数学前沿

A graph is edge-transitive if its automorphism group acts transitively on its edge set. This paper is devoted to a systematic study of cubic edge-transitive graphs of order $\ell p^2$, where $p$ is a prime and $\ell$ is an even integer. We first show that either $\Gamma$ is the unique cubic edge-transitive graph of order $18$, or its full automorphism group $\mathrm{Aut}(\Gamma)$ contains a normal Sylow $p$-subgroup $P$ which is isomorphic to either $Z_{p^2}$ or $Z_{p}^2$. For the case $P\cong Z_{p^2}$, we prove that $\Gamma$ is either arc-regular or edge-regular, and provide a complete classification of such graphs. For the case $P\cong Z_{p}^2$, we prove that $\Gamma$ is a normal cover of certain known graphs. This work generalizes previous classifications by Zhou and Feng (2024), Lu et al.~(2004), and others. As an application, we obtain a classification of cubic edge-transitive graphs of order $\ell p^2$ in the case where $\ell$ is square-free.

We study the structure of amenable subrelations of treeable measure-class-preserving equivalence relations, applying the Radon--Nikodym topography perspective initiated in our companion paper [TTD25]. For a nowhere smooth amenable subrelation, we show that the ends selected by each class in Adams's end selection theorem are precisely those which are nonvanishing along that class. From this, we derive a number of structure results about nowhere smooth amenable subrelations of treeable equivalence relations. For example, we show that every nowhere smooth amenable subrelation is contained in an essentially unique maximal amenable subrelation. This yields an anti-treeability criterion, which we use to show that every nonamenable locally compact second countable group with noncompact amenable radical is not treeable, and that every nonamenable nonunimodular treeable group has a compact open subgroup.

Assuming the absence of interior local extrema for first positive Neumann eigenfunctions on bounded smooth simply connected planar domains, as reported in \cite{SmoothInput}, we extend the hot spots conclusion to simply connected domains of finite area with compact Sobolev embedding and simple first positive Neumann eigenvalue. The supremum and infimum are approached in arbitrarily thin boundary layers. For bounded domains and eigenfunctions continuous up to the boundary, both extrema are attained on the boundary. We also transfer polynomial Neumann heat-kernel bounds uniformly along a conformal exhaustion and construct explicit bounded and unbounded examples, including domains with cusps and a domain with infinitely many corners and bottlenecks. Some of these examples admit no polynomial Neumann heat-kernel bound.

For a graph $G$, characterizing the structural behavior of achieving the minimal local antimagic chromatic number $\chi_{la}(G) = 2$ remains a fundamental open problem in the study of vertex-distinguishing edge labelings. This paper establishes complete characterizations for a significant structural class of disconnected graphs, specifically addressing disconnected one-point unions of cycles with at most three components. By introducing a robust algebraic approach, we show that these specific graph architectures form distinct, manageable solution families under a local antimagic 2-coloring framework. Crucially, our structural framework provides the theoretical basis for extending these complete characterizations to an arbitrary number of components, demonstrating unique edge assignments up to isomorphism. These results offer key insights into the structural properties governing low-chromatic antimagic invariants and offer significant progress toward a unified classification of graphs with $\chi_{la}(G) = 2$.

We give a construction of a Hamiltonian diffeomorphism of the equal-area symplectic product $S^2\times S^2$ with two fixed points. The argument starts with a four-fixed-point map, passes to rescaled cotangent coordinates near the antidiagonal, and reduces the local fixed-point equations to two planar saddle problems. A relative modification merges each pair of saddles, and mixed generating functions are used to transfer the modification to the original map. We include detailed explanations of the normal form along a level arc, the Moser equation, uniform existence of mixed generating functions, the order of parameter choices, and the global graph argument. This construction gives a counterexample to the degenerate version of Arnold's conjecture proposed by Arnold\cite{Arnold86} when the manifold is standard $S^2\times S^2$. This work is done with ChatGPT. The author gave the idea to study the limit behavior of $\Phi_\rho$. GPT suggested a key idea of using second generating function to construct this Hamiltonian. Also, most of the computation is done by GPT.

