Papers

This paper presents a comprehensive study of the characteristic polynomial of matrix tuples and finite dimensional group representations. Among other things, several key concepts are introduced, including the minimal polynomial, algebraic extension, spectral index and multiplicity, and characteristic graph. Notably, this framework leads to a complete resolution of Kippenhahn's conjecture by providing necessary and sufficient conditions for its validity.

We bound the total variation distance in the Kubilius model for sequences with positive level of distribution. We obtain a result that we expect is qualitatively optimal. As a special case, it recovers a recent result of Ford on shifted primes, with a slightly simplified proof. In the classical case considered by Kubilius, our theorem gives a simple proof of the optimal bound discovered by Tenenbaum, up to factors of $x^{o(1)}$ and $u^{o(u)}$.

In this paper, we study the fundamental group of compact de cent 4-dimensional spacetimes with lightlike hypersurface curvature. Such manifolds are foliated by totally geodesic null hypersurfaces. We prove that when the leaves of the foliation are compact then the fun damental group of the spacetime is polycyclic. We also prove that a non totally vicious 4-dimensional compact pp-wave spacetime has polycyclic fundamental group.

Flexible reflectors (FRs) have emerged as a low-cost and energy-efficient solution for reshaping electromagnetic propagation environments across a wide range of applications. This paper investigates FR-swarm-assisted target localization in scenarios where line-of-sight (LoS) paths are unavailable. By leveraging the virtual LoS paths created by the FRs, a simple yet accurate estimator is proposed for localization under severe blockage conditions. To characterize the performance limits of the proposed scheme, we derive the Cramer-Rao lower bound (CRLB) and use it to optimize the positions and orientations of the FRs. Furthermore, by accounting for random FR deployment, we characterize the CRLB distribution and reveal how different network configurations affect localization accuracy. Simulation results demonstrate that the developed scheme closely approaches the CRLB performance, while the derived analytical results provide useful guidelines for FR deployment and network design.

We investigate proper actions of arbitrary semigroups on separable locally compact metric spaces, where point orbits are allowed to escape to infinity. An action is pointwise uniquely ergodic when every compact orbit closure supports exactly one invariant probability measure and non-compact orbit closures support none. The associated ergodic map therefore assigns the selected probability measure to non-escaping points and the zero subprobability to escaping ones. Under the hypothesis that compact orbit closures admit at least one invariant measure, we establish that the weak* continuity of this ergodic map together with a vanishing at infinity condition is equivalent to the mean ergodicity of the Koopman representation on the space of continuous functions vanishing at infinity. In consequence, every such function and every finite signed measure split uniquely into invariant components and limits of coboundaries. The corresponding projections are obtained by integration against the ergodic map. Because this operator-theoretic characterization avoids explicit averaging schemes, it remains applicable even to semigroups without Følner sequences. When restricted to countable, discrete, bicancellative, and left amenable semigroups, these properties are shown to be equivalent to the uniform convergence of Følner averages and the weak-star continuity of their dual limits, extending classical results for group actions on compact spaces. Furthermore, we identify the space of ergodic measures with a compactified ergodic quotient, prove that the invariant measure simplex is Bauer, and show that these structural properties descend through proper factor maps. The theoretical framework is complemented by dynamical examples, including a continuously pointwise ergodic subshift that exhibits discontinuous entropy along the ergodic map.

We study a two–dimensional discrete nonlinear Schrödinger equation with saturable nonlinearity on the lattice $\mathbb Z^2$. Using a variational approach based on the Nehari manifold, we establish the existence of nontrivial periodic ground states on finite lattices and establish the existence of exponentially localized ground states in $\ell^2(\mathbb Z^2)$. A principal result is the rigorous passage from periodic to localized states: we show that, up to lattice translations, periodic ground states converge strongly in $\ell^2(\mathbb Z^2)$ to a localized ground state as the lattice periods tend to infinity. The analysis combines variational methods, spectral properties of the discrete Laplacian, and concentration–compactness techniques adapted to the two–dimensional discrete setting. We further derive qualitative properties of the resulting solutions, including positivity and exponential localization, and establish a conditional orbital stability result within the Grillakis–Shatah–Strauss framework. Numerical computations illustrate the theoretical results and confirm the predicted convergence and localization behavior.

