Papers

We prove that minimal-volume hyperplane sections of the regular simplex through its centroid are parallel to a facet. The proof combines variational methods with Fourier-analytic techniques and zero-diminishing arguments to show that every critical normal vector has at most three distinct non-zero coordinates. Analysis of the two- and three-value cases then yields the sharp lower bound.

In this work, we provide a new perspective on semiparametric efficiency theory. In particular, we reformulate the questions of optimal efficiency and its attainment as a variational problem: minimize variance over estimating functions subject to robust unbiasedness constraints. We develop this theory for marginal-integral functionals without requiring pathwise differentiability. We study estimating functions whose expectations remain equal to the target when any one specified nuisance component is misspecified and the others are correct. We characterize the infimum of their variances through conditional variance minimization. For maxima of affine functions of treatment-specific conditional means, we obtain explicit optimal weights and construct cross-fitted estimators that attain the bound under conditions on nuisance estimation and the probability of near ties. For analogous maxima based on jointly observed quantities, the optimal weights use the full conditional covariance matrix. We also identify conditions under which the variance bound agrees with a classical convolution bound for parametric perturbations that preserve ties to first order. Examples include optimal policy values, $L^1$ calibration error, Balke–Pearl bounds, and mediation parameters. In an application to the National Longitudinal Survey of Young Men, covariance weighting reduces the median estimated variance of the cross-fitted estimating function relative to equal weighting by 12.9% for the lower Balke–Pearl endpoint and 16.5% for the upper, with improvements in all 20 repeated cross-fitting splits.

We determine the defining ideal of the simple affine vertex algebra $L_{2-\ell}(\mathfrak{so}_{2\ell})$ for every odd $\ell\ge5$. Perše's quadratic singular vector alone generates the maximal ideal of the universal affine vertex algebra at this level. Together with the established even-rank presentation, this gives a complete parity-dependent description of this type-$D$ collapsing family at $k=2-\ell$: one quadratic generator in odd rank, and a quadratic generator together with two Pfaffian generators in even rank. The proof establishes a rank reduction for the quadratic quotients under minimal Drinfeld–Sokolov reduction, valid in both parities. Nonvanishing of reduction on nonzero graded subquotients then lifts simplicity along the odd-rank chain from the known base case $D_3\cong A_3$ at level $-1$.

We investigate forward and backward smoothing effects in Lebesgue spaces $L^p$ and $\mathcal{M}^p:=L^{p,\infty}$ for the Cauchy problem associated to the nonlinear and nonlocal fractional diffusion equation $\partial_t u+(-\Delta)^{\frac\sigma2}|u|^{m-1}u=0$ in $\mathbb{R}^N$, $0<\sigma<2$, in the very fast range $0<m\le m_c:=\frac{N-\sigma}{N}$. We prove that very weak solutions have an $\mathcal{M}^p$–$L^\infty$ smoothing effect if $p>p^*:=\frac{N}{\sigma}(1-m)$, and we construct counterexamples showing the failure of any $L^p$–$\mathcal{M}^q$ forward ($q>p$) smoothing effect if $1< p< p^*$, $m<m_c$ or $L^1$–$\mathcal{M}^q$ ($q>1$) if $m=m_c$. We also prove a backward $\mathcal{M}^p$–$L^1$ smoothing effect whenever $1\le p<p^*$, $m<m_c$, and we construct counterexamples showing that there is no $L^p$–$\mathcal{M}^q$ backward ($q<p$) smoothing effect if $p> p^*$. Regarding the threshold value $p=p^*$, we prove that all solutions starting in $\mathcal{M}^{p^*}$ become extinct in finite time, and show the failure of any $\mathcal{M}^{p^*}$–$\mathcal{M}^q$ smoothing before extinction for any $q\neq p^*$. The construction of counterexamples is based on new uniqueness and comparison results for very weak solutions, combined with the existence of self-similar solutions with suitable properties. The same approach yields new counterexamples for both forward and backward smoothing effects also in the local case $\sigma=2$.

