Papers

We prove a fractal analogue of the weighted theorems of Khintchine and Schmidt for products of self-similar measures on the real line whose defining iterated function systems share a common contraction ratio. The proof translates the counting problem into homogeneous dynamics and builds on recent work of Bénard, He and Zhang, who treated the unweighted case. Our main new ingredient is a cone upgrade: an effective double equidistribution theorem for translates of product self-similar measures, uniform over diagonal elements whose weights range over a fixed cone in the positive Weyl chamber, deduced from effective equidistribution for a single weight. This yields a partial effective form of a theorem of Khalil, Luethi and Weiss.

For a finite abelian group $G$ and $k \geq 1$, the generalized Davenport constant $D_k(G)$ is the least $\ell$ such that every sequence over $G$ of length at least $\ell$ has $k$ pairwise disjoint nonempty zero-sum subsequences. A theorem of Freeze and Schmid gives $D_k(C_5^3) \geq 5k+10$ for every $k \geq 2$. We prove the matching upper bound: $D_4(C_5^3)=30$, and hence $D_k(C_5^3)=5k+10$ for every $k \geq 2$, so the Freeze–Schmid bound is attained by $C_5^3$ from $k=2$ onward, as it is by $C_2^3$ and unlike $C_3^3$. The proof is finite and computer-assisted. The remaining case reduces to showing that every zero-sum sequence of length $31$ over $C_5^3$ contains a nonempty zero-sum subsequence of length at most five. A saturation argument confines the multiplicities of a hypothetical counterexample to $\{1,2,4\}$, its support pattern to one of $60$ solutions of two linear equations, and its geometry to one of $78$ rank/plane branches normalized to a standard basis; an exhaustive search exhausts every branch with no survivor. The search was carried out by three independently written implementations, and the branch cover was regenerated by separate programs from the lemmas alone; two further machine-verified values, $D_3(C_5^3)=25$ and $s_{\leq 6}(C_5^3)=24$, enter the second statement, and their records accompany the paper.

We revisit the kinetic electrostatic electron nonlinear (KEEN) waves studied by Afeyan et. al in 2014 using a hybrid flow-mapping strategy that combines the characteristic mapping method (CMM) with numerical flow iteration (NuFi). The study extends the classical setup to dynamical ions, comparing their impact on the long-time KEEN dynamics with the static-ion case. To this end, we extend the CMM-NuFI framework with a multi-map strategy, assigning one map to the ion and one to the electron characteristic flow. The resulting problem exhibits a wide separation of spatial and temporal scales, driven by the fine structures generated by the ponderomotive force and by the large ion-to-electron mass ratio, which renders it computationally prohibitive for conventional grid-based methods. Our multi-map CMM-NuFI method efficiently resolves these disparate scales, enabling long-time simulations of KEEN dynamics with fully dynamical ions.

We investigate the restoration of isotropy for anisotropic diffusions on pre-Sierpiński carpets in the plane, a problem previously studied by Barlow, Hattori, Hattori and Watanabe \cite{BHHW} in which they obtained a weak homogenization property for the ratio of effective resistances of the anisotropic diffusions. We establish the weak convergence of these anisotropic diffusions to Brownian motion on the Sierpiński carpet. As a consequence, we give an affirmative answer to the strong homogenization conjecture raised in \cite[p.3]{BHHW}.

Testing whether two independent samples arise from the same underlying distribution is a fundamental statistical problem. We propose a new class of two-sample distribution tests based on a family of probability metrics $\Delta_{n,p}$, constructed from repeatedly integrated quantile functions. On their respective domains, these metrics are proved to be genuine distributional distances. The case $n=1$ recovers the $p$-Wasserstein distance, which requires finite $p$-th moments; for $n\geq2$, the proposed metrics are well defined and require only finite first moments. The asymptotic properties of the plug-in statistic are established, including strong consistency and limiting distributions under the null and fixed alternatives. A permutation calibration for finite-sample inference is also proposed. We further derive an asymptotic power function under local alternatives. Finally, the finite-sample performance of the proposed tests is examined through simulation studies, and their reduced sensitivity to extreme upper-tail observations is illustrated through a real data application.

