Papers

A stationary Poisson hyperplane process in $\mathbb{R}^d$ is characterized by an intensity parameter and an even probability measure on the unit sphere called the directional distribution. In this work, we investigate the stability of the zero cell, i.e., the random convex polytope of the induced hyperplane tessellation containing the origin, under perturbations of the directional distribution. Our results provide quantitative bounds establishing local Hölder continuity of the distribution of the zero cell with respect to Wasserstein metrics on the space of random convex bodies and probability distributions on the unit sphere. As an application, we establish stability bounds for density estimators constructed from Poisson hyperplane tessellations. In particular, we bound the expected total variation distance between the zero-cell-based estimated probability measures in terms of the Wasserstein distance between their directional distributions.

For each integer $k \geq 3$ Hermann Karcher identified a complete singly periodic minimal surface $\Xi_k$ (unique up to similarity) with $2k$ ends asymptotic to the union of $k$ planes intersecting equiangularly along a single line and with genus zero in the quotient by a fundamental translation. Writing $\Xi_{k,m}$ for the quotient of $\Xi_k$ by translation through $m \geq 1$ fundamental periods, we study the Morse index and nullity of the subfamilies $\Xi_{k,2}$ and $\Xi_{3,m}$. In the $m=2$ case we prove for all $k \geq 3$ that $\Xi_{k,2}$ has Morse index $4k-3$ and nullity $3$. In the $k=3$ case we prove that there exists a real number $\alpha^* \in (1/3,1/2)$ such that for all $m \geq 1$ the Morse index of $\Xi_{3,m}$ is $6m- 4 \lfloor m\alpha^* \rfloor - 3$ and its nullity is $3$ unless $m\alpha^*$ is an integer, in which case its nullity is $7$. Both proofs exploit the symmetries of each surface and the Dirichlet-to-Neumann map on a half period to reduce the problem to Fourier-analytic computations on the unit circle (after a conformal change).

We describe the asymptotics of Chebyshev polynomials on an analytic Jordan arc in the plane. This gives an affirmative answer to a conjecture of Christiansen-Simon-Zinchenko, based on predictions of Widom from 1969. The proof combines weighted Faber polynomials with extremal signatures, discrete orthogonal polynomials and a Marcinkiewicz-Zygmund sampling inequality, and yields Szegő-Widom asymptotics for the Chebyshev polynomials themselves.

We study the fundamental gap of the Dirichlet $p$-Laplacian on bounded convex domains. We first prove the sharp one-dimensional estimate $\Gamma_p(I_D,V)\ge (p-1)(2^p-1)(\pi_p/D)^p$, $I_D=(-D/2,D/2)$, for all $1<p<\infty$ and convex potentials $V$, with equality exactly for constant $V$. An analysis of collapsing convex domains shows that $p=2$ is exceptional: for $p\ne2$, the higher-dimensional gap is not governed by the sharp one-dimensional constant. For $V\equiv0$, define $G_{p,N}:=\inf_{\Omega\subset\mathbb R^N}\operatorname{diam}(\Omega)^p\Gamma_p(\Omega,0)$ over bounded convex domains. We prove $G_{2,N}=3\pi^2$ and $G_{p,N}=0$ for $1<p<2$, $N\ge2$, while for every $p>2$, $G_{p,N}>0$ and the infimum is attained. We obtain explicit dimension-dependent bounds and $G_{p,N}\asymp_p N^{p-2}$ as $N\to\infty$, while for fixed $N$, $\lim_{p\downarrow2}G_{p,N}=3\pi^2$ and $\lim_{p\to\infty}G_{p,N}^{1/p}=4$. We also characterize the collapse as $p\downarrow2$: minimizing domains degenerate, and every limiting normalized ground-state measure is supported on a unit segment and, up to rigid motions, equals $2\cos^2(\pi t)\,dt$ on $(-1/2,1/2)$. The proof establishes log-concavity of the first $p$-eigenfunction with convex potential, a degenerate weighted Poincaré inequality for the ground-state measure, and a lower gap bound in terms of the first eigenvalue and the associated weighted Poincaré constant. These estimates also give the compactness and equality analysis for the collapsing minimizers.