We show that every $C^2$ immersion of the circle $\mathbb{S}^1$ into itself has low emergence.

The active inference framework provides a principled approach to modeling sentient behavior. In this framework perception and action selection are treated in a unified way. The resulting agents form an internal generative model of the relevant dynamics of the world in order to infer their future observations, their internal states and to select actions. We combine this modeling framework with the Integrated Information Theory of consciousness and are therefore able to analyze the active inference agents from the perspective of integrated information. The Integrated Information Theory aims at quantifying the level of consciousness of a system by assessing its capability to integrate information. Here, we define a measure of integrated information for the generative model by making an additional structural assumption. Experiments with simulated agents reveal a correlation between integrated information measures and the free energy of the active inference agents that increases with the size of the generative model.

We establish the optimal stability of Dirichlet boundary value problem for the regional (fractional) $p$-Laplacian $(-\Delta)^s_{p,\Omega}$ with $0<s\leq 1$, $\frac{1}{s}<p<\infty$ and $\Omega\subset \mathbb{R}^d$ bounded Lipschitz. More precisely, if $u_s \in W^{s,p}(\Omega)$ satisfies $(-\Delta)^s_{p,\Omega} u_s = f_s$ in $\Omega$ and $u_s = g_s$ on $\partial \Omega$, then under appropriate condition on the date $f_s$ and $g_s$ we show that $\|u_s - u_1\|_{W^{s,p}(\Omega)} \to 0 \quad \text{as } s \to 1^-.$ We also obtain an analogous optimal stability of the normalized Dirichlet eigenpairs $(\lambda_s,\varphi_s)$ associated with $(-\Delta)^s_{p,\Omega}$.

TOGEARI constructs a compressed contact correction for condensed finite-element equations with incremental potential contact (IPC). A core-conditioned interaction matrix ranks contact combinations by their mechanical responses; a raw-factor proxy acquires only the retained responses. For a symmetric positive-definite core, the reference analysis gives the exact spectrum, scaled inverse error and optimal worst omitted interaction. The selected responses are maintained through signed contact changes using supported operator images and a small coarse matrix. A complementary Schur spectrum and perturbation bounds connect this construction to the full current linear equation, damped local Newton steps and compatible episode reuse. On a 325,260-coordinate finger/cup problem, eight selected directions reduce Krylov work from eleven columns to five, matching full contact. Two reversed-order load studies give matching anchored displacements and accepted normal reactions, with 28.8-29.3 percent less warm time and 5.1-5.3 percent less complete time than per-step direct refactorization. A completed twelve-episode run saves 18.8 percent of total time against direct and 2.8 percent against a held factor. Full response caching has comparable time with 7.5 times the response storage. A compressed repeat terminates at an energy line search; separate diagnostics identify constitutive cancellation and terminal-backtracking sensitivity. Larger-contact, held-out-load and frictional comparisons show where compression or preparation loses its advantage. The complete Newton equation remains the residual operator throughout; the convergence guarantees retain their stated reference and local hypotheses.

A near Goldbach graph is a simple undirected graph whose vertex set consists of all positive even integers and there is an edge between two vertices $a,b$ if and only if $\frac{a+b}{2}, \frac{|a-b|}{2}$ are either odd primes or $1$. A finite near Goldbach graph $G(n)$ has the vertex set $\{x\in 2\mathbb{N}\, :\, x\leq 2n\}$ with the same adjacency rule. In this paper, we obtain two exact formulas for the degree of the even positive integer $x$ in $G(x/2)$. We compute a function $\eta(x)=\prod\limits_{p\mid x,\, p>2} \frac{p-1}{p-2}\, \frac{xe^{-0.183407}}{(\log\, x)^2}$ that approximates the degree of $x$ in $G(x/2)$ for a large even positive integer $x$. Finally, we introduce the concept of a nearly independent set of events and show that if the set of divisibility events for a large even integer $x$ is nearly independent, then $x$ can be expressed as the sum of two odd primes.