We construct a weak Lie 3-group $T_2 \mathbb B^{F_1}_n$ from the 2-category of gerbes over $\mathbb R^n / \mathbb Z^n$ and the $\mathbb R^n/ \mathbb Z^n$-action on it by pullback along translations. We also construct a homotopy equivalence between $T_2 \mathbb B^{F_1}_n$ and a different Lie 3-group $T_2 \mathbb D^{F_1}_n$, which admits a dimensional reduction to the homotopy equivalence of Lie 2-groups introduced by Nikolaus and Waldorf to model half-geometric T-duality. This is a first step towards establishing a higher form of T-duality for 2-gerbes over torus fibrations, relevant to supergravity and M-theory.

The statistical analysis of heavy-tailed data has received considerable attention because extreme observations frequently arise in many practical applications. The Pareto type-I distribution is a fundamental heavy-tailed model used in economics, finance, actuarial science, insurance, reliability, and extreme value analysis. In this paper, we propose novel goodness-of-fit tests for the Pareto distribution using a mean residual life characterization. The test statistic is constructed using U-statistic theory, and its asymptotic behaviour is established under both the null and alternative hypotheses. Its finite-sample performance is evaluated through Monte Carlo simulations using maximum-likelihood and method-of-moments estimation and compared with existing tests. The results show that the proposed test controls the nominal significance level and performs competitively in terms of power across a broad range of alternatives. Finally, the proposed methodology is illustrated using the Danish fire insurance loss and pollution datasets.

We analyze the numerical approximation of time-harmonic scattering by highly heterogeneous penetrable obstacles. These problems are especially challenging in the high-frequency regime, where the size of the scatterer $L$ is much larger than the wavelength, i.e., the wavenumber $k$ is such that $kL \gg 1$. Here, we further consider the situation where the scatterer contains different materials, with a characteristic size $\varepsilon$ such that $k\varepsilon \ll 1$. We propose a high-order multiscale finite element method, and provide an error analysis that is explicit in both $k$ and $\varepsilon$. Crucially, our error estimates suggest that using a high-order method should reduce the computational cost for large frequencies, which is corroborated by numerical examples.

In this letter, we study finite-horizon linear-quadratic Stackelberg difference games with coupled-affine state-control inequality constraints. Under the stated assumptions, we show that generalized open-loop Stackelberg equilibria admit an exact reformulation as an optimal control problem subject to a discrete-time linear complementarity system. Eliminating the dynamic variables yields a large-scale linear complementarity quadratic program, together with an explicit recovery map for the follower strategy. This reformulation enables the numerical computation of these equilibria. We illustrate the proposed approach using a constrained network-flow game.

Minimum-weight satisfiability asks for a satisfying Boolean assignment of minimum cost, while many fuzzy decision and knowledge systems operate with graded positive and negative relations rather than crisp literals. This paper introduces a bipolar fuzzy minimum-weight satisfiability problem in which positive and negative clause–variable relations, target clause levels, and variable states are represented in the unit interval. The clause requirements are expressed by bipolar fuzzy relational equalities defined with an arbitrary continuous t-norm. Classical minimum-weight CNF satisfiability is recovered as a crisp special case, independently of the selected t-norm, and an optimal Boolean solution is shown to exist in that specialization even though the proposed formulation uses continuous variables. For the graded problem, clause satisfaction is characterized through admissible variable domains and effective clause-witness sets. This structure leads to an exact clause-witness branch-and-bound method with preprocessing, propagation, lower-point closure, and a disjoint-witness packing bound. Computational experiments verify the solver against explicit witness enumeration, mixed-integer baselines, and external SATLIB formulas; on a focused graded ablation set, the packing bound reduces the median number of explored nodes by about 45 percent relative to the single-row witness bound.

We present a unifying framework to construct local Fortin operators for conforming mixed finite element pairs for the Stokes equations. The operators are constructed to satisfy the divergence-preservation property, local stability and approximation properties, and certain trace-preservation properties. For the latter, some of the finite element pairs require an enrichment of the velocity space. We present the construction for the $P_d-P_0$ element, the Bernardi-Raugel element, the conforming Crouzeix-Raviart element, a modified MINI element and generalised Taylor-Hood elements. Furthermore, we discuss implications of the existence of such a Fortin operator beyond inf-sup stability. These include applications to non-Newtonian fluid flow, problems with inhomogeneous Dirichlet boundary conditions, as well as uniform inf-sup stability relevant for domain approximation and moving domains.

