Papers

We develop a spectral cut-off and time-slicing construction for non-autonomous Hamiltonian evolution equations. Let \(H_0\) be a positive self-adjoint reference operator with compact resolvent on a Hilbert space \(\Hilb\), and let $$ P_N=\mathbf{1}_{[0,N]}(H_0). $$ For a time-dependent family of generally unbounded symmetric Hamiltonians (H(t)), we consider the finite-dimensional cut-off Hamiltonians $$ H_N(t)=P_NH(t)P_N. $$ Their time-sliced propagators admit finite-dimensional state-sum representations and, when suitable configuration-space or phase-space kernels are available, oscillatory integral realizations. We establish commutator conditions implying uniform stability of the cut-off dynamics and construct the full unitary propagator as the strong limit of the finite-dimensional propagators. Additional \(H_0\)-regularity yields the quantitative estimate $$ \sup{s,t\in I} |U_N(t,s)P_Nu-U(t,s)u| \leq C(1+N)^{-\sigma}|u|{\mu+\sigma}. $$ For Hamiltonians that are H"older continuous of exponent \(\alpha\) in time, we also prove the joint spectral and time-slicing bound $$ |U{N,M}(t,s)P_Nu-U(t,s)u| \leq C\left((1+N)^{-\sigma}+(1+N)^\mu M^{-\alpha}\right) |u|_{\mu+\sigma}. $$ The assumptions are verified for time-dependent Schr"odinger operators and symmetric first-order pseudodifferential Hamiltonians. In the periodic case, the construction is compatible with finite-order Floquet–Magnus coefficients for unbounded operators.

Forcing was first introduced by Paul J. Cohen in his work on the independence of the Continuum Hypothesis. Other formulations of forcing appeared using Model Theory, Boolean-valued Models, and Topos Theory. There is a folkloric claim that these three approaches are equivalent, at least at the level of their mathematical content. In this work, we present some results not found in the literature toward establishing connections between these versions of forcing.

The thinning-based integer-valued autoregressive moving-average (INARMA) models are popular for count time series. Recently, types of INARMA models have also been developed for count random fields, i.e., for spatial count data located on a regular two-dimensional grid. This article provides a comprehensive survey on existing INARMA random fields, covering approaches with different thinning operators, first- and higher-order models, as well as unilateral and multilateral model structures.

This paper develops a robust fixed time optimization framework for constrained problems that guarantees exact constraint satisfaction and convergence to KKT points within fixed time , independent of initial conditions. The approach treats the Lagrange multipliers as control inputs, composed of an equivalent control and a switching control, with the system states representing the decision variables. An equivalent control steers the gradient flow to a local KKT point asymptotically for nonconvex objectives and to unique global optimum in fixed time for convex objectives. Constraint enforcement is achieved by embedding the equality constraints directly as a sliding manifold, with a fixed time switching control ensuring rapid and reliable feasibility. The framework further accounts for the matched disturbances, providing robustness guarantees that are theoretically characterized and illustrated using spherical constraints. Numerical studies on a 3-bus AC optimal power flow problem and distributed consensus=based parameter estimation problem demonstrate the effectiveness, scalability and robustness of proposed approach.

We address observation noise misspecification in Bayesian filtering of dynamical systems via generalised Bayesian inference. Mismatch in tail decay between the true data generating process and an assumed observation model, often indicated via outliers, can strongly impact Bayesian updates in Kalman filters and their ensemble approximations. Adjusting information processing in the analysis step by employing diffusion score matching achieves robustness while maintaining closed-form Gaussian updates and good uncertainty quantification. We provide theoretical properties of this novel Kalman filter in linear Gaussian state space systems covering conjugacy of the analysis step, robustness, covariance stability, and tuning. We extend the approach to non-linear signal dynamics by adapting ensemble Kalman filter variants. Simulation studies on 2D target tracking and the Lorenz-63 and 40D Lorenz-96 systems highlight a critical trade-off between robustness and stability in filtering.