Covariate-adaptive randomization procedures are widely used in clinical trials to improve covariate balance. In modern applications, experimenters often have access to many covariates, motivating the need for a theory of covariate-adaptive randomization procedures with a diverging number of covariates. This paper studies two classes of covariate-adaptive randomization procedures, referred to as imbalance-efficient covariate-adaptive randomization and imbalance-robust covariate-adaptive randomization, when the feature dimension diverges. We derive convergence rates for the imbalance of the covariates used in randomization. For both procedures, the imbalance is of a smaller order than that under complete randomization when the feature dimension is $o(n)$, whereas it is of the same order when the feature dimension is $\Omega(n)$. For imbalance-robust covariate-adaptive randomization, we further establish the asymptotic properties of the imbalance of additional covariates and use these results to derive the asymptotic distribution of the difference-in-means estimator for the average treatment effect and construct asymptotically valid confidence intervals. Furthermore, we provide extensive numerical and empirical studies to illustrate the practical relevance of our theoretical results.

We perform a linearized stability analysis of the Vlasov–Fokker–Planck equation obtained as the mean-field description of a stochastic spontaneous aggregation model with memory. Memory is represented by a chain of $K$ internal variables. We characterize the spatially homogeneous equilibria and derive a scalar dispersion relation for spatially inhomogeneous perturbations. When all relaxation rates are equal, we show that every Fourier mode that is unstable in the memoryless model possesses a unique critical relaxation parameter: sufficiently long memory stabilizes the mode, whereas it remains unstable for short memory. Although memory may destabilize individual modes associated with negative Fourier coefficients of the sensing kernel, we prove that, for radially symmetric distance-decreasing kernels, it cannot destabilize a homogeneous equilibrium that is stable in the memoryless model. Thus, memory cannot create an overall instability of an otherwise stable homogeneous state, although it may change the set of the unstable modes. We present a numerical example for the normalized top-hat kernel, demonstrating that increasing the effective memory length successively stabilizes the Fourier modes, with higher frequencies being stabilized before lower ones. This is consistent with the coarsening effect observed in recent particle simulations. Finally, under suitable nondegeneracy and regularity assumptions, we use the Crandall–Rabinowitz theorem to obtain branches of spatially inhomogeneous stationary solutions bifurcating from the homogeneous state.

The less-than-truckload (LTL) industry plays a vital role in enhancing the efficiency and sustainability of logistics systems, as LTL shipments offer greater consolidation opportunities than full-truckload shipments. Despite of this flexibility, the average cost of LTL shipments remains considerably higher due to less efficient operations and highly fragmented networks of small and medium-sized carriers. Building on our ongoing effort to develop a distributed and dynamic logistics hub network system grounded in the Physical Internet (PI) principles of modular containers and open resource sharing, this study focuses specifically on inter-hub and in-hub operations, with cooperation among multiple regional hub networks. Therefore, a shipment may traverse multiple cooperating hub networks. With respect to each hub network each shipment enters, it is defined by its expected arrival time at the entry hub and its latest arrival time at the exit hub. Based on the defined shipment information, we design a set of multi-hub operation planning protocols for distributed hub operators. In their operating networks, operators use our smartly designed protocol separately to plan in-hub shipments' assignments to destination-specific trailers and inter-hub trailers' dispatch schedules. With carefully designed interconnections between hub networks, the aggregated hub network system is well-positioned to achieve cooperative outcomes and fulfill shipment requests. We evaluate the effectiveness of the proposed protocol through a simulation-based experiment under multiple scenarios in an operator's multi-hub network. Overall, this research improves the practicality and robustness of PI-based networks and supports greater cooperation among hub networks toward more efficient and sustainable logistics systems.

We introduce synthetic buildings for each finite group $G$ and prime $p$. These are $G$-simplicial complexes, among which are the $p$-subgroups complex of K.S. Brown, and also the buildings of finite groups of Lie type in characteristic $p$. Synthetic buildings are useful because of special properties, including providing a formula for the cohomology of $G$. Our goal is to describe the kinds of synthetic buildings that can arise, and ideally to classify them. The minimal synthetic buildings play a special role and for most groups there are not very many of these. However, we show that it is possible to find sequences of finite groups with increasingly many minimal synthetic buildings as we move along the sequence, and also groups with minimal synthetic buildings of different dimensions. It is also possible to find groups with infinitely many non-minimal synthetic buildings of a given dimension. In other situations, we show that there are no minimal synthetic buildings of equivariant homotopy type distinct from those of Brown's complex and the complex consisting of a single point. We make a number of conjectures.

