Papers

Let $X$ be a Cantor space and let $Q\subset M_{fc}(X)$ be a compact Choquet simplex of atomless full-support probability measures. We introduce the associated Affine Evaluation Map (AEM), which assigns to each clopen set $A\subset X$ the affine function \[ \widehat A(\mu)=\mu(A),\qquad \mu\in Q, \] and study the geometric subset condition obtained by comparing these evaluation functions pointwise on $Q$. For a good geometric AEM we construct the ordered group \[ G_Q=C(X,\mathbb Z)/N_Q, \qquad N_Q=\left\{f\in C(X,\mathbb Z): \int f\,d\mu=0\ \text{for every }\mu\in Q\right\}, \] and show that it is a simple dimension group whose normalized state space is canonically $Q$. We prove that its order interval $[0,u]$ is precisely the clopen scale and that \[ J_Q=N_Q, \] where $J_Q$ is generated by the elementary relations $\mathbf 1_A-\mathbf 1_B$ with $\widehat A=\widehat B$. We also show that the full stabilizer $\mathcal H_Q$ has invariant-measure simplex exactly $Q$. Using the clopen-scale property and the Herman–Putnam–Skau realization theorem, we obtain a dimension-group proof that every good geometric AEM is realized by a minimal Cantor homeomorphism $T$ with $Q=M_T(X)$. For a Cantor minimal system we identify $G_Q$ with the classical dimension group modulo infinitesimals. We further distinguish goodness, fairness, ergodicity, and minimality of the measure stabilizer by explicit examples. Finally, we apply the AEM framework to minimal Cantor actions of countable amenable groups. If $Q=M_G(X)$, then $Q$ canonically defines a proper geometric AEM, and we clarify which parts of the preceding theory depend only on $Q$ and which are specifically $\mathbb Z$-dynamical. In particular, $Q$ is good if and only if there exists a minimal homeomorphism $T$ of $X$ such that \[ M_T(X)=M_G(X). \]

In the present paper it is discussed to which extend conformal mappings of special circular triangles, which are thoroughly examined by M. Bonk using analytic and classical special functions methods, can be defined and investigated geometrically. This is in analogy to the elliptic modular function, which admits an analytic and also a geometric definition.

Multiplicity factoring is usually formulated for child families at comparable scales. For children of mixed geometry, thickening at the shortest parent scale produces nonuniform inflation ratios, and a single worst-case replacement does not preserve the natural density normalization. We prove a multiplicity-factoring theorem for finite indexed convex parent–child families in $\mathbb{R}^4$ that accommodates arbitrary child shapes, scales, orientations, aspect ratios, and repetitions. The local geometry of each assigned family is encoded by a thickening-weighted Frostman coefficient and a mean-normalized inflation efficiency, both determined by the base family before any shading refinement. The resulting coarse density satisfies $\mathcal{L}^{-A_\varepsilon}(w_1/w_4)^\varepsilon \mathfrak{P}_\varepsilon\lambda^{K_\varepsilon}$, up to the stated parameter-dependent constant, where $\mathfrak{P}_\varepsilon$ is an explicit profile of the parent loads and local efficiencies. The proof thickens arbitrary measurable shadings, projects along a shortest parent direction, establishes an indexed three-dimensional convex-union estimate in the presence of collisions, and lifts the resulting density response back to four dimensions. A weighted Hölder inequality then assembles the nonuniform parent data, while a common cellular refinement regularizes the fine and coarse multiplicities and yields the stated parentwise multiplicity-product estimate. Under a relative convex Frostman hypothesis, the profile is expressed explicitly in the minimum, mean, and maximum inflation ratios. The comparable-scale regime follows as a specialization after a preliminary load selection. Thus the theorem supplies a structural factoring input for four-dimensional overlap arguments; deriving new Kakeya maximal or Hausdorff-dimension estimates would require additional analytic ingredients.

