AI for Science

Inspired by the classical Besov $p$-spaces defined via higher-order derivatives on the upper half-plane, we introduce Besov-type spaces on simply connected domains. We first prove that on quasidisks, the first-order Besov space is isomorphic to its higher-order counterparts, and that these higher-order spaces preserve conformal quasi-invariance. Based on this result, we characterize chord-arc domains in terms of the isomorphism between the first-order Besov space and the boundary Besov space. This extends recent results for the Dirichlet space ($p=2$) to the general case $1 < p < \infty$.

For an integer $r \ge 2$ and an order $n \equiv 1, 3 \pmod{6}$, write $δ_r(n)$ for the minimum, over all $r$-colourings $χ: \binom{[n]}{3} \to [r]$, of $\max_{\mathcal{S}} \mathrm{disc}(\mathcal{S}, χ)$, where the maximum is over labelled Steiner triple systems $\mathcal{S}$ of order $n$ and $\mathrm{disc}(\mathcal{S}, χ) = \max_c |\#\{T \in \mathcal{S} : χ(T) = c\} - |\mathcal{S}|/r|$. Following Gishboliner, Glock, and Sgueglia \cite{GishbolinerGlockSgueglia2025}, the bulk of the recent work on this quantity has been on lower bounds for $r \ge 3$ (proving $δ_r(n) = Ω(n^2)$) and on structural characterisation of the low-discrepancy 2-colourings. We give three small computational contributions in the small-$n$ regime $n \in \{7, 9, 13, 15, 19, 21\}$: An exact value of $δ_2(n)$ for each such $n$, matching the formula $δ_2(n) = \min_{x \in [0, n] \cap \mathbb{Z}} |x(n-x)/2 - n(n-1)/12|$ obtained by optimising the GGS Example 1.1 family. Rigorous for $n \in \{7, 9\}$ via exhaustive search over labelled STSs ($30$ resp. $840$ systems) and over all $2$-colourings; computational for $n \in \{13, 15, 19, 21\}$ by simulated-annealing search; A wide near-optimal basin: at $n = 9$, every two-colour-flip neighbour of the optimal Example~1.1 colouring that maintains discrepancy $1.0$ exists; about $34\%$ of two-flip perturbations preserve optimality; Random-colouring statistics for $r \in \{2, 3, 4\}$: $\langle\max_{\mathcal{S}}\mathrm{disc}\rangle$ grows linearly in $n$, in agreement with a heuristic Gaussian estimate $n / \sqrt{6r} \cdot \sqrt{2 \log K}$ over $K$ sampled labellings; the typical-case discrepancy is far below the GGS worst-case $Ω(n^2)$. We additionally state a conjectural exact formula for $δ_2(n)$ that holds for every $n \equiv 1, 3 \pmod{6}$.

This paper explores the dimension theory of non-Noetherian graded rings by introducing the class of Hilbert--Serre rings. We generalize Krull's dimension theorem and Smoke's dimension theorem by establishing the fundamental inequalities $\dim_{\mathrm{gr}}(R) \leq \dim(R) \leq \operatorname{GKdim}_k(R) \leq d(R)$ for any Hilbert--Serre ring $R$, where $d(R)$ is the pole order of its Poincaré series at $t=1$. Furthermore, we apply these results to initial algebras, proving that all these dimensions, including the transcendence degree, coincide for monomial algebras. Finally, we provide explicit examples demonstrating that these inequalities can be strict in general, even for integral domains.

Let $B(m, n)$ be the number of ways to colour a $2m \times 2n$ grid in black and white so that, in each row and each column, half of the cells are white and half are black. Bhattacharya conjectured that the exponent of $2$ in the prime factorisation of $B(m, n)$ equals $s_2(m)s_2(n)$, where $s_2(x)$ denotes the number of $1$s in the binary expansion of $x$. We confirm this conjecture in some infinite families of special cases; most significantly, when $m$ is of the form either $2^k$ or $2^k + 1$ and $n$ is arbitrary. The proof when $m = 2^k + 1$ is substantially more difficult, and in connection with it we develop some general techniques for the analysis of inequalities between binary digit sums.

Let $D$ be an oriented graph with skew adjacency matrix $S(D)$. Two oriented graphs $D$ and $C$ are said to share the same generalized skew spectrum if $S(D)$ and $S(C)$ have the same eigenvalues, and $J-S(D)$ and $J-S(C)$ also have the same eigenvalues, where $J$ is the all-ones matrix. Such graphs that are not isomorphic are generalized cospectral mates. We derive tight upper bounds on the number of non-isomorphic generalized cospectral mates an oriented graph can admit, based on arithmetic criteria involving the determinant of its skew-walk matrix. As a special case, we also provide a criterion for an oriented graph to be weakly determined by its generalized skew spectrum (WDGSS), that is, its only generalized cospectral mate is its transpose. These criteria relate directly to the controllability of graphs, a fundamental concept in the control of networked systems, thereby connecting spectral characterization of graphs to graph controllability.