Let $\mathcal I_n^B$ be the involutions of the hyperoctahedral group $\mathfrak B_n$, and let $\des^B$ denote the descent number with respect to the natural Coxeter order. We derive the fixed-point-refined Worpitzky identity \[ \sum_{n\ge0}\frac{\mathcal F_n(p,t)\,z^n}{(1-t)^{n+1}} =\sum_{m\ge0} \frac{(1+pz)^m\,t^m}{(1-pz)^{m+1}(1-z^2)^{m(m+1)}}, \quad \mathcal F_n(p,t)=\sum_{\pi\in\mathcal I_n^B}p^{\fixB(\pi)}t^{\des^B(\pi)}. \] Extracting the stratum with $j$ two-cycles and $f$ fixed positions yields a one-parameter deformation of the fixed-point-free Worpitzky series of Wan, Gao, Li and Yang. After the change of variables $x=t/(1+t)^2$, this deformation becomes a positive binomial transform. More precisely, if \[ P_j(x)=\sum_sD_{2j,s}x^s \] is the fixed-point-free $\gamma$-polynomial, then the transform coefficients $A_{j,r}(x)$ are determined by \[ \sum_{r\ge0}A_{j,r}(x)W^r =\sum_{s=0}^{j}D_{2j,s}x^s (1+4xW)^{2j-2s}(1+2W+4xW^2)^s, \] and the $\gamma$-polynomial of the $(j,f)$-stratum is \[ \Phi_{j,f}(x)=\sum_{r=0}^{f}\binom fr A_{j,r}(x). \] This manifestly positive transform is the main structural result of the paper. As consequences, every fixed cycle-type stratum is $\gamma$-positive and $\mathcal F_n(p,t)$ is coefficientwise $\gamma$-positive in the fixed-point variable $p$. The cases $f=0$ and $p=1$ recover, respectively, the fixed-point-free theorem of Wan–Gao–Li–Yang and the all-involution theorem of Cao–Liu. We also give explicit formulas for the first binomial layers and for the strata with one and two two-cycles.

We determine the ozone group of \(U_q^{+}(B_2)\). More precisely, if \(\ell=\operatorname{ord}(q^2)\) when \(q\) is a root of unity, we obtain $$ \operatorname{Oz}(U_q^{+}(B_2)) \cong \begin{cases} \mu_2 , & \text{if \(q\) is not a root of unity},\\[2mm] \mu_{\gcd(\ell,2)}, & \text{if \(q\) is a primitive \(m\)-th root of unity, \(m\geq5\).} \end{cases} $$ We also study several homological properties of \(U_q^{+}(B_2)\), showing that it is Artin–Schelter regular of global dimension \(4\), Auslander-regular, Cohen–Macaulay, strongly Noetherian, and skew Calabi–Yau. Using two different iterated Ore presentations, we show that its Nakayama automorphism is trivial for every $q \in \K^*$, so $U_q^{+}(B_2)$ is Calabi–Yau of dimension $4$; for $q$ not a root of unity this also follows from work of Goodearl and Yakimov. When $q$ is a primitive $m$-th root of unity with $m\geq5$, we deduce that the center is Gorenstein, we check directly that the ozone group acts with trivial homological determinant, and we describe the nonzero normal elements in terms of the ozone group.

We prove that every planar Jordan curve of finite \(p\)-variation, with \(1\leq p<2\), inscribes a rectangle of every prescribed similarity class. In particular, every such curve inscribes a square. The proof combines the recent criterion of Asano and Ike with a variation-controlled approximation argument. We show that for every \(q>p\), a Jordan curve of finite \(p\)-variation can be approximated in the \(q\)-variation topology by smooth Jordan embeddings. The construction uses simple polygonal interpolants of Boedihardjo and Geng, an elementary interpolation inequality between variation seminorms, and a variation-controlled smoothing of polygonal embeddings. Young integration then gives locally uniform convergence of the associated primitives.

