The graph linear canonical transform (GLCT) is presented as an extension of the graph Fourier transform (GFT) and the graph fractional Fourier transform (GFrFT), offering more flexibility as an effective tool for graph signal processing. In this paper, we introduce a GLCT based on chirp multiplication-chirp convolution-chirp multiplication decomposition (CM-CC-CM-GLCT), which irrelevant to sampling periods and without oversampling operation. Various properties and special cases of the CM-CC-CM-GLCT are derived and discussed. In terms of computational complexity, additivity, and reversibility, we compare the CM-CC-CM-GLCT and the GLCT based on the central discrete dilated Hermite function (CDDHFs-GLCT). Theoretical analysis demonstrates that the computational complexity of the CM-CC-CM-GLCT is significantly reduced. Simulation results indicate that the CM-CC-CM-GLCT achieves similar additivity to the CDDHFs-GLCT. Notably, the CM-CC-CM-GLCT exhibits better reversibility.

We analyse a dynamic control problem for scalar reaction-diffusion equations, focusing on the emulation of pattern formation through the selection of appropriate active controls. While boundary controls alone prove inadequate for replicating the complex patterns seen in biological systems, particularly under natural point-wise constraints of the system state, their combination with the regulation of the diffusion coefficient enables the successful generation of such patterns. Our study demonstrates that the set of steady-states is path-connected, facilitating the use of the staircase method. This approach allows any admissible initial configuration to evolve into any stationary pattern over a sufficiently long time while maintaining the system's natural bilateral constraints. We provide also examples of complex patterns that steady-state configurations can adopt.

In this paper, we propose universal proximal mirror methods to solve the variational inequality problem with Holder continuous operators in both deterministic and stochastic settings. The proposed methods automatically adapt not only to the oracle's noise (in the stochastic setting of the problem) but also to the Holder continuity of the operator without having prior knowledge of either the problem class or the nature of the operator information. We analyzed the proposed algorithms in both deterministic and stochastic settings and obtained estimates for the required number of iterations to achieve a given quality of a solution to the variational inequality. We showed that, without knowing the Holder exponent and Holder constant of the operators, the proposed algorithms have the least possible in the worst case sense complexity for the considered class of variational inequalities. We also compared the resulting stochastic algorithm with other popular optimizers for the task of image classification.

Derivative-free optimization (DFO) is a method that does not require the calculation of gradients or higher-order derivatives of the objective function, making it suitable for cases where the objective function is non-differentiable or the computation of derivatives is expensive. This communication discusses the importance of \(\Lambda\)-poisedness in DFO and the outliers detected by the Local Outlier Factor (LOF) on the optimization process. We discuss the relationship between \(\Lambda\)-poisedness in derivative-free optimization and outliers in local outlier factor.

We study the fundamental properties of pointwise semi-Lipschitz functions between asymmetric spaces, which are the natural asymmetric counterpart of pointwise Lipschitz functions. We also study the influence that partial symmetries of a given space may have on the behavior of pointwise semi-Lipschitz functions defined on it. Furthermore, we are interested in characterizing the pointwise semi-Lipschitz structure of an asymmetric space in terms of real-valued pointwise semi-Lipschitz functions defined on it. By using two algebras of functions naturally associated to our spaces of pointwise real-valued semi-Lipschitz functions, we are able to provide two Banach-Stone type results in this context. In fact, these results are obtained as consequences of a general Banach-Stone type theorem of topological nature, stated for abstract functional spaces, which is quite flexible and can be applied to many spaces of continuous functions over metric and asymmetric spaces.

A coding scheme for broadcast channels (BCs) is proposed that shifts the users' code blocks by different amounts of time and applies alternating Gelfand-Pinsker encoding. The scheme achieves all rate tuples in Marton's region for two receiver BCs without time-sharing or rate-splitting. Simulations with short polar codes show that the method reduces the gap to capacity as compared to time-sharing.

In this paper we study the direct resonance problem for Rayleigh seismic surface waves and obtain a constraint on the location of resonances and establish a forbidden domain as the main result. In order to obtain the main result we make a Pekeris-Markushevich transformation of the Rayleigh system with free surface boundary condition such that we get a matrix Schr\"odinger-type form of it. We obtain parity and analytical properties of its fundamental solutions, which are needed to prove the main theorem. We construct a function made up by Rayleigh determinants factors, which is proven to be entire, of exponential type and in the Cartwright class and leads to the constraint on the location of resonances.

We prove a parametric jet interpolation theorem for symplectic holomorphic automorphisms of $\mathbb{C}^{2n}$ with parameters in a Stein space. Moreover, we provide an example of an unavoidable set for symplectic holomorphic maps.

In this paper we study $\gamma$ partial correction over a $t$-user arbitrarily varying multiple-access channel (AV-MAC). We first present necessary channel conditions for the $\gamma$ partially correcting authentication capacity region to have nonempty interior. We then give a block length extension scheme which preserves positive rate tuples from a short code with zero probability of $\gamma$ partial correction error, noting that the flexibility of $\gamma$ partial correction prevents pure codeword concatenation from being successful. Finally, we offer a case study of a particular AV-MAC satisfying the necessary conditions for partial correction.

We compute the fluctuation moments $\alpha_{m_1,\dots,m_r}$ of a Complex Wigner Matrix $X_N$ given by the limit $\lim_{N\rightarrow\infty}N^{r-2}k_r(Tr(X_N^{m_1}),\dots,Tr(X_N^{m_r}))$. We prove the limit exists and characterize the leading order via planar graphs that result to be trees. We prove these graphs can be counted by the set of non-crossing partitioned permutations which permit us to express the moments $\alpha_{m_1,\dots,m_r}$ in terms of simpler quantities $\overline{\kappa}_{m_1,\dots,m_r}$ which we call the pseudo-cumulants. We prove the pseudo-cumulants coincide with the higher order free cumulants up to $r=4$ which permit us to find the higher order free cumulants $\kappa_{m_1,\dots,m_r}$ associated to the moment sequence $\alpha_{m_1,\dots,m_r}$ up to order 4.

The main purpose of this paper is to introduce and study minimal and maximal ideals defined on ideal topological spaces. Also, we define and investigate the concepts of ideal quotient and annihilator of any subfamily of $2^X$, where $2^X$ is the power set of $X.$ We obtain some of their fundamental properties. In addition, several relationships among the above notions have been discussed. Moreover, we get a new topology, called sharp topology via the sharp operator defined in the scope of this study, finer than the old one. Furthermore, a decomposition of the notion of open set has been obtained. Finally, we conclude our work with some interesting applications.

Mixed Poisson distributions provide a flexible approach to the analysis of count data with overdispersion, zero inflation, or heavy tails. Since the Poisson mean must be nonnegative, the mixing distribution is typically assumed to have nonnegative support. We show this assumption is unnecessary and real-valued mixing distributions are also possible. Informally, the mixing distribution merely needs to have a light (subexponential) left tail and a small amount of probability mass on negative values. We provide several concrete examples, including the mixed Poisson-extreme stable family, where the mixing distribution has a power law tail.

In this article we study the first, the second and the third homology groups of the elementary group $\textrm{E}_2(A)$, where $A$ is a commutative ring. In particular, we prove a refined Bloch-Wigner type exact sequence over a semilocal ring (with some mild restriction on its residue fields) such that $-1\in (A^{\times})^2$ or $|A^{\times}/(A^{\times})^2|\leq 4$.

This paper investigates a specific instance of the static repositioning problem within station-based bike-sharing systems. Our study incorporates operational and damaged bikes, a heterogeneous fleet, and multiple visits between stations and the depot. The objective is to minimize the weighted sum of the deviation from the target number of bikes for each station, the number of damaged bikes not removed, and the total time used by vehicles. To solve this problem, we propose a matheuristic approach based on a randomized multi-start algorithm integrated with an integer programming model for optimizing the number of operatives and damaged bikes that will be moved between stations and/or the depot (loading instructions). The algorithm's effectiveness was assessed using instances derived from real-world data, yielding encouraging results. Furthermore, we adapted our algorithm to a simpler problem studied in the literature, achieving competitive outcomes compared to other existing methods. The experimental results in both scenarios demonstrate that this algorithm can generate high-quality solutions within a short computational time.

We propose a generalization of the adaptive N-Intertwined Mean-Field Approximation (aNIMFA) model studied in \emph{Achterberg and Sensi} \cite{achterbergsensi2022adaptive} to a heterogeneous network of communities. In particular, the multigroup aNIMFA model describes the impact of both local and global disease awareness on the spread of a disease in a network. We obtain results on existence and stability of the equilibria of the system, in terms of the basic reproduction number~$R_0$. Under light constraints, we show that the basic reproduction number~$R_0$ is equivalent to the basic reproduction number of the NIMFA model on static networks. Based on numerical simulations, we demonstrate that with just two communities periodic behaviour can occur, which contrasts the case with only a single community, in which periodicity was ruled out analytically. We also find that breaking connections between communities is more fruitful compared to breaking connections within communities to reduce the disease outbreak on dense networks, but both strategies are viable to networks with fewer links. Finally, we emphasise that our method of modelling adaptivity is not limited to SIS models, but has huge potential to be applied in other compartmental models in epidemiology.

The behaviour of the generalized Hilbert operator associated with a positive finite Borel measure $\mu$ on $[0,1)$ is investigated when it acts on weighted Banach spaces of holomorphic functions on the unit disc defined by sup-norms and on Korenblum type growth Banach spaces. It is studied when the operator is well defined, bounded and compact. To this aim, we study when it can be represented as an integral operator. We observe important differences with the behaviour of the Ces\`aro-type operator acting on these spaces, getting that boundedness and compactness are equivalent concepts for some standard weights. For the space of bounded holomorphic functions on the disc and for the Wiener algebra, we get also this equivalence, which is characterized in turn by the summability of the moments of the measure $\mu.$ In the latter case, it is also equivalent to nuclearity. Nuclearity of the generalized Hilbert operator acting on related spaces, such as the classical Hardy space, is also analyzed.

A significant correlation between Nelson algebras and Heyting algebras has been explored by several scholars, including Cignoli, Fidel, Vakarelov, and Sendlewski. This connection is integral to the concept of twist structures, whose origins can be traced back to the work of Kalman. In this paper, we obtain an expansion of the Fidel-Vakarelov construction, applying it to monadic Godel algebras (or monadic prelinear Heyting algebras). This extension leads to the emergence of a new variety, which we aptly term monadic prelinear Nelson algebras.

Let $R$ be a standard graded, finitely generated algebra over a field, and let $M$ be a graded module over $R$ with all Bass numbers finite. Set $(-)^{(n)}$ to be the $n$-th Veronese functor. We compute the Bass numbers of $M^{(n)}$ over the ring $R^{(n)}$ for all prime ideals of $R^{(n)}$ that are not the homogeneous maximal ideal in terms of the Bass numbers of $M$ over $R$. As an application to local cohomology modules, we determine the Bass numbers of $H_{I\cap R^{(n)}}^i(R^{(n)})$ over the ring $R^{(n)}$ in the case where $H_I^i(R)$ has finite Bass numbers over $R$ and $I$ is a graded ideal.

Higher-order notions of Kreweras complementation have appeared in the literature in the works of Krawczyk, Speicher, Mastnak, Nica, Arizmendi, Vargas, and others. While the theory has been developed primarily for specific applications in free probability, it also possesses an elegant, purely combinatorial core that is of independent interest. The present article aims at offering a simple account of various aspects of higher-order Kreweras complementation on the basis of elementary arithmetic, (co)algebraic, categorical and simplicial properties of noncrossing partitions. The main idea is to see noncrossing partitions as providing an interesting noncommutative analogue of the interplay between the divisibility poset and the multiplicative monoid of positive integers. Just as the divisibility poset can be regarded as the decalage of the multiplicative monoid, we exhibit the lattice of noncrossing partitions as the decalage of a partial monoid structure on noncrossing partitions encoding higher-order Kreweras complements. While our results may be considered known, some of the viewpoints can be regarded as novel, providing an efficient approach both conceptually and computationally.

Watkins's conjecture suggests that for an elliptic curve $E/\mathbb{Q}$, the rank of the group $E(\mathbb{Q})$ of rational points is bounded above by $\nu_2 (m_E)$, where $m_E$ is the modular degree associated with $E$. It is known that Watkins's conjecture holds on average. This article investigates the conjecture over certain thin families of elliptic curves. For example, for prime $\ell$, we quantify the elliptic curves featuring a rational $\ell$-torsion that satisfies Watkins's conjecture. Additionally, the study extends to a broader context, investigating the inequality $\mathrm{rank}(E(\mathbb{Q}))+M\leq \nu_2(m_E)$ for any positive integer $M$.

A method of constructing Markov chains on finite state spaces is provided. The chain is specified by three constraints: stationarity, dependence and marginal distributions. The generalized Pythagorean theorem in information geometry plays a central role in the construction. An algorithm for obtaining the desired Markov chain is described. Integer-valued autoregressive processes are considered for illustration.

In this paper, we focus on a new lower bound quantum cluster algebra which is generated by the initial quantum cluster variables and the quantum projective cluster variables of an acyclic quantum cluster algebra with principle coefficients. We show that the new lower bound quantum cluster algebra coincides with the corresponding acyclic quantum cluster algebra. Moreover, we establish a class of formulas between these generators, and obtain the dual PBW basis of this algebra.