The celebrated result of Johansson, Kahn and Vu determined the threshold order for clique factors in random graphs, and subsequent work identified the sharp threshold and the corresponding hitting-time phenomenon. In this paper we study the probability that there is no $K_r$-factor above the threshold and, more generally, the probability that the largest $K_r$-matching covers less than $n-s$ vertices of $G(n,p)$. For every fixed $r\ge3$, throughout the range $$n^{-2/r}(\log n)^{1/\binom r2}\ll p\ll n^{-2/(r+1)},\qquad n-s\in r\mathbb Z,\qquad s=o(n),$$ we prove $$\mathbb P\bigl(\phi_r^s(G(n,p))=0\bigr)=\exp\left(-\Theta_r\!\left((s+1)\frac{\mu_r(n,p)}n\right)\right),$$ where $\phi_r^s(G)$ is the number of $K_r$-matchings covering exactly $n-s$ vertices and $\mu_r(n,p):=\binom nrp^{\binom r2}$. The lower bound is given by $s+1$ vertices which lie in no copy of $K_r$. For the upper bound we develop an iterable one-root version of the Johansson–Kahn–Vu method. As a structural consequence, we show that the remainder of $G(n,p)$ outside every maximal $K_r$-matching has an almost-perfect $K_{r-1}$-matching throughout the sparse clique window. Independently, we prove a central limit theorem for the maximum $K_r$-matching number. Combining these inputs and a structural theorem for $r=2$ from our earlier work, we prove a central limit theorem for the chromatic number of very dense random graphs: for every $r\ge2$ and $n^{-2/r}(\log n)^{1/\binom r2}\ll p\ll n^{-2/(r+1)},$ $$\frac{\chi(G(n,1-p))-\mathbb E\chi(G(n,1-p))}{\sqrt{\mu_{r+1}(n,p)}/r}\xrightarrow{\mathrm d}\mathcal N(0,1),\qquad\operatorname{Var}\bigl(\chi(G(n,1-p))\bigr) \sim\frac{\mu_{r+1}(n,p)}{r^2}.$$ This settles the Surya–Warnke conjecture throughout the interior of every clique window with $r\ge2$, strengthening its concentration prediction to a Gaussian limit with asymptotically exact variance.

In this paper, we investigate the Lipschitz stability of the master field and its functional derivative with respect to the measure variable for the master equation in mean field games. A key novelty of this work is the derivation of a nonlocal parabolic partial differential equation governing the functional measure derivative along a characteristic measure flow. Based on this equation, we establish a Carleman estimate for the difference of the measure derivatives associated with two sufficiently regular classical solutions. We also establish a Carleman estimate for the difference of the corresponding master fields, yielding a terminal-to-interior Lipschitz stability estimate along a characteristic measure flow. Together with stability of the associated measure flows, these estimates are used to control the resulting source terms. We then obtain a terminal-to-interior Lipschitz stability estimate for both the master field and its functional measure derivative in terms of the discrepancies of the terminal data. The result provides quantitative control of the master field with measure-variable regularity, and develops a Carleman-based framework for the stability analysis of master fields and their measure derivatives in mean field game master equations.

We construct compact, exponentially stable $C_0$-semigroups $S$ on Hilbert spaces $X$ such that, for every $T>0$, the stochastic convolution $U_g(t)=\int_0^t S(t-s)g(s)\,\mathrm{d}\beta_s $ is unbounded on $[0,T]$ with positive probability for some predictable $g\in L^\infty(\Omega;L^2(0,T;X))$, where $\beta$ is a real Brownian motion. One example is even an analytic semigroup. For the other, the negative generator $A$ has sectorial angle $\pi/2$ and a bounded $H^\infty$-calculus. Our main tool is a necessary condition: for exponentially stable semigroups, an $L^2$-maximal estimate on $\mathbb R_+$ forces a lower square-function estimate for the negative generator. We deduce this implication from a new identity involving the stochastic convolution, the subordinated Poisson semigroup, and a stopped Brownian motion. Combining this condition with Schauder multipliers on a conditional trigonometric basis yields counterexamples to the maximal estimate. An extrapolation argument then produces the integrands with unbounded stochastic convolutions. Finally, an energy-adapted chaining argument gives continuity and maximal estimates for arbitrary $C_0$-semigroups under the stronger condition $g\in L^2([0,T]; L^\infty(\Omega;X))$.