We study a first-order stochastic transport equation driven by Brownian transport noise and a nonlinear state-dependent Poisson jump term. The drift vector field is merely integrable and satisfies the subcritical Krylov–Röckner condition. The dependence of the jump coefficient on the solution creates a nontrivial coupling between the solution value and its spatial gradient. To handle this coupling, we construct a stochastic characteristic system for the position, the solution value, and the gradient, and derive a characteristic representation by means of an Itô–Wentzell formula with jumps. For singular drifts, we combine smooth approximation, the Zvonkin transformation, estimates for stochastic flows, and stochastic Gronwall inequalities to pass to a weakly differentiable limit. Uniqueness is established directly in the weakly differentiable class through a renormalized energy estimate for the difference of two solutions, including the contribution of the Poisson compensator. Under suitable integrability and differentiability assumptions on the jump coefficient, we prove the existence and uniqueness of weakly differentiable solutions. We further show that the first-order spatial Sobolev regularity of the initial datum is preserved by the evolution.

We prove comparison inequalities for Dirichlet-to-Neumann maps corresponding to different non-positive Helmholtz parameters. For convex domains our bounds are sharp, and the resulting eigenvalue inequalities partially confirm an earlier conjecture, which we show does not hold in full generality. We further obtain geometry-dependent versions for arbitrary sufficiently regular domains, together with extensions to compact Riemannian manifolds with boundary. We also discuss analogous questions for metric graphs.

We study a long-run risk-sensitive portfolio problem with proportional transaction costs in a continuous-time market whose log-prices are given as a Lévy process, and rebalancing is possible only at random moments of investment opportunities. Using a Schauder fixed-point argument, we solve the ergodic Bellman equation without any mixing assumptions. Under additional full-support and growth conditions, the solution to the Bellman equation is unique, with the unique continuous maximiser. We also prove vanishing risk-aversion asymptotics towards the risk-neutral (Kelly) problem and convergence of a dyadic-time-grid approximation of the random intervention moments. Numerical examples illustrate the results.

We study the classical anisotropic Calderón problem associated to the elliptic equation $\text{div}(\sigma\nabla u)=0$, where the complex admittivity $\sigma$ is of the form $\sigma=A(\cdot,a(\cdot))$ in a domain $\Omega\subset\mathbb{R}^n$, $n\ge3$. We establish boundary stability estimates for $\sigma$ and its derivatives of arbitrary order from a local Neumann-to-Dirichlet map. Our results extend those of Comm. Partial Differential Equations, 34 (2009) from the real-valued conductivity setting to complex anisotropic admittivities.

In this paper, we present a general method of deducing the determinantal expressions for a polynomial and its derivative. As illustrations, we provide three determinantal expressions for the derivative of the Eulerian polynomial. Using a functional equation discovered by Gessel,we also establish the determinantal expressions for the second-order Eulerian polynomial and its derivative.

This paper investigates the construction of linear codes via abelian codes over finite fields. By exploiting the algebraic structure of multivariate polynomial quotient rings, we derive lower bounds on the minimum distance using a generalized shift bound, which extends the classical van Lint-Wilson bound for cyclic codes. Several infinite families of abelian codes are explicitly constructed, including binary and ternary cases that extend previously known cyclic constructions. To find more abelian codes with good parameters, we apply a genetic algorithm that searches over defining sets represented as binary chromosomes of cyclotomic cosets; the fitness function compares the computed minimum distance against the best known linear code (BKLC) bounds. The search yields multiple record-breaking codes over F_3 and F_4, with improvements over Grassl's tables. Furthermore, the nested structure of these codes enables the application of Construction X, yielding additional linear codes with improved parameters. The results demonstrate that abelian codes, combined with heuristic search, form a viable way for discovering linear codes with unknown parameters.

