In the present article, an approach to find the exact solution of the fractional Fokker-Planck equation is presented. It is based on transforming it to a system of first-order partial differential equation via Hopf transformation, together with implementing the extended unified method. On the other hand, reduction of the fractional derivatives to non autonomous ordinary derivative. Thus the fractional Fokker-Planck equation is reduced to non autonomous classical ones. Some explicit solutions of the classical, fractional time derivative Fokker-Planck equation, are obtained . It is shown that the solution of the Fokker-Planck equation is bi-Gaussian's. It is found that high friction coefficient plays a significant role in lowering the standard deviation. Further, it is found the fractionality has stronger effect than fractality. It is worthy to mention that the mixture of Gaussian's is a powerful tool in machine learning. Further, when varying the order of the fractional time derivatives, results to slight effects in the probability distribution function. Also, it is shown that the mean and mean square of the velocity vary slowly.

In this short note, we given a new proof of Mitchell's theorem that $L_{T\left(n\right)} K(Z) \cong 0$ for $n > 2$. Instead of reducing the problem to delicate representation theory, we use recently established hyperdescent technology for chromatically-localized algebraic K-theory.

Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that are missing a finite number of "exceptional" degrees. In this note we sketch the construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explicit description of the operators and polynomials in question.

In this paper, we study interval partition diffusions with Poisson--Dirichlet$(\alpha,\theta)$ stationary distribution for parameters $\alpha\in(0,1)$ and $\theta\ge 0$. This extends previous work on the cases $(\alpha,0)$ and $(\alpha,\alpha)$ and builds on our recent work on measure-valued diffusions. We work on spaces of interval partitions with $\alpha$-diversity. These processes can be viewed as diffusions on the boundary of a branching graph of integer compositions. The additional order and diversity structure of such interval partitions is essential for applications to continuum random tree models such as stable CRTs and limit structures of other regenerative tree growth processes, where intervals correspond to masses of spinal subtrees (or spinal bushes) in spinal order and diversities give distances between any two spinal branch points. We further show that our processes can be extended to enter continuously from the Hausdorff completion of our state space.

This article describes the efforts of the SFSU-Colombia Combinatorics Initiative to build a research and learning community between California and Colombia. It seeks to broaden and deepen representation in mathematics, based on four underlying principles: 1. Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries. 2. Everyone can have joyful, meaningful, and empowering mathematical experiences. 3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs. 4. Every student deserves to be treated with dignity and respect.

Everyone can have joyful, meaningful, and empowering academic experiences; but no single academic experience is joyful, meaningful, and empowering to everyone. How do we build academic spaces where every participant can thrive? Audre Lorde advises us to use our differences to our advantage. bell hooks highlights the key role of building community while addressing power dynamics. Rochelle Guti\'errez emphasizes the importance of welcoming students' full humanity. This note discusses some efforts to implement these ideas in a university classroom, focusing on the first day of class.

This article concerns on the existence of multiple solutions for a new Kirchhoff-type problem with negative modulus. We prove that there exist three nontrivial solutions when the parameter is enough small via the variational methods and algebraic analysis. Moreover, our fundamental technique is one of the Mountain Pass Lemma, Ekeland variational principle, and Minimax principle.

In this study, we provide error estimates and stability analysis of deep learning techniques for certain partial differential equations including the incompressible Navier-Stokes equations. In particular, we obtain explicit error estimates (in suitable norms) for the solution computed by optimizing a loss function in a Deep Neural Network (DNN) approximation of the solution, with a fixed complexity.

We consider when the symmetric algebra of an infinite-dimensional Lie algebra, equipped with the natural Poisson bracket, satisfies the ascending chain condition (ACC) on Poisson ideals. We define a combinatorial condition on a graded Lie algebra which we call Dicksonian because it is related to Dickson's lemma on finite subsets of $\mathbb N^k$. Our main result is: Theorem. If $\mathfrak g$ is a Dicksonian graded Lie algebra over a field of characteristic zero, then the symmetric algebra $S(\mathfrak g)$ satisfies the ACC on radical Poisson ideals. As an application, we establish this ACC for the symmetric algebra of any graded simple Lie algebra of polynomial growth, and for the symmetric algebra of the Virasoro algebra. We also derive some consequences connected to the Poisson primitive spectrum of finitely Poisson-generated algebras.

The paper explores further the computation of the quaternionic numerical range of a complex matrix. We prove a modified version of a conjecture by So and Tompson. Specifically, we show that the shape of the quaternionic numerical range for a complex matrix depends on the complex numerical range and two real values. We establish under which conditions the bild of a complex matrix coincides with its complex numerical range and when the quaternionic numerical range is convex.

This paper considers the multi-agent linear least-squares problem in a server-agent network. In this problem, the system comprises multiple agents, each having a set of local data points, that are connected to a server. The goal for the agents is to compute a linear mathematical model that optimally fits the collective data points held by all the agents, without sharing their individual local data points. This goal can be achieved, in principle, using the server-agent variant of the traditional iterative gradient-descent method. The gradient-descent method converges linearly to a solution, and its rate of convergence is lower bounded by the conditioning of the agents' collective data points. If the data points are ill-conditioned, the gradient-descent method may require a large number of iterations to converge. We propose an iterative pre-conditioning technique that mitigates the deleterious effect of the conditioning of data points on the rate of convergence of the gradient-descent method. We rigorously show that the resulting pre-conditioned gradient-descent method, with the proposed iterative pre-conditioning, achieves superlinear convergence when the least-squares problem has a unique solution. In general, the convergence is linear with improved rate of convergence in comparison to the traditional gradient-descent method and the state-of-the-art accelerated gradient-descent methods. We further illustrate the improved rate of convergence of our proposed algorithm through experiments on different real-world least-squares problems in both noise-free and noisy computation environment.

Given groups $A$ and $B$, what is the minimal commutator length of the 2020th (for instance) power of an element $g\in A*B$ not conjugate to elements of the free factors? The exhaustive answer to this question is still unknown, but we can give an almost answer: this minimum is one of two numbers (simply depending on $A$ and $B$). Other similar problems are also considered.

In this paper we develop a numerical scheme based on quadratures to approximate solutions of integro-differential equations involving convolution kernels, $\nu$, of diffusive type. In particular, we assume $\nu$ is symmetric and exponentially decaying at infinity. We consider problems posed in bounded domains and in $\R$. In the case of bounded domains with nonlocal Dirichlet boundary conditions, we show the convergence of the scheme for kernels that have positive tails, but that can take on negative values. When the equations are posed on all of $\R$, we show that our scheme converges for nonnegative kernels. Since nonlocal Neumann boundary conditions lead to an equivalent formulation as in the unbounded case, we show that these last results also apply to the Neumann problem.

In this paper, we consider numerical simulations of the nonlocal optical response of metallic nanostructure arrays inside a dielectric host, which is of particular interest to the nanoplasmonics community due to many unusual properties and potential applications. Mathematically, it is described by Maxwell's equations with discontinuous coefficients coupled with a set of Helmholtz-type equations defined only on the domains of metallic nanostructures. To solve this challenging problem, we develop an efficient multiscale method consisting of three steps. First, we extend the system into the domain occupied by the dielectric medium in a novel way and result in a coupled system with rapidly oscillating coefficients. A rigorous analysis of the error between the solutions of the original system and the extended system is given. Second, we derive the homogenized system and define the multiscale approximate solution for the extended system by using the multiscale asymptotic method. Third, to fix the inaccuracy of the multiscale asymptotic method inside the metallic nanostructures, we solve the original system in each metallic nanostructure separately with boundary conditions given by the multiscale approximate solution. A fast algorithm based on the $LU$ decomposition is proposed for solving the resulting linear systems. By applying the multiscale method, we obtain the results that are in good agreement with those obtained by solving the original system directly at a much lower computational cost. Numerical examples are provided to validate the efficiency and accuracy of the proposed method.

Underwater wireless optical communication is one of the critical technologies for buoy-based high-speed cross-sea surface communication, where the communication nodes are vertically deployed. Due to the vertically inhomogeneous nature of the underwater environment, seawater is usually vertically divided into multiple layers with different parameters that reflect the real environment. In this work, we consider a generalized UWOC channel model that contains$N$ layers. To capture the effects of air bubbles and temperature gradients on channel statistics, we model each layer by a mixture Exponential-Generalized Gamma(EGG) distribution. We derive the PDF and CDF of the end-to-end SNR in exact closed-form. Then, unified BER and outage expressions using OOK and BPSK are also derived. The performance and behavior of common vertical underwater optical communication scenarios are thoroughly analyzed through the appropriate selection of parameters. All the derived expressions are verified via Monte Carlo simulations.

We generalise the homogenisation theorem in \cite{Gehringer-Li-homo,Gehringer-Li-tagged} for passive tracer in a fractional Gaussian field to fractional non-Gaussian fields. We also obtain the limit theorems of normalized functionals of Hermite-Volterra processes, extending the result in \cite{Diu-Tran} to power series with fast decaying coefficients. We obtain either convergence to a Wiener process, in the short-range dependent case, or to a Hermite process, in the long-range dependent case. Furthermore, we prove convergence in the multivariate case with both, short and long-range dependent components and give an application to homogenization of fast/slow systems.

The Encuentro Colombiano de Combinatoria (ECCO) is an international summer school that welcomes students and researchers with a wide variety of mathematical and personal experiences. ECCO has taught us a lot about what it might mean to truly find community and belonging in a mathematical space. The goal of this article is to share a few of the lessons that we have learned from helping to build it.

In this paper a double quasi Poisson bracket in the sense of Van den Bergh is constructed on the space of noncommutative weights of arcs of a directed graph embedded in a disk or cylinder $\Sigma$, which gives rise to the quasi Poisson bracket of G.Massuyeau and V.Turaev on the group algebra $\mathbf k\pi_1(\Sigma,p)$ of the fundamental group of a surface based at $p\in\partial\Sigma$. This bracket also induces a noncommutative Goldman Poisson bracket on the cyclic space $\mathcal C_\natural$, which is a $\mathbf k$-linear space of unbased loops. We show that the induced double quasi Poisson bracket between boundary measurements can be described via noncommutative $r$-matrix formalism. This gives a more conceptual proof of the result of N. Ovenhouse that traces of powers of Lax matrix form an infinite collection of noncommutative Hamiltonians in involution with respect to noncommutative Goldman bracket on $\mathcal C_\natural$.