We study a random walk driven by a particle system from a generic class, and establish a law of large numbers for the walk for almost all densities of the environment. To do so, we exploit the finite-ranged approximations of the environment from arXiv:2409.02096 in a new way, whereby the monotonicity (in the density) of the walker's displacement is leveraged to show the existence of an actual speed. This bypasses the constructions in arXiv:1906.03167 and generalises its Theorem 1.1, which applied to specific environments. We illustrate this with a family of particle systems where the particles have underlying drifts, namely mixtures of APCRWs (Asymmetric Poisson Cloud of Random Walks). In particular, when all particles have the same drift, we prove the LLN under any choice of parameters save one critical density of the environment. To our knowledge, this is the first time that such a conservative and slow-mixing environment with drift is treated outside of the non-nestling case (in which the walker is already assumed to travel strictly faster/slower than the drift arXiv:2205.00282).

We study a symmetry problem for the $h$-polynomials of edge rings of bipartite graphs. Let $G$ be a bipartite graph and write $h(\mathbb{k}[G];t)=h_0+h_1t+\cdots+h_st^s$. We prove that if $\Bbbk[G]$ is pseudo-Gorenstein and $h_1=h_{s-1}$, then $\Bbbk[G]$ is Gorenstein. Equivalently, under these assumptions the $h$-polynomial of $\Bbbk[G]$ is palindromic. The proof treats the $2$-connected case first by translating the numerical condition $h_1=h_{s-1}$ into a tight-separation condition for non-edges, and then passes to arbitrary bipartite graphs using the block decomposition. We also construct a blockwise minimal Gorenstein closure, obtained by adjoining all non-edges not separated by tight acceptable sets, and show that this construction preserves the next-to-leading coefficient of the $h$-polynomial.

We study signed generalized Stirling polynomials $P_k(m,x)$ arising in closed forms for Malmsten-type hyperbolic secant integrals. Their product structure is used to prove recurrences, gamma–polygamma formulas for $P_{m-s}(m,x)$, a central vanishing identity, a finite approximation to $\cosh \pi x$, and a limit formula for $\pi$. We identify these polynomials as signed residues of the equal-period Barnes multiple zeta function, derive their reflection formula, and obtain finite parity-cancellation and Stirling cycle-number identities. We also compare the centered polynomials with a classical Meixner–Pollaczek orthogonal family. We evaluate finite nested sums built from the sequence $\chi_n$. Fixing the lower bounds turns these sums into coefficient-counting problems: common lower bounds give binomial coefficients and staircase bounds give Catalan numbers. Combining these counts with known formulas for $\chi_j$ yields explicit hyperbolic-secant integral evaluations involving Catalan's constant, zeta values, and polygamma values. A Wolfram Language package accompanies the formulas.

We obtain positive lower bounds on the Hausdorff dimension of sets of real numbers given by expressions of the form $\sum_{n=1}^\infty \frac{1}{a_n b_n}$, where $b_n$ satisfies some growth condition and $a_n$ lies in some set, possibly depending on $n$. As a consequence of our results, some of the irrational numbers arising from Erdős' celebrated construction from 1976 are not Liouville numbers.

Classical Laguerre spectral approximations are highly effective on the half-line when the target function is smooth in the usual polynomial scale. However, their accuracy deteriorates for nonsmooth functions. Such behavior appears naturally in fractional models, weakly singular integral equations, and semi-infinite-domain approximations with limited regularity near the origin. The main contribution of this work is the construction and analysis of a fractional Laguerre approximation framework tailored to nonsmooth functions on the half-line. We establish projection and interpolation error estimates in nonuniformly weighted Sobolev space. These estimates clarify how the fractional parameter adapts the approximation space to the regularity of nonsmooth functions and improves the resulting convergence behavior. We further introduce a generalized fractional Laguerre family with an additional algebraic parameter, which gives greater flexibility in controlling both the approximation space and the underlying weight. Numerical experiments confirm the theoretical estimates and demonstrate the advantage of the proposed functions over standard Laguerre-type approximations.