Mean field control games (MFCGs) provide a framework for studying non-cooperative games involving groups of cooperative players in the limit as both the number of players and the number of groups go to infinity.This class of games includes mean field games (MFGs) and mean field control (MFC) problems as special cases, corresponding to purely non-cooperative and purely cooperative settings, respectively. We incorporate interactions through the mean of actions, hence the terminology of extended MFCGs. We first show that the equilibrium control can be described using a coupled system of partial differential equations consisting of a Hamilton-Jacobi-Bellman equation and a Fokker-Planck equation. After introducing the general framework, we focus on a linear-quadratic (LQ) structure. We show that the equilibrium can be reduced to a system of ODEs, generalizing those obtained for MFGs and MFC problems. We then provide two numerical examples illustrating specific features of MFCGs.

Let \(K,T\subset\mathbb R^n\) be convex bodies with the origin in their interiors. If \(K=T^\circ\), then the identity \( K+T=K^\circ+T^\circ \) holds trivially. V.~Milman asked whether these trivial solutions are the only ones, that is, whether \( K+T=K^\circ+T^\circ \) forces \(K=T^\circ\). We show that the answer is positive for polytopes but negative for general convex bodies.

Building on the established one-particle formula for the Araki relative entropy of coherent states, we study how its value acquires a nontrivial dependence on the mass of the scalar field. For a localized vector $h$ belonging to the standard subspace $H_m$ of the one-particle Hilbert space, the known quadratic-form expression is: $S_{H_m}(h)=-\langle h,\log\delta_{H_m}\,h\rangle$. Our contribution is to construct explicitly a mass-indexed family of vectors of the wedge standard subspace on which this expression is evaluated. The mass-shell map $h_m=E_mf$ organizes four structural conditions on rapidity representatives—on-shell dependence, a controlled massless boundary value, decay for large real rapidity, and Bisognano–Wichmann strip analyticity—and we exhibit an entire rapidity wave function, built from a doubled light-cone phase, two sinc factors, and a Gaussian pair, that satisfies them together with the sharp localization criterion: Hardy-type $L^2$ control throughout the Bisognano–Wichmann strip and the exact Tomita boundary relation. The family therefore belongs to $H_m(\W_R)$ for every $m>0$, and its Araki relative entropy is finite and strictly positive, with an exact spectral representation that makes positivity manifest. The entropy is strongly suppressed at large mass, attains a maximum at intermediate mass in $1+1$ dimension, and converges to a finite value along the modular flow as $m\to0^+$. The construction extends fiberwise to $1{+}d$ dimensions through the transverse mass.

This paper presents an optimal interior penalty discontinuous Galerkin (IPDG) scheme for the planar biharmonic equation using piecewise polynomials of degree $k=3$ or $4$. In standard IPDG methods, large penalty parameters force the discrete solution into an overconstrained space, severely degrading accuracy—phenomenon known as numerical locking. To overcome this, our method enforces only vertex continuity and projects the jumps of the function and its normal derivative onto $\mathcal{P}^{k-3}$ and $\mathcal{P}^{k-2}$, respectively. Consequently, as the penalty parameters tend to infinity, the discrete solution is forced to lie in a constrained subspace $V_{h,\infty}^k$, which we identify as the optimal nonconforming finite element space $B_h^k$. This intrinsic connection fundamentally eliminates numerical locking. We prove optimal error estimates of $\mathcal{O}(h^{k-1})$ in a mesh-dependent energy norm and $\mathcal{O}(h^{k+1})$ in the $L^2$ norm. Numerical experiments on both convex and L-shaped domains confirm that the proposed scheme is robust and entirely locking-free, even for extremely large penalty parameters.