We consider simple autoregressive processes of type AR(1), whose stationary distribution is supported on a subset of the unit interval and is an affine transformation of a Bernoulli convolution. A new structural representation of the stationary distribution as a product of a power function with a log-periodic function near the origin is given, which gives structural insight to the stationary distribution on the whole unit interval by using a characteristic functional equation. This enables to prove that the stationary distribution of the AR(1)-process belongs to the domain of geometric partial attraction of a max-semistable law. We further prove uniform convergence of the distribution function of normalized maxima of the AR(1)-process to a certain power of the max-semistable law in the spirit of a merge theorem and point out connections to deterministic and random dynamical systems.

The Richelot $(2,2)$-isogeny graph $G_p$ of superspecial abelian surfaces in characteristic $p$ is the genus-2 analogue of the supersingular isogeny graph: its weighted adjacency matrix is a Brandt matrix for a compact inner form of $\mathrm{GSp}_4$, and it commutes with an Atkin-Lehner involution $R(\pi)$. Refining Ibukiyama's conjectures, we conjecture the automorphic origin and the $R(\pi)$-sign of every eigenvalue: Eisenstein, Saito-Kurokawa, Yoshida and paramodular eigenvalues, and pairs of type-Va eigensystems that only $R(\pi)$ separates. We compute $G_p$ at 33 primes, with up to 2176 vertices, and certify the predicted eigenvalues and signs in exact arithmetic; at $p=167$ and $173$, where the sign prediction first becomes non-trivial, the graph carries the predicted eigenvalue $-8$ with sign $-1$. For every prime we prove that each type-Va parameter contributes exactly one eigenline of each $R(\pi)$-sign, a closed formula for the trace of $R(\pi)$, and a Fricke sign bias. If no further CAP representations occur, the non-trivial spectrum lies in $[-8\sqrt{2}, 6+4\sqrt{2}]$, both ends being sharp under the conjecture, and only Saito-Kurokawa eigenvalues exceed $8\sqrt{2}$; at all 104 primes $p\le 601$ the extreme eigenvalue computed by Florit and Smith agrees with the largest Saito-Kurokawa one. Finally, the irrational roots of the residual square factor form one Galois orbit with full symmetric Galois group at each certified prime $p\ge 29$, and the spectrum appears to follow a Plancherel law.

We consider a sequence of discrete random variables whose generating functions are polynomials only having roots on the unit circle in the complex plane. For such sequences we prove, under mild assumptions, mod-$\phi$ convergence with explicit rate and limiting function. This allows us to recover, in a unified way, classical central limit theorems as well as finer asymptotic expansions. The general theory is applied to random permutations and random partitions.

We develop the kinematical and operator-algebraic foundations of a groupoid-based relational quantum field theory (RQFT) on curved spacetimes. Indeed, the usual group-based quantum reference frame (QRF) formalism is not directly suited to generic curved Lorentzian backgrounds as global symmetry groups are typically absent or too small. We formulate a notion of QRF for a continuous groupoid. This yields a groupoid relativization map and relational observables. We show that a localization limit recovers the ordinary non-relational description. We construct canonical sharp groupoid QRFs, which form the groupoidal counterpart of the ideal group QRFs based on $L^2(G)$. We further prove that the groupoid QRF construction reduces to the standard operational QRF formalism for locally compact groups. The action groupoid QRFs are torsor QRFs only for specific classes of fields of positive operator-valued measures (POVMs), and the torsor relativization map only applies to constant operator fields of system observables. We review the foundations of RQFT in Minkowski spacetime. We prove new results that further link RQFT to Wightman QFT. We show that covariant POVMs are $\mu$-continuous with respect to quasi-invariant $\sigma$-finite positive Borel measures $\mu$. Thus, relational quantum fields can be understood as the smearing of pointwise-defined kernels with respect to the QRF's statistics. We develop RQFT in curved spacetime, where we argue that the correct replacement for the Poincaré group is the Poincaré groupoid of the spacetime. We also indicate how the framework extends further to internal gauge symmetry and relational gauge-covariant quantum field theory via Atiyah groupoids, providing a first step towards formulating a relational quantum Yang-Mills field theory.

