also improvements and strengthening of some results already existing. We give simpler proofs of some known facts or we add details usually omitted in the existing literature. The introduction contains a separate part devoted to a brief review of the main spectral analysis methods used so far for block Jacobi operators.

For all odd positive integers $m$, we construct $\mu$-homogeneous solutions to the thin obstacle problem in $\mathbb{R}^3,$ with $\mu\in(m,m+1)$. For $m$ large, $\mu-m$ converges to $1$, so $\mu\neq m+\tfrac 1 2$. The restriction to odd values of $m$ is necessary: we show that, for all $n\ge 2$, there are no $\mu$-homogeneous solutions to the thin obstacle problem in $\mathbb{R}^n$ with $\mu \in \bigcup_{k\ge 0}(2k,2k+1)$. These examples also apply to $2$-valued $C^{1,1/2}$ stationary harmonic functions or $\mathbb{Z}/2\mathbb{Z}$-eigenfunctions of the laplacian on the sphere.

Introducing a moment map whose zero locus is the group of symplectomorphisms of the real four-dimensional torus, we exhibit a gradient flow that can be made into a strictly parabolic flow by mean of a DeTurck trick (famously known for its use in the study of the Ricci flow), showing the local existence and regularity for the solutions of this flow and hence showing that the group of symplectomorphisms of the real four-dimensional torus is locally contractible. This work follows the ideas introduced by Yann Rollin in [3], even though the moment map picture comes from different considerations.

In this paper, we introduce the notions of quasi-triangular and factorizable perm bialgebras, based on notions of the perm Yang-Baxter equation and $(R, \mathrm{ad})$-invariant condition. A factorizable perm bialgebra induces a factorization of the underlying perm algebra and the double of a perm bialgebra naturally admits a factorizable perm bialgebra structure. The notion of relative Rota-Baxter operators of weights on perm algebras is introduced to characterize solutions of the perm Yang-Baxter equation, whose skew-symmetric parts are $(R, \mathrm{ad})$-invariant. These operators are in one-to-one correspondence with linear transformations fulfilling a Rota-Baxter-type identity in the case of quadratic perm algebras. Furthermore, we introduce the notion of quadratic Rota-Baxter perm algebras of weights, demonstrate that a quadratic Rota-Baxter perm algebra of weight $0$ induces a triangular perm bialgebra, and establish a one-to-one correspondence between quadratic Rota-Baxter perm algebras of nonzero weights and factorizable perm bialgebras.

In this paper, we study the local connectivity and Hausdorff dimension for the boundaries of the bounded hyperbolic components in the space $\mathcal P_d$ of polynomials of degree $d\geq 3$. It is shown that for any non disjoint-type bounded hyperbolic component $\mathcal H\subset \mathcal P_d$, the locally connected part of $\partial\mathcal H$, along each regular boundary strata, has full Hausdorff dimension $2d-2$. An essential innovation in our argument involves analyzing how the canonical parameterization of the hyperbolic component--realized via Blaschke products over a mapping scheme--extends to the boundary. This framework allows us to study three key aspects of $\partial \mathcal H$: the local connectivity structure, the perturbation behavior, and the local Hausdorff dimensions.

In this paper, we deal with the following generalized vector equilibrium problem: Let $X, Y$ be topological vector spaces over reals, $D$ be a nonempty subset of $X$, $K$ be a nonempty set and $\theta$ be origin of $Y$. Given multi-valued mapping $F: D\times K\rightrightarrows Y$, can be formulated as the problem, find $\bar x\in D$ such that $$\mbox{GVEP}(F, D, K)\,\,\,\,\,\,\theta\in F(\bar x, y)\ \mbox{for all}\ y\in K.$$ We prove several existence theorems for solutions to the generalized vector equilibrium problem when $K$ is an arbitrary nonempty set without any algebraic or topological structure. Furthermore, we establish that some sufficient conditions ensuring the existence of a solution for the considered conditions are imposed not on the entire domain of the bifunctions but rather on a self-segment-dense subset. We apply the obtained results to variational relation problems, vector equilibrium problems, and common fixed point problems.

We prove the dilation principle for the relative Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal group and using the dilation principle we prove the Quillen's analog of the local-global principle for the group. Applying the relative local-global principle, we prove the solvability and nilpotency of the ${\rm SK_1}$-analog of the orthogonal groups and study the homotopy and commutativity principle for odd elementary orthogonal groups.

An explicit formula for the bracket of the exceptional simple Lie algebra E8 based on triality and oct-octonions is obtained, following the Barton-Sudbery description of E8.

We provide a new proof of the analogue of the Artin-Springer theorem for groups of type $\mathsf{D}$ that can be represented by similitudes over an algebra of Schur index $2$: an anisotropic generalized quadratic form over a quaternion algebra $Q$ remains anisotropic after generic splitting of $Q$, hence also under odd degree field extensions of the base field. Our proof is characteristic free and does not use the excellence property.

In this paper, we will introduce Bell numbers $D(n)$ of type $D$ as an analogue to the classical Bell numbers related to all the partitions of the set $[n]$. Then based on a signed set partition of type $D$, we will construct the recurrence relations of Bell numbers $D(n)$. In addition, we deduce the exponential generating function for $D(n)$. Finally, we will provide an explicit formula for $D(n)$.

In this paper, we propose a novel machine learning-based method to solve the acoustic scattering problem in unbounded domain. We first employ the Dirichlet-to-Neumann (DtN) operator to truncate the physically unbounded domain into a computable bounded domain. This transformation reduces the original scattering problem in the unbounded domain to a boundary value problem within the bounded domain. To solve this boundary value problem, we design a neural network with a subspace layer, where each neuron in this layer represents a basis function. Consequently, the approximate solution can be expressed by a linear combination of these basis functions. Furthermore, we introduce an innovative alternating optimization technique which alternately updates the basis functions and their linear combination coefficients respectively by training and least squares methods. In our method, we set the coefficients of basis functions to 1 and use a new loss function each time train the subspace. These innovations ensure that the subspace formed by these basis functions is truly optimized. We refer to this method as the alternately-optimized subspace method based on neural networks (AO-SNN). Extensive numerical experiments demonstrate that our new method can significantly reduce the relative $l^2$ error to $10^{-7}$ or lower, outperforming existing machine learning-based methods to the best of our knowledge.

We propose and implement modern computational methods to enhance catastrophe excess-of-loss reinsurance contracts in practice. The underlying optimization problem involves attachment points, limits, and reinstatement clauses, and the objective is to maximize the expected profit while considering risk measures and regulatory constraints. We study the problem formulation, paving the way for practitioners, for two very different approaches: A local search optimizer using simulated annealing, which handles realistic constraints, and a branch & bound approach exploring the potential of a future speedup via quantum branch & bound. On the one hand, local search effectively generates contract structures within several constraints, proving useful for complex treaties that have multiple local optima. On the other hand, although our branch & bound formulation only confirms that solving the full problem with a future quantum computer would require a stronger, less expensive bound and substantial hardware improvements, we believe that the designed application-specific bound is sufficiently strong to serve as a basis for further works. Concisely, we provide insurance practitioners with a robust numerical framework for contract optimization that handles realistic constraints today, as well as an outlook and initial steps towards an approach which could leverage quantum computers in the future.

It is well known that an Anosov diffeomorphism $T$ enjoys linear response of its SRB measure with respect to infinitesimal perturbations $T_2$. For a fixed observation function $c$, we develop a theory to optimise the response of the SRB-expectation of $c$. Our approach is based on the response of the transfer operator on the anisotropic Banach spaces of Gou\"ezel-Liverani. For any non-constant $c$ we prove that the optimising perturbation $T_2$ exists and is unique, and we provide explicit expressions for the Fourier coefficients of $T_2$. We develop an efficient Fourier-based numerical scheme to approximate the optimal vector field $T_2$, along with a proof of convergence. The utility of our approach is illustrated in two numerical examples, by localising SRB measures with small, optimally selected, perturbations.

This is a survey of contact homology and its applications to the study of contact manifolds. It is a small tribute to Yasha Eliashberg's huge generosity with his countless explanations of his deep mathematical insights all along his career. It is also the author's wishful thinking that this text could be useful to students and young mathematicians for learning about some of the holomorphic curves based invariants in contact geometry.

We introduce a method for studying reduction types of arithmetic curves and wildly ramified base change. We give new proofs of earlier results of Lorenzini and Obus-Wewers, and resolve a question of Lorenzini on the Euler characteristic of the resolution graph of a $p$-cyclic arithmetic surface quotient singularity. Our method consists of constructing a simultaneous skeleton for the associated cover of Berkovich analytifications and applying a skeletal Riemann-Hurwitz formula.

We consider a finite number of orientation preserving $C^2$ interval diffeomorphisms and apply them randomly in such a way that the expected Lyapunov exponents at the boundary points are positive. We prove the exponential mixing, with respect to the unique stationary measure supported on the interior of the interval. The key step is to show the exponential synchronization in average.

In this paper, we study the unitary Dyson Brownian motion through a partial differential equation approach recently introduced for the real Dyson case. The main difference with the real Dyson case is that the spectrum is now on the circle and not on the real line, which leads to particular attention to comparison principles. First we recall why the system of particles which are the eigenvalues of unitary Dyson Brownian motion is well posed thanks to a containment function. Then we proved that the primitive of the limit spectral measure of the unitary Dyson Brownian motion is the unique solution to a viscosity equation obtained by primitive the Dyson equation on the circle. Finally, we study some properties of solutions of Dyson's equation on the circle. We prove a L $\infty$ regularization. We also look at the long time behaviour in law of a solution through a study of the so-called free entropy of the system. We conclude by discussing the uniform convergence towards the uniform measure on the circle of a solution of the Dyson equation.