Functional flow matching learns to generate random functions, such as velocity fields or solutions of partial differential equations, by continuously transforming a simple probability distribution into a target distribution. In practice, however, training and sampling use only finitely many grid values or coefficients. Does refining these observations recover the intended function-space model, and how do observation, learning, and numerical errors affect the generated distribution? We address these questions by first proving that the velocity targets used for training converge in mean square when the finite reconstructions converge to the underlying functions and remain uniformly bounded. The observations need not be nested: each refinement may replace earlier sensor locations. We also give quantitative bounds for orthogonal projections and extend the result to point observations under suitable regularity and reconstruction assumptions. We then bound the discrepancy between the generated and target distributions in Wasserstein distance, separating observation, learning, numerical, and endpoint errors without requiring uniqueness of the population finite-dimensional dynamics. For a normalized quadrature neural operator representing the learned velocity field, we establish stability and learning bounds with constants independent of the number of sensors. Under fixed model dimension and boundedness assumptions, the statistical excess risk decreases at an inverse-sample-size rate, up to logarithmic factors. A Gaussian example combines these estimates into an explicit sampling-error rate.

We disprove the cap-sufficiency proposal in Wright's Problem 16 for $1<p<(\sqrt{13}-1)/2$ and for $p>3$. For each fixed exponent in these ranges, one smooth strictly convex domain has finite original directional cap integral, while the lacunary maximal operator of complete arclength fails restricted weak type. We also give a geometric sufficient condition for strong $L^p$ boundedness, for every finite $1<p<\infty$, on an arbitrary given bounded convex domain. It requires one logarithmic integral of a curvature lower envelope, without a differentiability assumption on the boundary or a restriction on the number of curvature zeros. Both results concern unnormalised true arclength, one fixed interior origin and all integer dyadic dilations.

We consider a supercritical two-type continuous-time linear birth-death process with mutation and selection, in which wild-type individuals give rise to mutant offspring with a larger net-growth rate. We previously had obtained first-order law of large number results for the driver site frequency spectrum. We return to obtain a second-order central limit theorem, in which we show stable convergence of a scaling of the deviation of the SFS from its first-order limit. We provide a central limit theorem at both large detection times and sizes. Then, we apply these results to obtain confidence intervals for a class of consistent estimators for the fitness advance.

We study minimizers of the Keplerian action functional with homogeneous potential of type $1/|u|^\alpha$ on periodic loops with fixed winding number $k \in \mathbb{Z}\setminus\{0\}$. This extends the classical theorems of Gordon and Venturelli, closing the previously open case $\alpha \in (1,2)$, $|k|\geq 2$. We show that the $k$-circular solution loses local minimality below the threshold $\alpha = 2-1/|k|^2$, and that minimizers are collision-free above $\alpha = 2-1/|k|$. Combining these results with the existence of precessing periodic orbits in between, we obtain a full classification of the minimizer in terms of $\alpha$ and $k$, which we also verify numerically for $k=2$.

Centering on quasi-Albanese morphisms, we prove several structural theorems for Calabi-Yau pairs.

We study perturbed exponential systems and unions of shifted frequency lattices through shifted-sinc operators and finite-dimensional fiberizations. For the auxiliary binary local-shift model, we identify the exact pattern-dependent invariant $$\rho_{\delta,s}=\|P_-T_{2\delta}P_+\|,$$ which gives the optimal squared synthesis bounds $1\pm\rho_{\delta,s}$. For periodic patterns, the synthesis operator reduces exactly to a single constant Vandermonde-type matrix, yielding finite-dimensional basis criteria and optimal bounds. We further prove that the worst binary lower bound is the infimum of its periodic counterparts; at the critical value $\delta=1/4$, a step pattern is governed by a half-shifted Hilbert matrix and has vanishing lower bound, although every fixed periodic pattern remains invertible. We also combine exact bounds for structured patterns with a support-independent perturbation estimate, obtaining explicit stability neighborhoods. For finite unions of bounded intervals and finitely many shifted lattices, folding modulo one yields finitely many rectangular phase matrices whose singular values and ranks give the exact bounds and complete Riesz-sequence, frame, and Riesz-basis criteria. This framework also yields finite Hadamard constraints for periodic spectra and an exact reduction of rationally commensurate multirate sampling.