A stratified space is a topological space equipped with a \emph{stratification}, which is a decomposition or partition of the topological space satisfying certain extra conditions. More recently, the notion of poset-stratified space, i.e., topological space endowed with a continuous map to a poset with its Alexandrov topology, has been popularized. Both notions of stratified spaces are ubiquitous in mathematics, ranging from investigations of singular structures in algebraic geometry to extensions of the homotopy hypothesis in higher category theory. In this article we study the precise mathematical relation between these different approaches to stratified spaces.

We survey free magmas and we explore the structure of their submagmas. By equipping the cyclic free magma with a second distributive operation we obtain a ringoid-like structure with some primitive arithmetical properties. A submagma is $k$-maximal when there are only $k-1$ submagmas between it and the free magma itself. These two tools, arithmetic and maximality, allow us to study the lattice of the submagmas of a free magma.

A magma is called equidecomposable when the operation is injective, or, in other words, if $x+y=x'+y'$ implies that $x=x'$ and $y=y'$. A magma is free iff it is equidecomposable and graded, hence the notion of equidecomposability is very related to the notion of freeness although it is not sufficient. We study main properties of such magmas. In particular, an alternative characterization of freeness, which uses a weaker condition, is proved. We show how equidecomposable magmas can be split into two disjoint submagmas, one of which is free. Certain tranformations on finite presentations permit to obtain a reduced form which allows us identify all the finite presented equidecomposable magmas up to isomorphisms.

In this article, we introduce the fractional medians, give an expression of the set of all fractional medians in terms of non-increasing rearrangements and then investigate mapping properties of the fractional maximal operators defined by such medians. The maximal operator is a generalization of that in Stromberg. It turns out that our maximal operator is a more smooth operator than the usual fractional maximal operator. Further, we give another proof of the embedding from $BV$ to $L^{n/(n-1),1}$ due to Alvino by using the usual medians.

Our main result is that chaos in dimension $n+1$ is a one-dimensional geometrical object embedded in a geometrical object of dimension $n$ which corresponds to a $n$ dimensional object which is either singular or non-singular. Our main result is then that this chaos occurs in the first case as either on an isolated or non-isolated singularity. In the first case this chaos is either boundary chaos or spherical chaos which is what happens also in the non-singular case. In the case of an isolated singular geometry one has chaos which can either be boundary, spherical or tubular chaos. We furthermore prove that the prime numbers display quantum behaviour.

In this article, we provide two-weight inequalities for the heat flow on the whole space by applying the sparse domination. For power weights, such inequalities were given by several authors. Owing to the sparse domination, we can treat general weights in Muckenhoupt classes. As a application, we present the local and global existence results for the Hardy-H\'enon parabolic equation, which has a potential belonging to a Muckenhoupt class.

We mathematically show an equality between the index of a Dirac operator on a flat continuum torus and the $\eta$ invariant of the Wilson Dirac operator with a negative mass when the lattice spacing is sufficiently small. Unlike the standard approach, our formulation using $K$-theory does not require the Ginsparg-Wilson relation or the modified chiral symmetry on the lattice. We prove that a one-parameter family of continuum massive Dirac operators and the corresponding Wilson Dirac operators belong to the same equivalence class of the $K^1$ group at a finite lattice spacing. Their indices, which are evaluated by the spectral flow or equivalently by the $\eta$ invariant at finite masses, are proved to be equal.

In this paper, we extend the twisted spectral large sieve inequalities of Deshouillers, Iwaniec, Luo, and Young from $\mathrm{SL}_2 (\mathbb{Z})$ onto $\mathrm{PGL}_2 (\mathbb{Z}[i])$.

In this paper we establish $q$-orthogonality relation for the continuous $q$-ultraspherical polynomials, which was considered by Gasper. A new $q$-beta integral with seven parameters is evaluated.

This paper presents a study of separation axioms and sobriety of bitopological spaces from the point of view of fuzzy topology via identifying bitopological spaces with topological spaces valued in the Boolean algebra of four elements. A system of separation axioms is proposed making use of Boolean-valued specialization order of bitopological spaces; The relationship between d-sobriety of bitopological spaces proposed by Jung and Moshier and sobriety of fuzzy topological spaces is studied; A Hofmann-Mislove theorem for bitopological spaces is established.

In this paper, we are concerned with divergence form, higher-order parabolic systems in a cylindrical domain with a finite number of subdomains. We establish $L_\infty$ and Schauder estimates of solutions when the leading coefficients and the non-homogeneous term exhibit piecewise Dini mean oscillation and piecewise H\"{o}lder continuity, respectively. To the best of our knowledge, our results are new for higher-order elliptic and parabolic systems.

In this paper, we study the multi-dimensional reflected backward stochastic differential equation driven by $G$-Brownian motion ($G$-BSDE) with a multi-variate constraint on the $G$-expectation of its solution. The generators are diagonally dependent on $Z$ and on all $Y$-components. We obtain the existence and uniqueness result via a fixed-point argumentation.

We found a special divergence structure for the $\sigma_k$-Yamabe operator and use it to get a monotonicity formula. We also get an interior $L^{\infty}$ estimate via its $L^1$ norm for the $\sigma_k$-Yamabe operator when $1\le k \le \frac{n}{2}$. Combining these two tools, we prove the weak continuity of the $\sigma_k$-Yamabe measure with respect to convergence in measure.

In this paper, a third-order time adaptive algorithm with less computation, low complexity is provided for shale reservoir model based on coupled fluid flow with porous media flow. The algorithm combines the three-step linear time filters method for simple post-processing and the second-order backward differential formula (BDF2), is third-order accurate and provides, at no extra computational complexity. At the same time, the time filter method can also be used to damp non-physical oscillations inherent in the BDF2 method, ensuring stability. We proves the variable time stepsize second-order backward differential formula plus time filter (BDF2-TF) algorithm's stability and the convergence properties of the fluid velocity u and hydraulic head $\phi$ in the $L^2$ norm with an order of $O(k_{n+1}^3 + h^3)$. In the experiments, the adaptive algorithm automatically adjusts the time step in response to the varying characteristics of different models, ensuring that errors are maintained within acceptable limits. This algorithm addresses the issue that high-order algorithms may select inappropriate time steps, resulting in instability or reduced precision of the numerical solution, thereby enhancing calculation accuracy and efficiency. We perform three-dimensional numerical experiments to verify the BDF2-TF algorithm's effectiveness, stability, and third-order convergence. Simultaneously, a simplified model is employed to simulate the process of shale oil extraction from reservoirs, further demonstrating the algorithm's practical applicability.

We investigate arcs on a pair of pants and present an algorithm to compute the self-intersection number of an arc. Additionally, we establish bounds for the self-intersection number in terms of the word length. We also prove that the spectrum of self-intersection numbers of 2-low-lying arcs covers all natural numbers.

We carry on a more detailed investigation of the composition of locally solid convergences as introduced in \cite{ectv}, as well as the corresponding notion of idempotency considered in \cite{erz}. In particular, we study the interactions between these two concepts and various operations with convergences. Some results from \cite{kt} about unbounded modification of locally solid topologies are generalized to the level of locally solid idempotent convergences. A simple application of the composition allows us to answer a question from \cite{ectv} about minimal Hausdorff locally solid convergences. We also show that the weakest Hausdorff locally solid convergence exists on an Archimedean vector lattice if and only if it is atomic.

Greedy algorithms have been successfully analyzed and applied in training neural networks for solving variational problems, ensuring guaranteed convergence orders. However, their practical applicability is limited due to the subproblems, which often require an exhaustive search over a discrete dictionary and incur significant computational costs. This limitation becomes critical, especially in high-dimensional problems. In this paper, we propose a more practical approach of randomly discretizing the dictionary at each iteration of the greedy algorithm. We quantify the required size of the randomized discrete dictionary and prove that, with high probability, the proposed algorithm realizes a weak greedy algorithm, achieving optimal convergence orders. Through numerous numerical experiments, we demonstrate the advantage of using randomized discrete dictionaries over a deterministic one by showing orders of magnitude reductions in the size of the discrete dictionary, particularly in higher dimensions.

In this paper, we study a collection of mean-reflected backward stochastic differential equations driven by $G$-Brownian motions ($G$-BSDEs), where $G$-expectations are constrained in some time-dependent intervals. To establish well-posedness results, we firstly construct a backward Skorokhod problem with sublinear expectation, and then apply that in the study of doubly mean-reflected $G$-BSDEs involving Lipschitz and quadratic generators under bounded and unbounded terminal conditions. Also we utilize fixed-point argumentations and $\theta$-methods while solving these equations. Finally, we extend the results to multi-dimensional doubly mean-reflected $G$-BSDEs with diagonal generators.

We study the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term. For this aim, we establish decay estimates of the fractional heat semigroup in several uniformly local Zygumnd spaces. Furthermore, we apply the real interpolation method in uniformly local Zygmund spaces to obtain sharp integral estimates on the inhomogeneous term and the nonlinear term. This enables us to find sharp sufficient conditions for the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term.

We consider a periodic higher-order nonlinear Schr\"odinger equation with the nonlinearity $u^k \partial_x u$, where $k$ is a natural number. We prove the norm inflation in a subspace of the Sobolev space $H^s(\mathbb{T})$ for any $s \in \mathbb{R}$. In particular, the Cauchy problem is ill-posed in $H^s(\mathbb{T})$ for any $s \in \mathbb{R}$.

The notion of a $k$-11-representable graph was introduced by Jeff Remmel in 2017 and studied by Cheon et al.\ in 2019 as a natural extension of the extensively studied notion of word-representable graphs, which are precisely 0-11-representable graphs. A graph $G$ is $k$-11-representable if it can be represented by a word $w$ such that for any edge (resp., non-edge) $xy$ in $G$ the subsequence of $w$ formed by $x$ and $y$ contains at most $k$ (resp., at least $k+1$) pairs of consecutive equal letters. A remarkable result of Cheon at al.\ is that {\em any} graph is 2-11-representable, while it is unknown whether every graph is 1-11-representable. Cheon et al.\ showed that the class of 1-11-representable graphs is strictly larger than that of word-representable graphs, and they introduced a useful toolbox to study 1-11-representable graphs. In this paper, we introduce new tools for studying 1-11-representation of graphs. We apply them for establishing 1-11-representation of Chv\'{a}tal graph, Mycielski graph, split graphs, and graphs whose vertices can be partitioned into a comparability graph and an independent set.

In a recent article "Projective geometries, $Q$-polynomial structures, and quantum groups" Terwilliger (arXiv:2407.14964) defined a certain weighted adjacency matrix, depending on a free (positive real) parameter, associated with the projective geometry, and showed (among many other results) that it is diagonalizable, with the eigenvalues and their multiplicities explicitly written down, and that it satisfies the $Q$-polynomial property (with respect to the zero subspace). In this note we (i) Write down an explicit eigenbasis for this matrix. (ii) Evaluate the adjacency matrix-eigenvector products, yielding a new proof for the eigenvalues and their multiplicities. (iii) Evaluate the dual adjacency matrix-eigenvector products and directly show that the action of the dual adjacency matrix on the eigenspaces of the adjacency matrix is block-tridiagonal, yielding a new proof of the $Q$-polynomial property.

An affine version of the linear subspace concentration inequality as proposed by Wu is established for centered convex bodies. This generalizes results from Wu and Freyer, Henk, Kipp on polytopes to convex bodies.

In this paper, we study the stochastic heat equation driven by a multiplicative space-time $G$-white noise within the framework of sublinear expectations. The existence and uniqueness of the mild solution are proved. By generalizing the stochastic Fubini theorem under sublinear expectations, we demonstrate that the mild solution also qualifies as a weak solution. Additionally, we derive moment estimates for the solutions.

This paper is a first step in the study of nonstandard graded algebras having Poincar\'e duality and their Lefschetz properties. We prove the equivalence between the toric setup and the G-graded one, generalize Macaulay-Matlis duality, introduce Lefschetz properties and prove a Hessian criteria in the G-graded setup. We prove a special case of the Codimension One Conjecture of Cattani-Cox-Dickenstein.

A solution of the n-body problem in R^d is a relative equilibrium if all of the mutual distance between the bodies are constant. In other words, the bodies undergo a rigid motion. Here we investigate the possibility of partially rigid motions, where some but not all of the distances are constant. In particular, a {\em hinged} solution is one such that exactly one mutual distance varies. The goal of this paper is to show that hinged solutions don't exist when n=3 or n=4. For n=3 this means that if 2 of the 3 distances are constant so is the third and for n=4, if 5 of the 6 distances are constant, so is the sixth. These results hold independent of the dimension d of the ambient space.