In 2016, Reiher's clique density theorem determined the minimum number of copies of $K_t$ in a graph with a prescribed edge density. In this paper, we investigate its local version and prove a local clique density theorem in $H$-free graphs as follows. For integers $r$ and $t$ with $2\leq t\leq r-1$, any $r$-chromatic graph $H$, any real numbers $\gamma$ and $\alpha$ with $\frac{t-2}{2(t-1)}\leq\gamma\leq \frac{r-2}{2(r-1)}$ and $0\leq\alpha\leq 1$, we determine the maximum value $\beta:=\beta(r,t,\alpha,\gamma)$ such that for every $n$-vertex $H$-free graph $G$ with at least $\gamma n^2$ edges, every $\lceil\alpha n\rceil$-vertex subset in $G$ contains at least $(\beta-o(1))n^{t}$ copies of $K_t$. In particular, when $H=K_r$, every $\lceil\alpha n\rceil$-vertex subset contains at least $\lfloor\beta n^t\rfloor$ copies of $K_t$, which is an exact bound. For suitable choices of $\alpha$ and $\gamma$, namely, those for which all part ratios in the corresponding extremal construction are rational, this bound is attained for infinitely many values of $n$.

We study a family of the non-vacuum stationary axisymmetric type D metrics, where the two principal null directions are geodesic and shearfree. It is demonstrated that the conformal-to-Carter metric gives the most general form of this family, without any assumptions which are previously used by Ovcharenko and Podolský to arrive at this conclusion.

Let $K$ be an algebraically closed field of characteristic 0 and transcendence degree at least 2. Let $C\subset K^2$ be an irreducible curve defined over $K$ but not defined over the algebraic closure of $\mathbb Q$. There is $(x ,y)$ a $K$-point of $C$ such that $x$ and $y$ are algebraically independent. Moreover, if $C_0$ and $C_1$ are two such curves and there is a finite-to-finite algebraic correspondence between them defined over $K$, then there are corresponding $K$-points $(x_0,y_0)\in C_0$ and $(x_1,y_1)\in C_1$ such that $x_0$ and $y_0$ are algebraically independent and $x_1$ and $y_1$ are algebraically independent. We use the latter result to construct non-Archimedean real closed fields of transcendence degree $\kappa$ with no non-trivial automorphisms for all $2\le\kappa\le \aleph_1$.

Let $p$ be an odd prime and let $H_{p^3}=\operatorname{UT}_3(\mathbb{F}_p)$ be the Heisenberg group of order $p^3$ and exponent $p$. We prove $\mathsf{d}(H_{p^3})=3p-3$. The main ingredient of the proof is an order-value growth theorem. If $B$ is a noncollinear zero-sum sequence of $n$ nonzero vectors in $\mathbb{F}_p^2$, then the alternating areas obtained by ordering $B$ assume at least $\min(p,n-1)$ distinct values. Its proof is a short contraction induction: contract a suitable independent pair, replace the contracted vector in both orders, and apply Cauchy–Davenport. A polynomial relative-subsum theorem and a sharp representation-rigidity lemma then turn this local growth into a uniform spread bound. Combined with the standard product-one criterion for $H_{p^3}$, the spread bound yields the upper bound; the usual sequence $x^{p-1}y^{p-1}v^{p-1}$ gives the lower bound.

We study the random induced-$P_4$-free graph process. Let $e_1,\ldots,e_N$, where $N=\binom{n}{2}$, be a uniformly random ordering of the edges of $K_n$. Starting from the empty graph $G_0$, we add $e_{m+1}$ whenever $G_m+e_{m+1}$ contains no induced $P_4$, and otherwise leave the graph unchanged. We show that the terminal graph is a trivially perfect graph and we describe the structure and distribution of the connected components of the terminal graph $G_N$. Consequently, we derive the limiting values of several natural graph parameters. In particular, the terminal graph $G_N$ has $\Theta(n)$ edges.

We prove that $Bir_2(\mathbb{P}^4)$ has exactly nine irreducible components and that the locus of transformations with zero-dimensional base scheme is irreducible.