The energy-conserving sampling and weighting (ECSW) method is a hyperreduction method originally developed for accelerating the performance of Galerkin projection-based reduced-order models (PROMs) associated with large-scale finite element models, when the underlying projected operators need to be frequently recomputed as in parametric and/or nonlinear problems. In this paper, this hyperreduction method is extended to Petrov-Galerkin PROMs where the underlying high-dimensional models can be associated with arbitrary finite element, finite volume, and finite difference semi-discretization methods. Its scope is also extended to cover local PROMs based on piecewise-affine approximation subspaces, such as those designed for mitigating the Kolmogorov $n$-width barrier issue associated with convection-dominated flow problems. The resulting ECSW method is shown in this paper to be robust and accurate. In particular, its offline phase is shown to be fast and parallelizable, and the potential of its online phase for large-scale applications of industrial relevance is demonstrated for turbulent flow problems with $O(10^7)$ and $O(10^8)$ degrees of freedom. For such problems, the online part of the ECSW method proposed in this paper for Petrov-Galerkin PROMs is shown to enable wall-clock time and CPU time speedup factors of several orders of magnitude while delivering exceptional accuracy.

We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman's entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on the sharp logarithmic Sobolev inequality of gradient shrinking Ricci solitons and a Vitali-type covering argument.

In this paper, we consider the set of all domino tilings of a cubiculated region. The primary question we explore is: How can we move from one tiling to another? Tiling spaces can be viewed as spaces of subgraphs of a fixed graph with a fixed degree sequence. Moves to connect such spaces have been explored in algebraic statistics. Thus, we approach this question from an applied algebra viewpoint, making new connections between domino tilings, algebraic statistics, and toric algebra. Using results from toric ideals of graphs, we are able to describe moves that connect the tiling space of a given cubiculated region of any dimension. This is done by studying binomials that arise from two distinct domino tilings of the same region. Additionally, we introduce tiling ideals and flip ideals and use these ideals to restate what it means for a tiling space to be flip connected. Finally, we show that if $R$ is a $2$-dimensional simply connected cubiculated region, any binomial arising from two distinct tilings of $R$ can be written in terms of quadratic binomials. As a corollary to our main result, we obtain an alternative proof to the fact that the set of domino tilings of a $2$-dimensional simply connected region is connected by flips.

In this paper, we talk about parahoric Hitchin systems over smooth projective curves with structure group a semisimple simply connected group. We describe the geometry of generic fibers of parahoric Hitchin fibrations using root stacks. We work over an algebraically closed field with a mild assumption of the characteristic. All of these can be treated as a generalization of GLn case in [SWW19]

The Mat{\'e}rn family of covariance functions has played a central role in spatial statistics for decades, being a flexible parametric class with one parameter determining the smoothness of the paths of the underlying spatial field. This paper proposes a new family of spatial covariance functions, which stems from a reparameterization of the generalized Wendland family. As for the Mat{\'e}rn case, the new class allows for a continuous parameterization of the smoothness of the underlying Gaussian random field, being additionally compactly supported. More importantly, we show that the proposed covariance family generalizes the Mat{\'e}rn model which is attained as a special limit case. The practical implication of our theoretical results questions the effective flexibility of the Mat{\'e}rn covariance from modeling and computational viewpoints. Our numerical experiments elucidate the speed of convergence of the proposed model to the Mat{\'e}rn model. We also inspect the level of sparseness of the associated (inverse) covariance matrix and the asymptotic distribution of the maximum likelihood estimator under increasing and fixed domain asymptotics. The effectiveness of our proposal is illustrated by analyzing a georeferenced dataset on maximum temperatures over the southeastern United States, and performing a re-analysis of a large spatial point referenced dataset of yearly total precipitation anomalies

Unbalanced optimal mass transport (OMT) seeks to remove the conservation of mass constraint by adding a source term to the standard continuity equation in the Benamou-Brenier formulation of OMT. In this note, we show how the addition of the source fits into the vector-valued OMT framework.

We give a definition of quaternion Lie algebra and of the quaternification of a complex or a real Lie algebra. so*(2n), sp(n) and sl(n,H) become quaternion Lie algebras. Then we shall prove that a simple Lie algebra has the quaternification. For the proof we follow the well known argument due to Harich-Chandra, Chevalley and Serre to construct the simple Lie algebra from its corresponding root system. The root space decomposition of this quaternion Lie algebra will be given. Each root sapce of a fundamental root is 2-dimensional. For example the quaternion special linear algebra sl(n,H) is the quaternification of the complex special Lie algebra sl(n,C).

We present some results about the burgeoning research area concerning set theory of the kappa-reals. We focus on some notions of measurability coming from generalizations of Silver and Miller trees. We present analogies and mostly differences from the classical setting.

In this paper, we classify relatively minimal genus-$1$ holomorphic Lefschetz pencils up to smooth isomorphism. We first show that such a pencil is isomorphic to either the pencil on $\mathbb{P}^1\times \mathbb{P}^1$ of bi-degree $(2,2)$ or a blow-up of the pencil on $\mathbb{P}^2$ of degree $3$, provided that no fiber of a pencil contains an embedded sphere. (Note that one can easily classify genus-$1$ Lefschetz pencils with an embedded sphere in a fiber.) We further determine the monodromy factorizations of these pencils and show that the isomorphism class of a blow-up of the pencil on $\mathbb{P}^2$ of degree $3$ does not depend on the choice of blown-up base points. We also show that the genus-$1$ Lefschetz pencils constructed by Korkmaz-Ozbagci (with nine base points) and Tanaka (with eight base points) are respectively isomorphic to the pencils on $\mathbb{P}^2$ and $\mathbb{P}^1\times \mathbb{P}^1$ above, in particular these are both holomorphic.

We show presentations of automata groups generated by Cayley machines of finite groups of nilpotency class two and these automata groups are all cross-wired lamplighters.

Transfer operators such as Perron-Frobenius or Koopman operator play a key role in modeling and analysis of complex dynamical systems, which allow linear representations of nonlinear dynamics by transforming the original state variables to feature spaces. However, it remains challenging to identify the optimal low-dimensional feature mappings from data. The variational approach for Markov processes (VAMP) provides a comprehensive framework for the evaluation and optimization of feature mappings based on the variational estimation of modeling errors, but it still suffers from a flawed assumption on the transfer operator and therefore sometimes fails to capture the essential structure of system dynamics. In this paper, we develop a powerful alternative to VAMP, called kernel embedding based variational approach for dynamical systems (KVAD). By using the distance measure of functions in the kernel embedding space, KVAD effectively overcomes the theoretical and practical limitations of VAMP. In addition, we develop a data-driven KVAD algorithm for seeking the ideal feature mapping within a subspace spanned by given basis functions, and numerical experiments show that the proposed algorithm can significantly improve the modeling accuracy compared to VAMP.

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number $f$ and analyse their genus. We find the asymptotic distribution of genus in this set of numerical semigroups and show that it is a product of a Gaussian and a power series. We show that almost all numerical semigroups with Frobenius number $f$ have genus close to $\frac{3f}{4}$. We denote the number of numerical semigroups of Frobenius number $f$ by $N(f)$. While $N(f)$ is not monotonic we prove that $N(f)<N(f+2)$ for every $f$.

We study equivariant Iwasawa theory for two-variable abelian extensions of an imaginary quadratic field. One of the main goals of this paper is to describe the Fitting ideals of Iwasawa modules using $p$-adic $L$-functions. We also provide an application to Selmer groups of elliptic curves with complex multiplication.

We demonstrate the large deviation principle in the small noise limit for the three dimensional stochastic planetary geostrophic equations of large-scale ocean circulation. In this paper, we first prove the well-posedness of weak solutions to this system by the method of monotonicity. As we know, a recently developed method, weak convergent method, has been employed in studying the large deviations and this method is essentially based on the main result of \cite{ba2} which discloses the variational representation of exponential integrals with respect to the Brownian noise. The It\^{o} inequality and Burkholder-Davis-Gundy inequality are the main tools in our proofs, and the weak convergence method introduced by Budhiraja, Dupuis and Ganguly in \cite{ba3} is also used to establish the large deviation principle.

In this paper, we study the domination number of middle graphs. Indeed, we obtain tight bounds for this number in terms of the order of the graph. We also compute the domination number of some families of graphs such as star graphs, double start graphs, path graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs and friendship graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the domination number of middle graphs.

Information theoretic secret key agreement is impossible without making initial assumptions. One type of initial assumption is correlated random variables that are generated by using a noisy channel that connects the terminals. Terminals use the correlated random variables and communication over a reliable public channel to arrive at a shared secret key. Previous channel models assume that each terminal either controls one input to the channel, or receives one output variable of the channel. In this paper, we propose a new channel model of transceivers where each terminal simultaneously controls an input variable and observes an output variable of the (noisy) channel. We give upper and lower bounds for the secret key capacity (i.e., highest achievable key rate) of this transceiver model, and prove the secret key capacity under the conditions that the public communication is noninteractive and input variables of the noisy channel are independent.

The present paper, characterizes the invertibility and causality conditions of a periodic ARFIMA (PARFIMA) models. We first, discuss the conditions in the multivariate case, by considering the corresponding p-variate stationary ARFIMA models. Second, we construct the conditions using the univariate case and we deduce a new infinite autoregressive representation for the PARFIMA model, the results are investigated through a simulation study.

The existence and uniqueness of formal Puiseux series solutions of non-autonomous algebraic differential equations of the first order at a nonsingular point of the equation is proven. The convergence of those Puiseux series is established. Several new examples are provided. Relationships to the celebrated Painleve theorem and lesser-known Petrovic's results are discussed in detail.

We study emergent dynamics of the Lohe hermitian sphere(LHS) model which can be derived from the Lohe tensor model \cite{H-P2} as a complex counterpart of the Lohe sphere(LS) model. The Lohe hermitian sphere model describes aggregate dynamics of point particles on the hermitian sphere $\bbh\bbs^d$ lying in ${\mathbb C}^{d+1}$, and the coupling terms in the LHS model consist of two coupling terms. For identical ensemble with the same free flow dynamics, we provide a sufficient framework leading to the complete aggregation in which all point particles form a giant one-point cluster asymptotically. In contrast, for non-identical ensemble, we also provide a sufficient framework for the practical aggregation. Our sufficient framework is formulated in terms of coupling strengths and initial data. We also provide several numerical examples and compare them with our analytical results.