It has been known for a long time that the equation of motion for dissipative linear dynamical systems with constant coefficients cannot be derived from the classical principle of stationary action because the term proportional to velocity in the equation of motion leads to the time derivative of order one half in the Lagrangian. Thus, approaches utilising fractional calculus have been used; however, they suffer from deficiencies both from mathematical and physical points of view. We here present our version of such fractional calculus based approach that provides correct Euler-Lagrange and, ultimately, the Hamilton equations, energy change of the moving body, and an attempt for a geometric interpretation of how energy dissipates.

We study the stochastic resistive Hall–magnetohydrodynamic (Hall–MHD) system on bounded convex polyhedral domains, subject to nonlinear perfectly conducting boundary conditions. The system is driven by multiplicative Gaussian forcing in the momentum equation together with structured curl-type noise in the induction equation arising from stochastic perturbations of the resistive and Hall parts in the generalised Ohm law. We develop a fully discrete, linearly implicit, structure-preserving mixed finite element method based on a compatible discrete de Rham complex, which preserves the magnetic Gauss law exactly. We prove subsequential convergence of the discrete solutions to a finite-energy weak martingale solution, thereby obtaining a constructive existence result for the stochastic Hall–MHD system with curl-type noise. Numerical experiments illustrate the stochastic dynamics, exact divergence preservation, and convergence of the method.

We give the minimum of category theory necessary for understanding the definition and applications of Grothendieck topos. We state the basic properties of sites and sheaves, and give applications to the theory of moduli. We prove that for categories $\mathbf C$ with explicit stated properties, we can construct a category of schemes of objects in $\mathbf C$ which is a proper category rather than a $2$-category as in the case of stacks.

Following ideas of Kapustka-Kapustka, we use Lagrangian fibrations to construct twisted hyperholomorphic sheaves on products of hyperkähler manifolds of K3$^{[n]}$- and OG10-type. As applications, we prove the Lefschetz standard conjecture and the D-equivalence conjecture for hyperkähler manifolds of OG10-type.

This article contrasts the concepts of a proper condensate and the Onsager-Penrose criterion for Bose-Einstein condensation in the form of a debate between two proponents of the respective concepts. The prologue briefly introduces the two criteria. The epilogue contains remarks on properties of the underlying resolvent algebra that are physically relevant in this context.

We prove the conjecture of Donnelly et al. that a certain class of Littlewood-Richardson tableaux are equinumerous with peakless Motzkin paths of length $n$. Furthermore, we construct an explicit bijection between this class of tableaux and peakless Motzkin paths of length $n$ for all $n\ge 1$.

Recently we started building a bridge between two cases of Langlands duality. The latter is one of the most influential topics in mathematical research. It has many different appearances and influential subtopics. Yet there is a topic that until now seems unrelated to the Langlands program. That is the topic of invariant differential operators. That is strange since both items are deeply rooted in Harish-Chandra's representation theory of semisimple Lie groups. We started with the case of the group ~$SL(2n)$. In the present paper we deal with the group $SL(2n+1)$. The two mentioned groups are similar, but their representation theories is rather different.

We prove that every Kreiss bounded $C_0$-semigroup $(T_t)_{t\ge0}$ on a Hilbert space satisfies \[ \|T_t\|\le C(1+t)^{1-\varepsilon_K}, \qquad t\ge0, \] where $\varepsilon_K>0$ depends explicitly only on the Kreiss constant. The same conclusion is obtained for positive Kreiss bounded $C_0$-semigroups on $L^p$-spaces, $1<p<\infty$, and in discrete time for Kreiss bounded operators on Hilbert spaces and positive Kreiss bounded operators on $L^p$-spaces. We further obtain a non-quantitative polynomial gap for individually eventually positive Kreiss bounded $C_0$-semigroups on $L^p$-spaces. Finally we prove that every Kreiss bounded operator on a UMD Banach space has a polynomial gap below linear growth

We develop an intrinsic theory of weighted, inhomogeneous Besov spaces on general homogeneous Lie groups without recourse to group-specific arguments. Starting from a definition in terms of localised test functions, we establish an equivalent, wavelet-like, multiscale characterisation. This provides a unified mechanism for deriving Besov embeddings, a Taylor-remainder characterisation, Young-type product estimates, Schauder estimates for convolution semigroups, and a weighted Kolmogorov criterion for random distributions. We apply this framework to parabolic Anderson-type equations associated with positive Rockland operators and with singular initial data. For space-time noise, our results cover the Young regime. For purely spatial noise, we also establish well-posedness in the first singular regime using a variant of the Cole-Hopf transform for Rockland operators. Both applications rely on a mild sewing lemma that accommodates time-dependent Banach spaces and increments with a singularity at the initial time.