Let $G$ be a simple graph on $n$ vertices and $e(G)$ edges. Let $\mu_1\geq \cdots \geq \mu_{n-1}\geq \mu_n=0$ be the Laplacian eigenvalues of $G$. For $k=1, \ldots, n$, let $S_k(G)=\sum_{i=1}^{k}\mu_i$. Brouwers conjecture asserts that for any $k\in\{1,\ldots, n\}$, $S_k(G)\leq e(G)+\binom{k+1}{2}$. In [Bounding the sum of the largest Laplacian eigenvalues of graphs, Discrete Appl. Math., 170:95–103, (2014)], Rocha and Trevisan showed that the conjecture holds true for $1 \leq k \leq \lfloor g/5 \rfloor$, where $g$ denotes the girth of $G$. This bound on $k$ was later improved by Chen in [Improved results on Brouwers conjecture for sum of the Laplacian eigenvalues of a graph, Linear Algebra Appl., 557:327–338, (2018)], who established that the conjecture holds for $1 \leq k \leq \lfloor g/4 \rfloor$. In this article, we further strengthen these results by proving that the Brouwers conjecture holds for $1\leq k\leq\left\lfloor \frac{m(G)}{2}\right\rfloor,$ where $m(G)$ denotes the matching number of $G$. Since $m(G)\geq \lfloor g/2\rfloor$, the case constitutes a genuine improvement over the aforementioned results. As an application, we show that if $G$ is a graph of order $n$ and with a perfect matching, then the Brouwers conjecture holds for $1\leq k\leq \left\lfloor \frac{n}{4}\right\rfloor$. Finally, we provide a new perspective on verifying Brouwers conjecture by proving that $G$ satisfies the Brouwers conjecture if and only if for a fixed positive integer $h$, $\mathcal{S}_h(\overline{G})\leq e(\overline{G})+\binom{h+1}{2}$ holds whenever $\mathcal{S}_h(G)\leq e(G)+\binom{h+1}{2}$, where $\overline{G}$ denotes the complement of $G$.

The Seiberg-Witten prepotential of four-dimensional $\mathcal{N}=2$ gauge theory can be reached by two recursions of very different nature, and this thesis compares them explicitly. The first part is a self-contained review of the microscopic side: the ADHM construction as a hyperkähler quotient, the supersymmetric measure on the instanton moduli space, Seiberg-Witten theory, and the one- and two-instanton contributions computed by non-commutative resolution and localisation. The second part introduces the $\Omega$-background and Nekrasov's partition function, and uses the AGT correspondence to turn Zamolodchikov's recursion for Virasoro conformal blocks into a recursion for the instanton partition function. For pure $SU(2)$, this recursion is run up to five instantons, including the first corrections in $\epsilon_1,\epsilon_2$. In the last part, the same quantities are obtained from the Eynard-Orantin topological recursion on the Seiberg-Witten curve, which yields the Taylor expansion of the prepotential around an arbitrary point of the Coulomb branch. For pure $SU(2)$ we give the Bergman kernel and the $B$-periods of $\omega_{0,3}$, $\omega_{0,4}$ and $\omega_{1,1}$ in closed form. They reproduce the instanton series generated by Zamolodchikov's recursion, remain valid at strong coupling, where they detect the massless monopole, and at genus one give the first correction to the self-dual $\Omega$-background. The two recursions are thus complementary: one is exact in the vacuum expectation value and perturbative in the instanton number, the other is exact in the instanton parameter and perturbative around a fixed vacuum. We formulate a conjecture relating the $B$-periods of $\omega_{g,n}$ to the derivatives of the genus-$g$ free energy, and list open problems.

We present two results derived from the sharp log-Sobolev inequality for the uniform measure on a complete graph which concern the von Neumann entropy and the Entanglement of Formation of a state of finite and infinite-dimensional quantum systems. The first result is a sharp Lipschitz lower semicontinuity bound for the von Neumann entropy at any mixed state $\rho$ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality $\,S(\rho)-S(\sigma)\leq C_\rho(1-F(\rho,\sigma))\,$ valid for any state $\sigma$, where $C_{\rho}$ is a constant depending on the rank of $\rho$. The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state $\rho$ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality $\,E_F(\rho)-E_F(\sigma)\leq C_\rho(1-\mathrm{Tr}\rho\sigma)\,$ valid for any state $\sigma$, where $C_{\rho}$ is a constant depending on the Schmidt rank of $\rho$. In both cases the optimal constant $C_\rho$ is equal to the optimal constant $K_{d}$ in the log-Sobolev inequality for the complete graph with $d$ vertices: in the first case $d=\mathrm{rank}\rho$, in the second one $d=\mathrm{rank}\rho_A=\mathrm{rank}\rho_B$. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.

