Papers

In this paper, we study and characterise the natural embedding of the Ree-Tits octagons. Keywords: Ree-Tits Octagon, Metasymplectic space, generalised octagon, generalised polygon

Gaussian graphical model selection is usually studied under independent sampling, but in many applications the data arise as a single trajectory of a dependent stochastic process. We study exact recovery of the graph from one trajectory of random-scan Gaussian Glauber dynamics. Existing techniques for this problem either inherit the mixing time of the chain, which can be super-polynomial in the dimension $p$ without strong assumptions, or are suboptimal in the minimum normalized edge strength $\kappa$. We propose two algorithms that are mixing-free and attain the $\kappa^{-2}$ dependence of the information-theoretic lower bounds. Both instantiate a shared dueling-neighborhood search meta-algorithm with a local statistic built directly from the update sequence. For every fixed precision matrix and deterministic initialization, the first algorithm fits a least-squares regression at the updates of each node and has pointwise recovery horizon $\widetilde O(pd^{2}/\kappa^{2})$, where $d$ is the maximum degree. Its horizon depends logarithmically on a local conditioning quantity and on the initialization potential. The second algorithm is based on counting occurences of a specific update pattern and requires $\widetilde O(pd^{4}/\kappa^{2})$ updates, with no dependence on any condition number. The central technical challenge is that both statistics are built from dependent, non-stationary observations. Our analysis tackles this by demonstrating how to extract fresh Gaussian innovations from the update sequence, which yields mixing-free control of appropriate quantities. Neither the algorithms nor their analyses invoke stationarity, a spectral gap, or mixing conditions.

The compressible fluid-solid interaction (FSI) with a thermomechanical phase transition is formulated at large strains within the Eulerian frame. For the deviatoric part, the Jeffreys (also called anti-Zener) rheology with an additional viscosity is adopted. The core philosophy governing the mechanical solid-liquid transition is that the viscous (or viscoplastic) response is temperature-dependent and may fully degenerate to a viscoelastic fluid during thawing, so that there is no elastic response on the shear distortion. This behavior enables the free flow of the fluid, its subsequent freezing into a new configuration, and potential re-melting back into a fluid, allowing such cycles to repeat indefinitely. The classical Stefan problem, associated with the latent heat of the first-order (thawing-freezing) phase transition, is augmented by incorporating kinetic overheating and undercooling. The analysis by a time discretization with an appropriate truncation is applied to a higher-gradient modification of the original formulation, utilizing the concept of multipolar nonsimple continua.

Researchers often select top-performing options or winners, based on a data-driven criterion, such as treatments, models, or model features and then report effect estimates for the selected winners. Naive post-selection estimates, however, are known to suffer from the winner's curse, producing systematically overoptimistic results. We introduce a flexible conditional inference method that corrects for this overoptimism through an adaptive exponential randomization scheme. Our method achieves selection quality that closely matches that of standard top-k selection, while also yielding shorter confidence intervals than existing approaches. Furthermore, our approach applies broadly to nonparametric settings with asymptotically linear selection statistics, covering wide-ranging applications such as inference for the efficacy of the most promising treatments in clinical trials, the abilities of top-ranked models on leaderboards, and the importance of the most predictive features in a model.

We study doubly reflected backward stochastic differential equations with jumps and two completely separated right-continuous with left limits barriers in a filtration generated by an inhomogeneous Levy process. We establish existence and uniqueness results under a stochastic Lipschitz condition on the driver by means of a penalization method. We also prove a comparison principle and present two closely related applications. The first concerns the nonlinear valuation of an American game option in such a Levy market, while the second addresses the associated generalized Dynkin game under nonlinear expectation. Moreover, under suitable semicontinuity assumptions on the barriers, we establish the existence of a saddle point for the game.