The periodic non-Hermitian Baxter-Fendley $Z_N$ clock chain has lacked a complete finite-size spectral solution, whereas its open-chain counterpart admits a solution in terms of independent quasienergies. For the periodic model we show that the operator-valued matching polynomial associated with its cyclic Weyl algebra simultaneously generates a set of conserved quantities, including the Hamiltonian, and realizes a cyclic $\tau^{(2)}$ Yang-Baxter transfer matrix. Root-of-unity closure yields a finite system of polynomial spectral equations in each charge sector, which reproduces the complete finite-size energy spectrum counted with algebraic multiplicity. As a first application of this result, we show that Newton continuation of these equations provides a practical numerical route to the periodic ground-state energy without enumerating the full spectrum. For homogeneous chains the thermodynamic seam response yields a criterion for boundary-induced criticality; for $N=3$ it predicts two reciprocal critical couplings with singular ground-state curvature, in contrast to the single self-dual open boundary critical point.

We study the finite-genus algebro-geometric solutions of the Wadati-Konno-Ichikawa (WKI) equation with the saturable nonlinearity and long-time asymptotic behaviors of their short-range perturbations. First, for both the focusing and defocusing reductions, we formulate the finite-genus Baker-Akhiezer functions as explicitly solvable the matrix Riemann-Hilbert (RH) problems on the complex spectral plane and obtain theta-function representations together with the reconstruction formulae for the WKI field and the reciprocal coordinate. We then consider the Cauchy problem of the defocusing WKI equation on a finite-genus algebro-geometric background. We construct the scattering data and RH problem, and perform a Deift-Zhou nonlinear steepest descent analysis. The space-time plane is divided into two transition regions, a Zakharov-Manakov (ZM) region, and a fast-decay region. The leading term is a phase-shifted finite-genus WKI solution. The transition corrections are governed by a Painlevé-XXXIV model, while the ZM radiation is described by parabolic-cylinder functions. The reciprocal-coordinate asymptotics are obtained simultaneously.

A vertex-coloring of a graph is centered if every connected subgraph has a vertex with a unique color. A vertex-coloring of a graph is linear if every path in the graph has a vertex with a unique color. Let $\chi_{\mathrm{cen}}(G)$ and $\chi_{\mathrm{lin}}(G)$ be the minimum number of colors in a centered (resp. linear) coloring of $G$. We present a family of graphs witnessing that if $f$ is a nondecreasing function such that $\chi_{\mathrm{cen}}(G) \leq f(\chi_{\mathrm{lin}}(G))$ for every graph $G$, then $f(k) = \Omega(k^2 / \log k)$. The construction was found by OpenAI's GPT-5.6 Sol Pro.

In a graph $G$, a set of edges $F$ is called a forcing set if there exists a unique perfect matching $M$ such that $F \subseteq M$. Similarly, a set of edges $A$ is called an anti-forcing set if the graph with edge set $ E(G)\setminus A$ has a unique perfect matching. It is known that, given a bipartite graph $G$ of maximum degree~$3$ and a perfect matching $M$, the problem of deciding whether there exists a forcing set of size at most $k$ for $M$ is NP-complete. Moreover, given a bipartite graph $G$ of maximum degree~$4$ and a perfect matching $M$, the problem of deciding whether there exists an anti-forcing set of size at most $k$ for $M$ is NP-complete. Furthermore, given a bipartite graph of maximum degree~$5$, the problem of deciding whether there exists a perfect matching $M$ that can be made unique by a forcing set of size at most $k$ is also NP-complete. In contrast, the computational complexity of deciding whether there exists a perfect matching $M$ that can be made unique by an anti-forcing set of size at most $k$ is not known, even for general graphs. In this paper, we show that all of these problems remain NP-complete even when restricted to bipartite graphs of maximum degree~$3$.

We design randomized approximation schemes for the partition function of antiferromagnetic Ising models with uniform external field on random regular bipartite graphs. Our algorithm generalizes the approach of Kocurek, Oveis Gharan and Tjowasi (arXiv, 2026) for hard-core models on the same random graph model beyond the uniqueness threshold. We show that, as long as $\lambda$ is upper bounded by a constant and $\lambda(1 - \beta) \lesssim \Delta^{-1/2}$, an efficient randomized algorithm approximates the partition function with high probability. The algorithm first truncates configurations that are large on either side of the bipartition and then samples from Gibbs distributions conditioned on fixed sizes on one or both sides. To choose an optimal truncation bound, we establish concentration properties of the Gibbs distribution on random regular bipartite graphs. Then we apply high-dimensional expansion and prove trickle-down theorems to obtain fast samplers for the conditioned distributions.