In this paper, we give a classification of the minimum weight codewords of Schubert codes $C_\alpha(\ell, m)$ by settling the conjecture proposed by Ghorpade and Singh in 2018, for all values of $q$ and all $\alpha$. We use this classification to prove that a codeword of the Grassmann code $C(\ell, m)$ has the second minimum weight if and only if it is indexed by an element of $\bigwedge^{m-\ell}V$ that can be written as the product of a decomposable $(m-\ell-2)$-vector and an alternating $2$-vector of rank $4$. Finally, we give an enumeration of the second minimum weight codewords of the Grassmann code.

Gaussian kernel sums are the computational core of maximum mean discrepancies (MMDs), kernel gradient flows, Stein variational gradient descent (SVGD), and many other kernel methods. At the same time, softmax attention has received an extraordinary amount of hardware-aware code engineering, culminating in flash attention. We show that Gauss kernel sums with arbitrary, signed weights can be evaluated via flash attention: two small input augmentations turn the normalized softmax reduction into the unnormalized Gauss sum, without writing a single line of custom GPU code. For feature dimension D>8 in fp16, this approach beats compiled PyTorch code as well as PyKeOps kernels (often significantly) in speed, memory-overhead and accuracy. Indeed, its memory scaling remains linear.

Starting from the Riemann–$\Xi$ specialization of Weil's explicit formula, we construct finite real-symmetric Prime–Weil matrices $(\mathbf S)$ from pole, archimedean, and finite prime-power data. This construction is a finite-dimensional arithmetic model within the Hilbert–Pólya program, which seeks a self-adjoint spectral realization of the nontrivial zeta-zero parameters. Their off-diagonal entries form a Loewner-type divided-difference matrix with a rank-two displacement identity. We formulate the spectral quotient as a Hermitian definite generalized eigenproblem on the fixed zero-mean contrast space. This realization is invariant under positive affine rescaling, avoids ground-vector normalization and an ill-conditioned oblique projector, and preserves the finite quotient spectrum. For a dimension-matched zero-side matrix built from $N$ distinct positive ordinates $\gamma_k$, rational interpolation gives the exact contrast-pencil spectrum $\{\pm\gamma_1,\ldots,\pm\gamma_N\}$ and positive-parity square spectrum $\{\gamma_1^2,\ldots,\gamma_N^2\}$. Since the ordinates are inputs, this is a reconstruction theorem. Assuming RH, the same interpolation vector proves $\lambda_{\min}(\mathbf S)\to0$. At $N=L=13$, Lemke's ground-state quotient and the contrast pencil agree numerically and reproduce the first three zeta ordinates to the reported precision. The remaining problem is a relative prime-to-zero perturbation theorem with uniform control of the compressed metric. No proof of RH is claimed.

We develop a finite-corner framework for convex cones of normal maps between \(\mathcal B(\mathcal H)\) and \(\mathcal B(\mathcal K)\). The basic structural assumption is stability under finite cut–pad operations. We prove that every point-ultraweakly closed cut–pad stable cone is completely determined by its finite-dimensional corner cones, and conversely that every coherent family of closed finite-dimensional corner cones admits a unique global realization of this type. For a cut–pad stable cone that is not necessarily point-ultraweakly closed, reconstruction from the norm closures of its corner cones yields exactly its point-ultraweak closure. Combining this reconstruction principle with finite-dimensional separation and the Choi representation, we show that non-membership in a global cone is always detected on a single finite-dimensional corner by a finite-dimensional witness. We illustrate the framework for completely positive and decomposable maps; in the decomposable case, the Hermitian parts of the finite-corner witnesses are PPT.