This paper investigates a full-duplex orthogonal-frequency-division multiple access (OFDMA) based multiple unmanned aerial vehicles (UAVs)-enabled wireless-powered Internet-of-Things (IoT) networks. In this paper, a swarm of UAVs is first deployed in three dimensions (3D) to simultaneously charge all devices, i.e., a downlink (DL) charging period, and then flies to new locations within this area to collect information from scheduled devices in several epochs via OFDMA due to potential limited number of channels available in Narrow Band IoT, i.e., an uplink (UL) communication period. To maximize the UL throughput of IoT devices, we jointly optimizes the UL-and-DL 3D deployment of the UAV swarm, including the device-UAV association, the scheduling order, and the UL-DL time allocation. In particular, the DL energy harvesting (EH) threshold of devices and the UL signal decoding threshold of UAVs are taken into consideration when studying the problem. Besides, both line-of-sight (LoS) and non-line-of-sight (NLoS) channel models are studied depending on the position of sensors and UAVs. The influence of the potential limited channels issue in NB-IoT is also considered by studying the IoT scheduling policy. Two scheduling policies, a near-first (NF) policy and a far-first (FF) policy, are studied. It is shown that the NF scheme outperforms FF scheme in terms of sum throughput maximization; whereas FF scheme outperforms NF scheme in terms of system fairness.

We formulate a few conjectures on some hypothetical coherent sheaves on the stacks of arithmetic local Langlands parameters, including their roles played in the local-global compatibility in the Langlands program. We survey some known results as evidences of these conjectures.

The rate of the weak convergence in the fractional step method for the Arratia flow is established in terms of the Wasserstein distance between the images of the Lebesque measure under the action of the flow. We introduce finite-dimensional densities describing sequences of collisions in the Arratia flow and derive an explicit expression for them. With the initial interval discretized, the convergence of the corresponding approximations of the point measure associated with the Arratia flow is discussed in terms of such densities.

Multimarginal Optimal Transport (MOT) has recently attracted significant interest due to its many applications. However, in most applications, the success of MOT is severely hindered by a lack of sub-exponential time algorithms. This paper develops a general theory about "structural properties" that make MOT tractable. We identify two such properties: decomposability of the cost into either (i) local interactions and simple global interactions; or (ii) low-rank interactions and sparse interactions. We also provide strong evidence that (iii) repulsive costs make MOT intractable by showing that several such problems of interest are NP-hard to solve--even approximately. These three structures are quite general, and collectively they encompass many (if not most) current MOT applications. We demonstrate our results on a variety of applications in machine learning, statistics, physics, and computational geometry.

With a view to prove an Ohsawa-Takegoshi type $L^2$ extension theorem with $L^2$ estimates given with respect to the log-canonical (lc) measures, a sequence of measures each supported on lc centres of specific codimension defined via multiplier ideal sheaves, this article is aiming at providing evidence and possible means to prove the $L^2$ estimates on compact K\"ahler manifolds $X$. A holomorphic family of $L^2$ norms on the ambient space $X$ is introduced which is shown to "deform holomorphically" to an $L^2$ norm with respect to an lc-measure. Moreover, the latter norm is shown to be invariant under a certain normalisation which leads to a "non-universal" $L^2$ estimate on compact $X$. Explicit examples on $\mathbb{P}^3$ with detailed computation are presented to verify the expected $L^2$ estimates for extensions from lc centres of various codimensions and to provide hint for the proof of the estimates in general.

We revisit the classical singular control problem of minimizing running and controlling costs. The problem arises in inventory control, as well as in healthcare management and mathematical finance. Existing studies have shown the optimality of a barrier strategy when driven by the Brownian motion or L\'evy processes with one-side jumps. Under the assumption that the running cost function is convex, we show the optimality of a barrier strategy for a general class of L\'evy processes. Numerical results are also given.

In this paper, we compute explicitly the $q$-dimensions of highest weight crystals modulo $q^n-1$ for a quantum group of arbitrary finite type under certain assumption, and interpret the modulo computations in terms of the cyclic sieving phenomenon. This interpretation gives an affirmative answer to the conjecture by Alexandersson and Amini. As an application, under the assumption that $\lambda$ is a partition of length $<m$ and there exists a fixed point in $\mathsf{SST}_m(\lambda)$ under the action $\mathsf{c}$ arising from the crystal structure, we show that the triple $(\mathsf{SST}_m(\lambda), \langle \mathsf{c} \rangle, \mathsf{s}_{\lambda}(1,q,q^2, \ldots, q^{m-1}))$ exhibits the cycle sieving phenomenon if and only if $\lambda$ is of the form $((am)^{b})$, where either $b=1$ or $m-1$. Moreover, in this case, we give an explicit formula to compute the number of all orbits of size $d$ for each divisor $d$ of $n$.

We prove that a graph G embeds r-locally planarly in a pseudo-surface if and only if a certain matroid associated to the graph G is co-graphic. This extends Whitney's abstract planar duality theorem from 1932.

We prove that a graph has an r-bounded subdivision of a wheel if and only if it does not have a graph-decomposition of locality r and width at most two.

How can sparse graph theory be extended to large networks, where algorithms whose running time is estimated using the number of vertices are not good enough? I address this question by introducing 'Local Separators' of graphs. Applications include: 1. A unique decomposition theorem for graphs along their local 2-separators analogous to the 2-separator theorem; 2. an exact characterisation of graphs with no bounded subdivision of a wheel; 3. an analogue of the tangle-tree theorem of Robertson and Seymour, where the decomposition-tree is replaced by a general graph.

In this work we consider the defocusing nonlinear wave equation in one-dimensional space. We show that almost all energy is located near the light cone $|x|=|t|$ as time tends to infinity. We also prove that any light cone will eventually contain some energy. As an application we obtain a result about the asymptotic behaviour of solutions to focusing one-dimensional wave equation with compact-supported initial data.

We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset $\Omega\subset\mathbb{R}^N$ and a Banach space $V$, we compare the classical Sobolev space $W^{1,p}(\Omega, V)$ with the so-called Sobolev-Reshetnyak space $R^{1,p}(\Omega, V)$. We see that, in general, $W^{1,p}(\Omega, V)$ is a closed subspace of $R^{1,p}(\Omega, V)$. As a main result, we obtain that $W^{1,p}(\Omega, V)=R^{1,p}(\Omega, V)$ if, and only if, the Banach space $V$ has the Radon-Nikod\'ym property

We study the problem on how to get good lower estimates for the integral $$ \int_T^{T+H} |\zeta(\sigma+it)| dt, $$ when $H \ll 1$ is small and $\sigma$ is close to $1$, as well as related integrals for other Dirichlet series, by using ideas related to the Balasubramanian-Ramachandra method. We use kernel-functions constructed by the Paley-Wiener theorem as well as the kernel function of Ramachandra. We also notice that the Fourier transform of Ramachandra's Kernel-function is in fact a $K$-Bessel function. This simplifies some aspects of Balasubramanian-Ramachandra method since it allows use of the theory of Bessel-functions.

In $L_2({\mathbb R}^3;{\mathbb C}^3)$, we consider a selfadjoint operator ${\mathcal L}_\varepsilon$, $\varepsilon >0$, given by the differential expression $\mu_0^{-1/2}\operatorname{curl} \eta(\mathbf{x}/\varepsilon)^{-1} \operatorname{curl} \mu_0^{-1/2} - \mu_0^{1/2}\nabla \nu(\mathbf{x}/\varepsilon) \operatorname{div} \mu_0^{1/2}$, where $\mu_0$ is a constant positive matrix, a matrix-valued function $\eta(\mathbf{x})$ and a real-valued function $\nu(\mathbf{x})$ are periodic with respect to some lattice, positive definite and bounded. We study the behavior of the operator-valued functions $\cos (\tau {\mathcal L}_\varepsilon^{1/2})$ and ${\mathcal L}_\varepsilon^{-1/2} \sin (\tau {\mathcal L}_\varepsilon^{1/2})$ for $\tau \in {\mathbb R}$ and small $\varepsilon$. It is shown that these operators converge to the corresponding operator-valued functions of the operator ${\mathcal L}^0$ in the norm of operators acting from the Sobolev space $H^s$ (with a suitable $s$) to $L_2$. Here ${\mathcal L}^0$ is the effective operator with constant coefficients. Also, an approximation with corrector in the $(H^s \to H^1)$-norm for the operator ${\mathcal L}_\varepsilon^{-1/2} \sin (\tau {\mathcal L}_\varepsilon^{1/2})$ is obtained. We prove error estimates and study the sharpness of the results regarding the type of the operator norm and regarding the dependence of the estimates on $\tau$. The results are applied to homogenization of the Cauchy problem for the nonstationary Maxwell system in the case where the magnetic permeability is equal to $\mu_0$, and the dielectric permittivity is given by the matrix $\eta(\mathbf{x}/\varepsilon)$.

In this article, the author improves Baues's homotopy classification of maps between indecomposable $(n-1)$-connected $(n+2)$ dimensional finite CW-complexes $X,Y,n>3$, by finding a generating set for any abelian group $[X,Y]$. Using these generators, the author firstly finds splitting cofiber sequences which imply Zhu-Pan's decomposability result of smash products of the complexes, and secondly, obtains partial results on the groups of homotopy classes of self-homotopy equivalences of the complexes and some of their natural subgroups.

We define a two-parameter family of Gaussian Markov processes, which includes Brownian motion as a special case. Our main result is that any centered self-similar Gaussian Markov process is a constant multiple of a process from this family. This yields short and easy proofs of some non-Markovianity results concerning variants of fractional Brownian motion (most of which are known). In the proof of our main theorem, we use some properties of additive functions, i.e. solutions of Cauchy's functional equation. In an appendix, we show that a certain self-similar Gaussian process with asymptotically stationary increments is not a semimartingale.

In this paper, time-independent Hamiltonian systems are investigated via a Lie-group/algebra formalism. The (unknown) solution linked with the Hamiltonian is considered to be a Lie-group transformation of the initial data, where the group parameter acts as the time. The time-evolution generator (i.e. the Lie algebra associated to the group transformation) is constructed at an algebraic level, hence avoiding discretization of the time-derivatives for the discrete case. This formalism makes it possible to investigate the continuous and discrete versions of time for time-independent Hamiltonian systems and no additional information on the system is required (besides the Hamiltonian itself and the initial conditions of the solution). When the time-independent Hamiltonian system is integrable in the sense of Liouville, one can use the action-angle coordinates to straighten the time-evolution generator and construct an exact scheme (i.e. a scheme without errors). In addition, a method to analyse the errors of approximative/numerical schemes is provided. These considerations are applied to well-known examples associated with the one-dimensional harmonic oscillator.

We study a class of nonlinear elliptic problems with Dirichlet conditions in the framework of the Sobolev anisotropic spaces with variable exponent, involving an anisotropic operator on an unbounded domain $\Omega\subset \>I\!\!R^{N}\>(N \geq 2)\>$. We prove the existence of entropy solutions avoiding sign condition and coercivity on the lowers order terms.