We develop a methodology for constructing confidence sets for parameters of statistical models via a reduction to sequential prediction. Our key observation is that for any generalized linear model (GLM), one can construct an associated game of sequential probability assignment such that achieving low regret in the game implies a high-probability upper bound on the excess likelihood of the true parameter of the GLM. This allows us to develop a scheme that we call online-to-confidence-set conversions, which effectively reduces the problem of proving the desired statistical claim to an algorithmic question. We study two varieties of this conversion scheme: 1) analytical conversions that only require proving the existence of algorithms with low regret and provide confidence sets centered at the maximum-likelihood estimator 2) algorithmic conversions that actively leverage the output of the online algorithm to construct confidence sets (and may be centered at other, adaptively constructed point estimators). The resulting methodology recovers all state-of-the-art confidence set constructions within a single framework, and also provides several new types of confidence sets that were previously unknown in the literature.

The paper considers existence of spatially regular solutions for a class of linear Boltzmann transport equations. The related transport problem is an (initial) inflow boundary value problem. This problem is characteristic with variable multiplicity, that is, the rank of the boundary matrix (here a scalar) is not constant on the boundary. It is known that for these types of (initial) boundary value problems the full higher order Sobolev regularity cannot generally be established. In this paper we present Sobolev regularity results for solutions of linear Boltzmann transport problems when the data belongs to appropriate anisotropic Sobolev spaces whose elements are zero on the inflow and characteristic parts of the boundary.

Duffin and Schaeffer provided a famous counterexample to show that Khintchine's theorem fails without monotonicity assumption. Given any monotonically decreasing approximation function with divergent series, we construct Duffin--Schaeffer-type counterexamples by restricting the denominator. We also extend these constructions to the inhomogeneous setting. Our results resolve some natural questions arising from the works of Erd\H{o}s, Vaaler, and Yu.

Countably generated projective modules that are relatively big with respect to a trace ideal were introduced by P. P\v{r}\'ihoda, as an extension of Bass' uniformly big projectives. It has already been proved that there are a number of interesting examples of rings whose countably generated projective modules are always relatively big. In this paper, we increase the list of such examples, showing that it includes all right noetherian rings satisfying a polynomial identity. We also show that countably generated projective modules over locally semiperfect torsion-free algebras over $h$-local domains are always relatively big. This last result applies to endomorphism rings of finitely generated torsion-free modules over $h$-local domains. As a consequence, we can give a complete characterization of those $h$-local domains of Krull dimension $1$ for which every direct summand of a direct sum of copies of a single finitely generated torsion-free module is again a direct sum of finitely generated modules.

We give explicit, highly symmetric equations for the versal deformation of the singularity $L_{n+1}^n$ consisting of n+1 lines through the origin in n-dimensional affine space in generic position. These make evident that the base space of the versal deformation of $L_{n+1}^n$ is isomorphic to the total space for $L_{n}^{n-1}$, if n>4. By induction it follows that the base space is irreducible and Gorenstein. We discuss the known connection to a modular compactification of the moduli space of (n+1)-pointed curves of genus 1. For other elliptic partition curves it seems unfeasable to compute the versal deformation in general. It is doubtful whether the base space is Gorenstein. For rational partition curves we show that the base space in general has components of different dimensions.

In the online non-stochastic control problem, an agent sequentially selects control inputs for a linear dynamical system when facing unknown and adversarially selected convex costs and disturbances. A common metric for evaluating control policies in this setting is policy regret, defined relative to the best-in-hindsight linear feedback controller. However, for general convex costs, this benchmark may be less meaningful since linear controllers can be highly suboptimal. To address this, we introduce an alternative, more suitable benchmark--the performance of the best fixed input. We show that this benchmark can be viewed as a natural extension of the standard benchmark used in online convex optimization and propose a novel online control algorithm that achieves sublinear regret with respect to this new benchmark. We also discuss the connections between our method and the original one proposed by Agarwal et al. in their seminal work introducing the online non-stochastic control problem, and compare the performance of both approaches through numerical simulations.

We study spaces of matrices coming from irreducible representations of reductive groups over an algebraically closed field of characteristic zero and we completely classify those of constant corank one. In particular, we recover the examples coming from symmetric forms discovered in [BFL22].

The aim of this paper is to study the cohomology theory of Reynolds Lie algebras equipped with derivations and to explore related applications. We begin by introducing the concept of Reynolds LieDer pairs. Subsequently, we construct the associated cohomology. Finally, we investigate formal deformations, abelian extensions, and extensions of a pair of derivations, all interpreted through the lens of cohomology groups.

We determine the Balmer spectrum of dualisable objects in the stable module category for $\mathrm{H}_1\mathfrak{F}$ groups of type $\mathrm{FP}_{\infty}$ and show that the telescope conjecture holds for these categories. We also determine the spectrum of dualisable objects for certain infinite free products of finite groups. Using this, we give examples where the stable category is not stratified by the spectrum of dualisable objects and where the telescope conjecture does not hold.

This paper proposes a new hybrid high-order discretization for the biharmonic problem and the corresponding eigenvalue problem. The discrete ansatz space includes degrees of freedom in $n-2$ dimensional submanifolds (e.g., nodal values in 2D and edge values in 3D), in addition to the typical degrees of freedom in the mesh and on the hyperfaces in the HHO literature. This approach enables the characteristic commuting property of the hybrid high-order methodology in any space dimension and allows for lower eigenvalue bounds of higher order for the eigenvalue problem. The main results are quasi-best approximation estimates as well as reliable and efficient error control. The latter motivates an adaptive mesh-refining algorithm that empirically recovers optimal convergence rates for singular solutions.

You can invent striking and challenging problems with unique solution by building some symmetry into functional equations. Some are suitable for high school; others could generate college-level projects involving computer algebra. The problems are functional equations with group actions in the background. Interesting examples arise even from small finite groups. Whether a given problem ``works" with a given choice of constant coefficients depends on whether a related multilinear form is nonzero. These forms are essentially the classical group determinants studied by Frobenius in the nineteenth century.

We introduce a diagrammatic braided monoidal category, the quantum spin Brauer category, together with a full functor to the category of finite-dimensional, type $1$ modules for $U_q(\mathfrak{so}(N))$ or $U_q(\mathfrak{o}(N))$. This functor becomes essentially surjective after passing to the idempotent completion. The quantum spin Brauer category can be thought of as a quantum version of the spin Brauer category introduced previously by the authors. Alternatively, it is an enlargement of the Kauffman category, obtained by adding a generating object corresponding to the quantum spin module.

We show that the spectral gap of the first cohomological Laplacian $\Delta_1$ for $\operatorname{Sp}_{2n}(\mathbb{Z})$ follows once a slightly stronger assumption holds for some $\operatorname{Sp}_{2m}(\mathbb{Z})$, where $n\geq m$. As an application of this result, we provide explicit lower bounds for some quotients of $\operatorname{Sp}_{2n}(\mathbb{Z})$ for any $n\geq 2$.

Let $A(D)$ be an elliptic homogeneous linear differential operator with complex constant coefficients, $ \mu $ be a vector-valued Borel measure and $w$ be a positive locally integrable function on $\mathbb{R}^N$. In this work, we present sufficient conditions on $\mu$ and $w$ for the existence of solutions in the weighted Lebesgue spaces $L^p_w$ for the equation $A^{*}(D)f=\mu$, for $ 1\leq p<\infty $. Those conditions are related to a certain control of the Riesz potential of the measure $\mu$. We also present sufficient conditions for the solvability when $p=\infty$ adding a canceling condition on the operator. Our method is based on a new weighted $L^1$ Stein-Weiss type inequality on measures for a special class of vector fields.

We introduce an efficient method for decomposing the circuit variety of a given matroid $M$, based on an algorithm that identifies its minimal extensions. These extensions correspond to the smallest elements above $M$ in the poset defined by the dependency order. We apply our algorithm to several classical configurations: the V\'amos matroid, the unique Steiner quadruple system $S(3,4,8)$, the projective and affine planes, the dual of the Fano matroid, and the dual of the graphic matroid of $K_{3,3}$. In each case, we compute the minimal irreducible decomposition of their circuit varieties.

We prove the existence and uniqueness of weak solutions of the inhomogeneous incompressible Navier--Stokes equations without vacuum using the relative energy method. We present a novel and direct proof of the existence of weak solutions based on approximation with more regular solutions. The analysis we employ to justify the strong convergence reveals how to conclude the stability and uniqueness of weak solutions. To the best of our knowledge, these stability estimates are completely new. Furthermore, for the first time, we establish energy conservation for weak solutions.

The local exact controllability of the one-dimensional bilinear Schr{\"o}dinger equation with Dirichlet boundary conditions has been extensively studied in subspaces of H 3 since the seminal work of K. Beauchard. Our first objective is to revisit this result and establish the controllability in H 1 0 for suitable discontinuous control potentials. In the second part, we consider the equation in the presence of periodic boundary conditions and a constant magnetic field. We prove the local exact controllability of periodic H 1 -states, thanks to a Zeeman-type effect induced by the magnetic field which decouples the resonant spectrum. Finally, we discuss open problems and partial results for the Neumann case and the harmonic oscillator.

In this paper, we develop the theory of multiple Rota-Baxter modules over multiple Rota-Baxter algebras. We introduce left, right, and bimodule structures and construct free $\Omega$-operated modules with mixable tensor establishing free commutative multiple Rota-Baxter modules. We provide a necessary and sufficient condition for a free module to admit a free multiple Rota-Baxter module structure. Furthermore, we define projective and injective multiple Rota-Baxter modules, showing that their category has enough projective and injective objects to support derived $\mathrm{Hom}$ functors. Finally, we introduce the tensor product of multiple Rota-Baxter algebras and define flat multiple Rota-Baxter modules, proving that both free and projective modules satisfy the flatness property.