We study polynomial holomorphic $1$-forms in $\mathbb{C}^2$ that are homologically trivial along the fibers of meromorphic pencils of the form $ \phi = \frac{f^p}{g^q}, $ where $f,g$ are holomorphic functions (possibly polynomials) in general position and $(p,q)=1$. We first establish a homological characterization of relative exactness: if a polynomial $1$-form $\Omega$ has vanishing periods along every closed path contained in the fibers $\phi_c$, then $\Omega$ decomposes as $\Omega = a\omega_0 + dh,$ where $\omega_0 = p gdf - q f dg,$ for suitable polynomials $a$ and $h$. In the homogeneous case, degree constraints force $a$ to be constant. We then apply this integration principle to foliations leaving invariant plane curve singularities of cusp type \[ f^p + g^q = 0. \] Under a natural genericity (Morse type) condition, we prove a globalization theorem showing that homological triviality along the associated pencil implies that $\Omega$ is a polynomial cusp basic form, \[ \Omega = d(f^p + g^q) + \lambda (p gdf - q fdg), \qquad \lambda \in \mathbb{C}. \] In particular, such foliations admit Liouvillian first integrals of hypergeometric type. Our results provide a bridge between relative cohomology, the geometry of rational pencils, and the analytic structure of cusp foliations, yielding explicit normal forms and first integrals under homological hypotheses.

The ultrametric skeleton theorem extracts from every compact metric probability space a subset of ultrametric distortion $O(1/\varepsilon)$ that carries a measure whose balls are controlled by the $(1-\varepsilon)$-power of the original measure on dilated concentric balls. We prove a two-sided version for arbitrary compact metric spaces: for every ball centered on the skeleton, the skeleton measure also has a lower bound in terms of the original measure on a smaller nonconcentric ball contained in it. We also give a short proof of the original skeleton theorem and improve the dilation of its control balls from $\exp(O(1/\varepsilon^2))$ to $O(1/\varepsilon)$.

We construct a vector space whose defining characteristics are rooted in univariate regular variation of random variables. Specifically, the base vector space $\mathbb{V}_b$ consists of random variables whose limiting tail probabilities, when scaled by regularly varying functions of the form $b(s)=s^\alpha L(s)$, are finite. Defining a subspace ${\cal N}_b$ corresponding to random variables in $\mathbb{V}_b$ whose limiting tail probabilities are zero when normalized by $b(s)$ allows the base space $\mathbb{V}_b$ to be partitioned into equivalence classes. We define a vector space $\mathbb{W}_b$ consisting of these equivalence classes, and show its nonzero elements are equivalence classes of regularly varying random variables. We show that a natural norm exists for $\mathbb{W}_b$ if $\alpha > 1$. We show that the equivalence classes and convergence in norm are different than more familiar vector spaces of random variables. Turning our attention to extreme value modeling, we consider finite-dimensional subspaces of $\mathbb{W}_b$ whose basis vectors are jointly regularly varying. We show that in the case $\alpha = 2$, the previously defined tail pairwise dependence measure serves as an inner product. As any finite-dimensional space is complete, we can use the projection theorem to perform linear prediction.

Tarski's geometry of solids reconstructs point-like entities from concentric families of spherical regions rather than taking points as primitive. We formalize this reconstruction in Coq within a nominal mereological framework inspired by Leśniewski, and study how the regional geometry of solids induces a topology on reconstructed point-classes without adding new point individuals. Three of Tarski's postulates concerning solids and interior points, P2–P4, are derived as theorems. We refine Tarski's interior-point notion and define a regional interior operator satisfying the four Kuratowski interior axioms, together with a boundary operation and an encoding of RCC8 relations. We then pass to the setoid of ball representatives modulo same_center. Each ball generates a stable basic point-plural GBasicPointSet(Q), and these plurals form a basis for a metatheoretic topology on reconstructed point-classes. This topology is Hausdorff under Tarski's separation axiom Three_points and non-discrete under the local richness hypothesis BallCenterBundle. Finally, geometric neighbourhoods yield a closure operator GClosurePoint satisfying the four Kuratowski closure axioms and respecting extensional point-set equality. The formalization thus verifies the passage from a regional topology of solids to a Hausdorff topology and neighbourhood closure on reconstructed point-classes.