In this note, we propose a general structural approach to model a default time $\tau$ as the first-passage time (FPT) of a (“firm-value”) process $S$ below a (“debt”) barrier $K$ that comply with a pre-specified survival probability curve $G(t)=\Pr(\tau>t)$. Following an idea of Mbaye and Vrins (Mathematical Finance, 2022) applied to reduced-form models, our approach consists in two steps: choose a latent FPT model driven by a barrier $\tilde{K}$ and process $\tilde{S}$, and time-change those using a deterministic clock $\Theta$ to get $K_t=\tilde{K}_{\Theta(t)}$ and $S_t:=\tilde{S}_{\Theta(t)}$, leading to the final FTP model $(K,S,\Theta)$. As the market curve $G$ and the latent model $(\tilde{K},\tilde{S})$ are assumed to be given, the calibration step simply consists in finding the clock $\Theta$ such that the distribution of the FPT of $S$ below $K$ coincides with the survival curve $G$. We show that this is achievable for a broad class of specified curves $G$ and latent FTP models. The calibration amounts to a simple inversion of a function, which is almost immediate provided that the latent model is tractable enough. In particular, we show that the AT1P model of Brigo, Morini and Tarenghi - which is able to reproduce a broad range of CDS term-structures - can be regarded as the FPT of a time-changed drifted Brownian motion to a constant barrier: $\tilde{V}_t=\mu t+W_t$ and $\tilde{K}_t=k<0$. This connection offers an elegant interpretation for the instantaneous volatility function featured in AT1P and yields an immediate calibration of the latter to perfectly match a target survival curve.

This work is a mathematical study of the multiphasic formulation of the Vlasov equation, which consists in recasting this collisionless kinetic equation as a system of coupled pressureless Euler equations. This framework allows to consider solutions that are only measure-valued in the velocity variable and is thus relevant to tackle physical problems where rough velocity distributions, such as Dirac masses, naturally arise. We specifically address the case of nonlinear Vlasov equations where the force field is one derivative more regular than some moments in velocity of the solution. Under this key assumption, a unified theory of local well-posedness at finite regularity (typically in Sobolev spaces) is developed from scratch for the multiphasic formulation, yielding corresponding results for the Cauchy problem of the associated Vlasov equation at low regularity. We thoroughly apply this abstract theory to two classes of equations, namely Vlasov-Poisson type systems, and Vlasov-Navier-Stokes type systems. In addition to the justification of the monokinetic limit, it also leads to specific applications for each class of equations, allowing possibly rough velocity distributions. For Vlasov-Poisson type systems, we justify the semiclassical limit from Hartree equations, and we describe the nonlinear instability of homogeneous equilibria. For Vlasov-Navier-Stokes type systems, we establish nonlinear asymptotic stability near monokinetic profiles. New results allowing a separation between the regularity in space and in velocity are proven, and along the way, we also provide new proofs of known results and generalize them to the case of rough solutions. Finally, the flexibility of the framework is exploited to obtain extensions to more sophisticated systems such as the Vlasov-Poisson equation for ions, or the Vlasov equation coupled with the compressible Navier-Stokes system.

A formal solution to a partial differential equation of infinite order involving the action of Mahler-type operators in several variables is obtained. Such formal solutions is constructed by means of a fixed point argument applied to the integral representation of the deceleration operator in several variables that has been developed. The Laplace transform in two variables of the actual solution of a related problem in both Borel plane and Fourier space turns out to admit strong asymptotic expansion along adequate multidirections.