The small Alfv\'en number (denoted by $\varepsilon$) limit (one type of large parameter limits, i.e. singular limits) in magnetohydrodynamic (abbr. MHD) equations was first proposed by Klainerman--Majda in (Comm. Pure Appl. Math. 34: 481--524, 1981). Recently Ju--Wang--Xu mathematically verified that the \emph{local-in-time} solutions of three-dimensional (abbr. 3D) ideal (i.e. the absence of the dissipative terms) incompressible MHD equations with general initial data in $\mathbb{T}^3$ (i.e. a spatially periodic domain) tend to a solution of 2D ideal MHD equations in the distribution sense as $\varepsilon\to 0$ by Schochet's fast averaging method in (J. Differential Equations, 114: 476--512, 1994). In this paper, we revisit the small Alfv\'en number limit in $\mathbb{R}^n$ with $n=2$, $3$, and develop another approach, motivated by Cai--Lei's energy method in (Arch. Ration. Mech. Anal. 228: 969--993, 2018), to establish a new conclusion that the \emph{global-in-time} solutions of incompressible MHD equations (including the viscous resistive case) with general initial data converge to zero as $\varepsilon\to 0$ for any given time-space variable $(x,t)$ with $t>0$. In addition, we find that the large perturbation solutions and vanishing phenomenon of the nonlinear interactions also exist in the \emph{viscous resistive} MHD equations for small Alfv\'en numbers, and thus extend Bardos et al.'s results of the \emph{ideal} MHD equations in (Trans Am Math Soc 305: 175--191, 1988).

We prove a new characterization of metric Sobolev spaces, in the spirit of Brezis--Van Schaftingen--Yung's asymptotic formula. A new feature of our work is that we do not need Poincar\'e inequality which is a common tool in the literature. Another new feature is that we find a direct link between Brezis--Van Schaftingen--Yung's asymptotic formula and Cheeger's Lipschitz differentiability.

We investigate whether every computable member of a given class of structures admits a fully primitive recursive (also known as punctual) or fully P-TIME copy. A class with this property is referred to as punctually robust or P-TIME robust, respectively. We present both positive and negative results for structures corresponding to well-known representations of trees, such as binary trees, ordered trees, sequential (or prefix) trees, and partially ordered (poset) trees. A corollary of one of our results on trees is that semilattices and lattices are not punctually robust. In the main result of the paper, we demonstrate that, unlike Boolean algebras, modal algebras - that is, Boolean algebras with modality - are not punctually robust. The question of whether distributive lattices are punctually robust remains open. The paper contributes to a decades-old program on effective and feasible algebra, which has recently gained momentum due to rapid developments in punctual structure theory and its connections to online presentations of structures.

We study the behavior of volumes of divisors in a family. We show that the volume of a divisor on the generic fiber equals the infimum of its volumes on fibers over any dense subset of the base. As an application, we show that the volume function is upper semicontinuous in flat families with reduced and irreducible fibers. We also prove that for a $\mathbb{Q}$-Cartier divisor $B$ on a family of varieties $X\rightarrow T$, if $(X_t,|B_t|_{\mathbb{Q}})$ is $\epsilon$-lc and the volume of $B_t$ is a constant for densely many closed points $t\in T$, then the generic fiber $(X_\eta,|B_\eta|_{\mathbb{Q}})$ is also $\epsilon$-lc.

We introduce a new multiplication for the polytope algebra, defined via the intersection of polytopes. After establishing the foundational properties of this intersection product, we investigate finite-dimensional subalgebras that arise naturally from this construction. These subalgebras can be regarded as volumetric analogues of the graded M\"obius algebra, which appears in the context of the Dowling-Wilson conjecture. We conjecture that they also satisfy the injective hard Lefschetz property and the Hodge-Riemann relations, and we prove these in degree one.

We are concerned with the (stochastic) Lagrangian trajectories associated with Euler or Navier-Stokes equations. First, we construct solutions to the 3D Euler equations which dissipate kinetic energy with $C_{t,x}^{1/3-}$ regularity, such that the associated Lagrangian trajectories are not unique. The proof is based on the non-uniqueness of positive solutions to the corresponding transport equations, in conjunction with the superposition principle. Second, in dimension $d\geq2$, for any $1<p<2,\frac{1}{p}+\frac{1}{s}>1+\frac1d$, we construct solutions to the Euler or Navier-Stokes equations in the space $C_tL^p\cap L_t^1W^{1,s}$, demonstrating that the associated (stochastic) Lagrangian trajectories are not unique. Our result is sharp in 2D in the sense that: (1) in the stochastic case, for any vector field $v\in C_tL^p$ with $p>2$, the associated stochastic Lagrangian trajectory associated with $v$ is unique (see \cite{KR05}); (2) in the deterministic case, the LPS condition guarantees that for any weak solution $v\in C_tL^p$ with $p>2$ to the Navier-Stokes equations, the associated (deterministic) Lagrangian trajectory is unique. Our result is also sharp in dimension $d\geq2$ in the sense that for any divergence-free vector field $v\in L_t^1W^{1,s}$ with $s>d$, the associated (deterministic) Lagrangian trajectory is unique (see \cite{CC21}).

Let $X$ be a smooth rigid space with an action of a finite group $G$ satisfying that $X/G$ is represented by a rigid space. We construct sheaves of $p$-adic Cherednik algebras on the small \'etale site of the quotient $X/G$, and study some of their properties. The sheaves of $p$-adic Cherednik algebras are sheaves of Fr\'echet $K$-algebras on $X/G$, which can be regarded as $p$-adic analytic versions of the sheaves of Cherednik algebras associated to the action of a finite group on a smooth algebraic variety defined by P. Etingof. Furthermore, their sections on small enough $G$-invariant affinoid spaces are canonically Fr\'echet-Stein algebras. Along the way, we construct sheaves of infinite order twisted differential operators on $X$, we give a $G$-equivariant classification of the Atiyah algebras (Picard algebroids) on $X$, and study the category of co-admissible modules over a sheaf of infinite order twisted differential operators.

We use Kan injectivity to axiomatise concepts in the 2-category of topoi. We showcase the expressivity of this language through many examples, and we establish some aspects of the formal theory of Kan extension in this 2-category (pointwise Kan extensions, fully faithful morphisms, etc.). We use this technology to introduce fragments of geometric logic, and we accommodate essentially algebraic, disjunctive, regular, and coherent logic in our framework, together with some more exotic examples. We show that each fragment $\mathcal{H}$ in our sense identifies a lax-idempotent (relative) pseudomonad $\mathsf{T}^{\mathcal{H}}$ on $\mathsf{lex}$, the $2$-category of finitely complete categories. We show that the algebras for $\mathsf{T}^{\mathcal{H}}$ admit a notion of classifying topos, for which we deliver several Diaconescu-type results. The construction of classifying topoi allows us to define conceptually complete fragments of geometric logic.

It is well known that algebraic power series are differentially finite (D-finite): they satisfy linear differential equations with polynomial coefficients. The converse problem, whether a given D-finite power series is algebraic or transcendental, is notoriously difficult. We prove that this problem is decidable: we give two theoretical algorithms and a transcendence test that is efficient in practice.

We introduce the concept of a triangular decomposition for Banach and Fr\'echet-Stein algebras over $p$-adic fields, which allows us to define a category $\mathcal{O}$ for a wide array of topological algebras. In particular, we apply this concept to $p$-adic rational Cherednik algebras, which allows us to obtain an analytic version of the category $\mathcal{O}$ developed by Ginzburg, Guay, Opdam and Rouquier. Along the way, we study the global sections of $p$-adic Cherednik algebras on smooth Stein spaces, and determine their behavior with respect to the rigid analytic GAGA functor.

We show that the equations of motion of the rigid body about a fixed point in the Newtonian field with a quadratic potential are special reduction of period-one closure of the Darboux dressing chain for the Schr\"odinger operators with matrix potentials. Some new explicit solutions of the corresponding matrix system and the spectral properties of the related Schr\"odinger operators are discussed.

We introduce a formalism of Hochschild (co)-homology for $\mathcal{D}$-cap modules on smooth rigid analytic spaces based on the homological tools of Ind-Banach $\mathcal{D}$-cap modules. We introduce several categories of $\mathcal{D}$-cap bimodules for which this theory is well-behaved. Among these, the most important example is the category of diagonal $\mathcal{C}$-complexes. We give an explicit calculation of the Hochschild complex for diagonal $\mathcal{C}$-complexes, and show that the Hochschild complex of $\mathcal{D}$-cap is canonically isomorphic to the de Rham complex of $X$. In particular, we obtain a Hodge-de Rham spectral sequence converging to the Hochschild cohomology groups of $\mathcal{D}$-cap. We obtain explicit formulas relating the Hochschild cohomology and homology of a given diagonal $\mathcal{C}$-complex.

There is a high interest in accelerating multiscale models using data-driven surrogate modeling techniques. Creating a large training dataset encompassing all relevant load scenarios is essential for a good surrogate, yet the computational cost of producing this data quickly becomes a limiting factor. Commonly, a pre-trained surrogate is used throughout the computational domain. Here, we introduce an alternative adaptive mixture approach that uses a fast probabilistic surrogate model as constitutive model when possible, but resorts back to the true high-fidelity model when necessary. The surrogate is thus not required to be accurate for every possible load condition, enabling a significant reduction in the data collection time. We achieve this by creating phases in the computational domain corresponding to the different models. These phases evolve using a phase-field model driven by the surrogate uncertainty. When the surrogate uncertainty becomes large, the phase-field model causes a local transition from the surrogate to the high-fidelity model, maintaining a highly accurate simulation. We discuss the requirements of this approach to achieve accurate and numerically stable results and compare the phase-field model to a purely local approach that does not enforce spatial smoothness for the phase mixing. Using a Gaussian Process surrogate for an elasto-plastic material, we demonstrate the potential of this mixture of models to accelerate multiscale simulations.

In the context of Berglund-Huebsch mirror symmetry, we compute the eigenvalues of the Frobenius endomorphism acting on a p-adic version of Borisov's complex. As a result, we conjecture an explicit formula for the number of points of crepant resolutions of invertible Calabi-Yau orbifolds defined over a finite field.