In this paper, we investigate the problem of finding {\it bisections} (i.e., balanced bipartitions) in graphs. We prove the following two results for {\it all} graphs $G$: (1). $G$ has a bisection where each vertex $v$ has at least $(1/4 - o(1))d_G(v)$ neighbors in its own part; (2). $G$ also has a bisection where each vertex $v$ has at least $(1/4 - o(1))d_G(v)$ neighbors in the opposite part. These results are asymptotically optimal up to a factor of $1/2$, aligning with what is expected from random constructions, and provide the first systematic understanding of bisections in general graphs under degree constraints. As a consequence, we establish for the first time the existence of a function $f(k)$ such that for any $k\geq 1$, every graph with minimum degree at least $f(k)$ admits a bisection where every vertex has at least $k$ neighbors in its own part, as well as a bisection where every vertex has at least $k$ neighbors in the opposite part. Using a more general setting, we further show that for any $\varepsilon > 0$, there exist $c_\varepsilon, c'_\varepsilon > 0$ such that any graph $G$ with minimum degree at least $c_\varepsilon k$ (respectively, $c'_\varepsilon k$) admits a bisection satisfying: every vertex has at least $k$ neighbors in its own part (respectively, in the opposite part), and at least $(1 - \varepsilon)|V(G)|$ vertices have at least $k$ neighbors in the opposite part (respectively, in their own part). These results extend and strengthen classical graph partitioning theorems of Erdős, Thomassen, and Kühn-Osthus, while additionally satisfying the bisection requirement.

We define instanton sheaves of higher rank on smooth Fano threefolds X of Picard rank one and show that their topological classification depends on two integers, namely the rank n (or the half of it, if the Fano index of X is odd) and the charge k. We elucidate the value of the minimal charge k0 of slope-stable n-instanton bundles (except for Fano threefolds of index 1 and genus 3 or 4), as an integer depending only on the genus of X and on n and we prove the existence of slope-stable n-instanton bundles of charge k greater than k0. Next, we study the acyclic extension of instantons on Fano threefolds with curvilinear Kuznetsov component and give a monadic description when the intermediate Jacobian is trivial. Finally, we provide several features of a general element in the main component of the moduli space of intantons, such as and generic splitting over rational curves contained in X and stable restriction to a K3 section S of X, and give applications to Lagrangian subvarieties of moduli spaces of sheaves on S.

It is shown that any continuous function depending on several $p$-adic variables, each of which is defined on $\mathbb{Z}_{p}$, can be represented as a superposition of continuous functions of one $p$-adic variable. This statement is true for both functions with values in $\mathbb{R}$ and functions with values in $\mathbb{Q}_{p}$.

This paper investigates a real-valued topological invariant of 3-manifolds called topological volume. For a given 3-manifold M it is defined as the smallest volume of the complement of a (possibly empty) hyperbolic link in M. Various refinements of this invariant are given, asymptotically tight upper and lower bounds are determined, and all non-hyperbolic closed 3-manifolds with topological volume of at most 3.07 are classified. Moreover, it is shown that for all but finitely many lens spaces, the volume minimiser is obtained by Dehn filling one of the cusps of the complement of the Whitehead link or its sister manifold.

In [FJ07], Favre and Jonsson developed tools from valuative theory to study the dynamics of a dominant endomorphism of the complex affine plane. We extend this theory to the case of any affine surface, over any field. We give a new method to construct an eigenvaluation of an endomorphism. We generalize the result of Favre and Jonsson and show that the first dynamical degree of a dominant endomorphism of a normal affine surface is an algebraic integer of degree less or equal than 2. The general method is to construct a compactification where our endomorphisms admit a fixed point at infinity where the dynamical degree can be computed by studying the local dynamics at this point. We then apply this construction to the study of the dynamics of automorphisms where we are able to say much more. In particular, we obtain a new kind of rigidity result: the set of first dynamical degrees of loxodromic automorphisms of a given affine surface must be fully contained in the set of integers or in the set of algebraic integers of degree 2.

We define and investigate in details the class of so-termed {\it GUSC} rings, that are those rings whose non-invertible elements are uniquely strongly clean. These rings are a common non-trivial generalization of the so-called {\it USC} rings, introduced by Chen-Wang-Zhou in J. Pure \& Appl. Algebra (2009), which are rings whose elements are uniquely strongly clean. These rings also properly generalize the so-named {\it GUC} rings, defined by Guo-Jiang in Bull. Transilvania Univ. Braşov (2023), which are rings whose non-invertible elements are uniquely clean.