This monograph elucidates and extends many theorems and conjectures in analytic number theory and algebraic asymptotic analysis via the natural notions of degree and logexponential degree. The Riemann hypothesis, for example, is equivalent to the statement that the degree of the function $\pi(x)- \operatorname{li}(x)$ is $1/2$, where $\pi(x)$ is the prime counting function and $\operatorname{li}(x)$ is the logarithmic integral function. Part 1 of the text is a survey of analytic number theory, Part 2 introduces the notion of logexponential degree and uses it to extend results in algebraic asymptotic analysis, and Part 3 applies the results of Part 2 to the various functions that figure most prominently in analytic number theory. Central to the notion of logexponential degree are Hardy's logarithmico-exponential functions, which are real functions defined in a neighborhood of $\infty$ that can be built from $\operatorname{id}$, $\exp$, and $\log$ using the operations $+$, $\cdot$, $/$, and $\circ$. Such functions are natural benchmarks for the orders of growth of functions in analytic number theory. The main goal of Part 3 is to express the logexponential degree of various functions in analytic number theory in terms of as few "logexponential primitives" as possible. The logexponential degree of the function $e^\gamma \prod_{p\leq x}(1-1/p) -1/\log x$, for example, can be expressed in terms of that of $\pi(x)- \operatorname{li}(x)$ and vice versa (where $\gamma \approx 0.5772$ is the Euler-Mascheroni constant), despite the fact that very little is known about the logexponential degree of either function separately, even on condition of the Riemann hypothesis.

Let $G=(V,E)$ be a graph with four distinguished vertices, two sources $s_1, s_2$ and two sinks $t_1,t_2$, let $c:\, E \rightarrow \mathbb Z_+$ be a capacity function, and let ${\cal P}$ be the set of all simple paths in $G$ from $s_1$ to $t_1$ or from $s_2$ to $t_2$. A biflow (or $2$-commodity flow) in $G$ is an assignment $f:\, {\cal P}\rightarrow \mathbb R_+$ such that $\sum_{e \in Q \in {\cal P}}\, f(Q) \le c(e)$ for all $e \in E$, whose value is defined to be $\sum_{Q \in {\cal P}}\, f(Q)$. A bicut in $G$ is a subset $K$ of $E$ that contains at least one edge from each member of ${\cal P}$, whose capacity is $\sum_{e\in K}\, c(e)$. In 1977 Seymour characterized, in terms of forbidden structures, all graphs $G$ for which the max-biflow (integral) min-bicut theorem holds true (that is, the maximum value of an integral biflow is equal to the minimum capacity of a bicut for every capacity function $c$); such a graph $G$ is referred to as a Seymour graph. Nevertheless, his proof is not algorithmic in nature. In this paper we present a combinatorial polynomial-time algorithm for finding maximum integral biflows in Seymour graphs, which relies heavily on a structural description of such graphs.

Bilevel optimization is widely applied in many machine learning tasks such as hyper-parameter learning, meta learning and reinforcement learning. Although many algorithms recently have been developed to solve the bilevel optimization problems, they generally rely on the (strongly) convex lower-level problems. More recently, some methods have been proposed to solve the nonconvex-PL bilevel optimization problems, where their upper-level problems are possibly nonconvex, and their lower-level problems are also possibly nonconvex while satisfying Polyak-{\L}ojasiewicz (PL) condition. However, these methods still have a high convergence complexity or a high computation complexity such as requiring compute expensive Hessian/Jacobian matrices and its inverses. In the paper, thus, we propose an efficient Hessian/Jacobian-free method (i.e., HJFBiO) with the optimal convergence complexity to solve the nonconvex-PL bilevel problems. Theoretically, under some mild conditions, we prove that our HJFBiO method obtains an optimal convergence rate of $O(\frac{1}{T})$, where $T$ denotes the number of iterations, and has an optimal gradient complexity of $O(\epsilon^{-1})$ in finding an $\epsilon$-stationary solution. We conduct some numerical experiments on the bilevel PL game and hyper-representation learning task to demonstrate efficiency of our proposed method.

We analyze a combinatorial rule satisfied by the signs of principal minors of a real symmetric matrix. The sign patterns satisfying this rule are equivalent to uniform oriented Lagrangian matroids. We first discuss their structure and symmetries and then study their asymptotics, proving that almost all of them are not representable by real symmetric matrices. We offer several conjectures and experimental results concerning representable sign patterns and the topology of their representation spaces.

The signature of a $p$-weakly geometric rough path summarises a path up to a generalised notion of reparameterisation. The quotient space of equivalence classes on which the signature is constant yields unparameterised path space. The study of topologies on unparameterised path space, initiated in [CT24b] for paths of bounded variation, has practical bearing on the use of signature based methods in a variety applications. This note extends the majority of results from [CT24b] to unparameterised weakly geometric rough path space. We study three classes of topologies: metrisable topologies for which the quotient map is continuous; the quotient topology derived from the underlying path space; and an explicit metric between the tree-reduced representatives of each equivalence class. We prove that topologies of the first type (under an additional assumption) are separable and Lusin, but not locally compact or completely metrisable. The quotient topology is Hausdorff but not metrisable, while the metric generating the third topology is not complete and its topology is not locally compact. We also show that the third topology is Polish when $p=1$.

Optimizing a building's energy supply design is a task with multiple competing criteria, where not only monetary but also, for example, an environmental objective shall be taken into account. Moreover, when deciding which storages and heating and cooling units to purchase (here-and-now-decisions), there is uncertainty about future developments of prices for energy, e.g. electricity and gas. This can be accounted for later by operating the units accordingly (wait-and-see-decisions), once the uncertainty revealed itself. Therefore, the problem can be modeled as an adjustable robust optimization problem. We combine adjustable robustness and multicriteria optimization for the case of building energy supply design and solve the resulting problem using a column and constraint generation algorithm in combination with an $\varepsilon$-constraint approach. In the multicriteria adjustable robust problem, we simultaneously minimize worst-case cost regret and carbon emissions. We take into account future price uncertainties and consider the results in the light of information gap decision theory to find a trade-off between security against price fluctuations and over-conservatism. We present the model, a solution strategy and discuss different application scenarios for a case study building.

In this paper, we establish a general setup for studying incidence-preserving motions of projective geometric configurations of points and lines via a "projective rigidity matrix". The spaces of infinitesimal motions of a point-line configuration and dependencies amongst the point-line incidences can be interpreted as the kernel and co-kernel of this projective rigidity matrix, respectively. We also introduce a symmetry-adapted projective rigidity matrix for analysing symmetric configurations and their symmetry-preserving motions. The symmetry may be a point group or a more general symmetry, such as an autopolarity.

We consider vector valued weak solutions $u:\Omega_T\to \mathbb{R}^N$ with $N\in \mathbb{N}$ of degenerate or singular parabolic systems of type \begin{equation*} \partial_t u - \mathrm{div} \, a(z,u,Du) = 0 \qquad\text{in}\qquad \Omega_T= \Omega\times (0,T), \end{equation*} where $\Omega$ denotes an open set in $\mathbb{R}^{n}$ for $n\geq 1$ and $T>0$ a finite time. Assuming that the vector field $a$ is not of Uhlenbeck-type structure, satisfies $p$-growth assumptions and $(z,u)\mapsto a(z,u,\xi)$ is H\"older continuous for every $\xi\in \mathbb{R}^{Nn}$, we show that the gradient $Du$ is partially H\"older continuous, provided the vector field degenerates like that of the $p$-Laplacian for small gradients.

Movable antenna (MA) technology can flexibly reconfigure wireless channels by adjusting antenna positions in a local region, thus owing great potential for enhancing communication performance. This letter investigates MA technology enabled multiuser uplink communications over general Rician fading channels, which consist of a base station (BS) equipped with the MA array and multiple single-antenna users. Since it is practically challenging to collect all instantaneous channel state information (CSI) by traversing all possible antenna positions at the BS, we instead propose a two-timescale scheme for maximizing the ergodic sum rate. Specifically, antenna positions at the BS are first optimized using only the statistical CSI. Subsequently, the receiving beamforming at the BS (for which we consider the three typical zero-forcing (ZF), minimum mean-square error (MMSE) and MMSE with successive interference cancellation (MMSE-SIC) receivers) is designed based on the instantaneous CSI with optimized antenna positions, thus significantly reducing practical implementation complexities. The formulated problems are highly non-convex and we develop projected gradient ascent (PGA) algorithms to effectively handle them. Simulation results illustrate that compared to conventional fixed-position antenna (FPA) array, the MA array can achieve significant performance gains by reaping an additional spatial degree of freedom.

We study the disassortative stochastic block model with three communities, a well-studied model of graph partitioning and Bayesian inference for which detailed predictions based on the cavity method exist [Decelle et al. (2011)]. We provide strong evidence that for a part of the phase where efficient algorithms exist that approximately reconstruct the communities, inference based on maximum a posteriori (MAP) fails. In other words, we show that there exist modes of the posterior distribution that have a vanishing agreement with the ground truth. The proof is based on the analysis of a graph colouring algorithm from [Achlioptas and Moore (2003)].

We investigate first-passage percolation on the lattice $\Z^d$ for dimensions $d \geq 2$. Each edge $e$ of the graph is assigned an independent copy of a non-negative random variable $\tau$. We only assume $\P[\tau=0]0$ is explicit) for the probability of having two disjoint geodesics between two pairs of neighbouring vertices at distance $n$. Additionally, under more specific assumptions on the distribution of $\tau$, we obtain similar lower bounds for the probability of having two disjoint geodesics (except for their starting and ending points) between the same two vertices.

The present paper extends the theory of Adaptive Virtual Element Methods (AVEMs) to the three-dimensional meshes showing the possibility to bound the stabilization term by the residual-type error estimator. This new bound enables a stabilization-free a posteriori control for the energy error. Following the recent studies for the bi-dimensional case, we investigate the case of tetrahedral elements with aligned edges and faces. We believe that the AVEMs can be an efficient strategy to address the mesh conforming requirements of standard three-dimensional Adaptive Finite Element Methods (AFEMs), which typically extend the refinement procedure to non-marked mesh cells. Indeed, numerical tests on the Fichera corner shape domain show that this method can reduce the number of three-dimensional cells generated in the refinement process by about 30% with compared to standard AFEMs, for a given error threshold.

A recent discovery of Eldan and Gross states that there exists a universal $C>0$ such that for all Boolean functions $f:\{-1,1\}^n\to \{-1,1\}$, $$ \int_{\{-1,1\}^n}\sqrt{s_f(x)}d\mu(x) \ge C\text{Var}(f)\sqrt{\log \left(1+\frac{1}{\sum_{j=1}^{n}\text{Inf}_j(f)^2}\right)} $$ where $s_f(x)$ is the sensitivity of $f$ at $x$, $\text{Var}(f)$ is the variance of $f$, $\text{Inf}_j(f)$ is the influence of $f$ along the $j$-th variable, and $\mu$ is the uniform probability measure. In this note, we give an alternative proof that applies to biased discrete hypercube, and spaces having positive Ricci curvature lower bounds in the sense of Bakry and \'Emery.

In this paper, we introduce and study the class of $\phi$-$w$-P-flat modules, which can be seen as generalizations of both $\phi$-P-flat modules and $w$-P-flat modules. In particular, we obtain that the class of $\phi$-$w$-P-flat modules is covering. We also utilize the class of $\phi$-$w$-P-flat modules to characterize $\phi$-von Neumann regular rings, strong $\phi$-rings and $\phi$-PvMRs.

Based on the global a priori estimates in [Guo-Tice, J. Eur. Math. Soc. (2024)], we establish the well-posedness of a viscous fluid model satisfying the dynamic law for the contact line \begin{equation*} \mathscr{W}(\p_t\zeta(\pm\ell,t))=[\![\gamma]\!]\mp\sigma\frac{\p_1\zeta}{(1+|\p_1\zeta|^2)^{1/2}}(\pm\ell,t) \end{equation*} in 2D domain, where $\zeta(x_1,t)$ is a free surface with two contact points $\zeta(\pm\ell,t)$, $[\![\gamma]\!]$ and $\sigma$ are constants characterizing the solid-fluid-gas free energy, and the increasing $\mathscr{W}$ is the contact point velocity response function. Motivated by the energy-dissipation structure, our construction relies on the construction of a pressureless weak solution for the coupled velocity and free interface for the linearized problems, via a Galerkin approximation with a time-dependent basis and an artificial regularization for the capillary operator.

In this paper, we study reflected backward stochastic differential equations driven by rough paths (rough RBSDEs), which can be understood as a probabilistic representation of nonlinear rough partial differential equations (rough PDEs) or stochastic partial differential equations (SPDEs) with obstacles. The well-posedness in the sense of \cite{DF} is proved via a variant of Doss-Sussman transformation. Moreover, we show that our rough RBSDEs can be approximated by a sequence of penalized BSDEs with rough drivers. As applications, firstly we provide the equivalence between rough RBSDEs and the obstacle problem of rough PDEs. Secondly, we show the solution of rough RBSDE solves the corresponding optimal stopping problem.

In vector tomography (VT), the aim is to reconstruct an unknown multi-dimensional vector field using line integral data. In the case of a 2-dimensional VT, two types of line integral data are usually required. These data correspond to integration of the parallel and perpendicular projection of the vector field along integration lines. VT methods are non-invasive, non-intrusive and offer more information on the field than classical point measurements; they are typically used to reconstruct divergence-free (or source-free) velocity and flow fields. In this paper, we show that VT can also be used for the reconstruction of fields with non-zero divergence. In particular, we study electric fields generated by dipole sources in bounded domains which arise, for example, in electroencephalography (EEG) source imaging. To the best of our knowledge, VT has not previously been used to reconstruct such fields. We explain in detail the theoretical background, the derivation of the electric field inverse problem and the numerical approximation of the line integrals. We show that fields with non-zero divergence can be reconstructed from the longitudinal measurements with the help of two sparsity constraints that are constructed from the transverse measurements and the vector Laplace operator. As a comparison to EEG source imaging, we note that VT does not require mathematical modelling of the sources. By numerical simulations, we show that the pattern of the electric field can be correctly estimated using VT and the location of the source activity can be determined accurately from the reconstructed magnitudes of the field.