We revisit Zhuravlev's 1974 quantitative form of Selberg's sign-change method for the zeros of the Riemann zeta function on the critical line. Zhuravlev's work, originally in Russian and little known in the West, established an explicit positive proportion of such zeros. Using modern techniques we optimise the arithmetic mean value at the heart of the method and obtain the strongest result yet known from Selberg's method, proving that more than $6.78\%$ of the non-trivial zeros of the Riemann zeta function lie on the critical line.

In a directed graph, a stable (or acyclic) set is a set which does not induce a directed cycle. Given a directed graph and two acyclic sets of the same size, the Acyclic-Set-Reconfiguration with Directed Token Sliding (ASR) problem asks whether one can, by iteratively replacing a vertex by an adjacent one, transform the first set into the second one, while maintaining acyclicity during the transformation. In this work, we present a study of the complexity of ASR and provide an FPT algorithm for semi-complete digraphs, when parametrized by the size of the sets being reconfigured, the directed pathwidth (or equivalently, the directed treewidth) of the instance, and a newly introduced parameter called the iteration of the reconfiguration sequence. To the best of our knowledge, the reconfiguration of stable sets has only been studied on undirected graphs, or (more recently) on sparse directed graphs.

The para-Krawtchouk polynomials arose in the search for spin chains with perfect state transfer; the para-Racah, $q$-para-Racah and para-Bannai--Ito polynomials followed, obtained through singular truncations of families of the Askey scheme and used in turn to design spin chains. All are orthogonal on bi-lattices; the prefix goes back to the para-Krawtchouk case, whose spectrum is that of the parabose oscillator. The representations that these polynomials provide of the corresponding algebras --- Hahn, Racah, $q$-Racah, Bannai--Ito and complementary Bannai--Ito --- are shown to be reducible. Each decomposes into a direct sum of two irreducible modules of the same algebra, one attached to each of the two interlaced sub-lattices, and the parameters of the submodules are given. The para polynomials are thereby expressed in terms of the classical polynomials that the submodules carry, and the pairing $\lambda_n=\lambda_{N-n}$ of their eigenvalues is accounted for; the three features of the bi-lattice setting on which the reduction rests are identified. The properties of the para families are then presented in a unified way in the light of this decomposition, together with the constructions through which these families arise and the settings in which they have been put to use.

We formalize a new type system for Elixir, a dynamically typed functional programming language of growing popularity that runs on the Erlang virtual machine. Our system combines gradual typing with semantic subtyping to enable precise, sound, and practical static type analysis, without requiring any changes to Elixir's compilation pipeline or runtime. Type soundness is ensured by leveraging runtime checks -- both implicit, from the Erlang VM, and explicit, via developer-written guards. Central to our approach are two key innovations: the notion of "strong functions", which can be assigned precise types even when applied to inputs that may fall outside their intended domain; and a fine-grained analysis of guards that enables accurate type refinement for case expressions and guarded function definitions. While type information is erased before execution and not used by the compiler, our "safe erasure" gradual typing strategy maintains soundness and expressiveness without compromising compatibility or performance. This work lays the theoretical foundation for Elixir's new type system, outlines its integration into recent versions of the language, and demonstrates its effectiveness on large-scale industrial codebases.