We classify universal divisor-profile separation for coprime polynomial pairs of arbitrary degree and for all pairs of degree at most two. For \(A\subset\N\) and \(m\in\Z\), let \(d_A(m)\) count the members of \(A\) dividing \(m\). For coprime nonzero \(F,G\in\Z[x]\), universally unbounded separation between \(d_A(F(n))\) and \(d_A(G(n))\) occurs if and only if one of \(F,G\) is intersective, that is, has a root modulo every positive integer. More generally, an intersective factor separated from finitely many polynomial opponents yields simultaneous one-sided dominance against all of them. For pairs with common irreducible factors, let \(U,V\) be the products of the factors occurring only on the two respective sides. Universal separation forces \(UV\) to have a root modulo almost every prime; equivalently, its Galois action has no derangement. We resolve the remaining finite \(p\)-adic boundary for all pairs of degree at most two: separation holds exactly when \(UV\) and at least one of \(F,G\) are intersective, and the criterion is unchanged by contents or factor multiplicities. The same criterion holds, in arbitrary degree, for three linear support factors with arbitrary positive multiplicities. The proofs combine uniform almost-prime values on root progressions with an adaptive local-routing argument.

A well-known conjecture in complex geometry states that a compact Hermitian manifold with constant Chern holomorphic sectional curvature must be Kähler when the constant is nonzero and Chern flat when the constant is zero. The conjecture is known in complex dimension two and in several special classes in higher dimensions. For Hermitian metrics with Bismut-parallel torsion, the non-balanced case and the balanced threefold case were established by Chen–Zheng, while the balanced fourfold case was settled recently by Wang–Zheng. In this article, we prove the nonzero case for balanced Bismut-torsion-parallel Hermitian manifolds in every complex dimension. As a corollary, we confirm that every BTP Hermitian manifold with Chern holomorphic sectional curvature is a nonzero constant, then $g$ is Kähler.

The construction and the stability of bridges between mixture probability measures are investigated. Given a collection of Markov transitions bridging the components of two mixtures, together with a coupling of the mixture labels, we construct a bridge between the mixtures on an extended state space, and we show that this bridge inherits the entropic optimality of the component bridges. We then turn to Gaussian mixtures bridged by Gaussian Schrödinger bridges, whose parameters are generally unknown and must be estimated. We prove a $2$-Wasserstein continuity theorem between the exact bridge and its plug-in approximation. The analysis rests on a representation of the Riccati fixed point map in inverse coordinates; this map is globally $1$-Lipschitz and requires no matrix inversion. These results are applied to three estimation schemes, with $d$ the dimension and $N$ the number of samples. For Gaussian mixtures estimated by the Expectation-Maximization algorithm, the squared $2$-Wasserstein error is of order $(d\log N/N)^{1/2}+d^2\log N/N$ with probability at least $1-10N^{-1}$. For single Gaussian marginals estimated by their sample moments, it is of order $d^2/N$ in expectation. For the regularized empirical Monge map based on an $\epsilon$-inflation of the sample covariance, the mean squared error is of order $d^2\{(1+\epsilon^{-1})/N+\epsilon^2\}$; as soon as $N$ is a sufficiently large multiple of $d$, it reduces to $d^2/N+d\,\epsilon^2$, up to an exponentially small term. These estimates are illustrated numerically.

Two graphs $G$ and $H$ are homomorphism indistinguishable over a graph class $\mathcal{F}$ if they admit the same number of homomorphisms from every graph in $\mathcal{F}$. A wide range of relaxations of graph isomorphism arise this way: isomorphism itself over the class of all graphs [Lovász, Acta Math. Hung. 1967], equivalence under the $k$-dimensional Weisfeiler-Leman algorithm over the graphs of treewidth $\leq k$ [Dvořák, J. Graph Theory 2010], and quantum isomorphism over planar graphs [Mančinska-Roberson, FOCS 2020]. Since the class $\mathcal{F}$ is typically infinite, it is not clear a priori whether homomorphism indistinguishability over $\mathcal{F}$ is decidable; for planar graphs it is undecidable. Every class for which decidability was previously known is sparse. We give the first decidability results for dense graph classes: We introduce the dense Weisfeiler-Leman algorithm that decides homomorphism indistinguishability over the class of graphs of cliquewidth $\leq k$, the dense counterpart of treewidth. This relation was not previously known to be decidable. The algorithm colors $k$-tuples of vertex subsets rather than $k$-tuples of vertices. Beyond the class of all graphs of cliquewidth $\leq k$, we prove a general meta-theorem: homomorphism indistinguishability over every $\mathsf{CMSO}_1$-definable graph class of bounded cliquewidth is decidable, in randomized exponential time. For classes of bounded linear cliquewidth the bound improves to $\mathsf{PSPACE}$, and we show this is tight by exhibiting such a class for which the problem is $\mathsf{PSPACE}$-complete. These are the first general algorithms for homomorphism indistinguishability over dense graph classes.