We develop a systematic adiabatic perturbation theory for the integrable $F=1$ spinor nonlinear Schrödinger equation under nonvanishing boundary conditions, formulated entirely within the framework of the associated Riemann–Hilbert problem. In this setting, localized nonlinear excitations are characterized by discrete spectral data consisting of a complex eigenvalue and an associated polarization vector. For a general class of small perturbations preserving the background, we derive the perturbation-induced evolution of the scattering data directly at the level of the Riemann–Hilbert problem. In the one-soliton sector, this yields a closed finite-dimensional dynamical system governing the slow evolution of the effective soliton parameters, including the spectral variables, the soliton center and phase, the residue amplitude, and the internal polarization state. The latter evolves according to a constrained dynamical equation with no scalar analogue. For localized perturbations, the modulation equations are expressed in explicit integral form in terms of the one-soliton eigenfunctions, providing a fully computable description of the dynamics. In the limit of vanishing boundary conditions, the resulting system reduces to the perturbation theory obtained by E. V. Doktorov, et al, Phys. Rev. A 77 (2008), no. 4, 043617.

We propose an adaptive multigrid (MG) method for discontinuous Galerkin formulations of elliptic problems using Brandt's full approximation scheme (FAS). Unlike common approaches, this method achieves local $hp$-refinement of hexahedral meshes without the need for hanging nodes. The core component of the FAS-MG method is an overlapping Schwarz smoother, which is optionally accelerated by a Krylov method. This smoother is designed for unstructured curvilinear meshes but maintains a tensor-product structure for fast diagonalization. Numerical experiments demonstrate the exceptional efficiency of the FAS-MG method. Dedicated studies confirm its robustness against high aspect ratios, element deformation, and irregular mesh topology. We also verify its capability for dynamic parallel mesh adaptation using the wave-front benchmark of Červený, Dobrev, and Kolev (SIAM J. Sci. Comp. 41, 2019). Finally, we present preliminary results of extending the method to incompressible Navier-Stokes problems.

We explicitly determine the numerical spectrum of the ordinary, non-equivariant global invariant for $n$-dimensional K-polystable toric $\mathbb{Q}$-Fano varieties. Specifically, we prove that $$\left\{ \alpha(X)\mid X\text{ is an $n$-dimensional K-polystable toric $\mathbb{Q}$-Fano variety}\right \} = \mathbb{Q}\cap \left[\frac{1}{n+1},\frac{1}{2}\right].$$ This result establishes a complete toric realization theorem, which strengthens a question raised by Liu and Zhuang and refines the recent construction results of Liu and Zhu. The initial construction of the examples in this work was suggested by GPT-5.6 Sol, and was subsequently refined and rigorously developed by the author.

We investigate the dynamics and spectral stability of traveling kink and antikink solutions in a dissipative sine–Gordon equation with two distinct fourth–order regularization mechanisms: a mixed space–time (inertial) term and a purely spatial (elliptic) term. The model includes damping, bias forcing, and higher–order dissipative effects, and is motivated by refined descriptions of fluxon dynamics in long Josephson junctions. Using a collective–coordinate reduction, we derive a Melnikov–type condition for speed selection, yielding explicit predictions for asymptotic propagation speeds, which are validated by direct numerical simulations of the full partial differential equation. Spectral stability is analyzed using Evans function techniques adapted to the singular slow–fast structure induced by the regularization. By formulating the linearized problem on a consistent–splitting domain, we show that no additional point spectrum bifurcates from the origin. Near the edges of the essential spectrum, a square–root transformation is used to resolve branch singularities and establish analyticity in a lifted spectral variable. Numerical Evans function computations near $\lambda=0$ and near the essential spectrum edges confirm the analytical results, indicating absence of unstable eigenvalues for both kink and antikink solutions.

