Papers

Denert's statistic is a classical Mahonian statistic on permutations. Together with the excedance number, it forms an Euler–Mahonian pair, a result first conjectured by Denert and later proved by Foata and Zeilberger. Motivated by this classical result, we introduce the sorting-Denert statistic, a sorting-index analogue of Denert's statistic. It is obtained by replacing, in Denert's statistic, the inversion number of the non-excedance-letter subsequence with its sorting index. We prove that the sorting-Denert statistic, together with the excedance number, forms a new Euler–Mahonian pair.

Boston and Jones constructed a probabilistic model, called the Markov model, in order to predict the cycle structures of elements in Galois groups $G_n(f)$ associated to the $n$-th iterate of a quadratic postcritically finite polynomial $f$ over a number field. Goksel refined this model, introducing the 'even' Markov groups $M_n(f)$. These groups conjecturally contain a copy of $G_n(f)$, leading to questions about local-global conjugacies within the larger automorphism group $\operatorname{Aut}(T_n)$ of the first $n$ levels of the binary rooted tree coming from arboreal representations of the Galois groups. While the conjugacy results found by Goksel were restricted to the study of these automorphism groups, we generalise the findings to affine extensions of groups by permutation representations, using a cohomological argument. Furthermore, we provide a counterexample to the main conjecture proposed by Goksel, demonstrating that more work is required to resolve the underlying questions regarding Markov models.

The aim of this paper is to study the well-posedness and finite element approximation of the electrostatic shear Alfvén wave equations, a coupled system of two partial differential equations arising in plasma physics as a simplified sub-model of the drift-reduced Braginskii equations. To this end, anisotropic Sobolev spaces depending on the normalized magnetic field $\b$ are introduced, together with a Poincaré-type inequality along the integral curves of $\b$, which holds under a geometric directedness condition on the magnetic field. Using these tools, existence, uniqueness, and stability of a weak solution are established via the Faedo-Galerkin method. It is also shown that the geometric condition is satisfied in tokamak and stellarator configurations. A numerical scheme is then proposed, combining Lagrange finite elements in space with a Crank-Nicolson discretization in time. The scheme is shown to conserve a discrete energy exactly in the homogeneous case, and a priori error estimates are derived in the natural energy norm. Several numerical experiments are reported in two and three space dimensions, which confirm the theoretical results and indicate that the geometric condition on the magnetic field is necessary for the invertibility of the discrete system matrix.

The notion of conicality, recently introduced in [2403.00916], captures the extent to which the joint future of a finite set in spacetime uniquely determines the generating subset via its light cone structure. In the same paper it was mentioned that conicality holds for Minkowski spacetime of dimension $1+N$ with $N\geq 2$ and it has been conjectured that this property holds more generally. In this work, we show that neither homotopy with Minkowski space nor global hyperbolicity alone are sufficient for the spacetime to satisfy conicality. We then establish that causally simple, future cohesive spacetimes of dimension $1+N$ with $N\geq 2$ are conical. This class of spacetimes includes, in particular, TIP spacetimes, which can be understood as the timelike past of an observer. This is in line with the origin of causal modeling since the past of an observer is the natural domain for the description of experiments.

We show that the symplectic groups Sp(2n, K) over a field $K$ are 5-reflectional if $2n \ge 4$. If $n$ is even and $(2n, K) \ne (4, {\rm GF}(3))$ they are even 4-reflectional.

Let $\{\phi^*_n\}$ be the sequence of reflected orthogonal polynomials on the unit circle $\partial \mathbb{D}$ generated by a measure $\mu$ of Szegő class, and let $D_{\mu}$ be the Szegő function of $\mu$. We prove the uniform Cesàro asymptotics $$ \sup_{z \in \Gamma_\zeta}\Biggl(\frac{1}{n}\sum_{k = 0}^{n-1}\Bigl||\phi_k^*(z) D_{\mu}(z)|^2 - 1\Bigr|\Biggr) \to 0, \qquad n \to \infty, $$ for almost all Stolz angles $\Gamma_{\zeta}$, $\zeta\in \partial \mathbb{D}$. This extends a well-known asymptotic result of Máté, Nevai, and Totik (1991) from the local scale $O(1/n)$ near $\partial \mathbb{D}$ to the global scale $O(1)$. We also study asymptotic behavior of arguments of orthogonal polynomials and extend a classical theorem due to Grenander and Szegő using a new technique. As an application, we derive global asymptotic results for polynomial reproducing kernels under various assumptions on the orthogonality measure.