For each nonnegative integer $m$, we construct smooth symmetric $3\times 3$ coefficient matrices $A_m$ satisfying the fixed ellipticity bound \[ I\leq A_m\leq 2^{81}I \] for which the smooth solutions of uniformly elliptic equations in nondivergence form \[ \text{tr}(A_m(x)D^2 u_m)=A_m(x):D^2u_m=0\qquad\text{in }B_2\subset {\mathbb R}^3 \] have common Dirichlet data, satisfy $\|u_m\|_{L^\infty(B_2)}\leq1$, but \[ \lim_{m\to \infty}\|Du_m\|_{L^1(B_1)}=\infty. \] Thus, there is no interior $W^{1,1}$ estimate depending only on ellipticity in dimension three, and consequently no such $W^{1,p}$ estimate for any $p\geq1$. This resolves in the negative an open question raised by Nadirashvili, Tkachev, and Vlăduţ. The construction also gives a uniformly convergent limit $u\notin \text{BV}_{\rm loc}(B_1)$ for a measurable uniformly elliptic coefficient matrix obtained as an $L^1$ limit of the $A_m$.

We construct a simple $\lambda$-double Lie algebra on $k[t,t^{-1}]$ for every nonzero $\lambda\in k$. We then show that the analogous two-sided construction of weight zero is also simple. Both products admit natural coefficient realizations via row-and-column-finite operators associated with the finitary $\mathbb Z\times\mathbb Z$ matrix algebra.

We study two-weight weak-type estimates for the operator $S_v f = \mathcal{T}(fv)/v$, where $\mathcal{T}$ is the Hardy-Littlewood maximal operator or a Calderón-Zygmund operator (CZO) and $v$ is a weight. Concretely, under certain conditions on the weights involved, we prove that $S_v$ is bounded from $L^{1}(wv)$ to $L^{1,\infty} (uv)$. We also consider the corresponding inequalities when $\mathcal{T}$ is a higher-order commutator of a CZO. These types of results are inspired by the article of Sawyer in [21], (see also [17]).

In this note, we construct moduli spaces of super-representations of quivers by extending King's Geometric Invariant Theory (GIT) framework to the super setting. Using the even part of super general linear groups and a parity-shifting operator, we construct moduli spaces that topologically parameterize super-representations of a fixed super-dimension. We also outline the construction of super quiver varieties for super-representations of framed quivers (without doubling); and realise super-Grassmannians and super-flag varieties as geometric quotients within this setting.

We prove new comparison results between the Wasserstein distance and its sliced and max-sliced counterparts. First, we show that the Hölder exponent~$\frac{2}{d+2}$ obtained by Bobkov and Götze for the max-sliced 1-Wasserstein distance on the unit ball is optimal for every $d \geq 2$, settling a question raised in their work. Second, we show that sharper comparisons are possible under stronger structural assumptions: if $\nu$ is a discrete measure and the optimal coupling between $\mu$ and $\nu$ transports each point to a nearest atom of $\nu$, then $W_p(\mu, \nu) \leq C \sqrt{d}\, K \, \mathrm{SW}_{p,1}(\mu, \nu)$ for a universal constant $C$, where the complexity parameter $K$ is always at most the number of atoms $N$ and can be substantially smaller. This complements a similar bound due to Park and Slepčev. An analogous bound holds for the sliced Wasserstein distance based on $k$-dimensional projections. Finally, using a construction from geometric discrepancy theory due to Chen and Travaglini, we prove that the linear dependence on $K$ in this bound cannot be improved, up to polylogarithmic factors.