We study a model of networked atoms or molecules oscillating around their equilibrium positions. The model assumes the harmonic approximation of the interactions. We provide bounds for the total number of phonons, and for the specific heat, in terms of the average Wiener capacity, or resistance, of the network. Thanks to such bounds, we can distinguish qualitatively different behaviours in terms of the network structure alone.

This paper aims to determine the ring structure of the torus equivariant cohomology of odd-dimensional complex quadrics by computing the graph equivariant cohomology of their corresponding GKM graphs. We show that its graph equivariant cohomology is generated by three types of subgraphs in the GKM graph, which are subject to four different types of relations. Furthermore, we consider the relationship between the two graph equivariant cohomology rings induced by odd- and even-dimensional complex quadrics.

Recently, Li, Sheng and Tang introduced post-Hopf algebras and relative Rota-Baxter operators (on cocommutative Hopf algebras), providing an adjunction between the respective categories under the assumption that the structures involved are cocommutative. We introduce Yetter-Drinfeld post-Hopf algebras, which become usual post-Hopf algebras in the cocommutative setting. In analogy with the correspondence between cocommutative post-Hopf algebras and cocommutative Hopf braces, the category of Yetter-Drinfeld post-Hopf algebras is isomorphic to the category of Yetter-Drinfeld braces introduced by the author in a joint work with D. Ferri. This allows to explore the connection with matched pairs of actions and provide examples of Yetter-Drinfeld post-Hopf algebras. Moreover, we prove that the category of Yetter-Drinfeld post-Hopf algebras is equivalent to a subcategory of Yetter-Drinfeld relative Rota-Baxter operators. The latter structures coincide with the inverse maps of Yetter-Drinfeld 1-cocycles introduced by the author and D. Ferri, and generalise bijective relative Rota-Baxter operators on cocommutative Hopf algebras. Hence the previous equivalence passes to cocommutative post-Hopf algebras and bijective relative Rota-Baxter operators. Once the surjectivity of the Yetter-Drinfeld relative Rota-Baxter operators is removed, the equivalence is replaced by an adjunction and one can recover the result of Li, Sheng and Tang in the cocommutative case.

In this paper, the asymptotic behavior of a semilinear heat equation with long time memory and non-local diffusion is analyzed in the usual set-up for dynamical systems generated by differential equations with delay terms. This approach is different from the previous published literature on the long time behavior of heat equations with memory which is carried out by the Dafermos transformation. As a consequence, the obtained results provide complete information about the attracting sets for the original problem, instead of the transformed one. In particular, the proved results also generalize and complete previous literature in the local case.

The paper establishes the exponential turnpike property for a class of mean-field stochastic linear-quadratic (LQ) optimal control problems with periodic coefficients. It first introduces the concepts of stability, stabilizability, and detectability for stochastic linear systems. Then, the long-term behavior of the associated Riccati equations is analyzed under stabilizability and detectability conditions.Subsequently, a periodic mean-field stochastic LQ problem is formulated and solved. Finally, a linear transformation of the periodic extension of its optimal pair is shown to be the turnpike limit of the initial optimal control problem.

This article is twofold. In the first part of the article, we consider Brezis-Oswald problem involving the mixed anisotropic and nonlocal $p$-Laplace operator and establish existence, uniqueness, boundedness and strong maximum principle. Further, for some mixed anisotropic and nonlocal $p$-Laplace type equations, we obtain Sturmian comparison theorem, prove some comparsion results, nonexistence results, establish weighted Hardy type inequality and study a system of singular mixed anisotropic and nonlocal $p$-Laplace equations. One of the key ingredient is the combination of Picone identity already available in the separate local and nonlocal cases. In the second part of the article, we are mainly concerned with regularity estimates. To this end, in the elliptic setting, we establish weak Harnack inequality and semicontinuity results. Further, we consider a class of doubly nonlinear mixed anisotropic and nonlocal equations and prove semicontinuity results and pointwise behavior of solutions. These are based on suitable energy estimates and De Giorgi type lemmas and expansion of positivity. Finally, we establish various energy estimates which may be of independent interest.

We complete the study concerning the minimization of the positive principal eigenvalue associated with a weighted Neumann problem settled in a bounded regular domain $\Omega\subset \mathbb{R}^{N}$, $N\ge2$, for the weight varying in a suitable class of sign-changing bounded functions. Denoting with $u$ the optimal eigenfunction and with $D$ its super-level set, corresponding to the positivity set of the optimal weight, we prove that, as the measure of $D$ tends to zero, the unique maximum point of $u$, $P\in \partial \Omega$, tends to a point of maximal mean curvature of $\partial \Omega$. Furthermore, we show that $D$ is the intersection with $\Omega$ of a $C^{1,1}$ nearly spherical set, and we provide a quantitative estimate of the spherical asymmetry, which decays like a power of the measure of $D$. These results provide, in the small volume regime, a fully detailed answer to some long-standing questions in this framework.

We study the integrable structure and scaling limits of the conditioned eigenvector overlap of the symplectic Ginibre ensemble of Gaussian non-Hermitian random matrices with independent quaternion elements. The average of the overlap matrix elements constructed from left and right eigenvectors, conditioned to $x$, are derived in terms of a Pfaffian determinant. Regarded as a two-dimensional Coulomb gas with the Neumann boundary condition along the real axis, it contains a kernel of skew-orthogonal polynomials with respect to the weight function $\omega^{(\mathrm{over})}(z)=|z-\overline{x}|^2(1+|z-x|^2)e^{-2|z|^2}$, including a non-trivial insertion of a point charge. The mean off-diagonal overlap is related to the diagonal (self-)overlap by a transposition, in analogy to the complex Ginibre ensemble. For $x$ conditioned to the real line, extending previous results at $x=0$, we determine the skew-orthogonal polynomials and their skew-kernel with respect to $\omega^{(\mathrm{over})}(z)$. This is done in two steps and involves a Christoffel perturbation of the weight $\omega^{(\mathrm{over})}(z)=|z-\overline{x}|^2\omega^{(\mathrm{pre})}(z)$, by computing first the corresponding quantities for the unperturbed weight $\omega^{(\mathrm{pre})}(z)$. Its kernel is shown to satisfy a differential equation at finite matrix size $N$. This allows us to take different large-$N$ limits, where we distinguish bulk and edge regime along the real axis. The limiting mean diagonal overlaps and corresponding eigenvalue correlation functions of the point processes with respect to $\omega^{(\mathrm{over})}(z)$ are determined. We also examine the effect on the planar orthogonal polynomials when changing the variance in $\omega^{(\mathrm{pre})}(z)$, as this appears in the eigenvector statistics of the complex Ginibre ensemble.

In recent years, there has been much progress in the field of structural Ramsey theory, in particular in the study of big Ramsey degrees. In all known examples of infinite structures with finite big Ramsey degrees, there is in fact a single expansion of the structure, called a big Ramsey structure, which correctly encodes the exact big Ramsey degrees of every finite substructure simultaneously. The first half of the article collects facts about this phenomenon that have appeared in the literature into a single cohesive framework, thus offering a conceptual survey of big Ramsey structures. We present some original results indicating that the standard methods of proving finite big Ramsey degrees automatically yield big Ramsey structures, often with desirable extra properties. The second half of the article is a survey in the more traditional sense, discussing numerous examples from the literature and showing how they fit into our framework. We also present some general results on how big Ramsey degrees are affected by expanding structures with unary functions.

In this paper, we improve the error term in the prime geodesic theorem for the Picard manifold $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3 $. Instead of $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3 $, we establish a spectral large sieve inequality for symmetric squares over $\mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathrm{PSL}_2 (\mathbb{C}) $. This enables us to improve the bound $ O (T^{3+2/3+\varepsilon}) $ of Balkanova and Frolenkov into $ O (T^{3+1/2+\varepsilon}) $ for the second moment of symmetric square $L$-functions over $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3 $. The basic idea is to enlarge the spherical family $\Pi_c^{0} (T)$ of Maass cusp forms on $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3 $ into the family $ \Pi_c (T, \sqrt{T}) $ of cuspidal representations on $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathrm{PSL}_2 (\mathbb{C}) $.

We consider a PDE-constrained optimization problem of tracking type with parabolic state equation. The solution to the problem is characterized by the Karush-Kuhn-Tucker (KKT) system, which we formulate using a strong variational formulation of the state equation and a super weak formulation of the adjoined state equation. This allows us to propose a preconditioner that is robust both in the regularization and the diffusion parameter. In order to discretize the problem, we use Isogeometric Analysis since it allows the construction of sufficiently smooth basis functions effortlessly. To realize the preconditioner, one has to solve a problem over the whole space time cylinder that is elliptic with respect to certain non-standard norms. Using a fast diagonalization approach in time, we reformulate the problem as a collection of elliptic problems in space only. These problems are not only smaller, but our approach also allows to solve them in a time-parallel way. We show the efficiency of the preconditioner by rigorous analysis and illustrate it with numerical experiments.

We prove that any $\tau$-tilting finite incidence algebra of a finite poset is representation-finite, and that any $\mathbf{g}$-tame incidence algebra of a finite simply connected poset is tame. As the converse of these assertions are known to hold, we obtain characterizations of $\tau$-tilting finite incidence algebras and $\mathbf{g}$-tame simply connected incidence algebras. Both results are proved using the theory of concealed algebras. The former will be deduced from the fact that tame concealed algebras are $\tau$-tilting infinite, and to prove the latter, we show that wild concealed algebras are not $\mathbf{g}$-tame. We conjecture that any incidence algebra of a finite poset is wild if and only if it is not $\mathbf{g}$-tame, and prove a result showing that there are relatively few possible counterexamples. In the appendix, we determine the representation type of a $\tau$-tilting reduction of a concealed algebra of hyperbolic type.

A new approach to the evanescent part of a two-dimensional weak-stationary stochastic process with the past given by a half-plane is proceed. The classical result due to Helson and Lowdenslager divides a two-parametric weak-stationary stochastic process into three parts. In this paper we describe the most untouchable one - the evanescent part. Moreover, we point out how this part depends on the shape of the past.

Whitney proved that 3-connected planar graphs can be embedded uniquely on the sphere. In general, such a graph may also have embeddings on different surfaces. Enami focused on cubic graphs and showed that a 3-connected cubic planar graph can be re-embedded on a surface of non-negative Euler characteristic if and only if the dual graph contains a specific subgraph. Motivated by applications to triangulated surfaces, we investigate strong re-embeddings and show that these are characterised by a subset of Enami's subgraphs. Additionally, we provide criteria for when a graph does not possess a strong re-embedding on the projective plane, the torus or the Klein bottle.

This paper solves a recursive optimal stopping problem with Poisson stopping constraints using the penalized backward stochastic differential equation (PBSDE) with jumps. Stopping in this problem is only allowed at Poisson random intervention times, and jumps play a significant role not only through the stopping times but also in the recursive objective functional and model coefficients. To solve the problem, we propose a decomposition method based on Jacod-Pham that allows us to separate the problem into a series of sub-problems between each pair of consecutive Poisson stopping times. To represent the value function of the recursive optimal stopping problem when the initial time falls between two consecutive Poisson stopping times and the generator is concave/convex, we leverage the comparison theorem of BSDEs with jumps. We then apply the representation result to American option pricing in a nonlinear market with Poisson stopping constraints.

This manuscript studies the preventive replacement policy for a series or parallel system consisting of n independent or dependent heterogeneous components. Firstly, for the age replacement policy, Some sufficient conditions for the existence and uniqueness of the optimal replacement time for both the series and parallel systems are provided. By introducing deviation costs, the expected cost rate of the system is optimized, and the optimal replacement time of the system is extended. Secondly, the periodic replacement policy for series and parallel systems is considered in the dependent case, and a sufficient condition for the existence and uniqueness of the optimal number of periods is provided. Some numerical examples are given to illustrate and discuss the above preventive replacement policies.

We introduce modified energies that are suitable to get upper bounds on the high Sobolev norms for solutions to the $1$D periodic NLS. Our strategy is rather flexible and allows us to get a new and simpler proof of the bounds obtained by Bourgain in the case of the quintic nonlinearity, as well as its extension to the case of higher order nonlinearities. Our main ingredients are a combination of integration by parts and classical dispersive estimates.

Symmetries and reductions of some algebraic equations are considered. Transformations that preserve the form of several algebraic equations, as well as transformations that reduce the degree of these equations, are described. Illustrative examples are provided. The obtained results and solutions can be used as test problems for numerical methods of solving algebraic equations.