For graph $G$, a connected graph $H$ of order $n$ is $G$-good if $r(G,H)=(\chi(G)-1)(n-1)+s(G)$, where $\chi(G)$ is the chromatic number of $G$ and $s(G)$ is the minimum size of a color class in a $\chi(G)$-coloring of $G$. Let $K_{\alpha_{1},\ldots ,\alpha_{p},n}$ be the complete $(p+1)$-partite graph with partite sets of sizes $\alpha_1,\ldots,\alpha_p,n$. Burr, Faudree, Rousseau and Schelp (1983) showed that $K_{\alpha_1,\ldots,\alpha_p,n}$ are $(K_2+mK_1)$-good for large $n$. We determine graphs $G$ such that $K_{\alpha_{1},\ldots ,\alpha_{p},n}$ are $G$-good for large $n$. The characterization depends on $\mathrm{snd}(\alpha_i)$, the smallest non-divisor of $\alpha_i$, where $1\le i\le p$.

Let $G_n$ be the coprime graph on $\{1,\ldots,n\}$. We prove that the mixed vertex-coloring coprime Ramsey number satisfies \[ \Rcop(k_1,\ldots,k_c)=p_{\sum_{i=1}^c(k_i-1)}, \] where $p_m$ is the $m$-th prime. The proof is elementary: the prime clique $\{1\}\cup\{p\le n:p\text{ prime}\}$ gives the upper bound by pigeonhole, while a prime-bin partition gives the matching lower bound by coloring each composite with a bin containing one of its prime divisors. We reserve $\Rcop$ for this vertex-coloring parameter; the edge-coloring parameter on the same host graph is denoted $\Redge$. The same certificate viewpoint yields several extensions, including a support-disjointness generalization, a polynomial-time certificate-extraction primitive, and an exact reduction of the edge-coloring variant to classical Ramsey numbers: $\Redge(k_1,\ldots,k_c)=p_{\Rcl(k_1,\ldots,k_c)-1}$. These two formulas are rank transfers from the same clique-label certificate. We also prove that the balanced two-color diagonal threshold equals the unrestricted threshold $p_{2k-2}$ for all $k\ge2$, via a deterministic prime-bin split requiring only the weak inequality $2p_m<p_{2m}<3p_m$; for fixed $c$, a Hall argument plus a standard Selberg–Delange estimate gives eventual multicolor balanced certificates.

We consider the non-Hermitian XY spin chain with open boundary conditions when the anisotropy parameter is extended to complex values. By analyzing the quasi-Hamiltonian matrix, we demonstrate that the free-fermion structure of the quasi-energy spectrum coincides with that of the Hermitian model and construct the corresponding biorthogonal fermionic basis away from exceptional points (EPs). We make use of an explicit Chebyshev-polynomial representation of the open-boundary eigenvectors in which the quasi-energy $\varepsilon$ is the natural spectral variable. This quasi-energy polynomial form is particularly useful at EPs, because EPs correspond to repeated roots of the same boundary polynomial, making the construction of generalized eigenvectors by $\varepsilon$-differentiation transparent. At EPs, where the quasi-Hamiltonian becomes defective, we derive the Jordan normal form and construct the associated generalized eigenvectors, which yields the correct counting of independent many-body eigenstates. We further show that EPs act as branch points in the complex anisotropy plane, leading to the characteristic permutation of eigenenergies and eigenstates upon encirclement. The branch-cut structure of the biorthogonal eigenstates provides direct evidence for the exchange of eigenstates when an EP is encircled. These results provide an analytically controlled many-body platform for studying EP physics and non-Hermitian topology beyond momentum-space descriptions.

Drawing on well-known results from the theory of canonical extensions and the theory of categories enriched over a quantale, we define canonical extensions of quantale-enriched categories and establish their basic properties.