This paper explores a construction of the elliptic classes of the Springer resolution using the periodic Hecke module. The module is established by employing the Poincaré line bundle over the product of the abelian variety of elliptic cohomology and its dual. Additionally, we introduce the elliptic twisted group algebra, which acts on the periodic module. The construction of the elliptic twisted group algebra is such that the Demazure-Lusztig (DL) operators with dynamical parameters are rational sections. We define elliptic classes as rational sections of the periodic module, and give explicit formulas of the restriction to fixed points. Our main result shows that a natural assembly of the DL operators defines a rational isomorphism between the periodic module and the one associated to the Langlands dual root system. This isomorphism intertwines the (opposite) elliptic classes with the fixed point basis in the dual system.

We introduce a Green function and analogues of other related kernels for finite and infinite networks whose edge weights are complex-valued admittances with positive real part. We provide comparison results with the same kernels associated with corresponding reversible Markov chains, i.e., where the edge weights are positive. Under suitable conditions, these lead to comparison of series of matrix powers which express those kernels. We show that the notions of transience and recurrence extend by analytic continuation to the complex-weighted case even when the network is infinite. Thus, a variety of methods known for Markov chains extend to that setting.

We prove the method of power iteration for matrices with at most finite entries from the Levi-Civita field $\mathcal C$ under the assumption that there exists an eigenvalue with the strictly largest in absolute value complex part. In this case the weak convergence of a start vector to the eigenvector, that corresponds to the largest eigenvalue, is proven. Further, we prove that the Rayleigh quotient of the largest eigenvector also converges weakly to the corresponding eigenvalue. As a corollary, the same holds for matrices and polynomials over the Puiseux series field. In addition to that, we deliver an implementation of our method in Python.

We use the measure theoretical entropy from Vivas-Sirvent and the topological entropies from Dinh-Sibony and Kelly-Tennant to prove that the equidistribution measures $μ_-$ and $μ_+$ found by the author maximize entropy for the $1$-parameter family $\lbrace\mathcal{F}_a\rbrace_a$ studied by Bullett-Lomonaco. We find as well a measure of maximal entropy for the composition of two deleted covering correspondences studied by Bullett.

Associated to any root datum, there is an elliptic affine Hecke algebra defined by Ginzburg, Kapranov, and Vasserot. In this study, we establish a Fourier-Mukai functor from the representation category of the elliptic affine Hecke algebra to the corresponding category associated with the Langlands dual root datum. To achieve this connection, we employ the elliptic Hecke algebra with dynamical parameters as an intermediary.

We prove that a smooth, complex plane curve $C$ of odd degree can be defined by a polynomial with real coefficients if and only if $C$ is isomorphic to its complex conjugate. Counterexamples are known for curves of even degree. More generally, we prove that a plane curve $C$ over an algebraically closed field $K$ of characteristic $0$ with field of moduli $k_{C}\subset K$ is defined by a polynomial with coefficients in $k'$, where $k'/k_{C}$ is an extension with $[k':k_{C}]\le 3$ and $[k':k_{C}]\mid \operatorname{deg} C$.

We investigate properties of spectrum of normalized Laplacian $\mathcal L$ for finite graphs over non-Archimedean ordered fields. We prove a Cheeger's inequality for first non-zero eigenvalue. Then we describe properties of the operator $\mathcal P=I-\mathcal L$, which is a generalization of transition operator. We show that Cheeger estimate $α_1\preceq \sqrt{1-h^2}$ for the second largest eigenvalue of $\mathcal P$ is crucial for investigation of the convergence of analogue of random walk to equilibrium over a non-Archimedean ordered fields. We consider examples over the Levi-Civita field.

Let $f$ be a primitive form of weight $2k+j-2$ for $SL_2(Z)$, and let $P$ be a prime ideal of the Hecke field of $f$. We denote by $Sp_m(Z)$ the Siegel modular group of degree $m$. Suppose that $k$ is congruent to $0$ modulo $4$, $j$ is congruent to $0$ modulo $4$, and that $P$ divides the algebraic part of $L(k+j,f)$. Put ${\bf k}=(k+j/2,k+j/2,j/2+4,j/2+4)$. Then under certain easily checkable conditions, we prove that there exists a Hecke eigenform $F$ in the space of modular forms of weight $(k+j,k)$ for $Sp_2(Z)$ such that $[I_2(f)]^{\bf k}$ is congruent to $A^{(I)}_4(F)$ modulo $P$. Here, $[I_2(f)]^{\bf k}$ is the Klingen-Eisenstein lift of the Saito-Kurokawa lift $I_2(f)$ of $f$ to the space of modular forms of weight ${\bf k}$ for $Sp_4(Z)$, and $A^{(I)}_4(F)$ is a certain lift of $F$ to the space of cusp forms of weight ${\bf k}$ for $Sp_4(Z)$. As an application, we prove Harder's conjecture on the congruence between the Hecke eigenvalues of $F$ and some quantities related to the Hecke eigenvalues of $f$. This version gives proofs of Lemmas 7.2 and 7.3 and Corollaries 7.4 and 7.5 in the paper arXiv:2306.07582v2.