Multi-electrode Arrays (MEAs) enable tissue-level electrophysiological studies of human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs) by recording extracellular field potentials. Recent computational models have improved MEA simulations by coupling detailed electrode descriptions with the Bidomain framework, a parabolic-elliptic system of nonlinear PDEs coupled with a stiff ionic model. The standard numerical strategy relies on operator splitting techniques that decouple the PDE and ODE components. In most implementations, the ionic subsystem is treated explicitly while the diffusive operator is handled implicitly, resulting in a first-order implicit-explicit (IMEX) time discretization. Although computationally convenient, this approach limits temporal accuracy and may reduce efficiency in large-scale simulations. In this work, we investigate higher-order IMEX Runge-Kutta schemes within a Strang-based operator splitting, specifically tailored for the MEA model of hiPSC-CMs monolayers. First- and second-order schemes are compared in terms of computational cost and global error against a high-fidelity reference solution obtained with a very small time step. For a fixed computational cost, the second-order schemes achieve errors that are 2-3 orders of magnitude smaller than those of the first-order method. These results demonstrate that higher-order IMEX integration significantly improves the accuracy-to-cost ratio of MEA simulations, providing a practical and reliable approach for large-scale electrophysiological studies.

Although there are many important contributions on compressible and incompressible fluid-particle interaction models respectively, how to connect the two-type fluid-particle models via the low Mach number limit remains a challenging open problem. In this paper, we resolve it for the compressible isentropic fluid-particle model (Euler–Vlasov–Fokker–Planck (Euler–VFP) system) in the whole space $\mathbb{R}^3$. First, we establish the global-in-time a priori estimates of strong solutions that are uniform with respect to the Mach number $\varepsilon$ near the global Maxwellian. The proof relies on a refined energy method that combines the relaxation structure $b^\varepsilon-u^\varepsilon$ induced by the fluid-particle interaction and the symmetrized acoustic structure of the compressible Euler part in the model. Under the assumption of well-prepared initial data, we derive a global-in-time uniform error estimate in the $H^2$ framework between the solution of the compressible Euler–VFP system and that of the limiting incompressible Euler–VFP system. A key point is to introduce the corrected acoustic variable $ q^\varepsilon-\varepsilon [P'(1)]^{-1}\pi$, which captures the pressure corrector in the low Mach number limit. This also allows us to exploit the exact cancellation of the singular acoustic terms and to close the global-in-time error estimate. The damping term $b^\varepsilon-u^\varepsilon$, which is absent in the pure Euler equations, plays an essential role in recovering the relative velocity dissipation and in controlling the coupled fluid-particle dynamics. As a consequence, we prove the low Mach number limit of the compressible Euler–VFP system with the convergence rate $\mathcal O(\varepsilon)$ in the time-continuous $H^2$ topology.

Finding large lattice subgraphs in sparse Erdős–Rényi random graphs is a classical problem at the interface of random graph theory and algorithms. General bounded-degree embedding and universality theorems give powerful results for broad graph families, but when specialized to square grids they operate at densities substantially larger than the grid-emergence scale. In this paper we exploit the specific geometry of the square grid. We introduce the Quarantined Local Search, a three-phase local algorithm that separates the construction of an initial boundary from the later corner-closure process and controls adaptive negative exposure through bounded pair-test histories. We prove that, for every fixed $\delta\in (0,1)$, there exists $C_\delta>0$ such that the algorithm embeds a $k\times k$ square grid with $k^2\le (1-\delta)n$ in $G(n,p)$ with high probability whenever $p\ge C_\delta\sqrt{\ln k/n}$. Thus, for $k^2=\Theta(n)$, a density of order $\sqrt{\log n/n}$ is sufficient, a factor of order $\sqrt{\log n}$ above the corresponding $n^{-1/2}$ emergence scale.

This paper investigates reversed product properties of two elements in rings. Motivated by Cline's formula, we characterize reversible and $\ast$-reversible rings in terms of group invertible elements, EP elements, and the transfer behaviour of generalized inverses for reversed products. We prove that a unital ring $R$ is reversible if and only if $ab\in R^{\sharp}$ yields $ba\in R^{\sharp}$. For an involutive ring $R$, $R$ is $\ast$-reversible precisely whenever $ab\in R^{\mathrm{EP}}$ implies $b^{\ast}a\in R^{\mathrm{EP}}$. Several counterexamples are constructed to differentiate these ring classes, and the mutual inclusion relations among them are also discussed.

