Papers

Each one-dimensional strong affine representation of the polycyclic monoid $\mathcal{P}_n$ is induced by a complete system of residues modulo $n$. We completely characterize these representations in the case when the system of residues is an arithmetic sequence. We accomplish this by introducing a closure operator on the set of primitive words over $\{0,1,\dots,n-1\}$, and we also describe the lattice of closed sets. This characterization covers all one-dimensional strong affine representations of $\mathcal{P}_2$.

Laser-based thermal processing, such as laser powder bed fusion, requires tight regulation of the peak surface temperature: heat accumulates where the moving source re-enters previously heated material, driving the temperature out of its process window and causing defects. High-fidelity thermal models capture this physics but are too slow for online optimization, which motivates fast, differentiable, and generalizable surrogates. We develop and validate a complete surrogate-based control pipeline that regulates the maximum surface temperature of a moving laser on a 304-stainless-steel substrate. We also determine conditions under which our surrogate can be trusted inside the control loop by probing its out-of-distribution limits. A key component of our surrogate is a multi-step deep operator network bespoke for moving sources: its branch subnetwork encodes the future power and trajectory (position and velocity) sequence, while its trunk encodes the current peak temperature and the temperature at the future laser locations, yielding a one-shot five-step prediction. By way of illustration, we use this surrogate as a smooth (algebraic-rectifier) nonlinear program inside a receding-horizon model predictive controller solved in CasADi/IPOPT. The surrogate forward pass is over thousand times faster than the equivalent finite-difference steps. We show that aggregate open-loop accuracy is necessary but not sufficient for control-readiness: two surrogates with near-identical offline error behave drastically differently in closed loop. A controlled two-ensemble data design reduces a 91 K path-corner underprediction failure to 1.4 K, and a calibrated one-sided constraint margin of 13 K yields zero violations of the true upper bound on all tested paths.

In J.~Phys.~A 31, 3493 (1998), Berry provides an analytic solution of the problem of determining the diffracted beam intensities for atoms incident upon a grating given by the potential, $v(x,y):=iV_0(e^{i\mathfrak{K}\,y}-1)$ for $0\leq x\leq\ell$ and $v(x,y):=0$ otherwise, where $V_0, \mathfrak{K}$, and $\ell$ are positive real parameters. This result, which relies on the paraxial approximation, readily applies to the diffraction of transverse electric (TE) waves by the nonmagnetic optical grating given by the relative permittivity, $\hat\varepsilon(x,y):=1-v(x,y)/k^2$, where $k$ is the incident wavenumber. We show that Berry's grating belongs to a larger class of possibly magnetic diffraction gratings whose scattering problem for both TE and transverse magnetic (TM) waves are exactly solvable. Our analysis makes use of a recently developed dynamical formulation of stationary scattering that is based on the idea of mapping the scattering problem to the quantum dynamics generated by an effective non-Hermitian Hamiltonian operator. {We use this formulation to derive explicit analytic expressions for the diffracted beam amplitudes that are valid beyond paraxial approximation.} As a concrete example, we offer the exact solution of the scattering problem for TE and TM waves incident upon a generalization of Berry's grating.

We resolve in the negative a conjecture of Frantzikinakis and Kuca concerning the vanishing of multiple correlation sequences in nilsystems. Specifically, we prove the existence of an ergodic $3$-step nilsystem $(G/\Gamma, \mu_{G/\Gamma}, R_\alpha)$ and bounded functions $f_0, f_1, f_2 \in L^\infty(\mu_{G/\Gamma})$ orthogonal to the Conze–Lesigne factor $L^2(G/G_3\Gamma)$, whose associated multiple correlation sequence $$a(n) = \int_{G/\Gamma} f_0(x) f_1(\alpha^n x) f_2(\alpha^{2n} x) \, d\mu_{G/\Gamma}(x)$$ does not decay to zero. The same counterexample also refutes another conjecture of Frantzikinakis and Kuca and a conjecture of Leibman. To construct this counterexample, we develop a framework for Fourier analysis on $G/\Gamma$ where $G$ is the free $3$-step nilpotent Lie group on $4$ generators, a methodology that extends naturally to general nilsystems.