The Dynamic Mode Decomposition (DMD) has been consolidated as a basic tool for data-driven analysis of dynamical systems, allowing simultaneous identification of coherent structures and their dynamics from time-resolved measurements. However, with a linear regression at its core, DMD is unable to produce accurate models from recordings of dynamics that are inherently nonlinear, such as the response to large perturbations and the evolution on chaotic attractors. Recent approaches attempt to simultaneously fit the linear and nonlinear contributions to the dynamics by performing a regression onto a physically motivated model structure. However, although the resulting nonlinear models can produce accurate short-term predictions, their linearization does not necessarily agree with that of the original system. In this work, we introduce a novel data-driven method — nonlinearity-subtracted DMD (NSDMD) — that focuses on producing an accurate linearization of a system when the nonlinear contribution to its dynamics are available while the linear part is not. This scenario is encountered, for example, when the nonlinear terms in the governing equations are known, while the linear operator contains uncertain material properties or it accounts for the closure of unresolved dynamics. This also arises when the data is generated by a black-box simulation code that is able to output the nonlinearity, but not the action of the linear operator on the snapshots. NSDMD leverages data snapshots of the nonlinearity to explicitly account for the purely nonlinear contributions to the dynamics and formulate a regression problem that finds a low-rank approximation of the underlying linear operator. We demonstrate the approach on several numerical examples, showcasing its improved capabilities for data-driven linear analysis of chaotic, partially observed, advection-dominated, and high-dimensional dynamics.

Let p > 3 be a prime, and let S_p denote the geometric isomorphism classes of supersingular elliptic curves in characteristic p whose j-invariants lie in F_p. For each negative fundamental discriminant -D for which p is inert in Q(sqrt(-D)), let m_i(D), i in S_p, be the integral coefficients of the corresponding Gross vector. We prove that the vectors (m_i(D) mod 2)_{i in S_p} span F_2^{S_p}. The key step reduces the parity of the representation numbers of Gross's ternary lattices to representation by rank-two sublattices perpendicular to Frobenius. Using Ibukiyama's explicit maximal orders, the resulting primitive binary forms are identified with those occurring in the Xiao–Zhou–Deng–Qu parametrization of supersingular elliptic curves over F_p. Class field theory and Chebotarev's theorem then allow the individual supersingular coordinates to be isolated. As a consequence, if E/Q has prime conductor p and positive Mordell–Weil rank, then every coefficient of its Brandt eigenvector indexed by S_p is even, proving a conjecture of Kazalicki and Kohen. Thus an odd coefficient at a rational supersingular class is an algebraic certificate of rank 0. Combining this parity theorem with formulas of Mestre and Gross–Kudla, we also prove that the modular degree of every positive-rank elliptic curve of prime conductor and root number +1 is divisible by 4. Consequently, Watkins' conjecture holds for all such curves of rank 2.

In this paper, we introduce the notion of a distinguished variety in the domain $\mathbf{\Theta}_n$. One of the main results of the paper is a determinantal representation for every distinguished variety in $\mathbf{\Theta}_n$. We also show that the closure of every distinguished variety is polynomially convex. Furthermore, we obtain a dilation and a functional model for a class of pure $\mathbf{\Theta}_n$-contractions. Finally, we show that for a $\mathbf{\Theta}_n$-contraction $\mathbf{T}=(T_1,\dots,T_n)$ such that $T_n^*$ is a pure contraction, there exists an algebraic variety in $\mathbf{\Theta}_n$ for which the von Neumann inequality holds on the intersection of the closure of the variety with the distinguished boundary of $\mathbf{\Theta}_n$.

This study develops a weighted framework for measuring the discrepancy between two nonnegative lifetime distributions through cumulative past extropy. We propose two measures, referred to as the weighted cumulative past extropy inaccuracy (WCPEI) and the weighted cumulative past extropy Kullback-Leibler divergence (WCPED). The generalized weight function is considered in this study. We investigate a number of theoretical properties of these measures. Empirical distribution function-based nonparametric estimator is subsequently constructed for the weighted cumulative past extropy inaccuracy ration (WCPEIR). Its finite-sample behavior is studied through Monte Carlo simulation experiments for different sample sizes. To illustrate the practical relevance of the WCPED, two applications are considered. First, an extropy-based goodness-of-fit procedure for testing uniformity is developed using the proposed divergence measure. Its power is then compared with that of several established uniformity tests under a variety of alternatives. Second, an image analysis application is presented in which the proposed measure is employed to assess changes in the distributions of pixel intensities when the image resolution is altered. The framework is further extended to a dynamic setting by conditioning on the lifetime information available up to a specified time point. This leads to the dynamic weighted cumulative past extropy inaccuracy (DWCPEI) and dynamic weighted cumulative past extropy divergence (DWCPED). Their theoretical properties are derived, and the corresponding nonparametric estimation procedures are proposed. The finite-sample performance of these estimators is evaluated through simulation studies using R software.