We consider delay differential equations of the form $ y'(t)=-ay(t)+bf(y(t-1)) $ with positive parameters $a,b$ and a unimodal $f:[0,\infty)\to [0,1]$. It is assumed that the nonlinear $f$ is close to a function $g:[0,\infty)\to [0,1]$ with $g(\xi)=0$ for all $\xi>1$. The fact $g(\xi)=0$ for all $\xi>1$ allows to construct stable periodic orbits for the equation $x'(t)=-cx(t)+dg(x(t-1))$ with some parameters $d>c>0$. Then it is shown that the equation $ y'(t)=-ay(t)+bf(y(t-1)) $ also has a stable periodic orbit provided $a,b,f$ are sufficiently close to $c,d,g$ in a certain sense. The examples include $f(\xi)=\frac{\xi^k}{1+\xi^n}$ for parameters $k>0$ and $n>0$ together with the discontinuous $g(\xi)=\xi^k$ for $\xi\in[0,1)$, and $g(\xi)=0$ for $\xi>1$. The case $k=1$ is the famous Mackey--Glass equation, the case $k>1$ appears in population models with Allee effect, and the case $k\in(0,1)$ arises in some economic growth models. The obtained stable periodic orbits may have complicated structures.

This work addresses stochastic optimal control problems where the unknown state evolves in continuous time while partial, noisy, and possibly controllable measurements are only available in discrete time. We develop a framework for controlling such systems, focusing on the measure-valued process of the system's state and the control actions that depend on noisy and incomplete data. Our approach uses a stochastic optimal control framework with a probability measure-valued state, which accommodates noisy measurements and integrates them into control decisions through a Bayesian update mechanism. We characterize the control optimality in terms of a sequence of interlaced Hamilton Jacobi Bellman (HJB) equations coupled with controlled impulse steps at the measurement times. For the case of Gaussian-controlled processes, we derive an equivalent HJB equation whose state variable is finite-dimensional, namely the state's mean and covariance. We demonstrate the effectiveness of our methods through numerical examples. These include control under perfect observations, control under no observations, and control under noisy observations. Our numerical results highlight significant differences in the control strategies and their performance, emphasizing the challenges and computational demands of dealing with uncertainty in state observation.

We prove that a homomorphism between free groups of finite rank equipped with the bi-invariant word metrics is a quasi-isometry if and only if it is an isomorphism.

In this work we consider the Lagrangian properties of a random version of the Arnold-Beltrami-Childress (ABC) in a three-dimensional periodic box. We prove that the associated flow map possesses a positive top Lyapunov exponent and its associated one-point, two-point and projective Markov chains are geometrically ergodic. For a passive scalar, it follows that such a velocity is a space-time smooth exponentially mixing field, uniformly in the diffusivity coefficient. For a passive vector, it provides an example of a universal ideal (i.e. non-diffusive) kinematic dynamo.

We show that the word problem for any 3-manifold group is solvable in time $O(n\log^3 n)$. Our main contribution is the proof that the word problem for admissible graphs of groups, in the sense of Croke and Kleiner, is solvable in $O(n\log n)$; this covers fundamental groups of non-geometric graph manifolds. Similar methods also give that the word problem for free products can be solved "almost as quickly" as the word problem in the factors.

Stochastic systems are used to model a variety of phenomena in which noise plays an essential role. In these models, one potential goal is to determine if noise can induce transitions between states, and if so, to calculate the most probable escape path from an attractor. In the small noise limit, the Freidlin-Wentzell theory of large deviations provides a variational framework to calculate these paths. This work focuses on using large deviation theory to calculate such paths for stochastic gradient systems with non-gradient perturbations. While for gradient systems the most probable escape paths consist of time-reversed heteroclinic orbits, for general systems it can be a challenging calculation. By applying Melnikov theory to the resulting Euler-Lagrange equations recast in Hamiltonian form, we determine a condition for when the optimal escape path is the heteroclinic orbit for the perturbed system. We provide a numerical example to illustrate how the computed most probable escape path compares with the theoretical result.

Let $\xi$ be the standard normal random vector in $\mathbb{R}^{k}$. Under some mild growth and smoothness assumptions on any increasing $P, Q : [0, \infty) \mapsto [0, \infty)$ we show $(P,Q)$ complex hypercontractivity $$ Q^{-1}(\mathbb{E} Q(|T_{z}f(\xi)|))\leq P^{-1}(\mathbb{E}P(|f(\xi)|)) $$ holds for all polynomials $f:\mathbb{R}^{k} \mapsto \mathbb{C}$, where $T_{z}$ is the hermite semigroup at complex parameter $z, |z|\leq 1$, if and only if \begin{align*} \left|\frac{tP''(t)}{P'(t)}-z^{2}\frac{tQ''(t)}{Q'(t)}+z^{2}-1\right|\leq \frac{tP''(t)}{P'(t)}-|z|^{2}\frac{tQ''(t)}{Q'(t)}+1-|z|^{2} \end{align*} holds for all $t>0$ provided that $F''>0$, and $F'/F''$ is concave, where $F = Q\circ P^{-1}$. This extends Hariya's result from real to complex parameter $z$. Several old and new applications are presented for different choices of $P$ and $Q$. The proof uses heat semigroup arguments, where we find a certain map $C(s)$, which interpolates the inequality at the endpoints. The map $C(s)$ itself is composed of four heat flows running together at different times.

In the event of a total loss of thrust a pilot must identify a reachable landing site and subsequently execute a forced landing. To do so this, they must estimate which region on the ground can be reached safely in gliding flight. We call this the gliding reachable region (GRR). To compute the GRR, we employ an optimal control formulation aiming to reach a point in space while minimizing altitude loss. A simplified model of the aircraft's dynamics is used, where the effect of turns is neglected. The resulting equations are discretized on a grid and solved numerically. Our algorithm for computing the GRR is fast enough to run in real time during flight, it accounts for ground obstacles and wind, and for each point in the GRR it outputs the path to reach it with minimal loss of altitude. A related problem is estimating the minimal altitude an aircraft needs to glide to a given airfield in the presence of obstacles. This information enables pilots to plan routes that always have an airport within gliding distance. We formalize this problem using an optimal control formulation based on the same aircraft dynamics model. The resulting equations are solved with a second algorithm that outputs the minimal re-entry altitude and the paths to reach the airfield from any position while avoiding obstacles. The algorithms we develop are based on the Ordered Upwind Method and the Fast Marching Method.

Machine learning techniques have recently been of great interest for solving differential equations. Training these models is classically a data-fitting task, but knowledge of the expression of the differential equation can be used to supplement the training objective, leading to the development of physics-informed scientific machine learning. In this article, we focus on one class of models called nonlinear vector autoregression (NVAR) to solve ordinary differential equations (ODEs). Motivated by connections to numerical integration and physics-informed neural networks, we explicitly derive the physics-informed NVAR (piNVAR) which enforces the right-hand side of the underlying differential equation regardless of NVAR construction. Because NVAR and piNVAR completely share their learned parameters, we propose an augmented procedure to jointly train the two models. Then, using both data-driven and ODE-driven metrics, we evaluate the ability of the piNVAR model to predict solutions to various ODE systems, such as the undamped spring, a Lotka-Volterra predator-prey nonlinear model, and the chaotic Lorenz system.

We build on our recent results on the Lipschitz dependence of the extreme spectral values of one-parameter families of pseudodifferential operators with symbols in a weighted Sj\"ostrand class. We prove that larger symbol classes lead to H\"older continuity with respect to the parameter. This result is then used to investigate the behavior of frame bounds of families of Gabor systems $\mathcal{G}(g,\alpha\Lambda)$ with respect to the parameter $\alpha>0$, where $\Lambda$ is a set of non-uniform, relatively separated time-frequency shifts, and $g\in M^1_s(\mathbb{R}^d)$, $0\leq s\leq 2$. In particular, we show that the frame bounds depend continuously on $\alpha$ if $g\in M^1(\mathbb{R}^d)$, and are H\"older continuous if $g\in M^1_s(\mathbb{R}^d)$, $0<s\leq 2$, with the H\"older exponent explicitly given.

We study exponentiable functors in the context of synthetic $\infty$-categories. We do this within the framework of simplicial Homotopy Type Theory of Riehl and Shulman. Our main result characterizes exponentiable functors. In order to achieve this, we explore Segal type completions. Moreover, we verify that our result is semantically sound.

These are extended lecture notes for a mini course at the Spring School on Non-Archimedean Geometry and Eigenvarieties held at Heidelberg University in March 2023. The goal of the course is to explain a modern take on the eigenvariety machine in the language of adic spaces. For this we build on the theory developed in the first week of the school. We also explain some basic selective non-archimedean functional analysis.

In \cite{nr} Narasimhan and Ramanan and in \cite{desing}, Seshadri constructed desingularisations of the moduli space $M^{ss}_{_{\text{SL}(2)}}$ of semistable $\SL(2)$-bundles on a smooth projective curve $C$ of genus $g \geq 3$. Seshadri's construction was even modular and canonical. In this paper, we construct a smooth modular compactification of the moduli of stable principal $H$-bundles when $H$ is a simply connected almost simple algebraic group of type ${\tt B}_{_\ell}, {\tt D}_{_\ell}, {\tt G}_{_2}, {\tt F}_{_4}~~or~~{\tt C}_{_3}$. These spaces give canonical desingularisations of the moduli space $M^{ss}_{_H}$ of semistable principal $H$-bundles and thereby, a comprehensive generalisation of \cite{desing}.

In this work, we present composite Bernstein polynomials as a direct collocation method for approximating optimal control problems. An analysis of the convergence properties of composite Bernstein polynomials is provided, and beneficial properties of composite Bernstein polynomials for the solution of optimal control problems are discussed. The efficacy of the proposed approximation method is demonstrated through a bang-bang example. Lastly, we apply this method to a motion planning problem, offering a practical solution that emphasizes the ability of this method to solve complex optimal control problems.

The goal of this paper is to prove the well-posedness of F. John's floating body problem in the case of a fixed object and for unsteady waves, in horizontal dimension $d=1$ and with a possibly emerging bottom. This problem describes the interactions of waves with a partially immersed object using the linearized Bernoulli equations. The fluid domain $\Omega$ therefore has corners where the object meets the free surface, which consists of various connected components. The energy space associated with this problem involves the space of traces on these different connected components of all functions in the Beppo-Levi space $\dot{H}^1(\Omega)$; we characterize this space, exhibiting non local effects linking the different connected components. We prove the well-posedness of the Laplace equation in corner domains, with mixed boundary conditions and Dirichlet data in this trace space, and study several properties of the associated Dirichlet-Neumann operator (self-adjointness, ellipticity properties, etc.). This trace space being only semi-normed, we cannot use standard semi-group theory to solve F. John's problem: one has to choose a realization of the homogeneous space (i.e. choose an adequate representative in the equivalence class) we are working with. When the fluid domain is bounded, this realization is obtained by imposing a zero-mass condition; for unbounded fluid domains, we have to choose a space-time realization which can be interpreted as a particular choice of the Bernoulli constant. Well-posedness in the energy space is then proved. Conditions for higher order regularity in times are then derived, which yield some limited space regularity that can be improved through smallness assumption on the contact angles. We finally show that higher order regularity away from the contact points can be achieved through weighted estimates.

We exhibit moduli spaces of slope stable vector bundles on general polarized HK varieties $(X,h)$ of type $K3^{[2]}$ which have an irreducible component of dimension $2a^2+2$, with $a$ an arbitrary integer greater than $1$. This is done by studying the case $X=S^{[2]}$ where $S$ is an elliptic $K3$ surface. We show that in this case there is an irreducible component of the moduli space of stable vector bundles on $S^{[2]}$ which is birational to a moduli space of sheaves on $S$. We expect that if the moduli space of sheaves on $S$ is a smooth HK variety (necessarily of type $K3^{[a^2+1]}$) then the following more precise version holds: the closure of the moduli space of slope stable vector bundles on $(X,h)$ in the moduli space of Gieseker-Maruyama semistable sheaves with its GIT polarization is a general polarized HK variety of type $K3^{[a^2+1]}$.

We examine the maximum dimension of a linear system of plane cubic curves whose $\mathbb{F}_q$-members are all geometrically irreducible. Computational evidence suggests that such a system has a maximum (projective) dimension of $3$. As a step towards the conjecture, we prove that there exists a $3$-dimensional linear system $\mathcal{L}$ with at most one geometrically reducible $\mathbb{F}_q$-member.

We study heuristic algorithms for job shop scheduling problems. We compare classical approaches, such as the shifting bottleneck heuristic with novel strategies using decision diagrams. Balas' local refinement is used to improve feasible solutions. Heuristic approaches are combined with Mixed Integer Programming and Constraint Programming approaches. We discuss our results via computational experiments.

We study the average edit distance between two random strings. More precisely, we adapt a technique introduced by Lueker in the context of the average longest common subsequence of two random strings to improve the known upper bound on the average edit distance. We improve all the known upper bounds for small alphabets. We also provide a new implementation of Lueker technique to improve the lower bound on the average length of the longest common subsequence of two random strings for all small alphabets of size other than $2$ and $4$.

A de Bruijn arrays code is a set of $r \times s$ binary doubly-periodic arrays such that each binary $n \times m$ matrix is contained exactly once as a window in one of the arrays. Such a set of arrays can be viewed as a two-dimensional generalization of a perfect factor in the de Bruijn graph. Necessary conditions for the existence of such arrays are given. Several direct constructions and recursive constructions for such arrays are given. A framework for a theory of two-dimensional feedback shift register which is akin to (one-dimensional) feedback shift registers is suggested.