We give a simple proof of the assertion claiming that the spectrum of the cone of a non-reduced plane curve can be determined only by its multiplicities along local irreducible components at each singular point as well as those along global ones together with the degrees of the latter (where the relation between global components and singular points is not needed) if the associated reduced plane curve have only ordinary singularities (for instance a line arrangement). Note that the last condition is strictly weaker than local homogeneity. As a corollary we can also get a simple proof of a formula which is equivalent to the one obtained earlier by the second-named author.

In this paper, we present an adaptive gradient descent method for geodesically convex optimization on a Riemannian manifold with nonnegative sectional curvature. The method automatically adapts to the local geometry of the function and does not use additional expensive computations other than calculation of the derivative of the Riemannian exponential. We prove the convergence of the method under the assumption of geodesic completeness. The performance of the method is illustrated by experiments on the sphere, the manifold of symmetric positive definite matrices equipped with the Bures-Wasserstein metric.

A fundamental question in information theory is to quantify the loss of information under a noisy channel. Partial orders and contraction coefficients are typical tools to that end, however, they are often also challenging to evaluate. For the special class of binary input symmetric output (BISO) channels, Geng et al. showed that among channels with the same capacity, the binary symmetric channel (BSC) and binary erasure channel (BEC) are extremal with respect to the more capable order. Here, we show two main results. First, for channels with the same KL contraction coefficient, the same holds with respect to the less noisy order. Second, for channels with the same Dobrushin coefficient, or equiv. maximum leakage or Doeblin coefficient, the same holds with respect to the degradability order. In the process, we provide a closed-form expression for the contraction coefficients of BISO channels. We also discuss the comparability of BISO channels and extensions to binary channels in general.

We establish numerical criteria for when a Noether-Lefschetz divisor on a moduli space of quasi-polarized K3 surfaces $F_{2d}$, or more generally on an orthogonal modular variety, generates an extremal ray in the cone of pseudoeffective divisors. In particular, for all d, we exhibit many extremal rays of the cone of pseudoeffective divisors of both $F_{2d}$ and any normal projective $\mathbb{Q}$-factorial compactification of $F_{2d}$ lying over its Baily-Borel compactification.

In this paper, we establish the asymptotic behavior of {\it supercritical} nearly unstable Hawkes processes with a power law kernel. We find that, the Hawkes process in our context admits a similar equation to that in \cite{MR3563196} for {\it subcritical} case. In particular, the rescaled Hawkes process $(Z^n_{nt}/n^{2\alpha})_{t\in[0,1]}$ converges in law to a kind of integrated fractional Cox Ingersoll Ross process with different coefficients from that in \cite{MR3563196}, as $n$ tends to infinity.

It is shown that novel relations between multiple zeta values and single-variable multiple polylogarithms at 1/2 (delta values) can be derived by comparing two distinct, yet a priori equal, series formulae for the Drinfeld associator (from the Knizhnik-Zamolodchikov connection). In particular, we demonstrate that two new relations are found by comparing the fifth order terms of each series formula.

This paper builds rigorous analysis on the root-exponential convergence for the lightning schemes via rational functions in approximating corner singularity problems with uniform exponentially clustered poles proposed by Gopal and Trefethen. The start point is to set up the representations of $z^\alpha$ and $z^\alpha\log z$ in the slit disk and develop results akin to Paley-Wiener theorem, from which, together with the Poisson summation formula, the root-exponential convergence of the lightning plus polynomial scheme with an exact order for each clustered parameter is established in approximation of prototype functions $g(z)z^\alpha$ or $g(z)z^\alpha\log z$ on a sector-shaped domain, which includes $[0,1]$ as a special case. In addition, the fastest convergence rate is confirmed based upon the best choice of the clustered parameter. Furthermore, the optimal choice of the clustered parameter and the convergence rate for corner singularity problems in solving Laplace equations are attested based on Lehman and Wasow's study of corner singularities and along with the decomposition of Gopal and Trefethen. The thorough analysis provides a solid foundation for lightning schemes and rational approximation. Ample numerical evidences demonstrate the optimality and sharpness of the estimates.

A cyclic Riemannian Lie group is a Lie group $G$ equipped with a left-invariant Riemannian metric $h$ that satisfies $\oint_{X,Y,Z}h([X,Y],Z)=0$ for any left-invariant vector fields $X,Y,Z$. The initial concept and exploration of these Lie groups were presented in Monatsh. Math. \textbf{176} (2015), 219-239. This paper builds upon the results from the aforementioned study by providing a complete description of cyclic Riemannian Lie groups and an in-depth analysis of their various curvatures.

We study how a smooth irreducible algebraic variety $X$ of dimension $n$ embedded in $\mathbb{C} \mathbb{P}^{m}$ (with $m \geq n+2$), which degree is $d$, can be recovered using two projections from unknown points onto unknown hyperplanes. The centers and the hyperplanes of projection are unknown: the only input is the defining equations of each projected varieties. We show how both the projection operators and the variety in $\mathbb{C} \mathbb{P}^{m}$ can be recovered modulo some action of the group of projective transformations of $\mathbb{C} \mathbb{P}^{m}$. This configuration generalizes results obtained in the context of curves embedded in $\mathbb{C} \mathbb{P}^3$ and results concerning surfaces embedded in $\mathbb{C} \mathbb{P}^4$. We show how in a generic situation, a characteristic matrix of the pair of projections can be recovered. In the process we address dimensional issues and as a result establish a necessary condition, as well as a sufficient condition to compute this characteristic matrix up to a finite-fold ambiguity. These conditions are expressed as minimal values of the degree of the dual variety. Then we use this matrix to recover the class of the couple of projections and as a consequence to recover the variety. For a generic situation, two projections define a variety with two irreducible components. One component has degree $d(d-1)$ and the other has degree $d$, being the original variety.

We establish formulas for the Hilbert series of the Feichtner--Yuzvinsky Chow ring of a polymatroid using arbitrary building sets. For braid matroids and minimal building sets, our results produce new formulas for the Poincar\'e polynomial of the moduli space $\overline{\mathcal{M}}_{0,n+1}$ of pointed stable rational curves, and recover several previous results by Keel, Getzler, Manin, and Aluffi--Marcolli--Nascimento. We also use our methods to produce examples of matroids and building sets for which the corresponding Chow ring has Hilbert series with non-log-concave coefficients. This contrasts with the real-rootedness and log-concavity conjectures of Ferroni--Schr\"oter for matroids with maximal building sets, and of Aluffi--Chen--Marcolli for braid matroids with minimal building sets.

We present a new method of linear regression based on principal components using Hilbert-space-valued covariates with unknown reproducing kernels. We develop a computationally efficient approach to estimation and derive asymptotic theory for the regression parameter estimates under mild assumptions. We demonstrate the approach in simulation studies as well as in data analysis using two-dimensional brain images as predictors.

We give an explicit description up to the third page of the Sinha homology mod 2 spectral sequence for the space of long knots in $\mathbb{R}^3$, that is conjecturally equivalent to the Vassiliev spectral sequence. The description arises from a multicomplex structure on the Fox Neuwirth chain complexes for euclidean configuration spaces. A computer assisted calculation reveals a non trivial third page differential from a 2-dimensional class, in contrast to the rational case.

For each $n\in N ^{\ast }$, we write $s_{n}=\left( 1,\ldots ,1,0\right) $ with $n$ times $1$. For each $a \in N$, we consider the binary representation $\left( a_{i}\right) _{i\in -N }$ of $a$ with $a_{i}=0$ for nearly each $i$; we denote by $\alpha _{n}(a)$ the number of integers $i$ such that $\left( a_{i}, \ldots ,a_{i+n} \right) =s_{n}$. We consider the curve $C_{n}=\left( S_{n,k}\right) _{k\in N ^{\ast }}$ which consists of consecutive segments of length $1$ such that, for each $k$, $S_{n,k+1}$ is obtained from $S_{n,k}$ by turning right if $k+\alpha _{n}(k)-\alpha _{n}(k-1)$ is even and left otherwise. $C_{1}$ is self-avoiding since it is the curve associated to the alternating folding sequence. In [1], M. Mend\`es France and J. Shallit conjectured that the curves $C_{n}$ for $n\geq 2$ are also self-avoiding. In the present paper, we show that this property is true for $n=2$. We also prove that $C_{2}$ has some properties similar to those which were shown in [2], [3] and [4] for folding curves.

This articles investigates physics-based passive imaging problem, wherein one infers an unknown medium using ambient noise and correlation of the noise signal. We develop a general backpropagation framework via the so-called extended adjoint state, suitable for any linear PDE; crucially, this approach reduces by half the number of required PDE solves. Applications to several different PDE models demonstrate the universality of our method. In addition, we analyze the nonlinearity of the correlated model, revealing a surprising tangential cone condition-like structure, thereby advancing the state of the art towards a convergence guarantee for regularized reconstruction in passive imaging.

Let $Z$ be a noetherian integral excellent regular scheme of dimension $2$. Let $Y$ be an integral normal scheme endowed with a finite flat morphism $Y\to Z$ of degree $2$. We give a description of Lipman's desingularization of $Y$ by explicit equations, leading to a desingularization algorithm for $Y$.

In this paper, we give an alternative proof of the Horowitz-Myers conjecture in dimension $3 \leq N \leq 7$. Moreover, we show that a metric that achieves equality in the Horowitz-Myers conjecture is locally isometric to a Horowitz-Myers metric.