We study clean energy transition planning for campus-scale integrated electricity-heat systems under both strategic level and operational level uncertainties. We formulate a multi-stage stochastic mixed-integer program that jointly optimizes investment and operational decisions for renewable generation, storage, and heat-transfer technologies whose costs and efficiencies evolve stochastically across stages. To account for short-term operational uncertainty, we further derive a robust reformulation based on box uncertainty sets for demand, renewable generation, and heat-transfer performance. To solve the resulting large-scale model, we develop a Benders decomposition algorithm with partial non-anticipativity relaxation. Investment variables and their non-anticipativity constraints are retained in the master problem while operational variables are assigned to scenario-wise subproblems where their non-anticipativity constraints are relaxed. Non-anticipativity of operational variables is restored only at termination through a smaller linear program. We prove that this correction step can only increase the objective function value by a finite bound. We improve the computational performance of the algorithm developed with valid inequalities and a two-phase cut-addition strategy. Using the proposed approach, we solve the Middle East Technical University campus case study at high temporal resolution within a reasonable computational budget, which is not otherwise possible with the extensive form or the classical Benders decomposition. We evaluate the resulting investment plans through rolling-horizon Monte Carlo simulations with an out-of-sample analysis and demonstrate that the added robustness can significantly improve the operational reliability of the transition plans with a moderate increase in total cost.

We study the zeros of functions in the Dirichlet space. Using extremal functions, we produce a necessary and sufficient condition for a sequence of points in the unit disk to be a zero set of the classical Dirichlet space. This Shapiro-Shields type condition involves kernels of the harmonic Dirichlet space associated with measures depending on the zero sequence.

We consider the problem of sequential prediction of an $m$-ary sequence, where at each epoch, (i) the environment selects an outcome from an $m$-ary alphabet, (ii) the learner selects a probability distribution over the same alphabet (unaware of the outcome generated by the environment), and finally, (iii) the learner incurs a cost that depends on the probability assigned to the outcome. The cost function we consider captures the complexity of predicting the outcome generated by the environment, in a scenario where the aforementioned prediction is performed via comparative queries to a lying oracle. We consider both stochastic and adversarial environments, propose algorithms for both settings, and establish logarithmic upper bounds on their regret.

We study the intrinsic notion of $\alpha$-stability for curves arising from the Beyond GIT approach to moduli spaces. We show that it recovers Deligne-Mumford stability for $9/11<\alpha\le1$, and more generally that the $\alpha$-semistable locus is contained in the known modular compactification of the Hassett-Keel proram for $\alpha>2/3-\varepsilon$. As applications, we also obtain new $\alpha$-stability results for smooth curves. Our approach combines slope inequalities with a degeneration theorem showing that every Gorenstein curve with a non-nodal singularity can be isotrivially degenerated to a Gorenstein curve with a $\mathbb G_m$-action.

Kostant–Kumar modules $K(\lambda,w,\mu)$ are submodules of a tensor product $V(\lambda)\otimes V(\mu)$ of irreducible highest weight modules over a symmetrizable Kac–Moody algebra, indexed by Weyl group elements $w$; their decomposition numbers $c^\nu_{\lambda\mu}(w)$ refine ordinary tensor product multiplicities. We study them module-theoretically. We show that $c^\nu_{\lambda\mu}(w)$ is computed by a natural quotient of the Kostant–Parthasarathy–Ranga Rao–Varadarajan multiplicity space, via orthogonal projection onto a Demazure module. For $\mathfrak{g}$ finite-dimensional semisimple or symmetric Kac–Moody, we present $K(\lambda,w,\mu)$ by generators and relations, extending the presentation of Demazure modules due to Joseph, Polo and Mathieu. We apply the presentation to obtain upper bounds on $c^\nu_{\lambda\mu}(w)$ and to study Schur positivity.