We introduce a family of generalized Schr\"oder polynomials $S_\tau(q,t,a)$, indexed by triangular partitions $\tau$ and prove that $S_\tau(q,t,a)$ agrees with the Poincar\'e series of the triply graded Khovanov-Rozansky homology of the Coxeter knot $K_\tau$ associated to $\tau$. For all integers $m,n,d\geq 1$ with $m,n$ relatively prime, the $(d,mnd+1)$-cable of the torus knot $T(m,n)$ appears as a special case. It is known that these knots are algebraic, and as a result we obtain a proof of the $q=1$ specialization of the Oblomkov-Rasmussen-Shende conjecture for these knots. Finally, we show that our Schr\"oder polynomial computes the hook components in the Schur expansion of the symmetric function appearing in the shuffle theorem under any line, thus proving a triangular version of the $(q,t)$-Schr\"oder theorem.

Unequal Error-Protecting (UEP) codes are error-correcting (EC) codes designed to protect some parts of the encoded data better than other parts. Here, we introduce a similar generalization of PIR codes that we call Unequal-Data-Demand (UDD) PIR codes. These codes are PIR-type codes designed for the scenario where some parts of the encoded data are in higher demand than other parts. We generalize various results for PIR codes to UDD codes. Our main contribution is a new approach to the Griesmer bound for linear EC codes involving an Integer Linear Programming (ILP) problem that generalizes to linear UEP codes and linear UDD PIR codes.

A copy of a graph $F$ is called an $F$-copy. For any graph $G$, the $F$-isolation number of $G$, denoted by $\iota(G,F)$, is the size of a smallest subset $D$ of the vertex set of $G$ such that the closed neighbourhood $N[D]$ of $D$ in $G$ intersects the vertex sets of the $F$-copies contained by $G$ (equivalently, $G-N[D]$ contains no $F$-copy). Thus, $\iota(G,K_1)$ is the domination number $\gamma(G)$ of $G$, and $\iota(G,K_2)$ is the vertex-edge domination number of $G$. We prove that if $F$ is a $k$-edge graph, $\gamma(F) = 1$ (that is, a vertex of $F$ is adjacent to all the other vertices of $F$), and $G$ is a connected $m$-edge graph, then $\iota(G,F) \leq \big\lfloor \frac{m+1}{k+2} \big\rfloor$ unless $G$ is an $F$-copy or $F$ is a $3$-path and $G$ is a $6$-cycle. This was recently posed as a conjecture by Zhang and Wu, who settled the case where $F$ is a star. The result for the case where $F$ is a clique had been obtained by Fenech, Kaemawichanurat and the present author. The bound is attainable for any $m \geq 0$ unless $m = k \leq 2$. New ideas, such as the consideration of divisibility, are introduced in the proof of the conjecture.

The decomposition locus of a tensor is the set of rank-one tensors appearing in a minimal tensor-rank decomposition of the tensor. For tensors lying on the tangential variety of any Segre variety, but not on the variety itself, we show that the decomposition locus consists of all rank-one tensors except the tangency point only. We also explicitly compute decomposition loci of all tensors belonging to tensor spaces with finitely many orbits with respect to the action of product of general linear groups.

Recent work showing the existence of conflict-free almost-perfect hypergraph matchings has found many applications. We show that, assuming certain simple degree and codegree conditions on the hypergraph $ \mathcal{H} $ and the conflicts to be avoided, a conflict-free almost-perfect matching can be extended to one covering all of the vertices in a particular subset of $ V(\mathcal{H}) $, by using an additional set of edges; in particular, we ensure that our matching avoids all of a further set of conflicts, which may consist of both old and new edges. This setup is useful for various applications, and our main theorem provides a black box which encapsulates many long and tedious calculations, massively simplifying the proofs of results in generalised Ramsey theory.

We consider the three-dimensional incompressible Euler equations in Sobolev conormal spaces and establish local-in-time existence and uniqueness in the half-space or channel. The initial data is Lipschitz having four square-integrable conormal derivatives and two bounded conormal derivatives. We do not impose any integrability or differentiability assumption for the normal derivative.

A second-order block coordinate descent method is proposed for the unconstrained minimization of an objective function with Lipschitz continuous Hessian. At each iteration, a block of variables is selected by means of a greedy (Gauss-Southwell) rule which considers the amount of first-order stationarity violation, then an approximate minimizer of a cubic model is computed for the block update. In the proposed scheme, blocks are not required to have a prefixed structure and their size is allowed to change during the iterations. For non-convex objective functions, global convergence to stationary points is proved and a worst-case iteration complexity analysis is provided. In particular, given a tolerance $\varepsilon$, we show that at most ${\cal O}(\varepsilon^{-3/2})$ iterations are needed to drive the stationarity violation with respect to the selected block of variables below $\varepsilon$, while at most ${\cal O}(\varepsilon^{-2})$ iterations are needed to drive the stationarity violation with respect to all variables below $\varepsilon$. Numerical results are finally provided.

If the laws and explanations for the combinatorial aspects of the Kronecker coefficients can be discovered through a large number of computations, then the solution to a long-standing open problem can be provided. The aim of our work is to compute the Kronecker coefficients for the representation ring of the symmetric group $S_6$. Specifically, the power formulas of irreducible representations of the symmetric group $S_6$ are computed using the character theory of finite groups. In addition, by decomposing tensor products of irreducible representations of $S_6$, we characterise the representation ring $\mathcal{R}(S_6)$, such as generators, its unit group, primitive idempotents and Casimir number. We also give another way, using Harrison center theory, to study the representation ring. Finally, we leave some open problems for future consideration.

We construct a bijection between marked bumpless pipedreams with reverse compatible pairs, which are in bijection with not-necessarily-reduced pipedreams. This directly unifies various formulas for Grothendieck polynomials in the literature. Our bijection is a generalization of a variant of the bijection of Gao and Huang in the unmarked, reduced case.

This paper investigates an optimal control problem associated with a two-dimensional multi-species Cahn-Hilliard-Keller-Segel tumor growth model, which incorporates complex biological processes such as species diffusion, chemotaxis, angiogenesis, and nutrient consumption, resulting in a highly nonlinear system of nonlinear partial differential equations. The modeling derivation and corresponding analysis have been addressed in a previous contribution. Building on this foundation, the scope of this study involves investigating a distributed control problem with the goal of optimizing a tracking-type cost functional. This latter aims to minimize the deviation of tumor cell location from desired target configurations while penalizing the costs associated with implementing control measures, akin to introducing a suitable medication. Under appropriate mathematical assumptions, we demonstrate that sufficiently regular solutions exhibit continuous dependence on the control variable. Furthermore, we establish the existence of optimal controls and characterize the first-order necessary optimality conditions through a suitable variational inequality.

We present an introduction to the field of statistical optimal transport, based on lectures given at \'Ecole d'\'Et\'e de Probabilit\'es de Saint-Flour XLIX.

Call a finite relational structure $k$-Slupecki if its only surjective $k$-ary polymorphisms are essentially unary, and Slupecki if it is $k$-Slupecki for all $k \geq 2$. We present conditions, some necessary and some sufficient, for a reflexive digraph to be Slupecki. We prove that all digraphs that triangulate a 1-sphere are Slupecki, as are all the ordinal sums $m \oplus n$ ($m,n \geq 2$). We prove that the posets $P = m \oplus n \oplus k$ are not 3-Slupecki for $m,n,k \geq 2$, and prove there is a bound $B(m,k)$ such that $P$ is 2-Slupecki if and only if $n > B(m,k)+1$; in particular there exist posets that are 2-Slupecki but not 3-Slupecki.

We stratify families of projective and very affine hypersurfaces according to their topological Euler characteristic. Our new algorithms compute all strata using algebro-geometric techniques. For very affine hypersurfaces, we investigate and exploit the relation to critical point computations. Euler stratifications are relevant in particle physics and algebraic statistics. They fully describe the dependence of the number of master integrals, respectively the maximum likelihood degree, on kinematic or model parameters.

We consider optimal control problems involving nonlinear ordinary differential equations with uncertain inputs. Using the sample average approximation, we obtain optimal control problems with ensembles of deterministic dynamical systems. Leveraging techniques for metric entropy bounds, we derive non-asymptotic Monte Carlo-type convergence rates for the ensemble-based solutions. Our theoretical framework is validated through numerical simulations on a harmonic oscillator problem and a vaccination scheduling problem for epidemic control under model parameter uncertainty.

Let $\{a_i\}_{i=1}^\ell$ be a strongly unimodal positive integer sequence with peak position $k$. The rank of such sequence is defined to be $\ell-2k+1$. Let $u(m,n)$ denote the number of sequences $\{a_i\}_{i=1}^\ell$ with rank $m$ and $\sum_{i=1}^{\ell} a_i=n$. Bringmann, Jennings-Shaffer, Mahlburg and Rhoades conjectured that $\{u(m,n)\}_m$ is strongly log-concave for any fixed $n$. Motivated by this conjecture, in this paper we prove the strongly unimodality of $\{u(m,n)\}_m$, that is $u(m,n)>u(m+1,n)$ for $m\ge 0$ and $n\ge \max\{6,{m+2\choose 2}\}$. This result gives supportive evidence for the above conjecture. Moreover, we find a combinatorial interpretation of $u(m,n)$, which leads to a new combinatorial interpretation of ${\rm ospt}(n)$. Furthermore, using this new combinatorial interpretation, a lower bound and an asymptotic formula on ${\rm ospt}(n)$ will be presented.

This work addresses the homotopical analysis of enveloping operads in a general cofibrantly generated symmetric monoidal model category. We show the potential of this analysis by obtaining, in a uniform way, several central results regarding the homotopy theory of operadic algebras.

By the introduction of locally constant prefactorization algebras at a fixed scale, we show a mathematical incarnation of the fact that observables at a given scale of a topological field theory propagate to every scale over euclidean spaces. The key is that these prefactorization algebras over $\mathbb{R}^n$ are equivalent to algebras over the little $n$-disc operad. For topological field theories with defects, we get analogous results by replacing $\mathbb{R}^n$ with the spaces modelling corners $\mathbb{R}^p\times\mathbb{R}^{q}_{\geq 0}$. As a toy example in $1d$, we quantize, once more, constant Poisson structures.

We investigate relaxations for a class of discrete bilevel programs where the interaction constraints linking the leader and the follower are linear. Our approach reformulates the upper-level optimality constraints by projecting the leader's decisions onto vectors that map to distinct follower solution values, each referred to as a state. Based on such a state representation, we develop a network-flow linear program via a decision diagram that captures the convex hull of the follower's value function graph, leading to a new single-level reformulation of the bilevel problem. We also present a reduction procedure that exploits symmetry to identify the reformulation of minimal size. For large networks, we introduce parameterized relaxations that aggregate states by considering tractable hyperrectangles based on lower and upper bounds associated with the interaction constraints, and can be integrated into existing mixed-integer bilevel linear programming (MIBLP) solvers. Numerical experiments suggest that the new relaxations, whether used within a simple cutting-plane procedure or integrated into state-of-the-art MIBLP solvers, significantly reduce runtimes or solve additional benchmark instances. Our findings also highlight the correlation between the quality of relaxations and the properties of the interaction matrix, underscoring the potential of our approach in enhancing solution methods for structured bilevel optimization instances.

A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and has a trapped endpoint. We show that the generating function enumerating GSAWs on a half-infinite strip of finite height is rational, and we give a procedure to construct a combinatorial finite state machine that allows one to compute this generating function. We then modify this procedure to compute generating functions for GSAWs under two probabilistic models. We perform Monte Carlo simulations to estimate the expected length and displacement for GSAWs on the quarter plane, half plane, full plane, and half-infinite strips of bounded height for which we cannot compute the generating function. Finally, we prove that the generating functions for Greek key tours (GSAWs on a finite grid that visit every vertex) on a half-infinite strip of fixed height are also rational, allowing us to resolve several conjectures.

Answering a question of Randal-Williams, we show the natural maps from split Steinberg modules of a Dedekind domain to the associated Steinberg modules are surjective.

This paper considers the decay in particle intensities for a translation invariant two species system of diffusing and reacting particles on $\mathbb{Z}^d$ for $d \geq 3$. The intensities are shown to approximately solve modified rate equations, from which their polynomial decay can be deduced. The system illustrates that the underlying diffusion and reaction rates can influence the exact polynomial decay rates, despite the system evolving in a supercritical dimension.

We consider a weighted sum of a series of independent Poisson random variables and show that it results in a new compound Poisson distribution which includes the Poisson distribution and Poisson distribution of order k. An explicit representation for its distribution is obtained in terms of Bell polynomials. We then extend it to a compound Poisson process and time fractional compound Poisson process (TFCPP). It is shown that the one-dimensional distributions of the TFCPP exhibit over-dispersion property, are not infinitely divisible and possess the long-range dependence property. Also, their moments and factorial moments are derived. Finally, the fractional differential equation associated with the TFCPP is also obtained.