We determine all positive integer solutions to $p^x+(2p+1)^y=z^2$, where $p$ is an odd prime and $|x-y|=1$. The unique solution is $(x,y,z)=(2,1,p+1)$ except when $p=7$, for which there is one additional solution, $(x,y,z)=(4,3,76)$. The proof reduces the nontrivial factorization case to an auxiliary equation, bounds its exponents using an AM–GM inequality, and isolates the exceptional solution with a divisibility argument modulo $2k^2$.

Given a certain specific, $M$ by $M$ matrix form of the Feshbach's effective (i.e., energy-dependent) Hamiltonian $H^{(M)}_{e\!f\!f}(E)$, the inverse-problem reconstruction of the full-space, $N$ by $N$ matrix Hamiltonian $H^{(N)}$ is considered and reduced to the solution of a coupled set of polynomial algebraic equations. Using computer-assisted symbolic manipulations, an explicit algebraic reconstruction of $H^{(N)}$ is found feasible at not too large $K = N-M$.

We study the $\mathbf{I}^*$-cohomology of a smooth real algebraic curve in terms of its real locus and its geometric genus. We notably extend results of Knebusch and Monnier to the twisted case. We further describe how to compute twisted Witt groups of surfaces, generalising work of Sujatha, and apply our results to the computation of the shifted and twisted Witt groups of the smooth real anisotropic quadric surface. Finally, we extend Monnier's investigation of the image of the global signature homomorphism for surfaces.

A finite group has the weak Erdos-Ko-Rado property if all of its transitive permutation actions have the EKR property. We characterize this property in terms of normal subgroups and chief factors. More precisely, we introduce a local intersection density and establish a normal-extension criterion which reduces the weak EKR property to difference-set conditions on the elementary abelian chief factors and the linear groups induced on them. For chief factors of rank one the condition is automatically satisfied, and for chief factors of rank two, this condition is equivalent to the induced linear group being intransitive on the one-dimensional subspaces. In the second part of the paper, we solve an open problem by determining the possible intersection densities of transitive permutation groups with a prescribed point stabilizer. We prove that, for every finite group $H$ of order $m\geq 4$ and every integer $n\geq m$, there exists a faithful transitive permutation group with point stabilizer isomorphic to $H$ and intersection density $n/m$.

We study Carlet's cyclic-additive conjecture for the Kasami almost perfect nonlinear (APN) function $F(x)=x^{4^k-2^k+1}$ on $GF(2^n)$, $\gcd(k,n)=1$: for the $2^{n-1}$-element set $\Delta=\{F(b)+F(b+1)+1: b\in GF(2^n)\}$ and all distinct nonzero $v_1,v_2\in GF(2^n)$, \[ \bigl|\{(x,y,z)\in\Delta^3 : v_1x+v_2y+(v_1+v_2)z=0\}\bigr| \;=\; 2^{2n-3}. \] This exact triple-count condition was first formulated by Carlet in his 2018 cyclic-additive difference-set framework; the Kasami instance was subsequently posed as an open problem at the NSUCRYPTO 2019 cryptographic olympiad, whose individual proposer was not publicly disclosed. We prove the conjecture for $k\bmod n\in\{1,2,n-2,n-1\}$, in particular a complete proof for $k=2$ ($d=13$) via a quadratic-form theory and an exact root-count reduction, and we verify it exhaustively by computer for every admissible $(n,k)$ with $n\le13$.