This paper extends the numerical method introduced by de Pinho et al. [22] and later developed in Nour and Zeidan [33, 37] to the nonautonomous case where the nonsmooth sweeping set depends explicitly on time. The moving character of the sweeping set creates substantial additional difficulties, since several geometric constants involved in the analysis may a priori depend on time. To overcome this issue, we establish uniform geometric estimates for the moving constraint sets and their boundaries. These estimates allow us to extend the discrete approximation scheme developed in [22, 33, 37] and prove its convergence toward admissible solutions of the original problem. Several examples illustrating the applicability of the proposed method are also presented.

We consider reversible interacting particle systems with conserved number of particles. A standard variational formulation describes the diffusion coefficient of such models as the infimum of a certain functional. The purpose of this paper is to derive a new, alternative, variational characterization, as the supremum of another functional. This is a more natural framework when one is interested in obtaining lower bounds on the diffusion coefficient. We present a specific example of a kinetically constrained lattice gas where this variational principle can be applied.

We consider the Schrödinger equation associated with the harmonic oscillator and show that if the initial data and its Fourier transform are dominated by Gaussian functions of widths $a>0$ and $b>0$, respectively, satisfying $ab<1$, then the evolved solution and its Fourier transform are dominated by a Gaussian of width $\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}- \sqrt{\left(\frac{1}{a}+\frac{1}{b}\right)^2-4}\right),$ for all times except for a discrete set, and for all times in one dimension. In the one-dimensional case, we prove that these estimates are sharp. Moreover, for a more restrictive class of initial data, we establish sharper time-dependent Gaussian bounds.

In this paper, we extend sequence-class methods from linear and multilinear theory to the Lipschitz setting, highlighting the substantial differences that arise from the lack of linearity. First, we establish a general criterion for lifting a Lipschitz mapping between Banach spaces to a Lipschitz operator between sequence spaces, and we use it to define the class $\Pi_{(Z,Y)}^{\mathrm{Lip}_0}$ of $(Z,Y)$-summing Lipschitz functions, where $Z$ and $Y$ are sequence classes. We then introduce the notion of Lipschitz stable sequence class and show that $[\Pi_{(Z,Y)}^{\mathrm{Lip}_0},\pi_{(Z,Y)}^{\mathrm{Lip}}]$ is a Banach Lipschitz ideal whenever the sequence classes satisfy this property. As applications, we present Rademacher type and cotype for Lipschitz functions and identify them with $(Z,Y)$-summing Lipschitz spaces for concrete choices of sequence classes. We prove that the type case forms a Banach Lipschitz ideal, whereas the cotype case does not, and we analyse composition and maximality for the type ideals.

We study critical points of the Ginzburg-Landau energy functional $$\mathcal{F}_\varepsilon[u] = \int_\Omega \Big[ \frac{1}{2}|\nabla u|^2 + \frac{1}{2\varepsilon^2} W(1 - |u|^2)\Big]\,dx, \quad u \in H^1(\Omega, \mathbb{R}^N),$$ with $\Omega \subset \mathbb{R}^M$, $\varepsilon>0$, $M,N \geq 2$ and general conditions on the non-negative potential $W$ allowing for super-quadratic behaviour near its zero set. Under a Dirichlet boundary data of unit-length on $\partial \Omega$, we prove the following maximum principle: every critical point $u_\varepsilon$ satisfies the global uniform bound $|u_\varepsilon| \leq 1$ in $\Omega$. Furthermore, if a family of critical points $(u_\varepsilon)$ converges (in energy) to a smooth $\mathbb{S}^{N-1}$-valued harmonic map in the limit $\varepsilon \to 0$, then we prove global uniform bounds for $(\Delta u_\varepsilon)_{\varepsilon>0}$ in $\Omega$ and, in particular, global Hölder convergence of the gradients $(\nabla u_\varepsilon)$ in $\Omega$ as $\varepsilon \to 0$.

Self-adjoint extensions of a symmetric operator are parametrised by Krein's formula, in which the $Q$-function interacts with another analytic function (the parameter). We obtain a characterisation of the eigenvalues, isolated or not, of a given self-adjoint extension in terms of these two functions. The setting is highly general, covering symmetric operators with arbitrary defect in a Hilbert or Pontryagin space. Of independent interest is our newly developed tool, the generalised value of a generalised Nevanlinna function, for which we give both a function-theoretic and an operator-theoretic description.

Given a von Neumann algebra with a semifinite normal faithful trace, the notion of a core decreasing operator is introduced. This construction depends on a family of commuting projections called an ordered core. With these tools in place, a version of the level function of a measurable operator is introduced. The down space of a strongly symmetric space of $\tau$-measurable operators is defined.