This paper presents a rigorous asymptotic analysis of the two-dimensional narrow escape problem for a Brownian particle initiated at a deterministic position x. While we address the mean escape time, the main novelty lies in our derivation of the asymptotic escape distribution. We derive explicit asymptotic formulas that characterize the dependence of the escape behavior on the relative configuration of the starting point and the window centers. These results remain valid as the starting position approaches the boundary windows and include refined error estimates. The formulas are obtained with two independent methods, each of them having their own interest. A central feature of this work is the rigorous treatment of the underlying mixed boundary value problems in the absence of a contact angle at the escape window boundaries. By leveraging recent results on the maximal regularity of the gradient of the solutions of such mixed boundary value problems, we ensure the analytical validity of our approach despite the inherent loss of regularity of these solutions at the escape window boundaries. Finally, we apply these general results to provide a full sharp asymptotic equivalents for regimes where window sizes scale at specific relative rates.

Let $K$ be a non-archimedean local field of residual characteristic $p\neq 2$. Let $G$ be a connected reductive group over $K$, let $\theta$ be an involution of $G$ over $K$, and let $H$ be the connected component of $\theta$-fixed subgroup of $G$ over $K$. By realizing the Steinberg representation of $G$ as the $G$-space of complex smooth harmonic cochains following the idea of Broussous--Court\`es, we study its space of distinction by $H$ as a finite dimensional complex vector space. We give an upper bound of the dimension, and under certain conditions, we show that the upper bound is sharp by explicitly constructing a basis using the technique of Poincar\'e series. Finally, we apply our general theory to the case where $G$ is a general linear group and $H$ a special orthogonal subgroup, which leads to a complete classification result.

Let $T(q,p)=\langle x,y\mid x^p=y^q\rangle$, where $p,q>1$ are relatively prime, and let $$R=\Hom(T(q,p),\mathrm{SL}_2(\mathbb C)).$$ We compute the integral homology of the singular locus of $R$. We show that this locus has complex dimension $2$ and is a disjoint union of $\mu=(p-1)(q-1)$ smooth affine surfaces, each homotopy equivalent to the two-dimensional sphere. Consequently, $$H_0(\Sing R;\Z)\cong H_2(\Sing R;\Z)\cong\Z^\mu,$$ and all remaining homology groups vanish. Thus both nonzero Betti numbers equal the first Betti number of the Milnor fiber of the plane curve singularity $z^p+w^q=0$, namely its Milnor number $\mu$.

We settle several questions about Hofer's geometry of the two-sphere. We prove that the space of equators has infinite Hofer diameter, construct simple commutators of arbitrarily large Hofer norm, and exhibit a Hamiltonian diffeomorphism whose iterates escape to infinity in Hofer distance from the set of autonomous diffeomorphisms. Finally, we construct a bi-Lipschitz group monomorphism from the free group of rank two, equipped with its conjugation-invariant generator norm, into the Hamiltonian group equipped with Hofer's metric. The proof rests on an interplay between Lagrangian Floer theory in the fourth symmetric product of the sphere and combinatorial group theory. This paper was written with substantial assistance from AI.

Let $M$ be a matroid, let $\ell:E(M)\to G$ be a labeling of its elements by an abelian group, and let $F\subseteq G$ be finite. A base is $F$-avoiding if the label sum of its elements is not in $F$. We prove that, for arbitrary costs, a min-cost $F$-avoiding base can be obtained from any min-cost base using at most $|F|$ exchanges. This resolves the weighted proximity conjecture by H\"orsch, Imolay, Mizutani, Oki, and Schwarcz. Our main technical tool is to replace the assumption of Schrijver--Seymour Conjecture in Liu and Xu's earlier work with Kemperman--Scherk Theorem. The proximity theorem combined with approaches in earlier work gives an FPT algorithm that finds a min-cost $F$-avoiding base parameterized by $|G|$. We additionally give an FPT algorithm parameterized by $|F|$, answering a question of H\"orsch et al.