This article develops a large-signal stability analysis for a sampled-data optimization-based secondary controller for distributed energy resources (DERs) in power systems. The induced closed loop combines nonlinear inverter power-flow dynamics, filtered active and reactive power measurements, constrained optimization updates, and interpolation-based actuation between sampling instants. We study this optimization-in-the-loop nonlinear sampled-data system beyond local linearization. The analysis provides computable bounds on the voltage, filtered reactive power, and the secondary control input. We further characterize steady-state operating points and establish how the optimizer objectives and constraints connect voltage regulation with equal per-unitized reactive power sharing. Finally, input-to-state stability of the frequency dynamics is established with respect to DER voltages and control inputs. These results provide a rigorous mathematical foundation for sampled-data optimization-based secondary control of DERs.

We propose a new way to construct one block design from another, thus linking some block designs in a new way. At the same time, we find relationships between block designs and systems of pairs. In particular, these relationships allow us to generate some systems of pairs.

Over an algebraically closed field of characteristic zero, we exhibit two symmetric algebras that are 3-representation-finite but are not orbit algebras of repetitive categories. The first is a 36-dimensional characteristic-zero lift $A$ of $Q(3A)^2_2$, the quaternion-type algebra that Böhmler and Marczinzik proved 3-representation-finite in characteristic 2; we construct an explicit 3-cluster-tilting module. We show that any orbit presentation of a connected symmetric algebra forces the algebra, or a connected cyclic cover of it, to admit a half-dimensional square-zero grading. Such gradings are detected by idempotent derivations, and a finite group of arrow characters constrains the possible covers. For $A$, only three double-cover candidates remain and the derivation obstruction excludes all of them. Thus $A$ is not an orbit algebra of any finite-dimensional algebra, regardless of global dimension, answering a question of Darpö and Iyama. The 84-dimensional 3-spherical weighted surface algebra is likewise 3-representation-finite and not an orbit algebra.

We derive complete asymptotic expansions for the binomial mass at a bounded lattice displacement and for the mean absolute deviation $E|X-Np|$, $X\sim Bin(N,p)$, with $0<p<1$. De Moivre's exact formula reduces the latter problem to the local mass at $\nu=\lceil Np\rceil$, so the coefficients depend on the oscillating displacement $h_N=\lceil Np\rceil-Np$. We show that the full expansion is governed by Bernoulli polynomials evaluated at this displacement; equivalently, the lattice correction is an Appell shift in the Stirling series. The calculation is based on a gamma-quotient expansion with unequal linear scalings, stated with uniformity in the bounded shift. In passing from the local mass to the mean absolute deviation, the elementary, non-Bernoulli part of the shift cancels term by term against the De Moivre prefactor, leaving coefficients that are pure Bernoulli polynomials. As consequences, the classical first correction of Frame and Johnson is embedded in the general coefficient sequence, and the Cesàro means of the oscillating coefficients are obtained from the multiplication theorem for Bernoulli polynomials. Finally, although an oscillating asymptotic expansion does not bound its function by truncation, De Moivre's identity together with Robbins's form of Stirling's formula yields an elementary two-sided bound of logarithmic width $O(N^{-2})$ in the interior; and at an integer mean the logarithmic expansion reduces to a sign-alternating series in odd powers of $N^{-1}$ which we prove to be enveloping, via a sign-definite Binet-kernel representation of the combined three-gamma Stirling remainder: successive truncations bracket the mean absolute deviation.

Let $\mathsf{B}_2(\Sigma_g)$ be the full braid group on two strings on a compact Riemann surface of genus $g$. We compute the number of finite cyclic, dihedral and extra-special quotients $\varphi \colon \mathsf{B}_2(\Sigma_g) \to G$, under the assumption that the quotient map $\varphi$ does not factor through $\pi_1(\operatorname{Sym^2}\Sigma_g)$. We then apply our algebraic results to the geometric problem of constructing smooth surfaces of general type as Galois covers of $\operatorname{Sym^2}(\Sigma_g)$ branched on the diagonal. In particular, we construct two $3$-dimensional families of minimal surfaces of general type with $p_g=7$, $q=4$ and $K^2=32$ such that members of different families have the same biregular invariants and the same Betti numbers, but different torsion part for the first homology group.