We introduce the calibration energy for oriented immersions into Euclidean space, quantifying the deviation from calibrated geometry. A key property is that this energy may remain finite for infinite-volume immersions, while a null-Lagrangian structure ensures that it has the same first variation as the volume functional. We establish an exact dissipation identity for the calibration energy along proper oriented mean curvature flows in arbitrary dimensions and codimensions, under a mild local-volume bound. In fact, our result covers a class of singular calibrations and singular initial data. Even in the smooth setting, this provides a new finite variational framework for mean curvature flow beyond the finite-volume regime. As a main application, we establish a general dynamical rigidity theorem for calibrated cones in arbitrary codimension: no singular calibrated cone can be desingularized by a proper oriented mean curvature flow. Our framework further yields novel rigidity theorems for solitons and convergence for two-dimensional immortal flows.

Let $A$ be a finite universal algebra. Then the orders of the $n$-generated free algebras $F_n$ in the variety (equational class) generated by $A$ satisfy G. Birkhoff's inequality: $|F_n|\le |A|^{|A|^n}$ for $n=1,2,\dots$ It follows that $\limsup_{n\to\infty}\sqrt[n]{\log |F_n|}\le |A|$. When $A$ is a finite group or a finite nonassociative algebra, we obtain a criterion for equality in this estimate; equivalently, a criterion for maximal growth of the sequence $\{|F_n|\}_{n=1}^{\infty}$.

Recently, Chen \cite{Chen2026} proved that Talagrand's Boolean convolution conjecture holds up to the dimension-free factor \((\log\log\eta)^{3/2}\), namely for every fixed \(\tau>0\), \[ \mu\{P_\tau f>\eta\|f\|_1\} \le C_\tau \frac{(\log\log\eta)^{3/2}}{\eta\sqrt{\log\eta}}, \qquad \eta>e^3. \] We revisit the terminal testing-discrepancy step in Chen's perturbed reverse-heat coupling. Chen estimates this discrepancy globally in terms of the remaining gap to the terminal level. We keep the same coupling and the same reverse-heat formulations, but localize the terminal discrepancy on each remaining-gap layer before summing the layers. This changes the fixed-time anti-concentration cost from order \((\log L)^{3/2}/\sqrt L\) to order \((\log L)/\sqrt L\), where \(L=\log\eta\). Consequently, we obtain a \((\log\log\eta)^{1/2}\) improvement as \[ \mu\{P_\tau f>\eta\|f\|_1\} \le C_\tau \frac{\log\log\eta}{\eta\sqrt{\log\eta}}, \qquad \eta>e^3. \]

Classical curvature formulas for Lie groups and Riemannian submersions are recast in terms of Lie groupoids with right-invariant source-fibre metrics and Riemannian Lie algebroids. We derive a sectional-curvature formula for such groupoids extending the Arnold-Milnor “1-2-3-4” formula. For transitive Riemannian Lie algebroids, we factor the base and vertical Levi-Civita connections through Lie algebroids of metric derivations and obtain Lie-algebroid analogues of O'Neill's curvature formulas. Applied to the action of the diffeomorphism group of the torus on densities, these formulas recover Arnold's curvature of the measure-preserving diffeomorphism group and give explicit Wasserstein sectional curvatures at the uniform density. Finite-dimensional examples include the rotational action algebroid of the round sphere and the WGS84 ellipsoid, and a smooth family of negative curvatures on non-abelian isotropy groups over the real projective line.

We propose a simple two-stage order selector for finite-order ARFIMA models. First, a preliminary log-periodogram estimate of the memory parameter is used to fractionally filter the data. Second, a Hannan–Rissanen residual construction is applied to the filtered series, and the autoregressive and moving-average orders are selected by a generalized information criterion over a growing candidate rectangle. The search bounds are allowed to satisfy \(P_n,Q_n\to\infty\), whereas the true orders remain fixed and finite. The penalty is allowed to be larger than the ordinary BIC penalty so that it dominates the error introduced by preliminary long-memory estimation and by the Hannan–Rissanen residual approximation. We prove a uniform residual-variance approximation over the growing rectangle and combine it with a population separation argument between the true finite ARMA representation and underfitted alternatives. The resulting generalized information-criterion selector is consistent.

