arXiv preprints from January 1, 2026 through July 21, 2026 — 04:35:09 EST
We consider supercritical long-range percolation on transitive graphs of polynomial growth. In this model, any two vertices $x$ and $y$ of the underlying graph $G$ connect by a direct edge with probability $1-\exp(-βJ(x,y))$, where $J(x,y)$ is a function that is invariant under the automorphism group of $G$, and we assume that $J$...
In operator-algebraic AQFT one routinely moves back and forth between two kinds of structure: inclusions of local algebras coming from inclusions of regions, and bimodules/intertwiners that implement the standard $L^2$-based constructions used to compare and compose observables. The obstruction to making this interplay genuinely...
In this paper we present a systematic study of continuous local iterated function systems. We prove local iterated function systems admit compact attractors and, under a contractivity assumption, construct their code space and present an extended shift that describes admissible compositions. In particular, the possible combinatorial...
We study $\mathbb Z_2\times\mathbb Z_2$ bi-graded Lie algebras. We describe their properties in relation to Lie superalgebras with some compatible structures. Then we focus on the approach to the Lie group--algebra correspondence based on Harish-Chandra pairs and provide some examples of application of it in the bi-graded setting.
We study the bond percolation on finite graphs induced by the level-sets of zero-average Gaussian free field on the associated metric graph above a given height (level) parameter $h \in \mathbb{R}$. We characterize the near- and off-critical phases of this model for any expanders family $\mathcal{G}_n = (V_n, E_n)$ with uniformly...
Large collections of matrices arise throughout modern machine learning, signal processing, and scientific computing, where they are commonly compressed by concatenation followed by truncated singular value decomposition (SVD). This strategy enables parameter sharing and efficient reconstruction and has been widely adopted across...
We prove the combinatorial invariance of the coefficient of $q$ in Kazhdan--Lusztig polynomials for arbitrary Coxeter groups. As a result, we obtain the Combinatorial Invariance Conjecture, of Lusztig and of Dyer, also for Bruhat intervals of length at most $6$. We also prove the Gabber--Joseph conjecture for the second-highest $Ext$...
We establish necessary and sufficient conditions for the existence of an LU factorization $A=LU$ for an arbitrary square matrix $A$, including singular and rank-deficient cases, without the use of row or column permutations. We prove that such a factorization exists if and only if the nullity of every leading principal submatrix is...
We relate Bruhat numbers in real Morse theory to cluster variables in braid varieties. This provides instances of wall-crossing coordinates in the study of Cerf diagrams.
This work investigates the initial value problem (IVP) for the two-parameter family of dispersive wave equations known as the Majda-McLaughlin-Tabak (MMT) model, which arises in the weak turbulence theory of random waves. The MMT model can be viewed as a derivative nonlinear Schrödinger (dNLS) equation where both the nonlinearity and...
Many large-scale testing procedures learn signal structure from the data to boost power. Direct data reuse can inflate Type-I error ("double dipping"), so a common remedy is masking: withholding some information during learning and using it for testing. Sample splitting masks by withholding observations for testing, while null...
We study the value of a two-player zero-sum game on a random matrix $M\in \mathbb{R}^{n\times m}$, defined by $v(M) = \min_{x\inΔ_n}\max_{y\in Δ_m}x^T M y$. In the setting where $n=m$ and $M$ has i.i.d. standard Gaussian entries, we prove that the standard deviation of $v(M)$ is of order $\frac{1}{n}$. This confirms an experimental...
The Compact Discontinuous Galerkin method was introduced by Peraire and Persson in (SIAM J. Sci. Comput., 30, 1806-1824, 2008). In this work, we present the stability and convergence analysis for the $hp$-version of this method applied to elliptic problems on polytopal meshes. Moreover, we introduce fast and practical algorithms that...
We study the topology of the real algebraic hypersurfaces in $\mathbb{P}^n$ that can be constructed via combinatorial patchworking using triangulations that are dilations by two of other triangulations. By examining the real critical points of the polynomials that define such hypersurfaces, we find some asymptotical upper bounds on...
Formanek made the conjecture that the minimal degree of the central polynomials for the $n\times n$ matrix algebra over a field of characteristic 0 is $(n^2+3n-2)/2$ and this is true for $n\leq 3$. For $n=4$ there are examples of central polynomials of degree $13=(4^2+3\cdot 4-2)/2$ and we do not know whether there are central...
We show that the composition of omega-series by surreal numbers, or more generally by elements of any confluent field of transseries, is monotonic in its second argument. In particular, omega-series and LE-series interpreted as functions have the intermediate value property. We also deduce a Taylor approximation theorem for...
The metric MinDist, introduced recently to quantify the distance of an arbitrary Rummy hand from a valid declaration, plays a central role in algorithmic hand evaluation and optimal play. Existing results show that the MinDist of any $13$-card Rummy hand from a single deck is bounded above by $9$. In this paper, we sharpen this bound...
In this paper, we continue the analysis of the effects of semiclassical sub principal controlled quasimodes, approximate solutions to P(h)u(h,b), depending on the subprincipal symbol b, which can give spectral insta bility (pseudospectrum). We consider a pseudodifferential operator, which has double zeros for the principal symbol, p....
The celebrated formula of Otter \emph{[Ann. of Math. (2) 49 (1948), 583--599]} asserts that the complete graph contains exponentially many non-isomorphic spanning trees. In this paper, we show that every connected almost regular graph with sufficiently large degree already contains exponentially many non-isomorphic spanning trees....
We present a physics-informed Wasserstein GAN with gradient penalty (WGAN-GP) for solving the inverse Chafee--Infante problem on two-dimensional domains with Dirichlet boundary conditions. The objective is to reconstruct an unknown initial condition from a near-equilibrium state obtained after 100 explicit forward Euler iterations of...
We study $\mathbb{S}_n$-equivariant motivic invariants of the moduli space $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$ of degree-$d$ maps from $n$-pointed curves of genus $g$ to $\mathbb{P}^r$. In particular, we obtain formulas for the Serre characteristic, which specializes to the Hodge--Deligne polynomial. Fixing $g, r \geq 1$, we prove...
Kernel-based methods offer a powerful and flexible mathematical framework for addressing histopolation problems. In histopolation, the available input data does not consist of pointwise function samples but of averages taken over intervals or higher-dimensional regions, and these mean values serve as a basis for reconstructing or...
The main contribution of this paper is the proof of the generic finiteness of Dziobek central configurations for a homogeneous potential and the derivation of a uniform upper bound for their number. By exploiting the isomorphism between the Veronese variety and the determinantal variety associated with the Dziobek conditions, we...
We show that, in the case of a prime $2$, the numerical triviality of $BP$-cycles modulo various powers of the (augmentation) invariant ideal $I(\infty)$ is controlled by pure symbols in $K^M_*/2$ over the flexible closure of the base field.
For two positive integers $m$ and $M$, we study the Davenport constant of the interval of integers $[\![ -m,M ]\!]$, that is the maximal length of a minimal zero-sum sequence composed of elements from $[\![ -m,M ]\!]$. We prove the conjecture that it is equal to $m+M- r$ where $r$ is the smallest integer which can be decomposed as a...