Operator splitting provides an attractive approach for the numerical integration of (coupled) port-Hamiltonian systems, as it allows the underlying system structure to be exploited at the level of the individual subproblems. However, the choice of the decomposition is not unique and may strongly affect both the computational efficiency and the preservation of the energy behavior of the original system. In this work, we investigate this interplay systematically and introduce energy consistency as a criterion for assessing splitting methods for port-Hamiltonian ordinary differential equations. We derive sufficient conditions under which a splitting based on a given decomposition inherits the energy behavior of the continuous system and use these conditions to analyze several decomposition strategies for coupled port-Hamiltonian systems. In particular, we compare decompositions that preserve the structure with approaches that exploit lower-dimensional subsystem dynamics or separated time scales. The analysis is complemented by numerical experiments using Strang splitting and its multiple-time-stepping extension. A scalable electro-thermal benchmark with fast electrical and slow thermal dynamics is employed to assess accuracy, energy behavior, and computational efficiency. The results demonstrate that preserving the port-Hamiltonian structure of the subflows is essential for energy-consistent splitting, whereas decompositions that exploit subsystem structure or time-scale separation can provide substantial computational advantages. In particular, the time-scale decomposition yields significant efficiency gains for systems with pronounced multirate characteristics, while structure-destroying decompositions may lead to undesirable energy behavior.

This paper studies physical-layer authentication (PLA) in the near-field regime under impersonation attacks. Unlike the far-field case, where the steering vector depends only on the angle of arrival (AoA), in the near field it depends on both angle and distance. We analyze the attack by minimizing the mean-square error (MSE) between the signal received from a legitimate transmitter Alice and the signal generated by an active attacker Eve. For a single-antenna Eve, we derive the optimal scalar precoder and show that, for an inter-element spacing no larger than half a wavelength and under the standard second-order Fresnel approximation, perfect impersonation is possible only if Eve has the same angle and the same distance from Bob as Alice. We then extend the analysis to a multi-antenna Eve and derive the optimal precoding vector. In this case, perfect impersonation is possible only if Alice steering vector belongs to the subspace spanned by Eve steering vectors. Simulation results show that, in the near field, a distance difference is sufficient to prevent a successful impersonation attack even when Alice and Eve have the same AoA, and confirm the analytical results.

A symmetric $(v,k,\lambda)$-design is said to be $f$-pyramidal, with $f<v-1$, under the action of a group $G$ if $G$ acts as an automorphism group fixing $f$ points and acting sharply transitively on the remaining ones. We show that, necessarily, $f\leq v -2(k-\lambda)$. In particular, for a Hadamard design with parameters $(v,k,\lambda)=(4m-1, 2m-1, m-1)$, for some $m\in\mathbb{N}$, it follows that $f\leq 2m-1$. Recently, working on the complement design, the family $P$ of Hadamard $(4m-1,2m-1,m-1)$-designs admitting an abelian $(2m-1)$-pyramidal automorphism group $G$ has been completely determined in the case $m=2^{k}>1$. In this paper, we generalize that result by showing that $P$ coincides with the family of Hadamard $(4m-1,2m-1,m-1)$-designs admitting a $(2m-1)$-pyramidal automorphism group, without assuming either that the group is abelian or that $m$ is a power of $2$.

The classic recombination equation, both in discrete and in continuous time, can be solved in a way that derives from the Markov chain of a partitioning process. Here, we revisit this structure from the point of view of the Markov embedding problem. In particular, we analyse when a discrete-time Markov matrix of recombination type can occur in a time-homogeneous Markov semigroup that is generated by a recombination rate matrix. En route, we also show that such rate matrices (or Markov generators) generally do not form a matrix algebra, but span a real Lie algebra.

We analyze the relation between connection matrices and Berry phases on the Siegel-Jacobi disk $\mathcal{D}^J_1$ and Siegel-Jacobi upper half-plane $\mathcal{X}^J_1$ and we explicitly compute the connection matrices and the Berry connections on these homogeneous spaces in different coordinates.