Full Vertex Operator Algebras (full VOA) are extensions of two commuting Vertex Operator Algebras, introduced to formulate compact two-dimensional conformal field theory. We define unitarity, polynomial energy bounds and polynomial spectral density for full VOA. Under these conditions and local $C_1$-cofiniteness of the simple full VOA, we show that the correlation functions of quasi-primary fields define tempered distributions and satisfy a conformal version of the Osterwalder-Schrader axioms, including the linear growth condition. As an example, we show that a family of full extensions of the Heisenberg VOA satisfies all these assumptions.

Joyce structures are a class of geometric structures which first arose in relation to holomorphic generating functions for Donaldson-Thomas invariants. They can be thought of as non-linear analogues of Frobenius structures, or as special classes of complex hyperkahler manifolds. We give a detailed introduction to Joyce structures, with particular focus on the geometry of the associated twistor space. We also prove several new results.

Borell's inequality states the existence of a positive absolute constant $C>0$ such that for every $1\leq p\leq q$ $$ \left(\mathbb E|\langle X, e_n\rangle|^p\right)^\frac{1}{p}\leq\left(\mathbb E|\langle X, e_n\rangle|^q\right)^\frac{1}{q}\leq C\frac{q}{p}\left(\mathbb E|\langle X, e_n\rangle|^p\right)^\frac{1}{p}, $$ whenever $X$ is a random vector uniformly distributed in any convex body $K\subseteq\mathbb R^n$ containing the origin in its interior and $(e_i)_{i=1}^n$ is the standard canonical basis in $\mathbb R^n$. In this paper, we will prove a discrete version of this inequality, which will hold whenever $X$ is a random vector uniformly distributed on $K\cap\mathbb Z^n$ for any convex body $K\subseteq\mathbb R^n$ containing the origin in its interior. We will also make use of such discrete version to obtain discrete inequalities from which we can recover the estimate $\mathbb E w(K_N)\sim w(Z_{\log N}(K))$ for any convex body $K$ containing the origin in its interior, where $K_N$ is the centrally symmetric random polytope $K_N=\operatorname{conv}\{\pm X_1,\ldots,\pm X_N\}$ generated by independent random vectors uniformly distributed on $K$ and $w(\cdot)$ denotes the mean width.

We consider typical properties of automorphisms from the spaces ${\bf Aut}$, ${\bf Aut}_\infty$, $\bf {\bf Mix}$, ${\bf Mix}_\infty$ and recall some unsolved problems.

This paper is the final step in solving the problem of starlikeness of Teichmuller spaces in Bers' embedding. This step concerns the case of finite dimensional Teichmuller spaces ${\mathbf T}(g, n)$ of positive dimension (corresponding to punctured Riemann surfaces of finite conformal type $(g, n)$ with $2g - 2 + n > 0$).

We study a class of nonlinear PDEs that admit the same bi-Hamiltonian structure as WDVV equations: a Ferapontov-type first-order Hamiltonian operator and a homogeneous third-order Hamiltonian operator in a canonical Doyle--Potemin form, which are compatible. Using various equivalence groups, we classify such equations in two-component and three-component cases. In a four-component case, we add further evidence to the conjecture that there exists only one integrable system of the above type. Finally, we give an example of the six-component system with required bi-Hamiltonian structure. To streamline the symbolic computation we develop an algorithm to find the aforementioned Hamiltonian operators, which includes putting forward a conjecture on the structure of the metric parameterising the first-order Hamiltonian operator.

It is generally believed (but not yet proved) that Ising spin glasses with nearest-neighbor interactions have a phase transition in three and higher dimensions to a low-temperature spin glass phase, but the nature of this phase remains controversial, especially whether it is characterized by multiple incongruent Gibbs states. Of particular relevance to this question is the behavior of the typical free energy difference restricted to a finite volume between two such putative Gibbs states, as well as the nature of the fluctuations of their free energy difference as the couplings within the volume vary. In this paper we investigate these free energy difference fluctuations by introducing a new kind of metastate which classifies Gibbs states through their edge overlap values with a reference Gibbs state randomly chosen from the support of the periodic boundary condition (PBC) metastate. We find that the free energy difference between any two incongruent pure states, regardless of the details of how they're organized into mixed states within the PBC metastate, converges to a Gaussian (or Gaussian-like) distribution whose variance scales with the volume, proving a decades-old conjecture of Fisher and Huse. The same conclusion applies, though with some additional restrictions, to both mixed Gibbs states and ground states. We discuss some implications of these results.

Due to its rigid foldability and predictable kinematics, the reverse fold is the fundamental mechanism behind some of the most well known origami kinematic structures, including the Miura Ori, Yoshimura, and waterbomb patterns. However, the reverse fold only has one parameter to control its behavior: the starting fold angle. In this paper I introduce an alternative to the traditional reverse fold, based on the spring into action pattern, called the spring joint. This novel rigidly foldable mechanism is able to couple multiple reverse folds into a compact space to amplify the kinematic output of a traditional reverse fold by up to ten times, and to add one parameter for each reverse fold, giving more programmatic control of origami structures. Methods of parameterizing both the starting angle, the path of travel, and the axis of motion are also introduced. Unfortunately, this versatility comes at the cost of a large buildup of layers, making the spring joint impractical for thick origami mechanisms. To solve this problem, I also introduce a modular alternative to the spring joint that has no additional layers, with the same kinematic properties. Both of these mechanisms are tested as replacements for the reverse fold in both traditional and custom origami structures.

This paper presents a neural network-based methodology for the decomposition of transport-dominated fields using the shifted proper orthogonal decomposition (sPOD). Classical sPOD methods typically require an a priori knowledge of the transport operators to determine the co-moving fields. However, in many real-life problems, such knowledge is difficult or even impossible to obtain, limiting the applicability and benefits of the sPOD. To address this issue, our approach estimates both the transport and co-moving fields simultaneously using neural networks. This is achieved by training two sub-networks dedicated to learning the transports and the co-moving fields, respectively. Applications to synthetic data and a wildland fire model illustrate the capabilities and efficiency of this neural sPOD approach, demonstrating its ability to separate the different fields effectively.

We construct $\mathbb{Z}_N$ orbifolds of the ten-dimensional heterotic string theories appropriate for implementing the stringy replica method for the calculation of quantum entanglement entropy. A novel feature for the heterotic string is that the gauge symmetry must be broken by a Wilson line to ensure modular invariance. We completely classify the patterns of symmetry breaking. We show that the tachyonic contributions in all cases can be analytically continued, with a finite answer in the domain $0<N \leq 1$, relevant for calculating entanglement entropy across the Rindler horizon. We discuss the physical implications of our results.

In this work we study the Friedmann-Lema\^{i}tre-Robertson-Walker cosmologies with arbitrary spatial curvature for the symmetric teleparallel theories of gravity, giving the first presentation of their coincident gauge form. Our approach explicitly starts with the cosmological Killing vectors and constructs the coincident gauge coordinates adapted to these Killing vectors. We then obtain three distinct spatially flat branches and a single spatially curved branch. Contrary to some previous claims, we show that all branches of connection in the covariant approach can be studied in this gauge-fixed formalism, which offers certain conceptual advantages. Interestingly, we find that the flat and negative spatially curved solutions in $f(Q)$ gravity can be seen as equivalent to the metric teleparallel $f(T)$ theories, demonstrating a deeper connection between these theories. This is accomplished by studying the connection equation of motion, which can be interpreted as a consistency condition in the gauge-fixed approach. Finally, we discuss the role of diffeomorphism invariance and local Lorentz invariance in these geometric modifications of gravity.

This study explores the mixed convection of CNT-water nanofluid within a square cavity containing heated cylinders under the influence of a magnetic field, focusing on three geometric configurations: a single heated cylinder, two heated cylinders, and two heated cylinders with a flow deflector. The impact of various parameters, including Reynolds number (Re), Richardson number (Ri), Hartmann number (Ha), wavy wall peaks (n), nanoparticle volume fraction ({\phi}), Hartmann angle ({\gamma}), rotational speed ({\omega}), and inclination angle ({\alpha}), on thermal and fluid dynamic behaviors is analyzed. Results reveal that MWCNT nanofluids consistently achieve higher Nusselt numbers than SWCNT nanofluids, indicating superior heat transfer capabilities. Introducing a second cylinder and a flow deflector enhances thermal interactions, while increasing Ha stabilizes the flow, improving thermal performance. Wavy wall peaks further enhance fluid mixing and heat transfer efficiency. Additionally, SWCNT nanofluids exhibit higher Bejan numbers, indicating a greater dominance of thermal entropy generation over fluid friction. These findings provide valuable insights for optimizing thermal management systems in engineering applications, highlighting the importance of selecting appropriate nanofluids, geometric configurations, and magnetic field parameters to achieve optimal thermal performance and fluid stability.

To advance our understanding of the magnetohydrodynamic (MHD) processes in liquid metals, in this paper we propose an approach combining the classical methods in the dynamo theory based on numerical simulations of the partial differential equations governing the evolution of the magnetic field with the statistical methods. In this study, we intend to answer the following ``optimization'' question: Can we find a statistical explanation what makes a flow to favour magnetic field generation in the linear regime (i.e. the kinematic dynamo is considered), where the Lorenz force is neglected? The flow is assumed to be steady and incompressible, and the magnetic field generation is governed by the magnetic induction equation. The behaviour of its solution is determined by the dominant (i.e. with the largest real part) eigenvalue of the magnetic induction operator. Considering an ensemble of 2193 randomly generated flows, we solved the kinematic dynamo problem and performed an attempt to find a correlation between the dominant eigenvalue and the standard quantities used in hydrodynamics -- vorticity and kinetic helicity. We have found that there is no visible relation between the property of the flow to be a kinematic dynamo and these quantities. This enables us to conclude that the problem requires a more elaborated approach to ``recognize'' if the flow is a dynamo or not; we plan to solve it using contemporary data-driven approach based on deep neural networks.

Attention-based transformers have been remarkably successful at modeling generative processes across various domains and modalities. In this paper, we study the behavior of transformers on data drawn from \kth Markov processes, where the conditional distribution of the next symbol in a sequence depends on the previous $k$ symbols observed. We observe a surprising phenomenon empirically which contradicts previous findings: when trained for sufficiently long, a transformer with a fixed depth and $1$ head per layer is able to achieve low test loss on sequences drawn from \kth Markov sources, even as $k$ grows. Furthermore, this low test loss is achieved by the transformer's ability to represent and learn the in-context conditional empirical distribution. On the theoretical side, our main result is that a transformer with a single head and three layers can represent the in-context conditional empirical distribution for \kth Markov sources, concurring with our empirical observations. Along the way, we prove that \textit{attention-only} transformers with $O(\log_2(k))$ layers can represent the in-context conditional empirical distribution by composing induction heads to track the previous $k$ symbols in the sequence. These results provide more insight into our current understanding of the mechanisms by which transformers learn to capture context, by understanding their behavior on Markov sources.

We study opinion dynamics over a directed multilayer network. In particular, we consider networks in which the impact of neighbors of agents on their opinions is proportional to their in-degree. Agents update their opinions over time to coordinate with their neighbors. However, the frequency of agents' interactions with neighbors in different network layers differs. Consequently, the multilayer network's adjacency matrices are time-varying. We aim to characterize how the frequency of activation of different layers impacts the convergence of the opinion dynamics process.

Use-dependent bias is a phenomenon in human sensorimotor behavior whereby movements become biased towards previously repeated actions. Despite being well-documented, the reason why this phenomenon occurs is not year clearly understood. Here, we propose that use-dependent biases can be understood as a rational strategy for movement under limitations on the capacity to process sensory information to guide motor output. We adopt an information-theoretic approach to characterize sensorimotor information processing and determine how behavior should be optimized given limitations to this capacity. We show that this theory naturally predicts the existence of use-dependent biases. Our framework also generates two further predictions. The first prediction relates to handedness. The dominant hand is associated with enhanced dexterity and reduced movement variability compared to the non-dominant hand, which we propose relates to a greater capacity for information processing in regions that control movement of the dominant hand. Consequently, the dominant hand should exhibit smaller use-dependent biases compared to the non-dominant hand. The second prediction relates to how use-dependent biases are affected by movement speed. When moving faster, it is more challenging to correct for initial movement errors online during the movement. This should exacerbate costs associated with initial directional error and, according to our theory, reduce the extent of use-dependent biases compared to slower movements, and vice versa. We show that these two empirical predictions, the handedness effect and the speed-dependent effect, are confirmed by experimental data.

This work introduces IsUMap, a novel manifold learning technique that enhances data representation by integrating aspects of UMAP and Isomap with Vietoris-Rips filtrations. We present a systematic and detailed construction of a metric representation for locally distorted metric spaces that captures complex data structures more accurately than the previous schemes. Our approach addresses limitations in existing methods by accommodating non-uniform data distributions and intricate local geometries. We validate its performance through extensive experiments on examples of various geometric objects and benchmark real-world datasets, demonstrating significant improvements in representation quality.