Understanding the ecological processes that regulate populations of the blacklegged tick, Ixodes scapularis, is essential for explaining regional differences in tick abundance and the risk of tick-borne diseases. In this study, we develop a stage-structured nonlinear system of difference equations by ordering and composing the seasonal biological events that govern tick and host life cycles. The model incorporates ratio-dependent host-attachment, allowing host-finding success to depend jointly on host availability and tick abundance. Parameter estimates from both northeastern and southeastern ecosystems in the United States are used to examine how regional differences in host community composition and questing behavior influence tick population dynamics. Analysis shows that a $+1$/$-1$ bifurcation pair occurs at the demographic persistence threshold, producing an exchange of stability between the extinction and positive fixed points together with the emergence of an unstable cohort-based 2-cycle. Quantitative results show that ratio-dependent host-finding success limits population growth by restricting successful feeding, while questing behavior alters both the frequency and composition of host encounters. Together, these mechanisms regulate tick persistence and long-term abundance, providing a demographic foundation for understanding variation in tick abundance and Lyme disease risk.

Weighted integrated observability inequalities for heat equations usually involve a small-time factor of the form $e^{-\gamma/t}$. We prove that this scale is not an artefact of Carleman or spectral methods: it is forced by the geometry of the observation set. Let $A$ be a nonnegative self-adjoint operator on sections of a finite-rank Euclidean vector bundle over a doubling metric measure space, satisfying ultracontractivity, Davies-Gaffney estimates (equivalently, finite speed of propagation for the wave equation) and a pointwise local Weyl law. If a weighted integrated observability inequality holds on a measurable set $\omega$, for a fixed horizon $T\in(0,+\infty]$ and an admissible weight $h$, then, for every $0<\kappa<\frac{1}{2}$, $$ h(t)\leq A_{T,\kappa}\exp\left(-\kappa\frac{\mathcal{L}(\omega)^2}{t}\right),\qquad 0<t<T, $$ where $\mathcal{L}(\omega)$ is the essential maximal distance to $\omega$, replaced by any finite radius when $\mathcal{L}(\omega)=+\infty$. Thus, for $h(t)=e^{-\gamma/t}$, necessarily $\gamma\geq\mathcal{L}(\omega)^2/2$. This settles, with the optimal threshold, the maximal-distance lower bound for the infinite-time constant left open in earlier work. In the control-norm convention, the fast-control rate is at least $\mathcal{L}(\omega)^2/4$, recovering Miller's bound. The proof rests on the spectral packet $(\cosh(r\sqrt A)-1)e^{-tA}$. A pointwise Weyl law gives its sharp lower growth, while finite propagation speed and a weak-kernel Kannai transmutation formula make it exponentially small on $\omega$. Without kernel continuity or compact resolvent, we develop pointwise spectral measures and weak wave kernels. The framework covers Laplace-type operators on compact Riemannian manifolds, coupled heat systems, Schrödinger operators on $\mathbb{R}^d$, equiregular sub-Laplacians and Grushin models, and $\delta'$-coupled Laplacians on metric graphs.

We introduce EuclideanDistanceDegree, a Macaulay2 package that implements symbolic and numerical methods for computing Euclidean Distance (ED) degrees. The package includes symbolic methods based on minors and conormal varieties as well as numerical methods for unit and generic ED degrees using tools from numerical algebraic geometry. We illustrate the package functionality with a range of examples in the paper and in the accompanying GitHub repository.

We derive an explicit representation for the distribution of a Lévy process with total stochastic resetting and use it to investigate the asymptotic behaviour of the corresponding transition densities. The proposed approach applies to several important classes of Lévy processes, including relativistic stable processes, and yields new asymptotic estimates for their transition densities.

Cops and Robbers is a well-studied pursuit-evasion game that provides insights into graph theory and theoretical computing. A central question is determining the minimum number of cops required to capture the robber, known as the cop number. We focus on reproducing an algorithm proposed by Petr, Portier, and Versteegen in 2022, which efficiently determines whether a graph is $k$-copwin. This paper presents a Python implementation of the $k$-copwin algorithm. In this work, we present our implementation in detail, clarify key aspects of the algorithm, and discuss its implications for future practical deployments.