The goals of the paper are 3-fold. First, we revisit the construction of imaginary Whittaker modules over untwisted affine Kac-Moody Lie algebras. These modules are obtained using the parabolic induction from irreducible Whittaker modules over the associated Heisenberg Lie algebras. We show that the infinite support condition for Whittaker functions on Heisenberg Lie algebras is essential for irreducibility: when the support is finite the modules becomes reducible, yielding infinite chains of submodules. We establish the irreducibility criterion for the induced modules over affine Lie algebras and construct a large family of such modules. In particular, we obtain a class of irreducible modules on which the derivation acts neither semisimply nor freely. Second, we consider quantum analogs of imaginary Whittaker modules and establish irreducibility for a family of such modules. Finally, we prove the irreducibility of a certain class of modules over $\mathcal{U}_q(A_1^{(1)})$, which are not quantum deformations of irreducible modules for the affine Kac-Moody Lie algebra $A_1^{(1)}$. Our results can be potentially extended to all types of untwisted quantum affine algebras, providing a pathway toward their classification.

We consider a quasi-birth-and-process (QBD) that duplicates itself at some fixed times within a tree that contains information about duplication times and potentially partially observed states. We analyse a continuous trait by discretising it to obtain the QBD level variable. Then, the phase variable is used to model the dynamics of the underlying environment. Here, we extend the framework of Soewongsono et al. to enable a more general analysis. We develop an efficient recursive algorithm for computing the likelihood of an observed tree under this model and construct several numerical examples to illustrate its application potential. Through our synthetic data examples, we show a range of potential behaviours that could be modelled with this approach. Further, we apply the framework to two empirical examples from comparative phylogenetics (the evolution of range area and body size traits across a phylogeny of 49 mammals) to gain different insights into the evolution of these continuous traits. In this setting duplication of the QBD represents speciation and continuous trait evolution is modelled in a discretised state space. In our empirical examples, we explore the impact of different parameter choices on the corresponding likelihood of observing a given phylogenetic tree and the observed levels at its tips.

We investigate the effects of the exponential potentials typical of non-supersymmetric strings in cosmologies whose spatial and internal slices are maximally symmetric spaces with curvatures labeled by a pair of integers $k$ and $k'$ ($=\pm 1$). We classify the solutions according to their singularity structure and asymptotic behavior and present a semi-quantitative picture of the generic dynamics in the physically most relevant cases with flat spatial slices. The analysis relies on exact solutions emerging when one of the effects dominates, special solutions arising when two or more effects are comparable, scaling asymptotics, and some numerical tests.

We construct a picture-valued invariant of classical knots by using the group $G_n^3$ and the plat closure of braids. We introduce a modification of the group $G_n^3$ and define a map from the braid group to framed 6-valent graphs with leaves. Each 6-valent vertex corresponds to the moment when three points become collinear. By using this construction, we obtain a knot invariant valued in equivalence classes of graphs modulo local moves. Our invariant provides a new graphical approach to the study of classical knots.

Integrated data and energy transfer (IDET) is a promising technique for supporting sustainable low-power wireless networks. To improve both communication reliability and energy transfer efficiency, this paper investigates a fluid index modulation (FIM) assisted IDET system, where the base station employs a two-dimensional fluid antenna system (FAS) and the receiver adopts a power-splitting architecture. In FIM, the information bits are delivered not only from the modulation symbols, but also the index of antenna position. Under finite-alphabet signaling, the average harvested power, bit error rate (BER), and achievable data rate are derived in closed form. A joint optimization problem is formulated to maximize the average harvested power subject to BER and achievable rate constraints by jointly optimizing the port selection, precoding vector, and power splitting ratio. An alternating optimization framework is developed, where the precoding vector and port selection are obtained via a Riemannian augmented Lagrangian method (RALM) and block coordinate descent (BCD) algorithm, respectively. Simulation results demonstrate that the proposed scheme achieves a superior rate-energy trade-off over benchmark schemes, while the proposed algorithm attains near-optimal performance with significantly lower complexity than exhaustive search.

In this article, we establish quantitative uniqueness results for a class of functions defined on a quasi-analytic compact manifold $X$ without boundary. This function class is characterized by the iterates of a positive elliptic linear differential operator on $X$ and, notably, encompasses all functions with a finite spectrum. By employing a relatively dense observable set, we extend classical Logvinenko-Sereda-type results to this quasi-analytic framework. Furthermore, we demonstrate that if these functions satisfy a doubling property, observability holds from any measurable set of positive measure. Our results generalise the propagation of smallness from finite sums of eigenfunctions to infinite sums with an appropriate energy-parameter decay, thereby extending recent findings by Kukavica-Li (Proc. Lond. Math. Soc., 2025) to the quasi-analytic setting.