We introduce block diagonal Carleson frames, i.e. singly generated dynamical frames in which the generating operator is block diagonal. We classify all such frames when the generating operator is a direct sum of Jordan blocks, the sizes of which are uniformly bounded. Furthermore, block diagonal Carleson frames are shown to enjoy the high redundancy properties observed for Carleson frames. When the generating operator has nonnegative spectrum, we provide a complete description of the redundancy of such frames under the assumption of the existence of natural density.

We study a boundary value problem for the $p$-Laplacian in the perforated domain $B(0,\rho)\setminus\Gamma\subset \mathbb{R}^d$, where $\rho>1$ and $\Gamma$ is the union of many small compact cavities placed near the unit sphere. The cavities are separated at scale $\varepsilon$, asymptotically equidistributed on the sphere, and have cardinality of order $\varepsilon^{1-d}$. The cavities have diameters of order $\alpha(\varepsilon)\varepsilon$, where $\alpha(\varepsilon)\to0$, and their relative $p$-capacity is comparable to the relative p-capacity of a ball of the same diameter. The solution is required to equal $1$ on all cavities and $0$ on $\partial B(0,\rho)$. We focus on the critical case $p=d>1$. We identify the critical scale through the parameter $\tau=\lim_{\varepsilon\downarrow0}[\varepsilon\log(1/\alpha(\varepsilon))]^{-1}\in[0,\infty]$. Thus, $\alpha(\varepsilon)=\exp[-(1+o(1))/(\tau\varepsilon)]$ when $0<\tau<\infty$. Away from the unit sphere, the solutions converge to $A_*U_\rho$, where $U_\rho(x)=\min\{1,1-\log |x|/\log\rho\}$ is the radial $d$-harmonic potential of the unit ball in $B(0,\rho)$. The constant $A_*$ equals $0$ when $\tau=0$, equals $1$ when $\tau=\infty$, and is explicit for $0<\tau<\infty$. We construct an explicit ansatz that approximates the solution for sufficiently small $\varepsilon$ in both $L^{\infty}$ and in terms of $d$-capacity.

For every integer $d \geq 2$, we show that there exists an irreducible, reduced curve which embeds in $\mathbb{P}^d$ but in no projective normal toric variety of dimension less than $d$. In particular, there exist reduced irreducible curves that embed in $\mathbb{P}^d$ but do not embed in any weighted projective space of dimension less than $d$.

We introduce a class of Laplace operators associated with quantum graphs in the operator-system framework. To this end, we investigate structural properties of quantum graphs related to the Schur product and transposition. Our main innovation is the identification of a specific inner product on the space of matrices with respect to which the projection onto the operator system underlying a quantum graph is orthogonal. This enables us to define a quantum analogue of the classical incidence operator and the associated Laplace operator. We compare this construction with a recently proposed alternative definition and examine the resulting Laplace operators for several families of quantum graphs. These results extend fundamental constructions from spectral graph theory to the operator-algebraic setting, providing a step toward a spectral theory of quantum graphs that generalizes classical graph-theoretic concepts while revealing new connections with quantum information theory and noncommutative geometry.

In this paper, we study cyclic and constacyclic codes over the finite chain ring R=Z4+iZ4, where i^2=-1. We prove that all constacyclic codes over R are equivalent to cyclic codes. An algorithm to obtain generators for all simple root constacyclic codes over R is presented. Using a Gray map we then obtain linear Z4 codes from constacyclic codes over R. We present new best linear Z4 codes found using this method.

The motion of liquids and soft matter is over-damped and `slow' when viscosity dominates. In this (low Reynolds-number) limit and when the system is isothermal, the equations of motion may be generated via Onsager's variational principle, which neglects inertia. This variational approach is immensely powerful, being used to obtain equations of motion for colloidal fluids, droplets on surfaces and much more. However, inertia can play a role, manifesting as vibrations and under-damped motion. Here we show how to extend this variational framework so that it remains valid for when damping/dissipation and inertia are both equally important.