Robustness analysis is an emerging field in the domain of uncertainty quantification. It consists of analysing the response of a computer model with uncertain inputs to the perturbation of one or several of its input distributions. Thus, a practical robustness analysis methodology should rely on a coherent definition of a distribution perturbation. This paper addresses this issue by exposing a rigorous way of perturbing densities. The proposed methodology is based the Fisher distance on manifolds of probability distributions. A numerical method to calculate perturbed densities in practice is presented. This method comes from Lagrangian mechanics and consists of solving an ordinary differential equations system. This perturbation definition is then used to compute quantile-oriented robustness indices. The resulting Perturbed-Law based sensitivity Indices (PLI) are illustrated on several numerical models. This methodology is also applied to an industrial study (simulation of a loss of coolant accident in a nuclear reactor), where several tens of the model physical parameters are uncertain with limited knowledge concerning their distributions.

It is found that $15$ different types of two-qubit $X$-states split naturally into two sets (of cardinality $9$ and $6$) once their entanglement properties are taken into account. We {characterize both the validity and entangled nature of the $X$-states with maximally-mixed subsystems in terms of certain parameters} and show that their properties are related to a special class of geometric hyperplanes of the symplectic polar space of order two and rank two. Finally, we introduce the concept of hyperplane-states and briefly address their non-local properties.

We define, for any special matching of a finite graded poset, an idempotent, regressive and order preserving function. We consider the monoid generated by such functions. The idempotents of this monoid are called special idempotents. They are interval retracts. Some of them realize a kind of parabolic map and are called special projections. We prove that, in Eulerian posets, the image of a special projection, and its complement, are graded induced subposets. In a finite Coxeter group, all projections on right and left parabolic quotients are special projections, and some projections on double quotients too. We extend our results to special partial matchings.

Let $\mathbb{D}$ be the unit disk and $\varphi\in L^p(\mathbb{D}, \mathrm{d}A)$, where $1\leq p\leq\infty$. For $z\in\mathbb{D}$, the Cauchy-transform on $\mathbb{D}$, denote by $\mathcal{P}$, is defined as follows: $$\mathcal{P}[\varphi](z)=-\int_{\mathbb{D}}\left(\frac{\varphi(w)}{w-z}+\frac{z\overline{\varphi(w)}}{1-\bar{w}z}\right)\mathrm{d}A(w).$$ The Beurling transform on $\mathbb{D}$, denote by $\mathcal{H}$, is now defined as the $z$-derivative of $\mathcal{P}$. In this paper, by using Hardy's type inequalities and Bessel functions, we show that $\|\mathcal{P}\|_{L^2\to L^2}=\alpha\approx1.086$, where $\alpha$ is a solution to the equation: $2J_0(2/\alpha)-\alpha J_1(2/\alpha)=0$, and $J_0$, $J_1$ are Bessel functions. Moreover, for $p>2$, by using Taylor expansion, Parseval's formula and hypergeometric functions, we also prove that $\|\mathcal{P}\|_{L^p\to L^{\infty}}=2(\Gamma(2-q)/\Gamma^2(2-\frac{q}{2}))^{1/q}$, where $q=p/(p-1)$ is the conjugate exponent of $p$, and $\Gamma$ is the Gamma function. Finally, applying the same techniques developed in this paper, we show that the Beurling transform $\mathcal{H}$ acts as an isometry of $L^2(\mathbb{D}, \mathrm{d}A)$.

We study the triangular array defined by the Graham--Knuth--Patashnik recurrence $T(n,k) = (\alpha n + \beta k + \gamma)\, T(n-1,k)+(\alpha' n + \beta' k + \gamma') \, T(n-1,k-1)$ with initial condition $T(0,k) = \delta_{k0}$ and parameters $\mathbf{\mu} = (\alpha,\beta,\gamma, \alpha',\beta',\gamma')$. We show that the family of arrays $T(\mathbf{\mu})$ is invariant under a 48-element discrete group isomorphic to $S_3 \times D_4$. Our main result is to determine all parameter sets $\mathbf{\mu} \in \mathbb{C}^6$ for which the ordinary generating function $f(x,t) = \sum_{n,k=0}^\infty T(n,k) \, x^k t^n$ is given by a Stieltjes-type continued fraction in $t$ with coefficients that are polynomials in $x$. We also exhibit some special cases in which $f(x,t)$ is given by a Thron-type or Jacobi-type continued fraction in $t$ with coefficients that are polynomials in $x$.

We prove that in dimension $d\le 3$ a modified density field of a stirring dynamics perturbed by a voter model converges to the stochastic heat equation.

The goal of this note is to establish non-tangential convergence results for Schr\"{o}dinger operators along restricted curves. We consider the relationship between the dimension of this kind of approach region and the regularity for the initial data which implies convergence. As a consequence, we obtain a upper bound for $p$ such that the Schr\"{o}dinger maximal function is bounded from $H^{s}(\mathbb{R}^{n})$ to $L^{p}(\mathbb{R}^{n})$ for any $s > \frac{n}{2(n+1)}$.

Let $p_{-r}(n)$ denote the $r$-coloured partition function, and $\sigma(n)=\sum_{d|n}d$ denote the sum of positive divisors of $n$. The aim of this note is to prove the following $$ p_{-r}(n)=\theta(n)+\,\sum_{k=1}^{n-1}\frac{r^{k+1}}{(k+1)!} \sum_{\alpha_1\,= k}^{n-1} \, \sum_{\alpha_2\,= k-1}^{\alpha_1-1} \cdots \sum_{\alpha_k\, = 1}^{\alpha_{k-1}-1}\theta(n-\alpha_1) \theta(\alpha_1 -\alpha_2) \cdots \theta(\alpha_{k-1}-\alpha_k) \theta(\alpha_k) $$ where $\theta(n)=n^{-1}\, \sigma(n)$, and its inverse $$\sigma(n) = n\,\sum_{r=1}^n \frac{(-1)^{r-1}}{r}\, \binom{n}{r}\, p_{-r}(n). $$

In this paper, a very accurate approximation method for the statistics of the sum of M\'{a}laga-$\mathcal{M}$ random variates with pointing error (MRVs) is proposed. In particular, the probability density function of MRV is approximated by a Fox's $H$-function through the moment-based approach. Then, the respective moment-generating function of the sum of $N$ MRVs is provided, based on which the average symbol error rate is evaluated for an $N $-branch maximal-ratio combining (MRC) receiver. The retrieved results show that the proposed approximate results match accurately with the exact simulated ones. Additionally, the results show that the achievable diversity order increases as a function of the number of MRC diversity branches.

In this paper we consider double covers of the projective space in relation with the problem of extensions of varieties, specifically of extensions of canonical curves to $K3$ surfaces and Fano 3-folds. In particular we consider $K3$ surfaces which are double covers of the plane branched over a general sextic: we prove that the general curve in the linear system pull back of plane curves of degree $k\geq 7$ lies on a unique $K3$ surface. If $k\leq 6$ the general such curve is instead extendable to a higher dimensional variety. In the cases $k=4,5,6$, this gives the existence of singular index $k$ Fano varieties of dimensions 8, 5, 3, and genera 17, 26, 37 respectively. For $k = 6$ we recover the Fano variety $\mathbf{P}(3, 1, 1, 1)$, one of only two Fano threefolds with canonical Gorenstein singularities with the maximal genus 37, found by Prokhorov. We show that the latter variety is no further extendable. We also study the extensions of smooth degree 2 sections of $K3$ surfaces of genus 3. In all these cases, we compute the co-rank of the Gauss--Wahl maps of the curves under consideration. Finally we observe that linear systems on double covers of the projective plane provide superabundant logarithmic Severi varieties.

Let $\pi$ be a set of primes. We show that $\pi$-separable groups have a conjugacy class of $\mathfrak F$-injectors for suitable Fitting classes $\mathfrak F$, which coincide with the usual ones when specializing to soluble groups.

The (interior) transmission eigenvalue problems are a type of non-elliptic, non-selfadjoint and nonlinear spectral problems that arise in the theory of wave scattering. They connect to the direct and inverse scattering problems in many aspects in a delicate way. The properties of the transmission eigenvalues have been extensively and intensively studied over the years, whereas the intrinsic properties of the transmission eigenfunctions are much less studied. Recently, in a series of papers, several intriguing local and global geometric structures of the transmission eigenfunctions are discovered. Moreover, those longly unveiled geometric properties produce some interesting applications of both theoretical and practical importance to direct and inverse scattering problems. This paper reviews those developments in the literature by summarizing the results obtained so far and discussing the rationales behind them. There are some side results of this paper including the general formulations of several types of transmission eigenvalue problems, some interesting observations on the connection between the transmission eigenvalue problems and several challenging inverse scattering problems, and several conjectures on the spectral properties of transmission eigenvalues and eigenfunctions, with most of them are new to the literature.

Game-theoretic upper expectations are joint (global) probability models that mathematically describe the behaviour of uncertain processes in terms of supermartingales; capital processes corresponding to available betting strategies. Compared to (the more common) measure-theoretic expectation functionals, they are not bounded to restrictive assumptions such as measurability or precision, yet succeed in preserving, or even generalising many of their fundamental properties. We focus on a discrete-time setting where local state spaces are finite and, in this specific context, build on the existing work of Shafer and Vovk; the main developers of the framework of game-theoretic upper expectations. In a first part, we study Shafer and Vovk's characterisation of a local upper expectation and show how it is related to Walley's behavioural notion of coherence. The second part consists in a study of game-theoretic upper expectations on a more global level, where several alternative definitions, as well as a broad range of properties are derived, e.g. the law of iterated upper expectations, compatibility with local models, coherence properties,... Our main contribution, however, concerns the continuity behaviour of these operators. We prove continuity with respect to non-increasing sequences of so-called lower cuts and continuity with respect to non-increasing sequences of finitary functions. We moreover show that the game-theoretic upper expectation is uniquely determined by its values on the domain of bounded below limits of finitary functions, and additionally show that, for any such limit, the limiting sequence can be constructed in such a way that the game-theoretic upper expectation is continuous with respect to this particular sequence.

Thomas Milton Liggett was a world renowned UCLA probabilist, famous for his monograph Interacting Particle Systems. He passed away peacefully on May 12, 2020. This is a perspective article in memory of both Tom Liggett the person and Tom Liggett the mathematician.

We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators. We adapt the notion of stochastic entropy solution and provide a new technical framework to prove the uniqueness. The existence proof relies on the vanishing viscosity method. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities and derive error estimate for the stochastic vanishing viscosity method. In addition, we develop uniqueness method "a la Kruzkov" for more general equations where the noise coefficient may depends explicitly on the spatial variable.

Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with $n$ vertices is $(1+o(1)) \frac{3n^2}{2\pi^2}$. This conjecture can be rephrased in terms of the spectral gap as follows: the spectral gap (algebraic connectivity) of a connected $k$-regular graph on $n$ vertices is at least $(1+o(1))\frac{2k\pi^2}{3n^2}$, and the bound is attained for at least one value of $k$. We determine the structure of connected quartic graphs on $n$ vertices with minimum spectral gap which enable us to show that the minimum spectral gap of connected quartic graph on $n$ vertices is $(1+o(1))\frac{4\pi^2}{n^2}$. From this result, the Aldous--Fill conjecture follows for $k=4$.

We study the asymptotic stability of the semi-discrete (SD) numerical method for the approximation of stochastic differential equations. Recently, we examined the order of $\mathcal L^2$-convergence of the truncated SD method and showed that it can be arbitrarily close to $1/2,$ see \textit{Stamatiou, Halidias (2019), Convergence rates of the Semi-Discrete method for stochastic differential equations, Theory of Stochastic Processes, 24(40)}. We show that the truncated SD method is able to preserve the asymptotic stability of the underlying SDE. Motivated by a numerical example, we also propose a different SD scheme, using the Lamperti transformation to the original SDE, which we call Lamperti semi-discrete (LSD). Numerical simulations support our theoretical findings.

Beliakova-Putyra-Wehrli studied various kinds of traces, in relation to annular Khovanov homology. In particular, to a graded algebra and a graded bimodule over it, they associate a quantum Hochschild homology of the algebra with coefficients in the bimodule, and use this to obtain a deformation of the annular Khovanov homology of a link. A spectral refinement of the resulting invariant was recently given by Akhmechet-Krushkal-Willis. In this short note we observe that quantum Hochschild homology is a composition of two familiar operations, and give a short proof that it gives an invariant of annular links, in some generality. Much of this is implicit in Beliakova-Putyra-Wehrli's work.

In this paper we propose an Approximate Weak stationarity ($AW$-stationarity) concept designed to deal with {\em Mathematical Programs with Cardinality Constraints} (MPCaC), and we proved that it is a legitimate optimality condition independently of any constraint qualification. Such a sequential optimality condition improves weaker stationarity conditions, presented in a previous work. Many research on sequential optimality conditions has been addressed for nonlinear constrained optimization in the last few years, some works in the context of MPCC and, as far as we know, no sequential optimality condition has been proposed for MPCaC problems. We also establish some relationships between our $AW$-stationarity and other usual sequential optimality conditions, such as AKKT, CAKKT and PAKKT. We point out that, despite the computational appeal of the sequential optimality conditions, in this work we are not concerned with algorithmic consequences. Our aim is purely to discuss theoretical aspects of such conditions for MPCaC problems.

In the paper we show that the monoid $\mathbf{I}\mathbb{N}_{\infty}$ of all partial cofinite isometries of positive integers does not embed isomorphically into the monoid $\mathbf{ID}_{\infty}$ of all partial cofinite isometries of integers. Moreover every non-annihilating homomorphism $\mathfrak{h}\colon \mathbf{I}\mathbb{N}_{\infty}\to\mathbf{ID}_{\infty}$ has the following property: the image $(\mathbf{I}\mathbb{N}_{\infty})\mathfrak{h}$ is isomorphic either to the two-element cyclic group $\mathbb{Z}_2$ or to the additive group of integers $\mathbb{Z}(+)$. Also we prove that the monoid $\mathbf{I}\mathbb{N}_{\infty}$ is not a finitely generated, and moreover monoid $\mathbf{I}\mathbb{N}_{\infty}$ does not contain its minimal generating set.

We characterise the weak$^*$ symmetric strong diameter $2$ property in Lipschitz function spaces by a property of its predual, the Lipschitz-free space. We call this new property decomposable octahedrality and study its duality with the symmetric strong diameter $2$ property in general. For a Banach space to be decomposably octahedral it is sufficient that its dual space has the weak$^*$ symmetric strong diameter $2$ property. Whether it is also a necessary condition remains open.

Over a field of characteristic $0$, we construct a minimal set of generators of the defining ideals of closures of nilpotent conjugacy class in the set of $n \times n$ matrices. This modifies a conjecture of Weyman and provides a complete answer to it.

Recent experimental and theoretical results show many molecules dissociate in a slow and complicated manner called roaming, that is due to a mechanism independent of the mechanisms for molecular and radical dissociation. While in most molecules the conventional molecular mechanism dominates roaming, acetaldehyde stands out by predominantly dissociating to products characteristic for roaming. This work contributes to the discussion of the prominence of roaming in (restricted) acetaldehyde from a dynamical systems perspective. We find two mechanisms consisting of invariant phase space structures that may lead to identical molecular products. One of them is a slow passage via the flat region that fits the term frustrated dissociation used to describe roaming and is similar to the roaming mechanism in formaldehyde. The other mechanism is fast and avoids the flat region altogether. Trajectory simulations show that the fast mechanism is significantly more likely than roaming.

It is proven that for any integer $g \ge 0$ and $k \in \{ 0, \ldots, 10 \}$, there exist infinitely many 5-regular graphs of genus $g$ containing a 1-factorisation with exactly $k$ pairs of 1-factors that are perfect, i.e. form a hamiltonian cycle. For $g = 0$, this settles a problem of Kotzig from 1964. Motivated by Kotzig and Labelle's "marriage" operation, we discuss two gluing techniques aimed at producing graphs of high cyclic edge-connectivity. We prove that there exist infinitely many planar 5-connected 5-regular graphs in which every 1-factorisation has zero perfect pairs. On the other hand, by the Four Colour Theorem and a result of Brinkmann and the first author, every planar 4-connected 5-regular graph satisfying a condition on its hamiltonian cycles has a linear number of 1-factorisations each containing at least one perfect pair. We also prove that every planar 5-connected 5-regular graph satisfying a stronger condition contains a 1-factorisation with at most nine perfect pairs, whence, every such graph admitting a 1-factorisation with ten perfect pairs has at least two edge-Kempe equivalence classes. The paper concludes with further results on edge-Kempe equivalence classes in planar 5-regular graphs.

In this review article, we report on some recent advances on the computational aspects of cohomology intersection numbers of GKZ systems developed in \cite{GM}, \cite{MH}, \cite{MT} and \cite{MT2}. We also discuss the relation between intersection theory and evaluation of an integral of a product of powers of absolute values of polynomials.

We provide an algorithm to generate trajectories of sparse stochastic processes that are solutions of linear ordinary differential equations driven by L\'evy white noises. A recent paper showed that these processes are limits in law of generalized compound-Poisson processes. Based on this result, we derive an off-the-grid algorithm that generates arbitrarily close approximations of the target process. Our method relies on a B-spline representation of generalized compound-Poisson processes. We illustrate numerically the validity of our approach.

In this work, we are concerned with the stability and convergence analysis of the second order BDF (BDF2) scheme with variable steps for the molecular beam epitaxial model without slope selection. We first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step constraint. Then we show that it preserves an energy dissipation law if the adjacent time-step ratios $r_k:=\tau_k/\tau_{k-1}<3.561.$ Moreover, with a novel discrete orthogonal convolution kernels argument and some new discrete convolutional inequalities, the $L^2$ norm stability and rigorous error estimates are established, under the same step-ratios constraint that ensuring the energy stability., i.e., $0<r_k<3.561.$ This is known to be the best result in literature. We finally adopt an adaptive time-stepping strategy to accelerate the computations of the steady state solution and confirm our theoretical findings by numerical examples.

In this paper we consider an Hartree-Fock type system made by two Schr\"odinger equations in presence of a Coulomb interacting term and a cooperative pure power and subcritical nonlinearity, driven by a suitable parameter $\beta \geq 0$. We show the existence of semitrivial and vectorial ground states solutions depending on the parameters involved. The asymptotic behavior with respect to the parameter $\beta$ of these solutions is also studied.

We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid torus, thus answering a question posed by Arnold. We similarly study spaces of unframed links in the 3-sphere, modulo rotations, and spaces of knots in the thickened torus. The subgroup of meridional rotations splits as a direct factor of the fundamental group of the space of an irreducible framed link. Its generators can be viewed as generalizations of the Gramain loop in the space of long knots. Taking the quotient by certain such rotations relates the spaces we study. All of our results generalize previous work of Hatcher and Budney. We provide many examples and explicitly describe generators of fundamental groups.

The stable module category of a selfinjective algebra is triangulated, but need not have any nontrivial $t$-structures, and in particular, full abelian subcategories need not arise as hearts of a $t$-structure. The purpose of this paper is to investigate full abelian subcategories of triangulated categories whose exact structures are related, and more precisely, to explore relations between invariants of finite-dimensional selfinjective algebras and full abelian subcategories of their stable module categories.

For the class of solvable groups of homeomorphisms of the line preserving orientation and containing a freely acting element, we establish the metabelianity of the quotient group $G/H_G$, where the elements of the normal subgroup $H_G$ are stabilizers of the minimal set. This fact is an important element in the classification theorem, used, in particular, in the study of the Thompson's group $F$.

The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra B to each formula. We show some basic results regarding the effect of the properties of B on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author's result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures - in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types.

We consider a distribution logistics scenario where a shipping operator, managing a limited amount of resources, receives a stream of collection requests, issued by a set of customers along a booking time-horizon, that are referred to a future operational period. The shipping operator must then decide about accepting or rejecting each incoming request at the time it is issued, accounting for revenues, but also considering resource consumptions. In this context, the decision process is based on dynamically finding the best trade-off between the immediate return of accepting the request and the convenience of preserving capacity to possibly exploit more valuable future requests. We give a dynamic formulation of the problem aimed at maximizing the operator revenues, accounting also for the operational distribution costs. Due to the "curse of dimensionality", the dynamic program cannot be solved optimally. For this reason, we propose a mixed-integer linear programming approximation, whose exact or approximate solutions provide the relevant information to apply some commonplace revenue management policies in the real-time decision-making. Adopting a capacitated vehicle routing problem as an underlying distribution application, we analyze the computational behaviour of the proposed techniques on a set of academic test problems.

We introduce discrete-time linear control system on connected Lie groups and present an upper bound for the outer invariance entropy of admissible pairs. In the case of solvable Lie groups the upper bound coincides with the outer invariance entropy.

The open XXZ spin chain with the anisotropy parameter $\Delta=-\frac12$ and diagonal boundary magnetic fields that depend on a parameter $x$ is studied. For real $x>0$, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in $x$ with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik-Zamolodchikov equations associated with the $R$-matrix and diagonal $K$-matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.