Each of the 12 known conjecturally optimal kissing configurations in dimensions 5, 6, and 7 carries a contact-graph polytope: a polytope whose vertices are the configuration points and whose edges are exactly its contacts. Eight of these polytopes are non-convex and distinct from their convex hulls, which include longer edges. Each polytope is radial about the sphere's center: its boundary meets every ray from the center exactly once. It is the unique radial polytope with those vertices and edges. Seven of the nine non-lattice configurations carry polytopes constructed from the lattice contact polytope in the same dimension by splitting some facets and reassembling the pieces. This split-and-reassemble construction describes the configurations by facets rather than by layers. The fold angles of the reassembled facets explain the inner products $1/(10-n)$, giving $1/5$, $1/4$, and $1/3$ in dimensions 5, 6, and 7, respectively. The contact graphs of all 12 configurations are infinitesimally rigid. We also prove that $Q_5$ and $R_5$ are infinitesimally jammed, extending the 10 previously established cases to all 12 configurations. All mathematical computations use exact arithmetic.

The paper studies finite-horizon ruin for an insurance company with several lines of business admitting heavy-tailed claims. The claim amount process of the insurance company is modeled as a multivariate increasing Lévy process possibly perturbed with fluctuations. A relevant insolvency event may involve the failure of a single line, of several lines at once, or of groups of lines linked through internal transfers, reinsurance, or a guarantee arrangement. These events have different geometries and need not occur on the same probability scale. Under asymptotic tail independence among claims from different lines, classical regular variation assigns zero limiting probability to a multi-line failure and is therefore unable to identify either the ruin probability or the corresponding solvency capital requirement. Using a multivariate Lévy process to model cumulative claims and regular variation on a nested sequence of subcones of the positive orthant to model dependence across lines, we provide the exact scaling rate of the ruin probability and the limit behavior. Additionally, explicit computations are presented for examples with independent business lines, common arrivals of independent claims, Gaussian copula dependent claims, Marshall-Olkin dependent claims, and bipartite insurance networks. Finally, we use Monte Carlo simulations to support our theoretical findings.

6G is expected to push ultra-reliable low-latency communication (URLLC) toward stringent residual-error targets for mission-critical services, where undetected errors and erasures carry fundamentally different costs. Block error rate (BLER) conflates block confusions (undetected errors) and block erasures, which have fundamentally different impacts on system reliability. This paper extends the confusion and erasure analysis of error-bounded decoders to square quadrature amplitude modulation (QAM) constellations in the finite blocklength (FBL) regime. To handle QAM's heterogeneous symbol energies - which make the per-pair Euclidean distance a distribution rather than a single value - we derive analytical lower and upper bounds on the block confusion rate by, respectively, collapsing this distribution to its root-mean-square (RMS) distance and averaging the pairwise confusion over it. These bounds are proven to be monotonically decreasing in both the average symbol energy and the blocklength, with the decrease rate governed by the constellation order. Numerical results confirm that as the signal-to-noise ratio (SNR) or redundancy increases, the confusion rate falls many orders of magnitude below the reliability target, leaving detectable erasures as the dominant residual error.

In this paper, we continue our recent study on anisotropic minimal surface equation with free boundary condition in Wang-Wei-Xia-Zhang (Arch Ration Mech Anal 250:42, 2026). The first main result solves the corresponding mixed boundary value problem. As an application, we prove the following Bernstein-type theorem: any anisotropic minimal graph over $\mathbb{R}^3_+$ with free boundary must be flat. These extend the classical results of Simon (Indiana Univ Math J 25:821–855, 1976; Math Z 154:265–273, 1977) to an appropriate free boundary setting.

We prove that every infinite compact abelian group $G$ admits a countably infinite proper set $E\subset\widehat G$ that is strongly small-2 but not Riesz. More precisely, $|\alpha|*|\beta|$ is absolutely continuous for all $\alpha,\beta\in M_E(G)$, whereas $M_E(G)$ contains a singular complex measure of total mass one with exact Fourier support $E$ whose convolution square belongs to $L^2(G)$. When the image of the doubling map on $\widehat G$ is infinite, $E$ may moreover be chosen so that $\#(E\cap(\gamma-E))<\infty$ for every $\gamma$ and $E\cap(-E)=\{0\}$. When that image is finite, we reduce the problem to a nonrectangular construction on the Walsh group, for which $E\cap(E+\gamma)$ is finite at every nonzero shift.