We study the population dynamics of a predator-prey system with two types in each species. Within a species, predator or prey, dynamics are described by a neutral competitive Lotka-Volterra model, i.e., birth, death and competition parameters are equal for both types. Additionally, we assume that the intra- and inter-type competition parameters are equal. The predator-prey interaction is defined by a matching-types model where predators of type $i$ exclusively interact with prey of type $i$. The individual-based model is described by a birth-death process with immigration, where immigration reflects mutations between the types of the same species. We completely describe the deterministic dynamics arising as a large population limit of this birth-death process. We find that depending on the parameters, potential equilibria are the coexistence of all four types, coexistence of a non-matching or matching pair of predators and prey, or the extinction of the predator or prey species resulting in a line of two-type equilibria. When mutations are sufficiently rare, then the predator-prey dynamics are described by successive jumps between the different deterministic equilibria on this mutational time scale. These jumps describe eco-evolutionary cycles of repeated prey or predator invasions and declines. When coexistence of all the types is possible, we show that these cycles accumulate on this time scale. Lastly, to prove that after the accumulation point the system converges to the coexistence equilibrium, we consider a slightly modified model with unequal intra- and inter-type competition parameters. This modified setting allows us to conclude that after the accumulation point all four populations remain macroscopic and converge to the coexistence equilibrium.

Several upper bounds as well as one lower bound for the operator norm of the generalized Hilbert operator $\mathcal{H}_b$ acting on weighted Bergman spaces $A_{\alpha}^p$ are established. Moreover, under some mild assumptions, we obtain the exact norm of $\mathcal{H}_b$ on $ A_{\alpha}^p$.

Classes of squarefree monomial ideals were intensively studied being an important connection between two main areas in mathematics: commutative algebra and combinatorics. Many algebraic invariants of the squarefree monomial ideals are given in terms of properties of the combinatorial associated objects (graphs, simplicial complexes etc.).

For a class of stochastic fractional diffusion equations, we prove the existence of Stratonovich solutions and obtain the precise values of the Lyapunov exponents governing the large $t$-asymptotics of their $p$-th moments for all real $p\in\{1\}\cup[2,\infty)$, as well as the large $p$-asymptotics of the $p$-th moments.

Diffusion-guided methods have recently emerged as effective tools for probing the intrinsic geometry of high-dimensional data. Understanding their accuracy requires quantifying both (i) finite-scale approximation error relative to the true local intrinsic dimension (LID) and (ii) estimation error from finite samples. We address these two questions for the FLIPD field (Kamkari et al., 2024; arXiv:2406.03537), defined at the population level through the logarithmic scale derivative of a Gaussian-smoothed density and admitting a Fokker–Planck representation. For $\alpha$-smooth densities on a regular $d$-dimensional manifold, we establish a uniform $O(\sigma^2)$ approximation bound relative to the manifold dimension $d$. Under expected integrated squared loss, we prove a minimax lower bound of order $(n\sigma^d)^{-1}$ for estimating the FLIPD field from $n$ observations when $n^{-1/(2\alpha+d)}\le\sigma\le\sigma_0$. A plug-in FLIPD estimator based on the empirical denoising score matching (DSM) solution attains a matching upper bound. Thus the minimax rate is of order $(n\sigma^d)^{-1}$ and it becomes $n^{-2\alpha/(2\alpha+d)}$ at its smallest scale.

We study the deformation stability of Kähler and balanced hyperbolicity. We prove that Kähler hyperbolicity is open under arbitrary small holomorphic deformations, but need not be closed, even in families of smooth projective surfaces. By contrast, balanced hyperbolicity is not open in general: in every complex dimension at least five, we construct a balanced hyperbolic central fibre with non-balanced nearby fibres. We establish criteria for the deformation stability of balanced hyperbolicity using Aeppli cohomology and Lefschetz theory. For principal holomorphic torus bundles over products of curves of genus at least two, we compute the relevant hyperbolic cohomology, obtain a numerical characterization of balanced hyperbolicity, and prove its openness under small deformations.

Loday's derived product is revisited in the setting of Q-manifolds, i.e., supermanifolds equipped with an odd vector field that `squares to zero'. We interpret the Grassmann odd product as a standard product upon shifting the grading, which implies the existence of a derived (noncommutative) associative $\mathbb{Z}_2$-graded algebra associated with any Q-manifold. We apply Connes' cyclic cohomology to the derived associative algebra, giving a new cohomology on Q-manifolds distinct from the standard cohomology: we refer to this as the derived cyclic cohomology of a Q-manifold. The derived cyclic cocycles are interpreted as classically BRST-invariant functionals within the BV–BFV–BRST formalism or generalised Ruelle–Sullivan currents when applied to regular foliations via their foliation Lie algebroids.