Lipid bilayer membranes are fundamental biological structures that serve as cellular boundaries, mediating transport, signaling, and maintaining structural integrity. This study introduces a novel mathematical model for open membranes immersed in Stokes flows, accounting for membrane elasticity, line tension at the open edge, and fluid-membrane interactions. The model is derived from an energy functional that incorporates Helfrich bending energy and a line energy associated with the open edge. By balancing dissipation in both the bulk fluid and the membrane surface, following the maximal dissipation principle, we derive the governing equations within an energy variational framework. Assuming axisymmetry and employing a boundary integral reduction, we transform the 3D problem into an effectively 1D problem, for which we develop a finite element-based numerical method to solve the resulting moving boundary problem. Several numerical examples are provided to validate the model and compare the results with existing studies.

Let $ \mathcal{L} = -\Delta + V $ be a Schr\"odinger operator acting on $ L^2(\mathbb{R}^n) $, where the nonnegative potential $ V $ belongs to the reverse H\"older class $ RH_q $ for some $ q \geq n/2 $. This article is primarily concerned with the study of endpoint boundedness for classical singular integral operators in the context of the space $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $, consisting of functions of vanishing mean oscillation associated with $ \mathcal{L} $. We establish the following main results: (i) the standard Hardy--Littlewood maximal operator is bounded on $\mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $; (ii) for each $ j = 1, \ldots, n$, the adjoint of the Riesz transform $ \partial_j \mathcal{L}^{-1/2} $ is bounded from $ C_0(\mathbb{R}^n) $ into $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $; and (iii) the approximation to the identity generated by the Poisson and heat semigroups associated with $ \mathcal{L} $ characterizes $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $ appropriately. These results recover the classical analogues corresponding to the Laplacian as a special case. However, the presence of the potential $ V $ introduces substantial analytical challenges, necessitating tools beyond the scope of classical Calder\'on--Zygmund theory. Our approach leverages precise heat kernel estimates and the structural properties of $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $ established by Song and the third author in [J. Geom. Anal. 32 (2022), no. 4, Paper No. 130, 37 pp].

We prove Bogomolov type inequalities for high Chern characters of semistable sheaves satisfying certain Frobenius semipositivity. The key ingredients in the proof are a high rank generalization of the asymptotic Riemann-Roch theorem and Langer's estimation theorem of the global sections of torsion free sheaves. These results give some Bogomolov type inequalities for semistable sheaves with vanishing low Chern characters. Our results are also applied to obtain inequalities of Chern characters of threefolds and varieties of small codimension in projective spaces and abelian varieties.

In this paper, we investigate distributed Nash equilibrium seeking for a class of two-subnetwork zero-sum games characterized by bilinear coupling. We present a distributed primal-dual accelerated mirror-descent algorithm that guarantees convergence. However, we demonstrate that this time-varying algorithm is not robust, as it fails to converge under even the slightest disturbances. To address this limitation, we introduce a distributed accelerated algorithm that employs a coordinated restarting mechanism. We model this new algorithm as a hybrid dynamical system and establish that it possesses structural robustness.

For a continuously differentiable Kolmogorov map defined from the nonnegative orthant to itself, a type-K competitive system is defined. Under the assumptions that the system is dissipative and the origin is a repeller, the global dynamics of such systems is investigated. A (weakly) type-K retrotone map is defined on a bounded set, which is backward monotone in some order. Under certain conditions, the dynamics of a type-K competitive system is the dynamics of a type-K retrotone map. Under these conditions, there exists a compact invariant set A that is the global attractor of the system on the nonnegative orthant exluding the origin. Some basic properties of A are established and remaining problems are listed for further investigation for general N-dimensional systems. These problems are completely solved for planar type-K competitive systems: every forward orbit is eventually monotone and converges to a fixed point; the global attractor A consists of two monotone curves each of which is a one-dimensional compact invariant manifold. A concrete example is provided to demonstrate the results for planar systems.

We consider the Kelvin-Helmholtz system describing the evolution of a vortex-sheet near the circular stationary solution. Answering previous numerical conjectures in the 90s physics literature, we prove an almost global existence result for small-amplitude solutions. We first establish the existence of a linear stability threshold for the Weber number, which represents the ratio between the square of the background velocity jump and the surface tension. Then, we prove that for almost all values of the Weber number below this threshold any small solution lives for almost all times, remaining close to the equilibrium. Our analysis reveals a remarkable stabilization phenomenon: the presence of both non-zero background velocity jump and capillarity effects enables to prevent nonlinear instability phenomena, despite the inherently unstable nature of the classical Kelvin-Helmholtz problem. This long-time existence would not be achievable in a setting where capillarity alone provides linear stabilization, without the richer modulation induced by the velocity jump. Our proof exploits the Hamiltonian nature of the equations. Specifically, we employ Hamiltonian Birkhoff normal form techniques for quasi-linear systems together with a general approach for paralinearization of non-linear singular integral operators. This approach allows us to control resonances and quasi-resonances at arbitrary order, ensuring the desired long-time stability result.

In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The application of finite element mesh makes the neural network element space satisfy the boundary value conditions directly on the complex geometric domains. The use of neural networks allows the accuracy of the approximate solution to reach the high level of neural network approximation even for the problems with singularities. We also provide the error analysis of the proposed method for the understanding. The proposed numerical method in this paper provides the way to enable neural network-based machine learning algorithms to solve a broader range of problems arising from engineering applications.

We say that a hereditary graph class $\mathcal{G}$ is \emph{clique-sparse} if there is a constant $k=k(\mathcal{G})$ such that for every graph $G\in\mathcal{G}$, every vertex of $G$ belongs to at most $k$ maximal cliques, and any maximal clique of $G$ can be intersected in at most $k$ different ways by other maximal cliques. We provide various characterisations of clique-sparse graph classes, including a list of five parametric forbidden induced subgraphs. We show that recent techniques for proving induced analogues of Menger's Theorem and the Grid Theorem of Robertson and Seymour can be lifted to prove induced variants in clique-sparse graph classes when replacing ``treewidth'' by ''tree-independence number''.

The projection filter is one of the approximations to the solution of the optimal filtering problem. It approximates the filtering density by projecting the dynamics of the square-root filtering density onto the tangent space of the square-root parametric densities manifold. While the projection filters for exponential and mixture families with continuous measurement processes have been well studied, the continuous-discrete projection filtering algorithm for non-conjugate priors has received less attention. In this paper, we introduce a simple Riemannian optimization method to be used for the Bayesian update step in the continuous-discrete projection filter for exponential families. Specifically, we show that the Bayesian update can be formulated as an optimization problem of $\alpha$-R\'enyi divergence, where the corresponding Riemannian gradient can be easily computed. We demonstrate the effectiveness of the proposed method via two highly non-Gaussian Bayesian update problems.

Primal-dual algorithm (PDA) is a classic and popular scheme for convex-concave saddle point problems. It is universally acknowledged that the proximal terms in the subproblems about the primal and dual variables are crucial to the convergence theory and numerical performance of primal-dual algorithms. By taking advantage of the information from the current and previous iterative points, we exploit two new proximal terms for the subproblems about the primal and dual variables. Based on two new proximal terms, we present a new primal-dual algorithm for convex-concave saddle point problems with bilinear coupling terms and establish its global convergence and O(1/N ) ergodic convergence rate. When either the primal function or the dual function is strongly convex, we accelerate the above proposed algorithm and show that the corresponding algorithm can achieve O(1/N^2) convergence rate. Since the conditions for the stepsizes of the proposed algorithm are related directly to the spectral norm of the linear transform, which is difficult to obtain in some applications, we also introduce a linesearch strategy for the above proposed primal-dual algorithm and establish its global convergence and O(1/N ) ergodic convergence rate . Some numerical experiments are conducted on matrix game and LASSO problems by comparing with other state-of-the-art algorithms, which demonstrate the effectiveness of the proposed three primal-dual algorithms.

We study an optimal partition problem on the sphere, where the cost functional is associated with the fractional $Q$-curvature in terms of the conformal fractional Laplacian on the sphere. By leveraging symmetries, we prove the existence of a symmetric minimal partition through a variational approach. A key ingredient in our analysis is a new H\"older regularity result for symmetric functions in a fractional Sobolev space on the sphere. As a byproduct, we establish the existence of infinitely many solutions to a nonlocal weakly-coupled competitive system on the sphere that remain invariant under a group of conformal diffeomorphisms and we investigate the asymptotic behavior of least-energy solutions as the coupling parameters approach negative infinity.

We pose the Kantorovich optimal transport problem as a min-max problem with a Nash equilibrium that can be obtained dynamically via a two-player game, providing a framework for approximating optimal couplings. We prove convergence of the timescale-separated gradient descent dynamics to the optimal transport plan, and implement the gradient descent algorithm with a particle method, where the marginal constraints are enforced weakly using the KL divergence, automatically selecting a dynamical adaptation of the regularizer. The numerical results highlight the different advantages of using the standard Kullback-Leibler (KL) divergence versus the reverse KL divergence with this approach, opening the door for new methodologies.

We introduce a fully-corrective generalized conditional gradient method for convex minimization problems involving total variation regularization on multidimensional domains. It relies on alternating between updating an active set of subsets of the spatial domain as well as of an iterate given by a conic combination of the associated characteristic functions. Different to previous approaches in the same spirit, the computation of a new candidate set only requires the solution of one prescribed mean curvature problem instead of the resolution of a fractional minimization task analogous to finding a generalized Cheeger set. After discretization, the former can be realized by a single run of a graph cut algorithm leading to significant speedup in practice. We prove the global sublinear convergence of the resulting method, under mild assumptions, and its asymptotic linear convergence in a more restrictive two-dimensional setting which uses results of stability of surfaces of prescribed curvature under perturbations of the curvature. Finally, we numerically demonstrate this convergence behavior in some model PDE-constrained minimization problems.