Hallucination is a central limitation of large language models (LLMs), and substantial effort has been devoted to understanding and mitigating it. Towards this, Kalai and Vempala (STOC 2024) introduced a probabilistic framework formalizing calibration and hallucination, and showed that, with high probability, calibrated LLMs hallucinate roughly at the rate of the "missing mass", a measure of how incomplete the training data is relative to its source. This raises two fundamental questions: (i) what property of a calibrated LLM makes hallucinations unavoidable? and (ii) can hallucinations be avoided by giving up calibration? We answer these questions by introducing a simpler property we call innovation that measures the tendency of a model to produce outputs outside the training data. We show that innovation is implied by the condition for hallucination identified by Kalai and Vempala, and, further, that it is an almost characterization of hallucination: hallucination implies innovation, and conversely, innovation implies hallucination with high probability. We also provide lower bounds on the hallucination rate based on the "innovation rate", and by relating innovation rate back to missing mass, we obtain new hallucination rate lower bounds based on missing mass that extend the results of Kalai and Vempala.

In this article we establish a large deviation principle for the empirical measures of a simple spatially inhomogeneous random walk on $\overline{\mathbb{Z}}$, the two-point compactification of $\mathbb{Z}$. The classical Donsker–Varadhan framework does not apply, since the random-walk kernel and the topology of $\overline{\mathbb{Z}}$ fall outside its standard assumptions. In certain regimes, the resulting rate function is non-convex on its effective domain. We also derive a large deviation principle for empirical means of observables $f:\mathbb{Z} \to \mathbb{R}^d$ admitting limits at $\pm\infty$. This result is optimal in the sense that in general, no large deviation principle holds for the larger class of bounded continuous functions on $\mathbb{Z}$.

Regev and Stephens-Davidowitz conjectured that the Gaussian mass $\Theta_\Lambda(t) = \sum_{x \in \Lambda} e^{-t\lVert x\rVert^2}$ of any integral lattice $\Lambda \subset \mathbb{R}^n$ is bounded above by $\Theta_{\mathbb{Z}^n}(t)$. For $n\ge 4$, we prove a saturation theorem for the natural scalar Poisson-summation certificates of this conjecture: any such certificate that is sharp at $\mathbb{Z}^n$ must interpolate the Gaussian, and have vanishing Fourier transform, at every nonzero point of integer squared norm. Applied to the lattice $E_8 \oplus \mathbb{Z}^{n-8}$, this rigidity is incompatible with the strict theta-series gap $\Theta_{\mathbb{Z}^8}(t) - \Theta_{E_8}(t) = \theta_2(it/\pi)^4\,\theta_4(it/\pi)^4 > 0$. Consequently, in dimensions $n \ge 8$, no scalar Poisson certificate can attain the sharp $\mathbb{Z}^n$ Gaussian mass bound. The same argument rules out the corresponding scalar certificate strategy for the stable-lattice formulation of the conjecture, and extends to orbit-constant graded families $\Lambda \mapsto h_\Lambda$; near-sharp sequences are similarly excluded under a uniform summability hypothesis.

Sampling is a fundamental algorithmic task in wide-ranging applications across multiple disciplines such as scientific computing, statistics and machine learning. In this paper, an efficient stochastic Runge-Kutta scheme is proposed to accelerate the Schrödinger-Föllmer sampler, designed for sampling from complex and high-dimensional multimodal distributions. The resulting stochastic Runge-Kutta Schrödinger-Föllmer sampler (SRKSFS) is proved to achieve a convergence rate of order $\mathcal{O} ( h^{3/2} |\ln h|)$ in the $L^2$-Wasserstein distance, considerably improving the order $\mathcal{O}(h)$ of the existing Euler type sampler. Obtaining the enhanced convergence rate is, however, not trivial, by noting that the drift of the diffusion process is not differentiable but only $\frac{1}{2}$-Hölder continuity with respect to the time variable. To address the difficulty, we rely on delicate error estimates to overcome the singularity due to time derivatives of the drift, at the expense of the logarithmic factor. Furthermore, the framework is extended to data-driven Schrödinger-Föllmer generation with empirical measures, enabling data-driven sampling without known density. A variety of numerical experiments are reported to validate the effectiveness of the proposed sampling algorithms.