We prove that for any two birationally equivalent del Pezzo surfaces of Picard rank one over a perfect field there is a birational map between them such that it is decomposed in a composition of at most two Sarkisov links.

The commutator of a pair of vector fields on a graph is not a vector field in general, but rather a second order differential operator. We investigate this departure from the classical case of vectors fields on a manifold by examining the geometry of balls of radius two, concentrating on the set of paths of length two connecting a given vertex with the center of the ball. There is a natural surjection from the space of sections of the second tangent bundle to the space of second order differential operators whose kernel reflects the geometry of these balls. Using this map we draw several conclusions about second order differential operators including canonical forms, formulas for their adjoints, and a necessary and sufficient condition for a commutator to be a vector field.

Demand responsiveness is an important consideration in public transport line planning, as network design and service quality influence passenger demand. However, accounting for this interaction further complicates an already challenging combinatorial optimization problem. To address this challenge, we propose a scalable Adaptive Large Neighborhood Search (ALNS) algorithm for large-scale line planning with endogenous demand. The algorithm jointly optimizes lines and frequencies while accounting for passenger mode choice, passenger assignment, and vehicle capacities. Candidate lines are generated dynamically throughout the search, and solutions are evaluated using an embedded evaluation procedure for passenger assignment and demand estimation, together with a dedicated local search procedure for frequency optimization. The proposed methodology is evaluated on the public transport network of Odense, Denmark, comprising approximately 1,800 origin-destination pairs. Computational results demonstrate the applicability of the approach to realistic, large-scale instances. The optimized networks concentrate resources on fewer, higher-frequency services, reducing average headways from approximately 41 minutes to 6.8-13 minutes while substantially increasing public transport ridership. Furthermore, the results show that network design is highly sensitive to assumptions regarding passenger behavior, highlighting the importance of carefully calibrated demand models when incorporating demand responsiveness into line planning.

This article proves norm inflation in the critical Sobolev space $H^{5/2}(\mathbb{R})$ for the Fokas-Olver-Rosenau-Qiao equation, which is a modified Camassa-Holm-type equation with cubic nonlinearity. This result complements the well-posedness theory for this equation, which was previously known to be locally well-posed in $H^{s}(\mathbb{R})$ for $s>5/2$. The proof relies on the construction of explicit initial data satisfying the previously known blow-up criteria for the Fokas-Olver-Rosenau-Qiao equation, a step that appears to be of independent interest.

We introduce and analyze Darcy flow through a saturated porous medium subject to bilateral constraints on the normal flux across a portion of the boundary. The problem is posed as the maximization of a velocity-based dual concave energy over a convex subset of $H(\mathrm{div};\Omega)$; Fenchel duality identifies a pressure-based predual formulation, yields strong duality, and provides convex optimality conditions with a complementarity structure on the constrained boundary. The primal–dual gap satisfies an a posteriori error identity, free of generic constants, valid for arbitrary admissible approximations. The duality structure is inherited by a Raviart–Thomas/Crouzeix–Raviart discretization, from which we derive a discrete error identity and a priori error decay rates under fractional regularity assumptions on the solution and the flux bounds. Numerical experiments, including adaptive refinement driven by localized primal–dual gap indicators, support the theory.

In this paper, we develop a linear, fully decoupled, and unconditionally energy-stable fully discrete finite element scheme for the Cahn–Hilliard–Navier–Stokes (CHNS) system by employing the scalar auxiliary variable (SAV) approach. Unlike existing SAV-based formulations that typically introduce multiple auxiliary variables or additional techniques to handle different nonlinearities, we introduce only one scalar auxiliary variable together with a novel update of the auxiliary variable to reformulate all nonlinear terms arising from the Cahn–Hilliard and Navier–Stokes equations, yielding an equivalent reformulation of the original CHNS system. An implicit–explicit (IMEX) Euler scheme is applied for temporal discretization, where the linear terms are treated implicitly and the nonlinear terms explicitly, while a finite element method is adopted for spatial discretization. The resulting fully discrete scheme can be efficiently decomposed into two linear subproblems and one scalar quadratic algebraic equation, which significantly simplifies the implementation. Furthermore, we prove that the proposed scheme satisfies an unconditional discrete energy dissipation law and establish its stability with respect to several relevant norms. Optimal-order $L^2$ error estimates are also derived for the fully discrete finite element approximation. Finally, a series of numerical experiments are presented to verify the theoretical results and demonstrate the efficiency of the proposed method.