In 1990, D. Chinea and C. Gonzalez gave a classification of almost contact metric manifolds into $2^{12}$ classes, based on the behaviour of the covariant derivative $\nabla^g\Phi$ of the fundamental $2$-form $\Phi$. This large number makes it difficult to deal with this class of manifolds. We propose a new approach to almost contact metric manifolds by introducing two intrinsic endomorphisms $S$ and $h$, which bear their name from the fact that they are, basically, the entities appearing in the intrinsic torsion. We present a new classification scheme for them by providing a simple flowchart based on algebraic conditions involving $S$ and $h$, which then naturally leads to a regrouping of the Chinea-Gonzalez classes, and, in each step, to a further refinement, eventually ending in the single classes. This method allows a more natural exposition and derivation of both known and new results, like a new characterization of almost contact metric manifolds admitting a characteristic connection in terms of intrinsic endomorphisms. We also describe in detail the remarkable (and still very large) subclass of $\mathcal{H}$-parallel almost contact manifolds, defined by the condition $(\nabla^g_X\Phi)(Y,Z)=0$ for all horizontal vector fields, $X,Y,Z\in\mathcal{H}$.

We prove some model theoretic equivalent forms of variants of collection principle in set theory on models of a very weak set theory.

This paper is devoted to Sobolev interpolation inequalities for spinors, with weights of Caffarelli-Kohn-Nirenberg (CKN) type. In view of the corresponding results for scalar functions, a natural question is to determine whether optimal spinors have symmetry properties, or whether spinors with symmetry properties are linearly unstable, in which case we shall say that symmetry breaking occurs. What symmetry means has to be carefully defined and the overall picture turns out to be richer than in the scalar case. So far, no symmetrization technique is available in the spinorial case. We can however determine a range of the parameters for which symmetry holds using a detailed analysis based mostly on spectral methods.

This paper presents a performance analysis of the Non-Primary Channel Access (NPCA) mechanism, a new feature introduced in IEEE 802.11bn to enhance spectrum utilization in Wi-Fi networks. NPCA enables devices to contend for and transmit on the secondary channel when the primary channel is occupied by transmissions from an Overlapping Basic Service Set (OBSS). We develop a Continuous-Time Markov Chain (CTMC) model that captures the interactions among OBSSs in dense WLAN environments when NPCA is enabled, incorporating new NPCA-specific states and transitions. In addition to the analytical insights offered by the model, we conduct numerical evaluations and simulations to quantify NPCA's impact on throughput and channel access delay across various scenarios. Our results show that NPCA can significantly improve throughput and reduce access delays in favorable conditions for BSSs that support the mechanism. Moreover, NPCA helps mitigate the OBSS performance anomaly, where low-rate OBSS transmissions degrade network performance for all nearby devices. However, we also observe trade-offs: NPCA may increase contention on secondary channels, potentially reducing transmission opportunities for BSSs operating there.

Homomorphic Encryption (HE) allows secure and privacy-protected computation on encrypted data without the need to decrypt it. Since Shor's algorithm rendered prime factorisation and discrete logarithm-based ciphers insecure with quantum computations, researchers have been working on building post-quantum homomorphic encryption (PQHE) algorithms. Most of the current PQHE algorithms are secured by Lattice-based problems and there have been limited attempts to build ciphers based on error-correcting code-based problems. This review presents an overview of the current approaches to building PQHE schemes and justifies code-based encryption as a novel way to diversify post-quantum algorithms. We present the mathematical underpinnings of existing code-based cryptographic frameworks and their security and efficiency guarantees. We compare lattice-based and code-based homomorphic encryption solutions identifying challenges that have inhibited the progress of code-based schemes. We finally propose five new research directions to advance post-quantum code-based homomorphic encryption.

How to allocate limited resources to projects that will yield the greatest long-term benefits is a problem that often arises in decision-making under uncertainty. For example, organizations may need to evaluate and select innovation projects with risky returns. Similarly, when allocating resources to research projects, funding agencies are tasked with identifying the most promising proposals based on idiosyncratic criteria. Finally, in participatory budgeting, a local community may need to select a subset of public projects to fund. Regardless of context, agents must estimate the uncertain values of a potentially large number of projects. Developing parsimonious methods to compare these projects, and aggregating agent evaluations so that the overall benefit is maximized, are critical in assembling the best project portfolio. Unlike in standard sorting algorithms, evaluating projects on the basis of uncertain long-term benefits introduces additional complexities. We propose comparison rules based on Quicksort and the Bradley--Terry model, which connects rankings to pairwise "win" probabilities. In our model, each agent determines win probabilities of a pair of projects based on his or her specific evaluation of the projects' long-term benefit. The win probabilities are then appropriately aggregated and used to rank projects. Several of the methods we propose perform better than the two most effective aggregation methods currently available. Additionally, our methods can be combined with sampling techniques to significantly reduce the number of pairwise comparisons. We also discuss how the Bradley--Terry portfolio selection approach can be implemented in practice.

Pinching antenna systems (PASS) have been proposed as a revolutionary flexible antenna technology which facilitates line-of-sight links via numerous low-cost pinching antennas with adjustable activation positions over waveguides. This letter proposes a two-timescale joint transmit and pinching beamforming design for the maximization of sum rate of a PASS-based downlink multi-user multiple input single output system. A primal dual decomposition method is developed to decouple the two-timescale problem into two sub-problems: 1) A Karush-Kuhn-Tucker-guided dual learning-based approach is proposed to solve the short-term transmit beamforming design sub-problem; 2) The long-term pinching beamforming design sub-problem is tackled by adopting a stochastic successive convex approximation method. Simulation results demonstrate that the proposed two-timescale algorithm achieves a significant performance gain compared to other baselines.

Calibration is crucial for ensuring the performance of phased array since amplitude-phase imbalance between elements results in significant performance degradation. While amplitude-only calibration methods offer advantages when phase measurements are impractical, conventional approaches face two key challenges: they typically require high-resolution phase shifters and remain susceptible to phase errors inherent in these components. To overcome these limitations, we propose a Rotating element Harmonic Electric-field Vector (RHEV) strategy, which enables precise calibration through time modulation principles. The proposed technique functions as follows. Two 1-bit phase shifters are periodically phase-switched at the same frequency, each generating corresponding harmonics. By adjusting the relative delay between their modulation timings, the phase difference between the $+1$st harmonics produced by the two elements can be precisely controlled, utilizing the time-shift property of the Fourier transform. Furthermore, the +1st harmonic generated by sequential modulation of individual elements exhibits a linear relationship with the amplitude of the modulated element, enabling amplitude ambiguity resolution. The proposed RHEV-based calibration method generates phase shifts through relative timing delays rather than physical phase shifter adjustments, rendering it less susceptible to phase shift errors. Additionally, since the calibration process exclusively utilizes the $+1$st harmonic, which is produced solely by the modulated unit, the method demonstrates consistent performance regardless of array size. Extensive numerical simulations, practical in-channel and over-the-air (OTA) calibration experiments demonstrate the effectiveness and distinct advantages of the proposed method.

An integration of satellites and terrestrial networks is crucial for enhancing performance of next generation communication systems. However, the networks are hindered by the long-distance path loss and security risks in dense urban environments. In this work, we propose a satellite-terrestrial covert communication system assisted by the aerial active simultaneous transmitting and reflecting reconfigurable intelligent surface (AASTAR-RIS) to improve the channel capacity while ensuring the transmission covertness. Specifically, we first derive the minimal detection error probability (DEP) under the worst condition that the Warden has perfect channel state information (CSI). Then, we formulate an AASTAR-RIS-assisted satellite-terrestrial covert communication optimization problem (ASCCOP) to maximize the sum of the fair channel capacity for all ground users while meeting the strict covert constraint, by jointly optimizing the trajectory and active beamforming of the AASTAR-RIS. Due to the challenges posed by the complex and high-dimensional state-action spaces as well as the need for efficient exploration in dynamic environments, we propose a generative deterministic policy gradient (GDPG) algorithm, which is a generative deep reinforcement learning (DRL) method to solve the ASCCOP. Concretely, the generative diffusion model (GDM) is utilized as the policy representation of the algorithm to enhance the exploration process by generating diverse and high-quality samples through a series of denoising steps. Moreover, we incorporate an action gradient mechanism to accomplish the policy improvement of the algorithm, which refines the better state-action pairs through the gradient ascent. Simulation results demonstrate that the proposed approach significantly outperforms important benchmarks.

In this paper we evaluate the performance of topological features for generalizable and robust classification of firn image data, with the broader goal of understanding the advantages, pitfalls, and trade-offs in topological featurization. Firn refers to layers of granular snow within glaciers that haven't been compressed into ice. This compactification process imposes distinct topological and geometric structure on firn that varies with depth within the firn column, making topological data analysis (TDA) a natural choice for understanding the connection between depth and structure. We use two classes of topological features, sublevel set features and distance transform features, together with persistence curves, to predict sample depth from microCT images. A range of challenging training-test scenarios reveals that no one choice of method dominates in all categories, and uncoveres a web of trade-offs between accuracy, interpretability, and generalizability.

We establish Nash equilibrium learning -- convergence of the population state to a suitably defined Nash equilibria set -- for a class of payoff dynamical mechanism with a first order modification. The first order payoff modification can model aspects of the agents' bounded rationality, anticipatory or averaging terms in the payoff mechanism, or first order Pad\'e approximations of delays. To obtain our main results, we apply a combination of two nonstandard system-theoretic passivity notions.

We propose a formal framework for understanding and unifying the concept of observers across physics, computer science, philosophy, and related fields. Building on cybernetic feedback models, we introduce an operational definition of minimal observers, explore their role in shaping foundational concepts, and identify what remains unspecified in their absence. Drawing upon insights from quantum gravity, digital physics, second-order cybernetics, and recent ruliological and pregeometric approaches, we argue that observers serve as indispensable reference points for measurement, reference frames, and the emergence of meaning. We show how this formalism sheds new light on debates related to consciousness, quantum measurement, and computational boundaries; by way of theorems on observer equivalences and complexity measures. This perspective opens new avenues for investigating how complexity and structure arise in both natural and artificial systems.