We present a Conjugate Gradient (CG) implementation of the probabilistic numerical solver BayesCG, whose error estimates are a fully integrated design feature, easy to compute, and competitive with the best existing estimators. More specifically, we extend BayesCG to singular prior covariances, derive recursions for the posterior covariances, express the posteriors as projections, and establish that BayesCG retains the minimization properties over Krylov spaces regardless of the singular priors. We introduce a possibly singular Krylov prior covariance, under which the BayesCG posterior means coincide with the CG iterates and the posteriors can be computed efficiently. Because of its factored form, the Krylov prior is amenable to low-rank approximation, which produces an efficient BayesCG implementation as a CG method. We also introduce a probabilistic error estimator, the `$S$-statistic'. Although designed for sampling from BayesCG posteriors, its mean and variance under approximate Krylov priors can be computed with CG. An approximation of the $S$-statistic by a `95 percent credible interval' avoids the cost of sampling altogether. Numerical experiments illustrate that the resulting error estimates are competitive with the best existing methods and are easy to compute.

Given a constant $k>1$, let $Z$ be the family of round spheres of radius $\textrm{artanh}(k^{-1})$ in the hyperbolic space $\mathbb{H}^3$, so that any sphere in $Z$ has mean curvature $k$. We prove a crucial nondegeneracy result involving the manifold $Z$. As an application, we provide sufficient conditions on a prescribed function $\phi$ on $\mathbb{H}^3$, which ensure the existence of a ${\cal C}^1$-curve, parametrized by $\varepsilon\approx 0$, of embedded spheres in $\mathbb{H}^3$ having mean curvature $k +\varepsilon\phi$ at each point.

Problems involving control of large ensmebles of structurally identical dynamical systems, called \emph{ensemble control}, arise in numerous scientific areas from quantum control and robotics to brain medicine. In many of such applications, control can only be implemented at the population level, i.e., through broadcasting an input signal to all the systems in the population, and this new control paradigm challenges the classical systems theory. In recent years, considerable efforts have been made to investigate controllability properties of ensemble systems, and most works emphasized on linear and some forms of bilinear and nonlinear ensemble systems. In this paper, we study controllability of a broad class of bilinear ensemble systems defined on semisimple Lie groups, for which we define the notion of ensemble controllability through a Riemannian structure of the state space Lie group. Leveraging the Cartan decomposition of semisimple Lie algebras in representation theory, we develop a \emph{covering method} that decomposes the state space Lie group into a collection of Lie subgroups generating the Lie group, which enables the determination of ensemble controllability by controllability of the subsystems evolving on these Lie subgroups. Using the covering method, we show the equivalence between ensemble and classical controllability, i.e., controllability of each individual system in the ensemble implies ensemble controllability, for bilinear ensemble systems evolving on semisimple Lie groups. This equivalence makes the examination of controllability for infinite-dimensional ensemble systems as tractable as for a finite-dimensional single system.

We consider the following data perturbation model, where the covariates incur multiplicative errors. For two $n \times m$ random matrices $U, X$, we denote by $U \circ X$ the Hadamard or Schur product, which is defined as $(U \circ X)_{ij} = (U_{ij}) \cdot (X_{ij})$. In this paper, we study the subgaussian matrix variate model, where we observe the matrix variate data $X$ through a random mask $U$: \begin{equation*} {\mathcal X} = U \circ X \; \; \; \text{ where} \; \; \;X = B^{1/2} {\mathbb Z} A^{1/2}, \end{equation*} where ${\mathbb Z}$ is a random matrix with independent subgaussian entries, and $U$ is a mask matrix with either zero or positive entries, where ${\mathbb E} U_{ij} \in [0, 1]$ and all entries are mutually independent. Subsampling in rows, or columns, or random sampling of entries of $X$ are special cases of this model. Under the assumption of independence between $U$ and $X$, we introduce componentwise unbiased estimators for estimating covariance $A$ and $B$, and prove the concentration of measure bounds in the sense of guaranteeing the restricted eigenvalue conditions to hold on the estimator for $B$, when columns of data matrix $X$ are sampled with different rates. Our results provide insight for sparse recovery for relationships among people (samples, locations, items) when features (variables, time points, user ratings) are present in the observed data matrix ${\mathcal X}$ with heterogenous rates. Our proof techniques can certainly be extended to other scenarios.

The invariant subspace problem is a well known unsolved problem in funtional analysis. While many partial results are known, the general case for complex, infinite dimensional separable Hilbert spaces is still open. It has been shown that the problem can be reduced to the case of operators which are norm limits of nilpotents. One of the most important subcases is the one of quasinilpotent operators, for which the problem has been extensively studied for many years. In this paper, we will prove that every quasinilpotent operator has a nontrivial invariant subspace. This will imply that all the operators for which the ISP has not been established yet are norm-limits of operators having nontrivial invariant subspaces.

In this paper, we first determine the optimal sets of $n$-means and the $n$th quantization errors for all $1\leq n\leq 6$ for two nonuniform discrete distributions with support the set $\{1, 2, 3, 4, 5, 6\}$. Then, for a probability distribution $P$ with support $\{\frac 1n : n\in \mathbb N\}$ associated with a mass function $f$, given by $f(x)=\frac 1 {2^k}$ if $x=\frac 1 k$ for $k\in \mathbb N$, and zero otherwise, we determine the optimal sets of $n$-means and the $n$th quantization errors for all positive integers up to $n=300$. Further, for a probability distribution $P$ with support the set $\mathbb N$ of natural number associated with a mass function $f$, given by $f(x)=\frac 1 {2^k}$ if $x=k$ for $k\in \mathbb N$, and zero otherwise, we determine the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$. At last we discuss for a discrete distribution, if the optimal sets are given, how to obtain the probability distributions.

To model discrete-time finite-state uncertain processes, we argue for the use of a global belief model in the form of an upper expectation that is the most conservative one under a set of basic axioms. Our motivation for these axioms, which describe how local and global belief models should be related, is based on two possible interpretations for an upper expectation: a behavioural one similar to Walley's, and an interpretation in terms of upper envelopes of linear expectations. We show that the most conservative upper expectation satisfying our axioms, that is, our model of choice, coincides with a particular version of the game-theoretic upper expectation introduced by Shafer and Vovk. This has two important implications: it guarantees that there is a unique most conservative global belief model satisfying our axioms; and it shows that Shafer and Vovk's model can be given an axiomatic characterisation and thereby provides an alternative motivation for adopting this model, even outside their game-theoretic framework. Finally, we relate our model to the upper expectation resulting from a traditional measure-theoretic approach. We show that this measure-theoretic upper expectation also satisfies the proposed axioms, which implies that it is dominated by our model or, equivalently, the game-theoretic model. Moreover, if all local models are precise, all three models coincide.

In this article we explore compactifications of cluster varieties of finite type in complex dimension two. Cluster varieties can be viewed as the spec of a ring generated by theta functions and a compactification of such varieties can be given by a grading on that ring, which can be described by positive polytopes [17]. In the examples we exploit, the cluster variety can be interpreted as the complement of certain divisors in del Pezzo surfaces. In the symplectic viewpoint, they can be described via almost toric fibrations over $\R^2$ (after completion). Once identifying them as almost toric manifolds, one can symplectically view them inside other del Pezzo surfaces. So we can identify other symplectic compactifications of the same cluster variety, which we expect should also correspond to different algebraic compactifications. Both viewpoints are presented here and several compactifications have their corresponding polytopes compared. The finiteness of the cluster mutations are explored to provide cycles in the graph describing monotone Lagrangian tori in del Pezzo surfaces connected via almost toric mutation [34].

S-Heun operators on linear and $q$-linear grids are introduced. These operators are special cases of Heun operators and are related to Sklyanin-like algebras. The Continuous Hahn and Big $q$-Jacobi polynomials are functions on which these S-Heun operators have natural actions. We show that the S-Heun operators encompass both the bispectral operators and Kalnins and Miller's structure operators. These four structure operators realize special limit cases of the trigonometric degeneration of the original Sklyanin algebra. Finite-dimensional representations of these algebras are obtained from a truncation condition. The corresponding representation bases are finite families of polynomials: the para-Krawtchouk and $q$-para-Krawtchouk ones. A natural algebraic interpretation of these polynomials that had been missing is thus obtained. We also recover the Heun operators attached to the corresponding bispectral problems as quadratic combinations of the S-Heun operators

In this work, we explore edge direction, transitivity, and connectedness of Cayley graphs of gyrogroups. More specifically, we find conditions for a Cayley graph of a gyrogroup to be undirected, transitive, and connected. We also show a relationship between the cosets of a certain type of subgyrogroups and the connected components of Cayley graphs. Some examples regarding these findings are provided.

Independence number, coloring number and related constants are investigated in the setting of oriented hypergraphs using the spectrum of the normalized Laplace operator.

We further develop the method of dressing the boundary for the focusing nonlinear Schr\"odinger equation (NLS) on the half-line to include the new boundary condition presented by Zambon. Additionally, the foundation to compare the solutions to the ones produced by the mirror-image technique is laid by explicitly computing the change of scattering data under the Darboux transformation. In particular, the developed method is applied to insert pure soliton solutions.

In this article, we offer a novel numerical approach for the solution of elastohydrodynamic lubrication line and point contact problems using a class of total variation diminishing (TVD) schemes on parallel computers. A direct parallel approach is presented by introducing a novel solver named as projected alternate quadrant interlocking factorization (PAQIF) by solving discrete variational inequality. For one-dimensional EHL case, we use weighted change in Newton-Raphson approximation to compute the Jacobian matrix in the form of a banded matrix by dividing two subregions on the whole computation domain. Such subregion matrices are assembled by measuring the ratio of diffusive coefficient and discrete grid length on the domain of the interest. The banded matrix is then processed to parallel computers for solving discrete linearized complementarity system using PAQIF algorithm. The idea is easily extended in two-dimensional EHL case by taking appropriate splitting in x and y alternating directions respectively. Numerical experiments are performed and analyzed to validate the performance of computed solution on serial and parallel computers.

Finite posets $R$ and $S$ are studied with $\# {\cal H}(P,R) \leq \# {\cal H}(P,S)$ for every finite poset $P$, where ${\cal H}(P,Q)$ is the set of order homomorphisms from $P$ to $Q$. It is shown that under an additional regularity condition, $\# {\cal H}(P,R) \leq \# {\cal H}(P,S)$ for every finite poset $P$ is equivalent to $\# {\cal S}(P,R) \leq \# {\cal S}(P,S)$ for every finite poset $P$, where ${\cal S}(P,Q)$ is the set of strict order homomorphisms from $P$ to $Q$. A method is developed for the rearrangement of a finite poset $R$, resulting in a poset $S$ with $\# {\cal H}(P,R) \leq \# {\cal H}(P,S)$ for every finite poset $P$. The results are used in constructing pairs of posets $R$ and $S$ with this property.