We solve the type $B$ case of the main open problem from Kuperberg's 1996 paper "Spiders for rank 2 Lie algebras." That is, we define a $\mathbb{C}(q)$-linear pivotal category $\mathbf{Web}(\mathfrak{so}_{2n+1})$ and prove that it is equivalent to the full subcategory of finite-dimensional representations of $U_q(\mathfrak{so}_{2n+1})$ tensor-generated by the fundamental representations. A consequence of our main result is an explicit construction of braid group symmetries for nonclassical finite-dimensional representations of the $\iota$quantum group ${U}^{\iota}_{-q^2}(\mathfrak{so}_m)$, which may be of independent interest.

Building on recent works that investigate positivity preserving matrix products, we {examine} the class of \JCP (\jcp) matrix products. A bilinear map on the Cartesian product of the space of n by n matrices with itself into m by m matrices is a \jcp matrix product if the natural linear map it induces on the tensor product of the space of n by n matrices with itself into m by m matrices is completely positive. In particular, a matrix product is \jcp if and only if its naturally associated Choi matrix is positive semidefinite. Similarly, a matrix product is \jcp if and only if it admits a Choi-Kraus representation. We use the Choi-Kraus representation of \jcp matrix products to study various basic properties, including positivity lower bounds, commutativity, units, causality, and separability. As examples, we apply our results to the Schur (Hadamard) product and the convolution product.

We consider the problem of parameter inference for ordinary differential equation (ODE) models with structural non-identifiability. Such models arise in a wide range of scientific fields, including control theory, systems biology, and public health. Structural non-identifiability occurs when distinct parameter values provide identical model outputs, resulting in lower-dimensional manifolds of observationally equivalent solutions in the parameter space. This poses challenges for Bayesian inference and Markov chain Monte Carlo (MCMC) methods, often leading to poor mixing and slow convergence. We develop two MCMC methods that use information from structural identifiability analysis. The first, Identifiability-Aware Geometric MCMC, constructs proposals that move within and between non-identifiable manifolds. The second, Identifiability-Aware Pseudo-Marginal MCMC, performs inference on the space of identifiable parameter combinations and reconstructs full parameter values. We show that both methods target the correct posterior distribution and are ergodic under standard conditions. Numerical examples demonstrate improved sampling efficiency and convergence compared with standard MCMC methods.

In this paper, we find the axiomatization for the topological square of S4.1. This is the first known topological square of a modal logic that differs from both the fusion and the Kripke product. We also prove the finite model property and decidability for this logic.

Solvent-based carbon capture can reduce CO2 emissions resulting from a continued reliance on fossil power plants for firm power. These capture processes remove CO2 from flue gases via a solvent. Careful design via process systems optimization can limit the overall cost of carbon capture, which is both capital- and energy intensive. As dispatchable power plants operate to meet varying load demand, the design process needs to account for varying operating points. However, optimizing the design over multiple operating points yields high computational complexity, which is why designs are often based on a single operating point in practice. Here, we identify optimal carbon capture process designs via stochastic optimization, reducing computational complexity through a data-driven approach-to-equilibrium model of the absorption and desorption processes. We represent variable flue gas conditions based on part-load operation data of a representative coal power plant. Accounting for this variability in the design substantially reduces equipment size and total plant cost by 6-9 % at the expense higher operating costs, yielding a reduction in total cost of carbon capture by 0.7-1.7 %. Given the capital intensity of carbon capture, variability of flue gas conditions therefore should be considered at the design stage, particularly if capture is deployed on plants subject to load following.