Many seismic imaging methods use wave field extrapolation operators to redatum sources and receivers from the surface into the subsurface. We discuss wave field extrapolation operators that account for internal multiple reflections, in particular propagator matrices, transfer matrices and Marchenko focusing functions. A propagator matrix is a square matrix that `propagates' a wave-field vector from one depth level to another. It accounts for primaries and multiples and holds for propagating and evanescent waves. A Marchenko focusing function is a wave field that focuses at a designated point in space at zero time. Marchenko focusing functions are useful for retrieving the wave field inside a heterogeneous medium from the reflection response at its surface. By expressing these focusing functions in terms of the propagator matrix, the usual approximations (such as ignoring evanescent waves) are avoided. While a propagator matrix acts on the full wave-field vector, a transfer matrix (according to the definition employed in this paper)`transfers' a decomposed wave-field vector (containing downgoing and upgoing waves) from one depth level to another. It can be expressed in terms of decomposed Marchenko focusing functions. We present propagator matrices, transfer matrices and Marchenko focusing functions in a consistent way and discuss their mutual relations. In the main text we consider the acoustic situation and in the appendices we discuss other wave phenomena. Understanding these mutual connections may lead to new developments of Marchenko theory and its applications in wave field focusing, Green's function retrieval and imaging.

We prove that a smooth, complex plane curve of odd degree can be defined by a polynomial with coefficients in $\mathbb{R}$ if and only if it is isomorphic to its complex conjugate; there are counterexamples in even degree. Over arbitrary base fields of characteristic $0$, we prove that a smooth plane curve of degree prime with $6$ can be defined by a polynomial with coefficients in the field of moduli. We also prove results about fields of moduli of algebraic cycles in $\mathbb{P}^{2}$. In particular, these apply to singular plane curves of arbitrary degree, too.

We study the behavior of the pressure function for Hölder continuous potentials on mixing subshifts of finite type. The classical theory of thermodynamic formalism shows that such pressure functions are convex, analytic and have slant asymptotes. We provide a sharp exponential lower bound on how fast the pressure function approaches its asymptotes. As a counterpart, we also show that there is no corresponding upper bound by exhibiting systems for which the convergence is arbitrarily slow. However, we prove that the exponential upper bound still holds for a generic Hölder potential. In addition, we determine that the pressure function satisfies a coarse uniform convexity property. Asymptotic bounds and quantitative convexity estimates are the first additional general properties of the pressure function obtained in the settings of Bowen and Ruelle since their groundbreaking work more than 40 years ago.

This paper introduces the trivial fiber topology on schemes. For one-dimensional base schemes, we use it to describe fibrant replacements in the stable motivic homotopy category and motivic infinite loop spaces. We also extend the Garkusha-Panin and Voevodsky strict $\mathbb{A}^{1}$-invariance theorems to one-dimensional base schemes. The trivial fiber topology plays a central role in the proof of refined localization results for motivic homotopy categories. Moreover, we extend Morel's $\mathbb{A}^{1}$-connectivity theorem on Nisnevich sheaves of stable motivic homotopy groups. These results open new vistas for computations of motivic invariants over deeper base schemes of arithmetic interest.

A method of constructing algebraic-geometric codes with many automorphisms arising from Galois points for algebraic curves is presented.

In Part I, the present authors introduced the notion of a quasi-Galois point, for investigating the automorphism groups of plane curves. In this second part, the number of quasi-Galois points for smooth plane curves is described. In particular, sextic or quartic curves with many quasi-Galois points are characterized.

In this paper, we prove a uniform estimate for the modulus of continuity of solutions to degenerate complex Monge--Ampère equation in big cohomology classes. This improves the previous results of Di Nezza--Lu and of the first author.

Let $G$ be a cyclic $p$-group or generalized quaternion group, $X\in π_0 S_G$ be a virtual $G$-set, and $V$ be a fixed point free complex $G$-representation. Under conditions depending on the sizes of $G$, $X$, and $V$, we construct a self map $v\colonΣ^V C(X)_{(p)}\rightarrow C(X)_{(p)}$ on the cofiber of $X$ which induces an equivalence in $G$-equivariant $K$-theory. These are transchromatic $v_{1,\vec{0}}$-self maps, in the sense that they are lifts of classical $v_1$-self maps for which the telescope $C(X)_{(p)}[v^{-1}]$ can have nonzero rational geometric fixed points.