We study an opinion dynamics model in which $n$ agents hold directed trust or distrust opinions about one another, represented as a matrix $M \in \{0,1\}^{n \times n}$ in which 1 represents trust and 0 represents distrust. A gossip event $(a, z, y)$ causes agent $z$ to adopt agent $a$'s opinion of $y$, provided that $z$ trusts $a$. We characterize the absorbing states of this process, i.e. the states in which no further opinion change can take place: we find that they are the states in which agents are partitioned into isolated factions, each faction containing a subset of core members who share mutual trust, while the remaining peripheral members trust all core members but receive no trust in return. This structure establishes a bijection between absorbing states on $[n]$ and pairs consisting of a set partition $\pi$ of $[n]$ together with a choice of non-empty subset of each faction of $\pi$. The number of such absorbing states is therefore given by OEIS A143405, with exponential generating function $\exp(\exp(x) \cdot (\exp(x) - 1))$. In addition, up to isomorphism, the count equals the number of plane partitions of $n$, given by OEIS A000219, recovering MacMahon's classical product formula $\prod_{k \geq 1} 1/(1 - x^k)^k$. Exhaustive computation for $n \leq 7$ confirms both counts.

We establish a family of parametric isoperimetric-type inequalities with multiple geometric quantities for closed convex curves. These inequalities hold under certain parameter conditions. We also prove the equality conditions. Some new inequalities and improved versions of proven inequalities are derived by adjusting the parameters. Such inequalities are stable under stronger conditions.

We prove several new results on the combinatorial structures of the unit spheres of the norms induced by Thurston's metric on the tangent and cotangent spaces of the Teichmüller space of a closed surface of negative Euler characteristic. These results include a formula for the dimension of every face of a unit sphere in the tangent space in terms of an invariant of the chain-recurrent lamination representing the face. We then prove that the combinatorial structure of such a unit sphere is independent of the underlying point in Teichmüller space. Provided the genus of the surface is $\ge$ 2, we show that there is a natural isomorphism between the extended mapping class group of the surface and the group of combinatorial automorphisms of such a unit sphere. In the case of genus 2, we obtain a natural epimorphism between the two groups whose kernel is the class of the hyperelliptic involution. Regarding the unit spheres of Thurston's metric in the cotangent spaces, we obtain a formula describing the codimensions of faces of such a sphere in terms of corresponding projective measured laminations. We then give a necessary and sufficient condition for a face to be exposed, and of a face to correspond to a projectively weighted multi-curve. Some of the results obtained answer open questions.

We generalize and prove the recursive relation \[ S_{\lambda}(V) = \sum_{L\subseteq V\text{ line}} S_{\lambda}(V \mathbin{/\mkern-5mu/} L) \] conjectured by I. G. Macdonald for his "7th variation" of the Schur functions. This variation is a family of polynomials over a finite field that mimic the (straight and skew) Schur polynomials using powers of the Frobenius.

We study both diffuse and sharp liquid-vapor interfaces. The equilibrium equation of fluids is derived by using the principle of virtual work in a domain including the interfaces. For diffuse interfaces, the surface tension coefficient depends on the density profile across the interface. For sharp interfaces, the liquid-vapor layer is mathematically represented by a geometric surface and its specific energy is a Dirac delta function at the surface. We compare the both approaches and find relations between the surface tension coefficient and parameters of the models.

We establish a lower bound for the number of non-CM Galois orbits of newforms in $S_k(N,\Psi)$ with non-trivial quadratic nebentypus $\Psi$ for sufficiently large weights. Extending the work of Dieulefait, Pacetti, and Tsaknias in the trivial nebentypus setting, we analyze the restrictions imposed by the quadratic character on local inertial types and determine the number of admissible Galois orbits of such types. We further prove that Atkin-Li pseudo-eigenvalues are Galois equivariant and hence, up to a natural equivalence relation, define a global Galois invariant. Together with existence results for newforms having prescribed local behavior, these invariants yield a lower bound for the number of non-CM Galois orbits by counting compatible pairs of local-global invariants. Finally, computations in small weights show that this lower bound is not always attained, indicating that certain local equivalences are not realized globally by Galois conjugation over the coefficient field of the modular form.