Imaging reconstruction of interferometric data is a hard ill-posed inverse problem. Its difficulty is increased when observing the Galactic Center, which is obscured by a scattering screen. This is because the scattering breaks the one-to-one correspondence between images and visibilities. Solving the scattering problem is one of the biggest challenges in radio imaging of the Galactic Center. In this work we present a novel strategy to mitigate its effect and constrain the screen itself using multiobjective optimization. We exploit the potential of evolutionary algorithms to describe the optimization landscape to recover the intrinsic source structure and the scattering screen affecting the data. We successfully recover both the screen and the source in a wide range of simulated cases, including the speed of a moving screen at 230 GHz. Particularly, we can recover a ring structure in scattered data at 86 GHz. Our analysis demonstrates the huge potential that recent advancements in imaging and optimization algorithms offer to recover image structures, even in weakly constrained and degenerated, possibly multi-modal settings. The successful reconstruction of the scattering screen opens the window to event horizon scale works on the Galactic Center at 86G Hz up to 116 GHz, and the study of the scattering screen itself.

We generalize the quantum CUSUM (QUSUM) algorithm for quickest change-point detection, analyzed in finite dimensions by Fanizza, Hirche, and Calsamiglia (Phys. Rev. Lett. 131, 020602, 2023), to infinite-dimensional quantum systems. Our analysis relies on a novel generalization of a result by Hayashi (Hayashi, J. Phys. A: Math. Gen. 34, 3413, 2001) concerning the asymptotics of quantum relative entropy, which we establish for the infinite-dimensional setting. This enables us to prove that the QUSUM strategy retains its asymptotic optimality, characterized by the relationship between the expected detection delay and the average false alarm time for any pair of states with finite relative entropy. Consequently, our findings apply broadly, including continuous-variable systems (e.g., Gaussian states), facilitating the development of optimal change-point detection schemes in quantum optics and other physical platforms, and rendering experimental verification feasible.

Ball, Liu, Mazor and Pass proved that the existence of key-agreement protocols is equivalent to the hardness of a certain problem about interactive Kolmogorov complexity. We generalize the statement and give a short proof of the difficult implication.

We present a Bayesian inversion-based digital twin that employs acoustic pressure data from seafloor sensors, along with 3D coupled acoustic-gravity wave equations, to infer earthquake-induced spatiotemporal seafloor motion in real time and forecast tsunami propagation toward coastlines for early warning with quantified uncertainties. Our target is the Cascadia subduction zone, with one billion parameters. Computing the posterior mean alone would require 50 years on a 512 GPU machine. Instead, exploiting the shift invariance of the parameter-to-observable map and devising novel parallel algorithms, we induce a fast offline-online decomposition. The offline component requires just one adjoint wave propagation per sensor; using MFEM, we scale this part of the computation to the full El Capitan system (43,520 GPUs) with 92% weak parallel efficiency. Moreover, given real-time data, the online component exactly solves the Bayesian inverse and forecasting problems in 0.2 seconds on a modest GPU system, a ten-billion-fold speedup.

Perceptual hashing is used to detect whether an input image is similar to a reference image with a variety of security applications. Recently, they have been shown to succumb to adversarial input attacks which make small imperceptible changes to the input image yet the hashing algorithm does not detect its similarity to the original image. Property-preserving hashing (PPH) is a recent construct in cryptography, which preserves some property (predicate) of its inputs in the hash domain. Researchers have so far shown constructions of PPH for Hamming distance predicates, which, for instance, outputs 1 if two inputs are within Hamming distance $t$. A key feature of PPH is its strong correctness guarantee, i.e., the probability that the predicate will not be correctly evaluated in the hash domain is negligible. Motivated by the use case of detecting similar images under adversarial setting, we propose the first PPH construction for an $\ell_1$-distance predicate. Roughly, this predicate checks if the two one-sided $\ell_1$-distances between two images are within a threshold $t$. Since many adversarial attacks use $\ell_2$-distance (related to $\ell_1$-distance) as the objective function to perturb the input image, by appropriately choosing the threshold $t$, we can force the attacker to add considerable noise to evade detection, and hence significantly deteriorate the image quality. Our proposed scheme is highly efficient, and runs in time $O(t^2)$. For grayscale images of size $28 \times 28$, we can evaluate the predicate in $0.0784$ seconds when pixel values are perturbed by up to $1 \%$. For larger RGB images of size $224 \times 224$, by dividing the image into 1,000 blocks, we achieve times of $0.0128$ seconds per block for $1 \%$ change, and up to $0.2641$ seconds per block for $14\%$ change.

This paper develops a framework for the Hamiltonian quantization of complex Chern-Simons theory with gauge group $\mathrm{SL}(2,\mathbb{C})$ at an even level $k\in\mathbb{Z}_+$. Our approach follows the procedure of combinatorial quantization to construct the operator algebras of quantum holonomies on 2-surfaces and develop the representation theory. The $*$-representation of the operator algebra is carried by the infinite dimensional Hilbert space $\mathcal{H}_{\vec{\lambda}}$ and closely connects to the infinite-dimensional $*$-representation of the quantum deformed Lorentz group $\mathscr{U}_{\mathbf{q}}(sl_2)\otimes \mathscr{U}_{\widetilde{\mathbf{q}}}(sl_2)$, where $\mathbf{q}=\exp[\frac{2\pi i}{k}(1+b^2)]$ and $\widetilde{\mathbf{q}}=\exp[\frac{2\pi i}{k}(1+b^{-2})]$ with $|b|=1$. The quantum group $\mathscr{U}_{\mathbf{q}}(sl_2)\otimes \mathscr{U}_{\widetilde{\mathbf{q}}}(sl_2)$ also emerges from the quantum gauge transformations of the complex Chern-Simons theory. Focusing on a $m$-holed sphere $\Sigma_{0,m}$, the physical Hilbert space $\mathcal{H}_{phys}$ is identified by imposing the gauge invariance and the flatness constraint. The states in $\mathcal{H}_{phys}$ are the $\mathscr{U}_{\mathbf{q}}(sl_2)\otimes \mathscr{U}_{\widetilde{\mathbf{q}}}(sl_2)$-invariant linear functionals on a dense domain in $\mathcal{H}_{\vec{\lambda}}$. Finally, we demonstrate that the physical Hilbert space carries a Fenchel-Nielsen representation, where a set of Wilson loop operators associated with a pants decomposition of $\Sigma_{0,m}$ are diagonalized.

Machine learning interatomic potentials (MLIPs) provide an effective approach for accurately and efficiently modeling atomic interactions, expanding the capabilities of atomistic simulations to complex systems. However, a priori feature selection leads to high complexity, which can be detrimental to both computational cost and generalization, resulting in a need for hyperparameter tuning. We demonstrate the benefits of active set algorithms for automated data-driven feature selection. The proposed methods are implemented within the Atomic Cluster Expansion (ACE) framework. Computational tests conducted on a variety of benchmark datasets indicate that sparse ACE models consistently enhance computational efficiency, generalization accuracy and interpretability over dense ACE models. An added benefit of the proposed algorithms is that they produce entire paths of models with varying cost/accuracy ratio.

We develop a wave mechanics formalism for qubit geometry using holomorphic functions and Mobius transformations, providing a geometric perspective on quantum computation. This framework extends the standard Hilbert space description, offering a natural interpretation of standard quantum gates on the Riemann sphere that is examined through their Mobius action on holomorphic wavefunction. These wavefunctions emerge via a quantization process, with the Riemann sphere serving as the classical phase space of qubit geometry. We quantize this space using canonical group quantization with holomorphic polarization, yielding holomorphic wavefunctions and spin angular momentum operators that recover the standard $SU(2)$ algebra with interesting geometric properties. Such properties reveal how geometric transformations induce quantum logic gates on the Riemann sphere, providing a novel perspective in quantum information processing. This result provides a new direction for exploring quantum computation through Isham's canonical group quantization and its holomorphic polarization method.

Reconfigurable intelligent surfaces (RISs) enhance wireless communication by creating engineered signal reflection paths in addition to direct links. This work presents a stochastic geometry framework using point processes (PPs) to model multiple randomly deployed RISs conditioned on their associated base station (BS) locations. By characterizing aggregated reflections from multiple RISs using the Laplace transform, we analytically assess the performance impact of RIS-reflected signals by integrating this characterization into well-established stochastic geometry frameworks. Specifically, we derive closed-form expressions for the Laplace transform of the reflected signal power in several deployment scenarios. These analytical results facilitate performance evaluation of RIS-enabled enhancements. Numerical simulations validate that optimal RIS placement favors proximity to BSs or user equipment (UEs), and further quantify the impact of reflected interference, various fading assumptions, and diverse spatial deployment strategies. Importantly, our analytical approach shows superior computational efficiency compared to Monte Carlo simulations.

In modern large-scale systems with sensor networks and IoT devices it is essential to collaboratively solve complex problems while utilizing network resources efficiently. In our paper we present three distributed optimization algorithms that exhibit efficient communication among nodes. Our first algorithm presents a simple quantized averaged gradient procedure for distributed optimization, which is shown to converge to a neighborhood of the optimal solution. Our second algorithm incorporates a novel event-triggered refinement mechanism, which refines the utilized quantization level to enhance the precision of the estimated optimal solution. It enables nodes to terminate their operation according to predefined performance guarantees. Our third algorithm is tailored to operate in environments where each message consists of only a few bits. It incorporates a novel event-triggered mechanism for adjusting the quantizer basis and quantization level, allowing nodes to collaboratively decide operation termination based on predefined performance criteria. We analyze the three algorithms and establish their linear convergence. Finally, an application on distributed sensor fusion for target localization is used to demonstrate their favorable performance compared to existing algorithms in the literature.