We study energy transport in a finite one-dimensional unpinned harmonic chain with stochastic nearest-neighbor momentum exchanges and Langevin heat baths at its endpoints. Such systems are known to exhibit superdiffusive transport driven by long-wavelength acoustic modes, leading to fractional macroscopic behavior. While fractional heat equations have been rigorously derived for infinite chains, the corresponding boundary conditions for finite systems in contact with heat baths remain unclear due to the nonlocality of the fractional Laplacian. Under the superdiffusive time scaling $t\sim n^{3/2}$, where $n$ is the system size, we prove that the averaged microscopic energy profile converges, as $n\to+\infty$, to a temperature field solving a fractional heat equation on $[0,1]$, with the generator given by a Neumann fractional Laplacian and additional nonlocal boundary terms induced by the heat baths. Our results provide a rigorous derivation of macroscopic boundary conditions for superdiffusive heat transport in open chains and introduce new boundary conditions for fractional Laplacians, that are motivated by a physical model.

In this paper we deal with parametric, linear-quadratic optimal control problems in which the solution can be uniquely characterized by the optimal final time adjoint state. As a motivating example, we establish theoretical results showing that for distributed control of the heat equation, the manifold of final time adjoints over the parameter space exhibits a slow decay of its Kolmogorov width if this was already the case for the parameter-dependent target states. Traditional linear reduced-order models would thus require a large reduced space in order to guarantee a sufficient accuracy, making them inefficient in this application. To overcome the limitation of linear models, we discuss a nonlinear surrogate based on U-Nets that maps parametric fields to approximate final time adjoints. We show that a suitable a posteriori error estimator remains applicable to the U-Net approximation and can be used to certify the surrogate results. Through two extensive numerical experiments, we show the potential of the U-Net surrogate and compare it with several linear and nonlinear methods from the literature. The results show that the U-Net consistently achieves the highest accuracy among the methods considered while requiring significantly fewer training samples.

We prove Sabidussi's compatibility conjecture. Let $G$ be a finite connected multigraph in which every vertex has even degree and the minimum degree is at least four, and let $T$ be an Euler tour of $G$. The edges of $G$ can be partitioned into circuits (connected $2$-regular subgraphs) so that no circuit contains two edges used consecutively anywhere in $T$. In fact, the edges can be four-coloured so that every such pair receives different colours and every colour class has even degree at every vertex. We use a counting argument based on the Chevalley-Warning theorem to show that a four-colouring with the required properties exists. Splitting each colour class into circuits then gives the desired compatible decomposition. Formalization in Lean 4 is also available in the author's github.

Function-space summation-by-parts (FSBP) operators enable conservative and energy-stable numerical methods for hyperbolic conservation laws based on general, non-polynomial approximation spaces. Recent works show that using generalized Gaussian quadrature significantly reduces the number of grid points required compared to existing constructions that have mostly focused on equidistant grids. In this paper, we compare open and closed FSBP operators constructed with generalized Gaussian quadratures and apply them to numerically solve hyperbolic conservation laws. Furthermore, to support open node distributions, we extend the FSBP framework by introducing function-space exact extrapolation operators and operationalize them in numerical schemes for solving hyperbolic conservation laws. Our numerical experiments include the one-dimensional linear advection, non-viscous Burgers, and compressible Euler equations of gas dynamics. We observe that applying FSBP operators in numerical schemes can improve efficiency and accuracy. Notably, we demonstrate these advantages in more challenging time-dependent settings compared to other recent works on Gaussian FSBP operators.

We develop a method to find new obstructions to the existence of homomorphisms between homogeneous C*-algebras with prescribed behavior on K-theory. As an application, we give a complete answer to a problem posed by Blackadar in 1993 concerning the existence of unital homomorphisms between algebras of matrix-valued functions on even spheres which are injective on K_0: we show that if n, d, k, r are natural numbers and k is not in the range n, n+1, ..., n+d-1, then there is no unital homomorphism from C (S^{2n}, M_r) to C (S^{2k}, M_{dr}) which is injective on K_0.

Every almost disjoint family $A$ of infinite subsets of $\omega$ gives rise to a scattered compact space $K_A$ and a family of Johnson-Lindenstrauss spaces JL$_p(A)$. We consider almost disjoint families arising from finitely branching trees and investigate whether they define homeomorphic compacta and isomorphic Banach spaces.