We give an explicit strong deformation retraction of the complement of a Fermat hypersurface of degree $d$ in the complex projective space of dimension $n$ into a $\Delta$-complex of real dimension~$n$. This retraction is performed in several steps, using the branched cover structure of the Fermat hypersurfaces to the degree $1$ case. Moreover, an implementation of the method in Sagemath is provided.

A transmitting intelligent surfaces (TISs) aided satellite indoor navigation system is investigated. By leveraging the unique features of TIS, we address the limitations of conventional global navigation satellite systems (GNSS) in providing reliable positioning services within indoor environments. To facilitate the extension of GNSS indoor signals, we establish an extended line-of-sight link using TIS which has the capability to change signal direction. A three-stage TIS-aided satellite indoor positioning algorithm (TSIPA), which utilizes the positions of TIS arrays and the angle of arrival, is proposed to locate indoor users. To evaluate the distribution of TIS arrays, we propose TIS position dilution of precision (TPDoP) to evaluate centroid deviation and utilize the root mean square error (RMSE) to represent compactness.

Protesting mildly against the notion of an exactly correct parametric model the view is adopted that the logistic regression equation is merely an approximation to the underlying, true function. The behaviour of likelihood based estimators is investigated in such a general framework. The maximum likelihood estimator is shown to be consistent for a certain least false parameter value minimising a weighted average of quantities that measure the distance from the true to the parametric model. Asymptotic normality is also demonstrated. Finally a number of additional remarks are offered, some pointing to natural generalisations and some to new questions for research, like weighted and local likelihood estimation methods.

In this paper we propose dynamic output-feedback controller synthesis methods for discrete-time linear time-invariant systems. The synthesis goal is either to achieve dissipativity with respect to a given quadratic supply rate, or to achieve given $H_2$ performance level. It is assumed that the autoregressive model of system dynamics is unknown, expect for the noisy disturbance term which is not part of the performance channel. Instead, we have a recorded trajectory of inputs and outputs which can be corrupted by an unknown but bounded disturbance. Methods are formulated in terms of linear matrix inequalities parametrized by a scalar variable, while in noiseless case they reduce to linear matrix inequalities. Within the considered setting, synthesis procedures are non-conservative.

The group isomorphism problem in computational complexity asks whether two finite groups given by their Cayley tables are isomorphic or not. Although polynomial-time isomorphism tests exist for many specific types of groups, no general polynomial-time algorithm is known, classes of solvable and nilpotent groups being the main obstacles. In 2012 Babai and Qiao gave a polynomial-time isomorphism test for the class of solvable groups admitting normal series with abelian Sylow factors. We generalize their result and give a polynomial-time isomorphism test for solvable A-groups, i.e. solvable groups with abelian Sylow subgroups. The algorithm heavily relies both on the computational methods developed by Babai and Qiao, and structural properties of A-groups.

Time-dependent partial differential equations (PDEs) often develop sharp fronts, localized peaks, and other moving structures that occupy only a small portion of the space–time domain but dominate the approximation error. This makes fixed or uniformly sampled collocation strategies inefficient for physics-informed neural networks (PINNs), especially in high dimensions and over long-time prediction intervals. We propose the predictive moving sample method (PMSM), which builds on the moving sample method (MSM) in \cite{xu2026moving} by replacing its full time domain iterative training with a progressive time-stepping strategy and simplifying the velocity-field loss to further reduce the per-step cost. To improve practicality for long-time prediction, we further introduce the windowed-reset predictive moving sample method (WR-PMSM), which restricts extension training to an active time window and periodically resets the reference state, thereby reducing the growth of optimization cost while preserving global consistency through a final refinement stage. Across four representative benchmarks, PMSM consistently outperforms both standard PINNs and the original MSM under matched collocation budgets. These results suggest that transporting samples according to residual dynamics provides an effective and practical route to neural network solvers for time-dependent PDEs.