Tests based on the $2$- and $\infty$-norm have received considerable attention in high-dimensional testing problems, as they are powerful against dense and sparse alternatives, respectively. The power enhancement principle of Fan et al. (2015) combines these two norms to construct tests that are powerful against both types of alternatives. Nevertheless, the $2$- and $\infty$-norm are just two out of the whole spectrum of $p$-norms that one can base a test on. In the context of testing whether a candidate parameter satisfies a large number of moment equalities, we construct a test that harnesses the strength of all $p$-norms with $p\in[2, \infty]$. As a result, this test consistent against strictly more alternatives than any test based on a single $p$-norm. In particular, our test is consistent against more alternatives than tests based on the $2$- and $\infty$-norm, which is what most implementations of the power enhancement principle target. We illustrate the scope of our general results by using them to construct a test that simultaneously dominates the Anderson-Rubin test (based on $p=2$) and tests based on the $\infty$-norm in terms of consistency in the linear instrumental variable model with many (weak) instruments.

Differential forms is a highly geometric formalism for physics used from field theories to General Relativity (GR) which has been a great upgrade over vector calculus with the advantages of being coordinate-free and carrying a high degree of geometrical content. In recent years, Geometric Algebra appeared claiming to be a unifying language for physics and mathematics with a high level of geometrical content. Its strength is based on the unification of the inner and outer product into a single geometric operation, and its easy interpretation. Given their similarities, in this article we compare both formalisms side-by-side to narrow the gap between them in literature. We present a direct translation including differential identities, integration theorems and various algebraic identities. As an illustrative example, we present the case of classical electrodynamics in both formalism and finish with their description of GR.

Sintering is a pivotal technology for processing ceramic and metallic powders into solid objects. A profound understanding of microstructure evolution during sintering is essential for manufacturing products with tailored properties. While various phase-field models have been proposed to simulate microstructure evolution in solid-state sintering, correctly incorporating the crucial densification mechanism, particularly grain motion, remains a challenge. The fundamental obstacle lies in the ad hoc treatment of the micromechanics of grain motion, where the thermodynamical driving force cannot be derived from the system's free energy. This work presents a novel phase-field-micromechanics model for sintering (PFMMS) that addresses this challenge. We define a unified energy law, under which the governing equations for microstructure evolution in sintering are derived using variational principles. Our approach ensures thermodynamic consistency, with the driving force for grain motion derived from the system's free energy. Consequently, the proposed PFMMS guarantees the evolution of microstructure in a direction that reduces the system's energy and eliminates non-densifying phenomena. We rigorously validate PFMMS against recent benchmarks of theoretical and numerical analysis. It is found that PFMMS captures intrinsic stress distribution along and beyond grain boundaries, exhibits system-size-independent shrinkage strain, and maintains thermodynamic equilibrium states. These features are fundamental requirements for a physically consistent sintering model.

Recent advances in molecular and genetic research have identified a diverse range of brain tumor sub-types, shedding light on differences in their molecular mechanisms, heterogeneity, and origins. The present study performs whole-brain connectome analysis using diffusionweighted images. To achieve this, both graph theory and persistent homology - a prominent approach in topological data analysis are employed in order to quantify changes in the structural connectivity of the wholebrain connectome in subjects with brain tumors. Probabilistic tractography is used to map the number of streamlines connecting 84 distinct brain regions, as delineated by the Desikan-Killiany atlas from FreeSurfer. These streamline mappings form the connectome matrix, on which persistent homology based analysis and graph theoretical analysis are executed to evaluate the discriminatory power between tumor sub-types that include meningioma and glioma. A detailed statistical analysis is conducted on persistent homology-derived topological features and graphical features to identify the brain regions where differences between study groups are statistically significant (p < 0.05). For classification purpose, graph-based local features are utilized, achieving a highest accuracy of 88%. In classifying tumor sub-types, an accuracy of 80% is attained. The findings obtained from this study underscore the potential of persistent homology and graph theoretical analysis of the whole-brain connectome in detecting alterations in structural connectivity patterns specific to different types of brain tumors.

Developing an effective evaluation metric is crucial for accurately and swiftly measuring LiDAR perception performance. One major issue is the lack of metrics that can simultaneously generate fast and accurate evaluations based on either object detection or point cloud data. In this study, we propose a novel LiDAR perception entropy metric based on the probability of vehicle grid occupancy. This metric reflects the influence of point cloud distribution on vehicle detection performance. Based on this, we also introduce a LiDAR deployment optimization model, which is solved using a differential evolution-based particle swarm optimization algorithm. A comparative experiment demonstrated that the proposed PE-VGOP offers a correlation of more than 0.98 with vehicle detection ground truth in evaluating LiDAR perception performance. Furthermore, compared to the base deployment, field experiments indicate that the proposed optimization model can significantly enhance the perception capabilities of various types of LiDARs, including RS-16, RS-32, and RS-80. Notably, it achieves a 25% increase in detection Recall for the RS-32 LiDAR.

We study symmetry reductions in the context of Euclidean Chern-Simons gauge theories to obtain lower dimensional field theories. Symmetry reduction in certain gauge theories is a common tool for obtaining explicit soliton solutions. Although pure Chern-Simons theories do not admit solitonic solutions, symmetry reduction still leads to interesting results. We establish relations at the semiclassical regime between pure Chern-Simons theories on $S^3$ and the reduced Quantum Field Theories, based on actions obtained by the symmetry reduction of the Chern-Simons action, spherical symmetry being the prominent one. We also discuss symmetry reductions of Chern-Simons theories on the disk, yielding $BF$-theory in two dimensions, which signals a curious relationship between symmetry reductions and the boundary conformal field theories. Finally, we study the Chern-Simons-Higgs instantons and show that under certain circumstances, the reduced action can formally be viewed as the action of a supersymmetric quantum mechanical model. We discuss the extent to which the reduced actions have a fermionic nature at the level of the partition function.

By utilizing recently developed tools for constructing gradient flows on Wasserstein spaces, we extend an analysis technique commonly employed to understand alternating minimization algorithms on Euclidean space to the Expectation Maximization (EM) algorithm via its representation as coordinate-wise minimization on the product of a Euclidean space and a space of probability distributions due to Neal and Hinton (1998). In so doing we obtain finite sample error bounds and exponential convergence of the EM algorithm under a natural generalisation of a log-Sobolev inequality. We further demonstrate that the analysis technique is sufficiently flexible to allow also the analysis of several variants of the EM algorithm.

Empirically-determined scaling laws have been broadly successful in predicting the evolution of large machine learning models with training data and number of parameters. As a consequence, they have been useful for optimizing the allocation of limited resources, most notably compute time. In certain applications, storage space is an important constraint, and data format needs to be chosen carefully as a consequence. Computer vision is a prominent example: images are inherently analog, but are always stored in a digital format using a finite number of bits. Given a dataset of digital images, the number of bits $L$ to store each of them can be further reduced using lossy data compression. This, however, can degrade the quality of the model trained on such images, since each example has lower resolution. In order to capture this trade-off and optimize storage of training data, we propose a `storage scaling law' that describes the joint evolution of test error with sample size and number of bits per image. We prove that this law holds within a stylized model for image compression, and verify it empirically on two computer vision tasks, extracting the relevant parameters. We then show that this law can be used to optimize the lossy compression level. At given storage, models trained on optimally compressed images present a significantly smaller test error with respect to models trained on the original data. Finally, we investigate the potential benefits of randomizing the compression level.

In the first part of this thesis, we make use of representation theory of groups and algebras to perform an irreducible decomposition of tensors in the context of metric-affine gravity. In particular, we consider the action of the orthogonal group O(1, d$-1$) on the Riemann tensor associated with an affine connection defined on a d-dimensional pseudo-Riemannian manifold. This connection, with torsion and non-metricity, is the characteristic ingredient of metric-affine theories of gravity. In the second part of this thesis, we construct the projection operators used for the aforementioned decomposition. They are realized in terms of the symmetric group algebra $\mathbb{C}\mathfrak{S}_n$ and of the Brauer algebra B$_n$(d) which are related respectively to the action of GL(d,$\mathbb{C}$) (and its real form GL(d,$\mathbb{R}$)) and to the action of O(d,$\mathbb{C}$) (and its real form O(1 , d$-1$)) on tensors via the Schur-Weyl duality. First of all, we give an alternative approach to the known formulas for the central idempotents of $\mathbb{C}\mathfrak{S}_n$. These elements provide a unique reducible decomposition, known as the isotypic decomposition. For our purposes, this decomposition is remarkably handy to arrive at the sought after irreducible decomposition with respect to GL(d,$\mathbb{R}$). Then, we construct the elements in B$_n$(d) which realize the isotypic decomposition of a tensor under the action of O(d,$\mathbb{C}$). This decomposition is irreducible under O(d,$\mathbb{C}$) when applied to an irreducible GL(d,$\mathbb{C}$) tensor of order $5$ or less. As a by product of the construction, we give a solution to the problem of decomposing an arbitrary tensor into its traceless part, doubly traceless part and so on. These results led to the development of several Mathematica packages linked to the \textit{xAct} bundle for tensor calculus in field theory.

We establish a number of results concerning conditions for minimum energy dissipation and advective travel time in porous and fractured media. First, we establish a pair of converse results concerning fluid motion along a streamline between two points of fixed head: the minimal advective time is achieved under conditions of constant energy dissipation, and minimal energy dissipation is achieved under conditions of constant velocity along the streamline (implying homogeneous conductivity in the vicinity of the streamline). We also show directly by means of variational methods that minimum advection time along a streamline with a given average conductivity is achieved when the conductivity is constant. Finally, we turn our attention to minimum advection time and energy dissipation in parallel and sequential fracture systems governed by the cubic law: for which fracture cross-section and conductivity are intimately linked. We show that, as in porous domains, flow partitioning between different pathways always acts to minimize system energy dissipation. Finally, we consider minimum advection time as a function of aperture distribution in a sequence of fracture segments. We show that, for a fixed average aperture, a uniform-aperture system displays the shortest advection time. However, we also show that any sufficiently small small perturbations in aperture away from uniformity always act to reduce advection time.

We consider optimal swarm control problems where two different classes of agents are present. Continuum idealizations of large-scale swarms are used where the dynamics describe the evolution of the spatially-distributed densities of each agent class. The problem formulation we adopt is motivated by applications where agents of one class are assigned to agents of the other class, which we refer to as demand and resource agents respectively. Assignments have costs related to the distances between mutually assigned agents, and the overall cost of an assignment is quantified by a Wasserstein distance between the densities of the two agent classes. When agents can move, the assignment cost can decrease at the expense of a physical motion cost, and this tradeoff sets up a nonlinear, infinite-dimensional optimal control problem. We show that in one spatial dimension, this problem can be converted to an infinite-dimensional, but decoupled, linear-quadratic (LQ) tracking problem when expressed in terms of the respective quantile functions. Solutions are given in the general one-dimensional case, as well as in the special cases of constant and periodically time-varying demands.

An observer with a finite lifetime $\mathcal{T}$ perceives the Minkowski vacuum as a thermal state at temperature $T_D = 2 \hbar/(\pi \mathcal{T})$, as a result of being constrained to a double-coned-shaped region known as a causal diamond. In this paper, we explore the emergence of thermality in causal diamonds due to the role played by the symmetries of conformal quantum mechanics (CQM) as a (0+1)-dimensional conformal field theory, within the de Alfaro-Fubini-Furlan model and generalizations. In this context, the hyperbolic operator $S$ of the SO(2,1) symmetry of CQM is the generator of the time evolution of a diamond observer, and its dynamical behavior leads to the predicted thermal nature. Our approach is based on a comprehensive framework of path-integral representations of the CQM generators in canonical and microcanonical forms, supplemented by semiclassical arguments. The properties of the operator $S$ are studied with emphasis on an operator duality with the corresponding elliptic operator $R$, using a representation in terms of an effective scale-invariant inverse square potential combined with inverted and ordinary harmonic oscillator potentials.

Causal diamonds are known to have thermal behavior that can be probed by finite-lifetime observers equipped with energy-scaled detectors. This thermality can be attributed to the time evolution of observers within the causal diamond, governed by one of the conformal quantum mechanics (CQM) symmetry generators: the noncompact hyperbolic operator $S$. In this paper, we show that the unbounded nature of $S$ endows it with a quantum instability, which is a generalization of a similar property exhibited by the inverted harmonic oscillator potential. Our analysis is semiclassical, including a detailed phase-space study of the classical dynamics of $S$ and its dual operator $R$, and a general semiclassical framework yielding basic instability and thermality properties that play a crucial role in the quantum behavior of the theory. For an observer with a finite lifetime $\mathcal{T}$, the detected temperature $T_D = 2 \hbar/(\pi \mathcal{T})$ is associated with a Lyapunov exponent $\lambda_L = \pi T_D/\hbar$, which is half the upper saturation bound of the information scrambling rate.

Geometrically frustrated assemblies where building blocks misfit have been shown to generate intriguing phenomena from self-limited growth, fiber formation, to structural complexity. We introduce a graph theory formulation of geometrically frustrated assemblies, capturing frustrated interactions through the concept of incompatible flows, providing a direct link between structural connectivity and frustration. This theory offers a minimal yet comprehensive framework for the fundamental statistical mechanics of frustrated assemblies. Through numerical simulations, the theory reveals new characteristics of frustrated assemblies, including two distinct percolation transitions for structure and stress, a crossover between cumulative and non-cumulative frustration controlled by disorder, and a divergent length scale in their response.