Memory encoded in the environment mediates interactions between active agents, from trail-following organisms to synthetic active matter depositing persistent tracks. Although such memory is known to strongly affect transport, its consequences for collective first-passage phenomena remain largely unexplored. Here we study $N$ one-dimensional random walkers interacting through a shared trail field. We characterize the $k^{\rm th}$ (among $N$) arrival time at a fixed target, and the probability that exactly $k$ walkers in $[0,1]$ reach one boundary before the other. For the broad class of self-interacting walkers with a saturating response to the trail, we derive exact expressions for the corresponding persistence exponents and splitting probabilities. Strikingly, despite the strong history-dependent correlations generated by the common environment, splitting probabilities are exactly identical whether walkers explore simultaneously or one after another. This invariance breaks down for nonsaturating trail interactions. Our results follow from an exact representation of the collective trail field and establish a framework for first-passage phenomena in systems coupled through persistent environmental memory.

We construct a punctured Wilson–evolving-set coupling for finite reversible Markov chains. The transformed evolving set carries a Markov trajectory up to its exit from a punctured domain, whereas the ordinary evolving set gives the corresponding probabilities after de-biasing. Applied to Wilson's algorithm with exponential killing, the final-point projection yields the root law and an ordered factorization of the probability that several prescribed vertices are roots. We identify this factorization with a probabilistic Schur-complement decomposition of the known determinantal root formula. The survival projection yields an evolving-set representation of hitting probabilities before killing. This representation gives a quantitative consequence which does not follow from the root process alone. On graphs of polynomial growth satisfying a Gaussian heat-kernel upper bound, in dimension larger than four, we obtain exponential localization at scale $q^{-1/2}$ for two-point forest connectivity, up to the natural finite-volume correction, and the bound $\mathbb E[|C_q(x)|]\leq Cq^{-2}$. Dirichlet eigenvalues in successively punctured domains also give product bounds for prescribed root events. We record a determinant-free Poisson-type concentration bound for the number of roots, while making explicit that the determinantal description gives the sharper Bernoulli decomposition. The complete graph is computed exactly and discrete tori, bottleneck graphs and the hypercube are treated as examples.

We consider statistical testing on the basis of incomplete observations with values in a separable Hilbert space, where the dimension is possibly large or even infinite. The general Hilbert space setting allows various data types as they arise in modern applications, in particular high dimensional and functional data. Possible Hilbert space testing problems are goodness-of-fit, symmetry, homogeneity and independence. We present an approach for modeling incomplete data that covers several problems in practice, e.g., ultra high dimensional random vectors with missing entries or partially observed stochastic processes. We identify a specific structure (independent and identically distributed) in the incomplete data that enables the analysis of statistical procedures with the help of suitable mathematical results (e.g., laws of large numbers and central limit theorems). Additionally, a general and novel concept for testing different hypotheses in this situation is suggested and sketched for the example of testing goodness-of-fit for normality.

We consider the space of $(n-1)$-colored graphs on a fixed set of $N$ vertices. Each edge position of the complete graph $K_N$ has $n$ possible states: the absence of an edge and $n-1$ colors. This gives a natural identification of the space of such graphs with the tensor power $(\mathbb C^n)^{\otimes m}$, where $m=\binom N2$, and defines on it the diagonal action of the Lie algebra $\mathfrak{gl}_n$, and, after restriction, the action of $\mathfrak{sl}_n$. For a fixed profile $\alpha=(\alpha_0,\dots,\alpha_{n-1})$, we consider the graph $J(m;\alpha)$ whose vertices are colored graphs of this profile and whose adjacency is defined by a single exchange of states in two edge positions. This graph is the transposition graph on the set of words with fixed profile, also known as the multislice. The main result is an expression of the adjacency operator in terms of the root operators of $\mathfrak{sl}_n$ and a derivation of its spectrum by means of the quadratic Casimir operator of $\mathfrak{gl}_n$ and the Schur–Weyl decomposition. It is proved that the adjacency operator belongs to the center of the algebra $\End_{S_m}(\mathcal C_\alpha)$. The contribution of each spectral block to the multiplicity of the corresponding eigenvalue is described in terms of a Kostka number and the dimension of a Specht module. For $n=2$, one obtains the classical Johnson graph and its known spectrum. As applications, a formula for the valency is established, connectivity is proved, the Hoffman bound for independent sets is obtained, and the three-state case is considered in detail; in this case the natural symmetrized subspace realizes the module $\Sym^m(\mathbb C^3)$.