We prove for the Euclidean space $\left(\mathbb{R}^n, \|.\|_2\right)$ the existence of subsets $A^n \subseteq \mathbb{R}^n$ with $d_H(\mathbb{R}^n, A^n)$ arbitrarily small, such that $d_{GH}(\mathbb{R}^n, A^n) = \sqrt{\frac{n+1}{2n}}d_{H}(\mathbb{R}^n, A^n)$, showing that the lower bound $d_{GH}(\mathbb{R}^n, A^n) \geq \sqrt{\frac{n+1}{2n}}d_H(\mathbb{R}^n, A^n)$ proved in arXiv:2607.18447 is tight for sets that are arbitrarily dense in the surrounding space. We also prove a similar statement for Riemannian manifolds and sufficiently dense subsets as a converse to arXiv:2309.16648.

We present a biparametric preconditioning technique for large, sparse, non-symmetric, and non-singular double saddle point problems. The preconditioner is induced using a stationary iteration method. It is also based on a two-parameterized block LU factorization of the coefficient matrix. To begin with, some convergence results of the iteration method are derived. Moreover, the spectral features exhibited by the preconditioned matrix are given. Finally, the presented preconditioner is conjuncted with the GMRES method. Numerical results demonstrate the satisfactory performance of the preconditioner.

A graph is $K_{r+1}$-saturated if it is $K_{r+1}$-free and adding any missing edge creates a copy of $K_{r+1}$. Kim, Kim, Kostochka, and O conjectured that $K_{r-1}\vee(n-r+1)K_1$ minimizes the spectral radius among all $n$-vertex $K_{r+1}$-saturated graphs. They proved the case $r=2$, and the cases $r=3$ and $r\in\{4,5\}$ were subsequently established by Kim, Kostochka, O, Shi, and Wang, and by Wang and Hou, respectively. We settle the conjecture for all $r\ge3$: if $n\ge r+1$ and $G$ is an $n$-vertex $K_{r+1}$-saturated graph, then \[ \rho(G)\ge \frac{r-2+\sqrt{(r-2)^2+4(r-1)(n-r+1)}}{2}, \] with equality if and only if $G\cong K_{r-1}\vee(n-r+1)K_1$. We also prove O's local two-walk conjecture for every $r\ge2$: \[ \sum_{w\in N_G(v)}d_G(w) \ge (r-2)d_G(v)+(r-1)(n-r+1) \qquad(v\in V(G)). \] If $G$ has no universal vertex, the inequality holds with the additional term $(r-1)(r-2)$ on the right-hand side. This constant is best possible uniformly in $n$ for every fixed $r$, and gives a strict improvement when $r\ge3$. A corresponding spectral bound follows.

We estimate the scale of an exponential lifetime from observations censored randomly from both sides, under an informative model in which each censoring law is a power of the lifetime survival function – a two-sided generalization of the Koziol–Green proportional hazards model. Substituting the empirical distribution function into the likelihood equation yields a closed-form pseudo-maximum-likelihood estimator of the scale that is computed in one pass over the order statistics and needs neither iteration nor numerical optimization. A digamma identity shows the estimating equation to be exactly, not merely asymptotically, Fisher-consistent. Strong consistency and asymptotic normality are established with no restriction on the censoring depth; the key is an identity that rewrites the underlying $L$-statistic without its unbounded score function and so removes the condition that classical limit theorems would impose. The influence function and the asymptotic variance are explicit in polygamma functions, so confidence intervals require no numerical integration. Against the information bound of the observed-data model the efficiency exceeds $98.6\%$ in the designs considered and stays above $96.2\%$ over a wide sweep. Only the entry half of the model is needed: the estimator remains exactly Fisher-consistent and strongly consistent whatever the right-censoring law, and exponentiality is shown to be the precise price of that freedom. The construction extends verbatim to a proportional hazards class with known baseline. Three data sets illustrate the procedure, including a cohort in which left censoring is genuine.