As electric vehicle (EV) adoption grows, quantifying the scheduling burden and economic cost of long-distance travel under the existing charging infrastructure becomes increasingly important for infrastructure planning and policy. This paper presents a scalable, optimization-based framework for scheduling EV charging stops along real-world charging stations and simulated long-distance personal vehicle trajectories across the United States using POLARIS. Taking the existing charging network as fixed input, the framework minimizes total detour and queuing costs for each vehicle while respecting plug capacity constraints at each station. The methodology proceeds in three phases: (i) infeasibility pruning via a forward-pass reachability heuristic, (ii) per-vehicle optimal charging schedule computation via dynamic programming on a directed acyclic graph, and (iii) capacity-aware iterative congestion resolution through a penalty-based heuristic that augments detour costs at congested stations, with a first-in, first-out queue fallback. Applied to approximately 2.7M origin–destination vehicle trajectories derived from a 1% sample of national personal travel demand within the POLARIS agent-based transportation simulation framework and covering 14,260 DC fast charging stations with 68,641 plugs from the Alternative Fuels Station Locator, the framework produces capacity-feasible schedules in under 1.3 hours on a 128-core high-performance computing cluster without requiring any commercial optimization solver. A three-tier economic analysis spanning operational costs, total cost of ownership, and amortized infrastructure investment is conducted to evaluate EV cost competitiveness relative to internal combustion engine vehicles across scenarios.

The random cut is one of the most fundamental shuffles in card-based cryptography: it rotates a sequence of face-down cards by a secret amount. Under this shuffle, two sequences of cards are indistinguishable if and only if they are cyclic shifts of each other. This motivates the question of whether, given two sequences of cards, inserting cards at matching positions can make them indistinguishable. A previous study shows that such an insertion is always possible when any cards may be inserted, as long as the two words are permutations of each other. This paper considers a stronger restriction: if the cards are binary, carrying only 0 or 1, can we insert only 0s to make the sequences indistinguishable? We call two words 0-cyclically equalizable if one can insert 0s into both sequences at matching positions so that the resulting words are cyclic shifts of each other. Our main result is that two binary words of equal length are 0-cyclically equalizable if and only if they have equal Hamming weight, that is, the same number of 1-bits. Since equal Hamming weight is clearly necessary, the content of the paper is to show that it is also sufficient. Our proof is constructive: we encode a pair of binary words as a single word over the four-letter alphabet {A, B, X, O}, reduce equalizability to a simpler condition in this encoding, and build the required insertion explicitly.

For a finite-dimensional normed space $V$ and a subset $X$ with finite Hausdorff distance from $V$, we prove that the Gromov–Hausdorff distance between $X$ and $V$ is at least the Hausdorff distance between $X$ and $V$, divided by twice the relative Jung constant of $V$. If $V$ furthermore satisfies a certain intersection property, we show a stronger result where the relative Jung constant can be replaced with its absolute version. Key words: Normed spaces, Jung constant, Hausdorff distance, Gromov–Hausdorff distance.

Ecological communities are shaped by the interplay between trophic interactions and infectious disease, yet how spatially mediated interactions influence disease-driven ecosystem dynamics remains poorly understood. Here, we develop a gravity-based eco-epidemiological framework for a tri-trophic food chain in which trophic interaction depends on species abundances and effective interaction distance. The disease-free food chain system supports a stable coexistence equilibrium, providing a baseline for investigating disease-induced ecological transitions. Introducing infection at the intermediate trophic level destabilizes this equilibrium through a Hopf bifurcation, leading to sustained oscillations, whereas infection at the top predator level results in a qualitatively different transition from persistence to extinction. By systematically varying the gravity coupling strength, we show that gravity-mediated trophic interactions regulate the thresholds separating these ecological regimes, while the trophic position of infection determines the nature of the transition. Together, these findings establish a unified framework for understanding how spatially mediated trophic interactions and infectious disease jointly govern ecosystem stability, providing new insights into disease-driven dynamics in ecological communities.