Optical wireless communication (OWC) is envisioned as a key enabler for immersive indoor data transmission in future wireless communication networks. However, multi-user interference management arises as a challenge in dense indoor OWC systems composed of multiple optical access points (APs) serving multiple users. In this paper, we propose a novel dual-function OWC system for communication and localization. Non-orthogonal multiple access (NOMA) with random linear network coding (RLNC) is designed for data transmission, where NOMA allows the serving of multiple users simultaneously through controlling the power domain, and RLNC helps minimize errors that might occur during signal processing phase. This setup is assisted with a light detection and localization system (LiDAL) that can passively obtain spatio-temporal indoor information of user presence and location for dynamic-user grouping. The designed LiDAL system helps to improve the estimation of channel state information (CSI) in realistic indoor network scenarios, where the CSI of indoor users might be noisy and/or highly correlated. We evaluate the performance of NOMA combined with RLNC by analyzing the probability of successful decoding compared to conventional NOMA and orthogonal schemes. In addition, we derive the Cramer-Rao Lower Bound (CRLB) to evaluate the accuracy of location estimation. The results show that the proposed RLNC-NOMA improves the probability of successful decoding and the overall system performance. The results also show the high accuracy of the unbiased location estimator and its assistant in reducing the imperfection of CSI, leading to high overall system performance.

This article investigates the application of pinching-antenna systems (PASS) in multiuser multiple-input single-output (MISO) communications. Two sum-rate maximization problems are formulated under minimum mean square error (MMSE) decoding, with and without successive interference cancellation (SIC). To address the joint optimization of pinching antenna locations and user transmit powers, a fractional programming-based approach is proposed. Numerical results validate the effectiveness of the proposed method and show that PASS can significantly enhance uplink sum-rate performance compared to conventional fixed-antenna designs.

We study a simple two-dimensional swarmalator model that incorporates higher-order phase interactions, uncovering a diverse range of collective states. The latter include spatially coherent and gas-like configurations, neither of which appear in models with only pairwise interactions. Additionally, we discover bistability between various states, a phenomenon that arises directly from the inclusion of higher-order interactions. By analyzing several of these emergent states analytically, both for identical and nonidentical populations of swarmalators, we gain deeper insights into their underlying mechanisms and stability conditions. Our findings broaden the understanding of swarmalator dynamics and open new avenues for exploring complex collective behaviors in systems governed by higher-order interactions.

Safety assessment of patients with one-dimensionally structured passive implants, like cranial plates or stents, exposed to low or medium frequency magnetic fields, like those generated in magnetic resonance imaging or magnetic hyperthermia, can be challenging, because of the different length scales of the implant and the human body. Most of the methods used to estimate the heating induced near such implants neglect the presence of the metallic materials within the body, modeling the metal as thermal seeds. To overcome this limitation, a novel numerical approach that solves three-dimensional and one-dimensional coupled problems is proposed. This method leads to improved results by modelling the thermal diffusion through the highly conductive metallic implants. A comparison of the proposed method predictions with measurements performed on a cranial plate exposed to the magnetic field generated by a gradient coil system for magnetic resonance imaging is presented, showing an improved accuracy up to 25 % with respect to the method based on thermal seeds. The proposed method is finally applied to a magnetic hyperthermia case study in which a patient with a cranial plate is exposed to the magnetic field generated by a collar-type magnetic hyperthermia applicator for neck tumour treatment, predicting a temperature increase in proximity of the implant that is 10 % lower than the one overestimated by relying on thermal seeds.

This textbook compiles the lecture notes from security courses taught at Oxford in the 2000s, at Royal Holloway in the 2010s, and currently in Hawaii. The early chapters are suitable for a first course in security. The middle chapters have been used in advanced courses. Towards the end there are also some research problems.

Geometric hitting set problems, in which we seek a smallest set of points that collectively hit a given set of ranges, are ubiquitous in computational geometry. Most often, the set is discrete and is given explicitly. We propose new variants of these problems, dealing with continuous families of convex polyhedra, and show that they capture decision versions of the two-level finite adaptability problem in robust optimization. We show that these problems can be solved in strongly polynomial time when the size of the hitting/covering set and the dimension of the polyhedra and the parameter space are constant. We also show that the hitting set problem can be solved in strongly quadratic time for one-parameter families of convex polyhedra in constant dimension. This leads to new tractability results for finite adaptability that are the first ones with so-called left-hand-side uncertainty, where the underlying problem is non-linear.

In this paper, we develop a wavelet-based theoretical framework for analyzing the universal approximation capabilities of neural networks over a wide range of activation functions. Leveraging wavelet frame theory on the spaces of homogeneous type, we derive sufficient conditions on activation functions to ensure that the associated neural network approximates any functions in the given space, along with an error estimate. These sufficient conditions accommodate a variety of smooth activation functions, including those that exhibit oscillatory behavior. Furthermore, by considering the $L^2$-distance between smooth and non-smooth activation functions, we establish a generalized approximation result that is applicable to non-smooth activations, with the error explicitly controlled by this distance. This provides increased flexibility in the design of network architectures.

Phase-field fracture models provide a powerful approach to modeling fracture, potentially enabling the unguided prediction of crack growth in complex patterns. To ensure that only tensile stresses and not compressive stresses drive crack growth, several models have been proposed that aim to distinguish between compressive and tensile loads. However, these models have a critical shortcoming: they do not account for the crack direction, and hence they cannot distinguish between crack-normal tensile stresses that drive crack growth and crack-parallel stresses that do not. In this study, we apply a phase-field fracture model, developed in our earlier work, that uses the crack direction to distinguish crack-parallel stresses from crack-normal stresses. This provides a transparent energetic formulation that drives cracks to grow in when crack faces open or slide past each other, while the cracks respond like the intact solid when the crack faces contact under normal compressive loads. We compare our approach against two widely used approaches, Spectral splitting and the Volumetric-Deviatoric splitting, and find that these predict unphysical crack growth and unphysical stress concentrations under loading conditions in which these should not occur. Specifically, we show that the splitting models predict spurious crack growth and stress concentration under pure crack-parallel normal stresses. However, our formulation, which resolves the crack-parallel stresses from the crack-normal stresses, predicts these correctly.

A simple closed-form formula for the period of a pendulum with finite amplitude is proposed. It reproduces the exact analytical forms both in the small and large amplitude limits, while in the mid-amplitude range maintains average error of 0.06% and maximum error of 0.17%. The accuracy should be sufficient for typical engineering applications. Its unique simplicity should be useful in a theoretical development that requires trackable mathematical framework or in an introductory physics course that aims to discuss a finite amplitude pendulum. A simple and formally exact solution of angular displacement for the full range of amplitudes is illustrated.

A generalised form of time-translation-invariance permits to re-derive the known generic phenomenology of ageing, which arises in many-body systems after a quench from an initially disordered system to a temperature $T\leq T_c$, at or below the critical temperature $T_c$. Generalised time-translation-invariance is obtained, out of equilibrium, from a change of representation of the Lie algebra generators of the dynamical symmetries of scale-invariance and time-translation-invariance. Observable consequences include the algebraic form of the scaling functions for large arguments of the two-time auto-correlators and auto-responses, the equality of the auto-correlation and the auto-response exponents $\lambda_C=\lambda_R$, the cross-over scaling form for an initially magnetised critical system and the explanation of a novel finite-size scaling if the auto-correlator or auto-response converge for large arguments $y=t/s\gg 1$ to a plateau. For global two-time correlators, the time-dependence involving the initial critical slip exponent $\Theta$ is confirmed and is generalised to all temperatures below criticality and to the global two-time response function, and their finite-size scaling is derived as well. This also includes the time-dependence of the squared global order-parameter. The celebrate Janssen-Schaub-Schmittmann scaling relation with the auto-correlation exponent is thereby extended to all temperatures below the critical temperature. A simple criterion on the relevance of non-linear terms in the stochastic equation of motion is derived, taking the dimensionality of couplings into account. Its applicability in a wide class of models is confirmed, for temperatures $T\leq T_c$. Relevance to experiments is also discussed.

We propose a model of the functional architecture of curvature sensible cells in the visual cortex that associates curvature with scale. The feature space of orientation and position is naturally enhanced via its oriented prolongation, yielding a 4-dimensional manifold endowed with a canonical Engel structure. This structure encodes position, orientation, signed curvature, and scale. We associate an open submanifold of the prolongation with the quasi-regular representation of the similitude group SIM (2), and find left-invariant generators for the Engel structure. Finally, we use the generators of the Engel structure to characterize curvature-sensitive receptive profiles .

We explore novel topological responses and axion-like phenomena in three-dimensional insulating systems with spacetime-dependent mass terms encoding domain walls. Via a dimensional-reduction approach, we derive a new axion-electromagnetic coupling term involving three axion fields. This term yields a topological current in the bulk and, under specific conditions of the axions, real-space topological defects such as magnetic-like monopoles and hopfions. Moreover, once one the axions acquires a constant value, a nontrivial boundary theory realizes a (2+1)-dimensional analog of the Witten effect, which shows that point-like vortices on the gapped boundary of the system acquire half-integer electric charge. Our findings reveal rich topological structures emerging from multi-axion theories, suggesting new avenues in the study of topological phases and defects.

As the field of representation learning grows, there has been a proliferation of different loss functions to solve different classes of problems. We introduce a single information-theoretic equation that generalizes a large collection of modern loss functions in machine learning. In particular, we introduce a framework that shows that several broad classes of machine learning methods are precisely minimizing an integrated KL divergence between two conditional distributions: the supervisory and learned representations. This viewpoint exposes a hidden information geometry underlying clustering, spectral methods, dimensionality reduction, contrastive learning, and supervised learning. This framework enables the development of new loss functions by combining successful techniques from across the literature. We not only present a wide array of proofs, connecting over 23 different approaches, but we also leverage these theoretical results to create state-of-the-art unsupervised image classifiers that achieve a +8% improvement over the prior state-of-the-art on unsupervised classification on ImageNet-1K. We also demonstrate that I-Con can be used to derive principled debiasing methods which improve contrastive representation learners.