We prove instances of the subnormaliser conjecture on character bijections for finite groups by deriving generic versions of such bijections for unipotent characters of nearly simple groups of Lie type. For this, we formulate an extension of $d$-Harish-Chandra theory to what we call generic subnormalisers, which are certain, usually disconnected, reductive subgroups of a simple algebraic group. For very good primes the generic bijections give rise to bijections to certain subgroups that should contain subnormalisers and thus be suitable for an eventual inductive approach. If the group in question has abelian Sylow $\ell$-subgroups for some prime~$\ell$ then our bijections satisfy the properties predicted by the subnormaliser conjecture, and moreover preserve character values up to sign, Brauer $\ell$-blocks, and are Galois equivariant.

In this article, we establish a quantitative uniqueness theorem for quasi-analytic functions defined on compact, connected Lie groups $G$ and on homogeneous spaces $G/H$, where $H$ is any closed subgroup of $G$. Our result extends classical Logvinenko-Sereda-type theorems to the setting of quasi-analytic functions on compact Lie groups and their homogeneous spaces. We introduce the quasi-analytic class of functions using iterates of the Casimir operator on $G$. This construction is justified by establishing that every function in this class possesses the strong unique continuation property. In particular, our result extends a result of P. Chernoff (Bull. Amer. Math. Soc., 1975) to the framework of compact Lie groups and their homogeneous spaces.

Our main results establish two rigidity phenomena in the class of time changes of a fixed product of Anosov flows. Our first result shows that two time changes having the same stabilizers for all periodic orbits are conjugate up to automorphism. The second rigidity result proves that if the stabilizers of periodic orbits can be simultaneously diagonalized, then the time change is conjugate to a product of flows up to automorphism. We allow our time changes to be Hölder continuous, which by structural stability implies that our results hold for $C^1$ perturbations of products of Anosov flows. We apply our main results to $C^1$ time changes of products of Anosov flows on $3$-dimensional manifolds. For such actions, we show that being totally Anosov, being conjugate to a product of flows and having the kernels of Lyapunov functionals not depend on the periodic orbits are all equivalent properties. We also build counterexamples to the Katok-Spatzier conjecture as time changes of products of any transitive Anosov flows, extending a result of Vinhage beyond the continuously accessible case.

We revisit the problem of sequentially testing the mean of bounded distributions in a level-$\alpha$ power-one framework. We study a $\mathrm{KL_{inf}}$-based sequential test that is known to attain the information-theoretic lower bound on the expected stopping time with exact constants as $\alpha \to 0$. Going beyond first-order asymptotics, we establish a central limit theorem (CLT) for the stopping time of this test. Our analysis proceeds in two steps. First, we prove a novel CLT for the $\mathrm{KL_{inf}}$ statistic itself, characterizing its fluctuations around its deterministic limit. We then leverage this result to show that the stopping time, centered appropriately and scaled by $\sqrt{\log(1/\alpha)}$, converges in distribution to a Gaussian limit with an explicit variance. This yields a second-order characterization of an asymptotically optimal sequential test for bounded distributions. Finally, we present numerical experiments that corroborate our theoretical findings.

This study addresses the boat river-crossing problem under non-uniform flow velocities by constructing three models: constant flow (Model 1), linear distribution (Model 2), and even-power function distribution (Model 3, adjustable via parameter n ). By using the vector addition, combined with the solutions of calculus and differential equations, the analytical expression of the ship's spatial trajectory under a fixed heading angle relative to the water flow is derived. For the shortest-time control problem, the Lagrange multiplier method is introduced to construct a constrained optimization model, and the analytical solution of the optimal heading angle that satisfies the boundary condition of reaching the direct opposite bank is solved. The research results provide theoretical support for the path planning of inland ship intelligent navigation systems, and the proposed multi-model analysis framework can effectively simulate the complex flow velocity distribution scenarios of real rivers.

Let $p>3$ be a prime and $b\ge 2$ an integer such that $p$ does not divide $b$. Then $1/p$ has a periodic digit expansion with respect to the basis $b$. The length $m$ of the period is the (multiplicative) order of $b$ mod $p$. In the case $m=(p-1)/2$, the frequency of each digit $k\in\{0,1,\ldots, b-1\}$ can be expressed in terms of generalized (first order) Bernoulli numbers. In some cases only Bernoulli numbers belonging to quadratic characters occur. This means that the frequencies can be written in terms of class numbers of imaginary quadratic number fields (the so-called class number case). In the present paper we classify the numbers $p$ and $b$ falling under the class number case. We also highlight one of the simplest examples not falling under this case ($p\equiv 1$ mod $4$, $b=10$) and the most complex example of the class number case. Moreover, we show that knowing the frequency of each $k$ is equivalent to knowing the respective Bernoulli numbers.