Machine learning and data analysis techniques have recently emerged as powerful tools for identifying patterns and formulating conjectures in mathematical research, most notably in the field of low-dimensional topology. In this paper, we initiate a systematic approach to handling mathematical data structured as (truncated) infinite $q$-series, or equivalently, infinite series of integers. To apply this data analysis pipeline, we construct a comprehensive dataset of $\widehat{Z}$-invariants (homological blocks) for plumbed 3-manifolds. We demonstrate that neural networks can reliably extract essential topological information, such as homology class and underlying graph structure, directly from the $q$-series coefficients. A central feature of our methodology is a focus on interpretability; by contrasting local gradient sensitivity with global feature relevance, we reveal that the networks learn to bypass complex topological rules in favor of specific spectral and geometric proxies. Finally, we apply this pipeline to probe homology cobordism, discovering a high-accuracy predictive relationship between the $\widehat{Z}$-invariant exponents and the Heegaard Floer $d$-invariant (correction term). These results suggest that $\widehat{Z}$-invariants capture subtle geometric information regarding cobordism equivalences, warranting a new direction for the study of quantum invariants.

Teng and Tian proved the following result. Let $k\ge 2$ and $t\ge 3$, and let $G$ be a $k$-connected graph of order $n$. If $n\ge 6k+1$ and $\delta(G)\ge \lceil(n+6)/2\rceil$ when $t=3$, while $n\ge 6k+7t-17$ and $\delta(G)\ge \lceil(n+t+2)/2\rceil$ when $t\ge 4$, then, for any two distinct vertices $u,v$ and every integer $s$ with $1\le s\le t$, there exist $s$ internally vertex-disjoint $(u,v)$-paths $P_1,\dots,P_s$ whose union spans $G$ and such that $G-E(P_1\cup\cdots\cup P_s)$ is $k$-connected. They asked whether the minimum-degree condition could be lowered to $\delta(G)\ge \lceil(n+t)/2\rceil$ for every $t\ge 3$. We answer this question affirmatively and further reduce the required order to $n\ge \max\{6k+9-3t,\,2k+t+3\}$.

By constructing explicit entire maps, we prove that every exceptional Morimoto–Nagano model $M_t^3$, $t>1$, can be realized as a compact real-analytic hypersurface in $\mathbb C^3$. This extends Isaev's realization results from restricted parameter ranges to all $t>1$ and provides a complete solution to a long-standing open problem in the Morimoto–Nagano classification of compact, simply connected, real-analytic hypersurfaces in complex Euclidean spaces that are homogeneous under the action of a connected Lie group of CR automorphisms.

For a discrete subspace $\Gamma$ of a homogeneous space $X$ and a positive definite reproducing kernel Hilbert space $\mathscr{H}(X)$, a convergence proof is given for series of partial Whitney functions over $\Gamma$ that arise in the span of coherent states in $\mathscr{H}(X)$.

We study the effect of random signed permutations on products of orthogonal matrices and their applications to fast dimension reduction. Let $A,B \in \mathbb{R}^{d\times d}$ be orthogonal matrices and let $\Sigma \in \mathbb{R}^{d\times d}$ be a uniformly random signed permutation matrix. We analyze the random orthogonal matrix \[ U=A \Sigma B, \] and show that, under mild assumptions on the size of the entries of $A$ and $B$, \[ \max_{i,j=1,\ldots,d} |U_{ij}| =O\left ( \sqrt{\frac{\log d}{d}}\right ) \] with high probability. As an application, we show that ORA, an analogue of the Kac walk in which every update is a $\pi/4$ rotation, reaches the same maximal entry scale after $O(d\log d)$ updates. This resolves a question of Jain et al. and improves the running time of their construction. We also show that parallel ORA reaches this scale after $O(\log d)$ rounds. We then study the random embedding \[ \Phi = \sqrt{\frac{d}{m}}\, P_I U D_{\xi'}, \] where $P_I$ restricts to $m$ coordinates and $\xi'$ is an independent Rademacher vector. We identify two parameters controlling norm preservation and show that, throughout the corresponding admissible range, $\Phi$ achieves optimal embedding dimension $m\asymp\varepsilon^{-2}\log(N)$. Finally, we extend the result to structured infinite models, including sparse vectors, low-rank matrices, and finite unions of subspaces.