The one-warehouse multi-store (OWMS) system is a fundamental inventory network in which a nonreplenishable warehouse allocates shared stock across multiple stores over time. Existing OWMS learning policies are built around a fixed target calibrated to the initial average resource rate, but such a fixed-target architecture cannot re-center after realized sales change the remaining resource available per future period. We develop Resource-Adaptive Primal-Dual Learning, a new learning framework that tracks the primal-dual resolving path with censored demand as the remaining-resource state evolves. In each period, the current resource rate indexes the target store allocations and dual variable, while censored sales provide gradient estimates for updating both. The analysis combines expected-sales geometry with a moving-target argument to yield logarithmic expected regret, improving on the state-of-the-art square-root-order guarantees of existing OWMS learning policies. The underlying design and analytical ideas may inform other online learning problems with depleting shared resources. Numerical experiments further demonstrate good finite-horizon performance of a practical variant across different horizon lengths and inventory regimes.

We construct symbolic dynamics for non-uniformly hyperbolic flows, in any dimension, possibly with fixed points. More precisely, for each $\chi>0$, we code a set which has full measure for every $\chi$-hyperbolic invariant probability measure that gives zero mass to the set of fixed points. As a main application, we prove that a three dimensional $C^\infty$ flow with positive topological entropy on a closed manifold has finitely many ergodic measures of maximal entropy. For flows in any dimension, we also provide applications to the number of periodic orbits and to the Bernoulli property for equilibrium states of Hölder continuous potentials. The main technical result of this paper is a new method for handling singularities of vector fields by modifying the Riemannian metric. This technique is analogous to a blowup, which allows many results for nonsingular vector fields to be directly applied to vector fields with singularities.

We show that the order dimension of the poset of regions in a simplicial hyperplane arrangement can exceed its rank, answering a question of Reading and Segovia. Examples are Coxeter arrangements \(H_4\) and \(E_6\), with \( \dim W(H_4) \geq 5\) and \( \dim W(E_6) \geq 7\).

In this paper, we study the nonlinear fractional Choquard equation \begin{equation}\label{eq:0.1a} (-\Delta)^su+Vu=(|x|^{-\gamma}*|u|^2)u \quad in \quad \mathbb{R}^N, \end{equation} where $0<\gamma<4$, $0<s<1$, $N\geq 4$ and $V\in C^1(\mathbb{R}^N)$ is a positive potential. Set $s_0=\frac {\gamma}{4}$. Under suitable assumptions on $V$, we prove that the equation admits a nonnegative ground state solution for $s\in(s_0,1)$, whereas no ground state solution exists for $0<s\le s_0$. Furthermore, we show that any ground state solution $u_s$ blows up and concentrates at a minimum point of $V$ as $s\downarrow s_0$. Finally, up to a subsequence, the ground state solution $u_s$ converges to a ground state solution of the classical Choquard equation as $s\uparrow 1$.

A fully discrete approximation of the dynamic von Kármán equations combines nonconforming Morley finite element methods for spatial discretization with an energy conserving modified unconditionally stable Newmark second- order time-stepping scheme. Brouwer's fixed-point theorem establishes existence of a solution to the fully discrete scheme and further uniqueness and stability estimates follow for small loads. Optimal order a priori error estimates in the piecewise energy norm with quadratic convergence in time are derived for the fully discrete scheme. The results of the numerical experiments validate the theoretical error bounds.

Environmental variability is a defining feature of natural ecosystems, yet many theories of ecological stability assume static environments. Here, we develop an analytical framework for local stability in complex ecological communities subjected to periodic environmental forcing. We show that, in slowly varying environments and under a large-system approximation, the maximal Lyapunov exponent can be approximated by the time-averaged rightmost spectral edge of the instantaneous interaction matrix, yielding explicit stability criteria for large ecological communities. Within the random-matrix interaction ensembles considered here, our minimal theoretical framework predicts a topology-dependent ordering of local perturbation stability under periodic temporal modulation in high-dimensional ecological networks, which is supported by numerical simulations. Beyond the adiabatic regime, rapid environmental oscillations dynamically stabilize otherwise unstable ecosystems, revealing a high-frequency rescue effect that is absent from static theories. These results extend ecological stability theory beyond autonomous systems and provide a general framework for understanding resilience in fluctuating environments.