Slow fluid antenna multiple access (sFAMA), enabled by the fluid antenna system (FAS), has recently emerged as a practical and low-complexity paradigm for supporting massive wireless connectivity. While existing studies have characterized its physical-layer performance under one-shot transmission, its interaction with retransmission protocols and the resulting networking performance remain largely unexplored. In this paper, we study a downlink hybrid automatic repeat request (HARQ)-assisted sFAMA framework, termed HARQ-sFAMA, in which each user performs distinguished port selection in every HARQ round and combines the received signals across multiple rounds to improve decoding reliability. We develop a comprehensive analytical framework to characterize the outage probability, average packet waiting time, and energy efficiency of the proposed system. The analysis reveals how HARQ exploits the spatial reconfigurability of FAS to simultaneously enhance reliability and improve queueing performance. Numerical results corroborate the theoretical analysis and demonstrate that the HARQ-sFAMA system significantly outperforms conventional one-shot sFAMA in terms of reliability, delay, and energy efficiency. These findings suggest that the integration of HARQ and sFAMA provides a promising pathway toward a practical and standards-compatible massive access solution for future wireless networks.

We show that MI connects subtly and elegantly the two best-known state-of-the-art independence test statistics: the $G^2$ and the Pearson's chi-square statistic $\chi^2$. Furthermore, we show that the MI connects directly those statistics by an elegant formula arising from a stochastic Taylor expansion of MI ($\delta$-method). MI second order term is precisely $\chi^2$ up to a scale factor. As a consequence, by this connection, the difference between $G^2$ and $\chi^2$ can be explicitly quantified.

We study fibrations of higher categories from the perspective of oriented category theory, a framework which accounts for lax phenomena in higher category theory via systematic enrichment in the Gray tensor product. We give several equivalent characterizations of fibrations of $(\infty,\infty)$-categories and oriented categories, and show that categories of fibrations naturally organize to form oriented categories. We study the interaction between fibrations and oriented pullbacks and construct higher-categorical versions of free fibrations and universal fibrations. The latter give rise to Grothendieck constructions for fibrations of $(\infty,\infty)$-categories and oriented categories.

A permutation $\pi \in S_n$ is (32-1)-avoiding when there do not exist $1 \leq i < i+1 < j \leq n$ such that $\pi_i > \pi_{i+1} > \pi_j$. We determine the maximum inversion number for (32-1)-avoiding permutations and count the number of permutations that achieve this maximum. We then provide a direct construction that enumerates these permutations.

$\texttt{codesign-mcdp}$ is a Python library for formulating and solving $\textit{Monotone Co-Design Problems}$ (MCDPs) in the framework of Censi (2015). A design problem is a relation between two posets, a functionality poset $F$ and a resource poset $R$; given a target functionality, the problem asks for the antichain of minimal resources needed to deliver it. Design problems compose under three operators (series, parallel, feedback), and the resulting class is closed under composition. The library implements the antichain calculus, six primitive design-problem types, the three composition operators, a Kleene fixed-point solver, and two high-level builders (an MCDPL-style declarative builder and a modular $\texttt{System}$ builder). Further layers add set-based and stochastic uncertainty, compositional online learning, and temporal, vector-state, and online co-design, alongside a suite of worked examples.

We study the partition function of five-dimensional $\mathcal{N}=1$ $U(N)$ supersymmetric Yang-Mills (SYM) theory with an adjoint hypermultiplet of mass $m_{\rm adj}$ on a toric Kähler surface $S$ times a circle of radius $\boldsymbol{\beta}$. Extending earlier work in $\mathcal{N}=2^*$ SYM theory on $S$, and in pure $\mathcal{N}=1$ SYM on $S\times \mathbb{S}^1_{\boldsymbol{\beta}}$, we find that the path integral localizes to an integral along the Cartan torus of the product of Nekrasov 5D partition functions for each affine patch. Restricting to the gauge group $U(2)$ for simplicity, the integrand has an infinite set of poles of degree at most $\chi(S)-2$. With a natural prescription for integrating around such poles, we find that the contributing poles are in one-to-one correspondence with the torus-fixed points in the moduli space of semi-stable torsion-free sheaves on $S$. Moreover, for non-even first Chern class, their contributions are independent of the equivariant parameters $\epsilon_1,\epsilon_2$ and add up to the $\chi_{y^2}$-genus of that moduli space, where $y^2=e^{-\boldsymbol{\beta} m_{\rm adj}}$, and hence coincide with the refined Vafa-Witten invariants. For even Chern class, the partition function depends on the equivariant parameters $\epsilon_1,\epsilon_2$ as well as $y$, and its relation to rational, refined Vafa-Witten invariants remains unclear.