Strain engineering is used to obtain desirable materials properties in a range of modern technologies. Direct nanoscale measurement of the three-dimensional strain tensor field within these materials has however been limited by a lack of suitable experimental techniques and data analysis tools. Scanning electron diffraction has emerged as a powerful tool for obtaining two-dimensional maps of strain components perpendicular to the incident electron beam direction. Extension of this method to recover the full three-dimensional strain tensor field has been restricted though by the absence of a formal framework for tensor tomography using such data. Here, we show that it is possible to reconstruct the full non-symmetric strain tensor field as the solution to an ill-posed tensor tomography inverse problem. We then demonstrate the properties of this tomography problem both analytically and computationally, highlighting why incorporating precession to perform scanning precession electron diffraction may be important. We establish a general framework for non-symmetric tensor tomography and demonstrate computationally its applicability for achieving strain tomography with scanning precession electron diffraction data.

Two proofs of the Koml\'os-Major-Tusn\'ady embedding theorems, one for the uniform empirical process and one for the simple symmetric random walk, are given. More precisely, what are proved are the univariate coupling results needed in the proofs, such as Tusn\'{a}dy's lemma. These proofs are modifications of existing proof architectures, one combinatorial (the original proof with many modifications, due to Cs\"{o}rg\~o, R\'{e}v\'{e}sz, Bretagnolle, Massart, Dudley, Carter, Pollard etc.) and one analytical (due to Sourav Chatterjee). There is one common idea to both proofs: we compare binomial and hypergeometric distributions among themselves, rather than with the Gaussian distribution. In the combinatorial approach, this involves comparing Binomial(n,1/2) distribution with the Binomial(4n,1/2) distribution, which mainly involves comparison between the corresponding binomial coefficients. In the analytical approach, this reduces Chatterjee's method to coupling nearest neighbour Markov chains on integers so that they stay close.

We study maps on the set of permutations of n generated by the R\'enyi-Foata map intertwined with other dihedral symmetries (of a permutation considered as a 0-1 matrix). Iterating these maps leads to dynamical systems that in some cases exhibit interesting orbit structures, e.g., every orbit size being a power of two, and homomesic statistics (ones which have the same average over each orbit). In particular, the number of fixed points (aka 1-cycles) of a permutation appears to be homomesic with respect to three of these maps, even in one case where the orbit structures are far from nice. For the most interesting such "Foatic" action, we give a heap analysis and recursive structure that allows us to prove the fixed-point homomesy and orbit properties, but two other cases remain conjectural.

We study systems of equations over graphs, posets and matroids. We give the criteria, when a direct power of such algebraic structures is equationally Noetherian. Moreover we prove that any direct power of a finite algebraic structure is weakly equationally Noetherian.

As hordes of data-hungry devices challenge its current capabilities, Wi-Fi strikes back with 802.11be, alias Wi-Fi 7. This brand-new amendment promises a (r)evolution of unlicensed wireless connectivity as we know it. With its standardisation process being consolidated, we provide an updated digest of 802.11be essential features, vouching for multi-AP coordination as a must-have for critical and latency-sensitive applications. We then get down to the nitty-gritty of one of its most enticing implementations-coordinated beamforming-, for which our standard-compliant simulations confirm near-tenfold reductions in worst-case delays.

We construct the free Lagrangian of the magnetic sector of Carrollian electrodynamics, which surprisingly, is not obtainable as an ultra-relativistic limit of Maxwellian Electrodynamics. The construction relies on Helmholtz integrability condition for differential equations in a self consistent algorithm working hand in hand with imposing invariance under infinite dimensional Conformal Carroll algebra (CCA). It requires inclusion of new fields in the dynamics and the system in free of gauge redundancies. We calculate two-point functions in the free theory based entirely on symmetry principles. We next add interaction (quartic) terms to the free Lagrangian, strictly constrained by conformal invariance and Carrollian symmetry. Finally, a successful dynamical realization of infinite dimensional CCA is presented at the level of charges, for the interacting theory. In conclusion, we calculate the Poisson brackets for these charges.

We have previously studied -in part I- the quantization of a mixed bulk-boundary system describing the coupled dynamics between a bulk quantum field confined to a spacetime with finite space slice and with timelike boundary, and a boundary observable defined on the boundary. Our bulk system is a quantum field in a spacetime with timelike boundary and a dynamical boundary condition -the boundary observable's equation of motion. Owing to important physical motivations, in part I, we have computed the renormalized local state polarization and local Casimir energy for both the bulk quantum field and the boundary observable in the ground state and in a Gibbs state at finite, positive temperature. In this work, we introduce an appropriate notion of coherent and thermal coherent states for this mixed bulk-boundary system, and extend our previous study of the renormalized local state polarization and local Casimir energy to coherent and thermal coherent states.

We formulate a new approach to solving the initial value problem of the shallow water-wave equations utilizing the famous Carrier-Greenspan transformation [G. Carrier and H. Greenspan, J. Fluid Mech. 01, 97 (1957)]. We use a Taylor series approximation to deal with the difficulty associated with the initial conditions given on a curve in the transformed space. This extends earlier solutions to waves with near shore initial conditions, large initial velocities, and in more complex U-shaped bathymetries; and allows verification of tsunami wave inundation models in a more realistic 2-D setting.

In this paper, we study a new Workforce Scheduling and Routing Problem, denoted Multiperiod Workforce Scheduling and Routing Problem with Dependent Tasks. In this problem, customers request services from a company. Each service is composed of dependent tasks, which are executed by teams of varying skills along one or more days. Tasks belonging to a service may be executed by different teams, and customers may be visited more than once a day, as long as precedences are not violated. The objective is to schedule and route teams so that the makespan is minimized, i.e., all services are completed in the minimum number of days. In order to solve this problem, we propose a Mixed-Integer Programming model, a constructive algorithm and heuristic algorithms based on the Ant Colony Optimization (ACO) metaheuristic. The presence of precedence constraints makes it difficult to develop efficient local search algorithms. This motivates the choice of the ACO metaheuristic, which is effective in guiding the construction process towards good solutions. Computational results show that the model is capable of consistently solving problems with up to about 20 customers and 60 tasks. In most cases, the best performing ACO algorithm was able to match the best solution provided by the model in a fraction of its computational time.

Numerous complex systems, both natural and artificial, are characterized by the presence of intertwined supply and/or drainage networks. Here we present a minimalist model of such co-evolving networks in a spatially continuous domain, where the obtained networks can be interpreted as a part of either the counter-flowing drainage or co-flowing supply and drainage mechanisms. The model consists of three coupled, nonlinear partial differential equations that describe spatial density patterns of input and output materials by modifying a mediating scalar field, on which supply and drainage networks are carved. In the 2-dimensional case, the scalar field can be viewed as the elevation of a hypothetical landscape, of which supply and drainage networks are ridges and valleys, respectively. In the 3-dimensional case, the scalar field serves as the chemical signal strength, in which vascularization of the supply and drainage networks occurs above a critical 'erosion' strength. The steady-state solutions are presented as a function of non-dimensional channelization indices for both materials. The spatial patterns of the emerging networks are classified within the branched and congested extreme regimes, within which the resulting networks are characterized based on the absolute as well as the relative values of two non-dimensional indices.

Road information such as road profile and traffic density have been widely used in intelligent vehicle systems to improve road safety, ride comfort, and fuel economy. However, vehicle heterogeneity and parameter uncertainty make it extremely difficult for a single vehicle to accurately and reliably measure such information. In this work, we propose a unified framework for learning-based collaborative estimation to fuse local road estimation from a fleet of connected heterogeneous vehicles. The collaborative estimation scheme exploits the sequential measurements made by multiple vehicles traversing the same road segment and let these vehicles relay a learning signal to iteratively refine local estimations. Given that the privacy of individual vehicles' identity must be protected in collaborative estimation, we directly incorporate privacy-protection design into the collaborative estimation design and establish a unified framework for privacy-preserving collaborative estimation. Different from patching conventional privacy mechanisms like differential privacy which will compromise algorithmic accuracy or homomorphic encryption which will incur heavy communication/computational overhead, we leverage the dynamical properties of collective estimation to enable inherent privacy protection without sacrificing accuracy or significantly increasing communication/computation overhead. Numerical simulations confirm the effectiveness and efficiency of our proposed framework.

In 1665, Huygens observed that two pendulum clocks hanging from the same beam became synchronized in antiphase after hundreds of swings. On the other hand, modern experiments with metronomes placed on a movable platform show that they tend to synchronize in phase, not antiphase. Here, using a simple model of coupled clocks and metronomes, we calculate the regimes where in-phase and antiphase synchronization are stable. Unusual features of our approach include its treatment of the escapement mechanism, a small-angle approximation up to cubic order, and a three-time scale asymptotic analysis.

Phase equilibrium calculation, also known as flash calculation, plays significant roles in various aspects of petroleum and chemical industries. Since Michelsen proposed his milestone studies in 1982, through several decades of development, the current research interest on flash calculation has been shifted from accuracy to efficiency, but the ultimate goal remains the same focusing on estimation of the equilibrium phase amounts and phase compositions under the given variable specification. However, finding the transition route and its related saddle points are very often helpful to study the evolution of phase change and partition. Motivated by this, in this study we apply the string method to find the minimum energy paths and saddle points information of a single-component VT flash model with the Peng-Robinson equation of state. As the system has strong stiffness, common ordinary differential equation solvers have their limitations. To overcome these issues, a Rosenbrock-type exponential time differencing scheme is employed to reduce the computational difficulty caused by the high stiffness of the investigated system. In comparison with the published results and experimental data, the proposed numerical algorithm not only shows good feasibility and accuracy on phase equilibrium calculation, but also successfully calculates the minimum energy path and and saddle point of the single-component VT flash model with strong stiffness.

We propose a novel stochastic network model, called Fractal Gaussian Network (FGN), that embodies well-defined and analytically tractable fractal structures. Such fractal structures have been empirically observed in diverse applications. FGNs interpolate continuously between the popular purely random geometric graphs (a.k.a. the Poisson Boolean network), and random graphs with increasingly fractal behavior. In fact, they form a parametric family of sparse random geometric graphs that are parametrized by a fractality parameter $\nu$ which governs the strength of the fractal structure. FGNs are driven by the latent spatial geometry of Gaussian Multiplicative Chaos (GMC), a canonical model of fractality in its own right. We asymptotically characterize the expected number of edges and triangle in FGNs. We then examine the natural question of detecting the presence of fractality and the problem of parameter estimation based on observed network data, in addition to fundamental properties of the FGN as a random graph model. We also explore fractality in community structures by unveiling a natural stochastic block model in the setting of FGNs.

In 2019, the European Union introduced two new actors in the European energy system: Renewable and Citizen Energy Communities (RECs and CECs). Modelling these two new actors and their effects on the energy system is crucial when implementing the European Legislation, incorporating energy communities (ECs) into the electric grid, planning ECs, and conducting academic research. This paper aims to bridge the gap between the letter of the law and numerical models of ECs. After introducing RECs and CECs, we list elements of the law to be considered by regulators, distribution system operators, EC planners, researchers, and other stakeholders when modelling ECs. Finally, we provide three case studies of EC models that explicitly include elements of the European Law.

We study the complexity of finding an optimal hierarchical clustering of an unweighted similarity graph under the recently introduced Dasgupta objective function. We introduce a proof technique, called the normalization procedure, that takes any such clustering of a graph $G$ and iteratively improves it until a desired target clustering of G is reached. We use this technique to show both a negative and a positive complexity result. Firstly, we show that in general the problem is NP-complete. Secondly, we consider min-well-behaved graphs, which are graphs $H$ having the property that for any $k$ the graph $H(k)$ being the join of $k$ copies of $H$ has an optimal hierarchical clustering that splits each copy of $H$ in the same optimal way. To optimally cluster such a graph $H(k)$ we thus only need to optimally cluster the smaller graph $H$. Co-bipartite graphs are min-well-behaved, but otherwise they seem to be scarce. We use the normalization procedure to show that also the cycle on 6 vertices is min-well-behaved.

The sphere partition function of Calabi-Yau gauged linear sigma models (GLSMs) has been shown to compute the exact Kaehler potential of the Kaehler moduli space of a Calabi-Yau. We propose a universal expression for the sphere partition function evaluated in hybrid phases of Calabi-Yau GLSMs that are fibrations of Landau-Ginzburg orbifolds over some base manifold. Special cases include Calabi-Yau complete intersections in toric ambient spaces and Landau-Ginzburg orbifolds. The key ingredients that enter the expression are Givental's I/J-functions, the Gamma class and further data associated to the hybrid model. We test the proposal for one- and two-parameter abelian GLSMs, making connections, where possible, to known results from mirror symmetry and FJRW theory.

Monte Carlo simulations of quantum field theories on a lattice become increasingly expensive as the continuum limit is approached since the cost per independent sample grows with a high power of the inverse lattice spacing. Simulations on fine lattices suffer from critical slowdown, the rapid growth of autocorrelations in the Markov chain with decreasing lattice spacing. This causes a strong increase in the number of lattice configurations that have to be generated to obtain statistically significant results. This paper discusses hierarchical sampling methods to tame this growth in autocorrelations. Combined with multilevel variance reduction, this significantly reduces the computational cost of simulations for given tolerances $\epsilon_{\text{disc}}$ on the discretisation error and $\epsilon_{\text{stat}}$ on the statistical error. For an observable with lattice errors of order $\alpha$ and an integrated autocorrelation time that grows like $\tau_{\mathrm{int}}\propto a^{-z}$, multilevel Monte Carlo (MLMC) can reduce the cost from $\mathcal{O}(\epsilon_{\text{stat}}^{-2}\epsilon_{\text{disc}}^{-(1+z)/\alpha})$ to $\mathcal{O}(\epsilon_{\text{stat}}^{-2}\vert\log \epsilon_{\text{disc}} \vert^2+\epsilon_{\text{disc}}^{-1/\alpha})$. Even higher performance gains are expected for simulations of quantum field theories in $D$ dimensions. The efficiency of the approach is demonstrated on two model systems, including a topological oscillator that is badly affected by critical slowdown due to freezing of the topological charge. On fine lattices, the new methods are orders of magnitude faster than standard sampling based on Hybrid Monte Carlo. For high resolutions, MLMC can be used to accelerate even the cluster algorithm for the topological oscillator. Performance is further improved through perturbative matching which guarantees efficient coupling of theories on the multilevel hierarchy.

We prove that any $n$-node graph $G$ with diameter $D$ admits shortcuts with congestion $O(\delta D \log n)$ and dilation $O(\delta D)$, where $\delta$ is the maximum edge-density of any minor of $G$. Our proof is simple, elementary, and constructive - featuring a $\tilde{\Theta}(\delta D)$-round distributed construction algorithm. Our results are tight up to $\tilde{O}(1)$ factors and generalize, simplify, unify, and strengthen several prior results. For example, for graphs excluding a fixed minor, i.e., graphs with constant $\delta$, only a $\tilde{O}(D^2)$ bound was known based on a very technical proof that relies on the Robertson-Seymour Graph Structure Theorem. A direct consequence of our result is that many graph families, including any minor-excluded ones, have near-optimal $\tilde{\Theta}(D)$-round distributed algorithms for many fundamental communication primitives and optimization problems including minimum spanning tree, minimum cut, and shortest-path approximations.

A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be expanded into a set of resource proof-structures: its Taylor expansion. We introduce a new criterion characterizing those sets of resource proof-structures that are part of the Taylor expansion of some MELL proof-structure, through a rewriting system acting both on resource and MELL proof-structures. As a consequence, we prove also the semi-decidability of the type inhabitation problem for MELL proof-structures.

In this paper we have chosen to work with two different approaches to solving the inverse problem of the calculus of variation. The first approach is based on an integral representation of the Lagrangian function that uses the first integral of the equation of motion while the second one relies on a generalization of the well known Noether's theorem and constructs the Lagrangian directly from the equation of motion. As an application of the integral representation of the Lagrangian function we first provide some useful remarks for the Lagrangian of the modified Emden-type equation and then obtain results for Lagrangian functions of (i) cubic-quintic Duffing oscillator, (ii) Li\'{e}nard-type oscillator and (iii) Mathews-Lakshmanan oscillator. As with the modified Emden-type equation these oscillators were found to be characterized by nonstandard Lagrangians except that one could also assign a standard Lagrangian to the Duffing oscillator. We used the second approach to find indirect analytic (Lagrangian) representation for three velocity-dependent equations for (iv) Abraham-Lorentz oscillator, (v) Lorentz oscillator and (vi) Van der Pol oscillator. For each of the dynamical systems from (i)-(vi) we calculated the result for Jacobi integral and thereby provided a method to obtain the Hamiltonian function without taking recourse to the use of the so-called Legendre transformation.

Bit threads are curves in holographic spacetimes that manifest boundary entanglement, and are represented mathematically by continuum analogues of network flows or multiflows. Subject to a density bound, the maximum number of threads connecting a boundary region to its complement computes the Ryu-Takayanagi entropy. When considering several regions at the same time, for example in proving entropy inequalities, there are various inequivalent density bounds that can be imposed. We investigate for which choices of bound a given set of boundary regions can be "locked", in other words can have their entropies computed by a single thread configuration. We show that under the most stringent bound, which requires the threads to be locally parallel, non-crossing regions can in general be locked, but crossing regions cannot, where two regions are said to cross if they partially overlap and do not cover the entire boundary. We also show that, under a certain less stringent density bound, a crossing pair can be locked, and conjecture that any set of regions not containing a pairwise crossing triple can be locked, analogously to the situation for networks.

We propose a new class of rough stochastic volatility models obtained by modulating the power-law kernel defining the fractional Brownian motion (fBm) by a logarithmic term, such that the kernel retains square integrability even in the limit case of vanishing Hurst index $H$. The so-obtained log-modulated fractional Brownian motion (log-fBm) is a continuous Gaussian process even for $H = 0$. As a consequence, the resulting super-rough stochastic volatility models can be analysed over the whole range $0 \le H < 1/2$ without the need of further normalization. We obtain skew asymptotics of the form $\log(1/T)^{-p} T^{H-1/2}$ as $T\to 0$, $H \ge 0$, so no flattening of the skew occurs as $H \to 0$.

Mobile communications have been undergoing a generational change every ten years or so. However, the time difference between the so-called "G's" is also decreasing. While fifth-generation (5G) systems are becoming a commercial reality, there is already significant interest in systems beyond 5G - which we refer to as the sixth-generation (6G) of wireless systems. In contrast to the many published papers on the topic, we take a top-down approach to 6G. We present a holistic discussion of 6G systems beginning with the lifestyle and societal changes driving the need for next generation networks, to the technical requirements needed to enable 6G applications, through to the challenges, as well as possibilities for practically realizable system solutions across all layers of the Open Systems Interconnection stack. Since many of the 6G applications will need access to an order-of-magnitude more spectrum, utilization of frequencies between 100 GHz and 1 THz becomes of paramount importance. We comprehensively characterize the limitations that must be overcome to realize working systems in these bands; and provide a unique perspective on the physical, as well as higher layer challenges relating to the design of next generation core networks, new modulation and coding methods, novel multiple access techniques, antenna arrays, wave propagation, radio-frequency transceiver design, as well as real-time signal processing. We rigorously discuss the fundamental changes required in the core networks of the future, such as the redesign or significant reduction of the transport architecture that serves as a major source of latency. While evaluating the strengths and weaknesses of key technologies, we differentiate what may be practically achievable over the next decade, relative to what is possible in theory. For each discussed system aspect, we present concrete research challenges.

Extracting informative and meaningful temporal segments from high-dimensional wearable sensor data, smart devices, or IoT data is a vital preprocessing step in applications such as Human Activity Recognition (HAR), trajectory prediction, gesture recognition, and lifelogging. In this paper, we propose ESPRESSO (Entropy and ShaPe awaRe timE-Series SegmentatiOn), a hybrid segmentation model for multi-dimensional time-series that is formulated to exploit the entropy and temporal shape properties of time-series. ESPRESSO differs from existing methods that focus upon particular statistical or temporal properties of time-series exclusively. As part of model development, a novel temporal representation of time-series $WCAC$ was introduced along with a greedy search approach that estimate segments based upon the entropy metric. ESPRESSO was shown to offer superior performance to four state-of-the-art methods across seven public datasets of wearable and wear-free sensing. In addition, we undertake a deeper investigation of these datasets to understand how ESPRESSO and its constituent methods perform with respect to different dataset characteristics. Finally, we provide two interesting case-studies to show how applying ESPRESSO can assist in inferring daily activity routines and the emotional state of humans.