We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \le x$ for which the Alladi–Erdős function $A(n) = \sum_{p^k \parallel n} k p$ takes values in a given residue class modulo q, where q varies uniformly up to a fixed power of $\log x$. We establish a similar result for the equidistribution of the Euler totient function $\phi(n)$ among the coprime residues to the ‘correct’ moduli q that vary uniformly in a similar range and also quantify the failure of equidistribution of the values of $\phi(n)$ among the coprime residue classes to the ‘incorrect’ moduli.

We prove two sharp anisotropic weighted geometric inequalities that hold for star-shaped and F-mean convex hypersurfaces in $\mathbb{R}^{n+1}$, which involve the anisotropic p-momentum, the anisotropic perimeter, and the volume of the region enclosed by the hypersurface. We also consider their quantitative versions characterized by asymmetry index and the Hausdorff distance between the hypersurface and a rescaled Wulff shape. As a corollary, we obtain the stability of the Weinstock inequality for the first non-zero Steklov eigenvalue for star-shaped and strictly mean convex domains.

Let X be a smooth, projective and geometrically connected curve defined over a finite field ${\mathbb {F}}_q$ of characteristic p different from $2$ and $S\subseteq X$ a subset of closed points. Let $\overline {X}$ and $\overline {S}$ be their base changes to an algebraic closure of ${\mathbb {F}}_q$. We study the number of $\ell $-adic local systems $(\ell \neq p)$ in rank $2$ over $\overline {X}-\overline {S}$ with all possible prescribed tame local monodromies fixed by k-fold iterated action of Frobenius endomorphism for every $k\geq 1$. In all cases, we confirm conjectures of Deligne predicting that these numbers behave as if they were obtained from a Lefschetz fixed point formula. In fact, our counting results are expressed in terms of the numbers of some Higgs bundles.

This paper initiates the explicit study of face numbers of matroid polytopes and their computation. We prove that, for the large class of split matroid polytopes, their face numbers depend solely on the number of cyclic flats of each rank and size, together with information on the modular pairs of cyclic flats. We provide a formula which allows us to calculate $f$-vectors without the need of taking convex hulls or computing face lattices. We discuss the particular cases of sparse paving matroids and rank two matroids, which are of independent interest due to their appearances in other combinatorial and geometric settings.

Let Λ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of Λ-mod with $n \geq 2$. From the viewpoint of higher homological algebra, a question that naturally arose in Ebrahimi and Nasr-Isfahani (The completion of d-abelian categories. J. Algebra 645 (2024), 143–163) is when $\mathcal{M}$ induces an n-cluster tilting subcategory of Λ-Mod. In this article, we answer this question and explore its connection to Iyama’s question on the finiteness of n-cluster tilting subcategories of Λ-mod. In fact, our theorem reformulates Iyama’s question in terms of the vanishing of Ext and highlights its relation with the rigidity of filtered colimits of $\mathcal{M}$. Also, we show that ${\rm Add}(\mathcal{M})$ is an n-cluster tilting subcategory of Λ-Mod if and only if ${\rm Add}(\mathcal{M})$ is a maximal n-rigid subcategory of Λ-Mod if and only if $\lbrace X\in \Lambda-{\rm Mod}~|~ {\rm Ext}^i_{\Lambda}(\mathcal{M},X)=0 ~~~ {\rm for ~all}~ 0 \lt i \lt n \rbrace \subseteq {\rm Add}(\mathcal{M})$ if and only if $\mathcal{M}$ is of finite type if and only if ${\rm Ext}_{\Lambda}^1({\underrightarrow{\lim}}\mathcal{M}, {\underrightarrow{\lim}}\mathcal{M})=0$. Moreover, we present several equivalent conditions for Iyama’s question which shows the relation of Iyama’s question with different subjects in representation theory such as purity and covering theory.

In this note, we formulate a ‘one-sided’ version of Wormald’s differential equation method. In the standard ‘two-sided’ method, one is given a family of random variables that evolve over time and which satisfy some conditions, including a tight estimate of the expected change in each variable over one-time step. These estimates for the expected one-step changes suggest that the variables ought to be close to the solution of a certain system of differential equations, and the standard method concludes that this is indeed the case. We give a result for the case where instead of a tight estimate for each variable’s expected one-step change, we have only an upper bound. Our proof is very simple and is flexible enough that if we instead assume tight estimates on the variables, then we recover the conclusion of the standard differential equation method.

We discuss, in a non-Archimedean setting, the distribution of the coefficients of L-polynomials of curves of genus g over $\mathbb{F}_q$. Among other results, this allows us to prove that the $\mathbb{Q}$-vector space spanned by such characteristic polynomials has dimension g + 1. We also state a conjecture about the Archimedean distribution of the number of rational points of curves over finite fields.

In the last two decades the study of random instances of constraint satisfaction problems (CSPs) has flourished across several disciplines, including computer science, mathematics and physics. The diversity of the developed methods, on the rigorous and non-rigorous side, has led to major advances regarding both the theoretical as well as the applied viewpoints. Based on a ceteris paribus approach in terms of the density evolution equations known from statistical physics, we focus on a specific prominent class of regular CSPs, the so-called occupation problems, and in particular on $r$-in-$k$ occupation problems. By now, out of these CSPs only the satisfiability threshold – the largest degree for which the problem admits asymptotically a solution – for the $1$-in-$k$ occupation problem has been rigorously established. Here we determine the satisfiability threshold of the $2$-in-$k$ occupation problem for all $k$. In the proof we exploit the connection of an associated optimization problem regarding the overlap of satisfying assignments to a fixed point problem inspired by belief propagation, a message passing algorithm developed for solving such CSPs.

We consider the hard-core model on a finite square grid graph with stochastic Glauber dynamics parametrized by the inverse temperature $\beta$. We investigate how the transition between its two maximum-occupancy configurations takes place in the low-temperature regime $\beta \to \infty$ in the case of periodic boundary conditions. The hard-core constraints and the grid symmetry make the structure of the critical configurations for this transition, also known as essential saddles, very rich and complex. We provide a comprehensive geometrical characterization of these configurations that together constitute a bottleneck for the Glauber dynamics in the low-temperature limit. In particular, we develop a novel isoperimetric inequality for hard-core configurations with a fixed number of particles and show how the essential saddles are characterized not only by the number of particles but also their geometry.

We give a criterion for separability of subgroups of certain outer automorphism groups. This answers questions of Hagen and Sisto, by strengthening and generalizing a result of theirs on mapping class groups.

Let $\mathbb{N}$ be the set of all non-negative integers. For any integer r and m, let $r+m\mathbb{N}=\{r+mk: k\in\mathbb{N}\}$. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_{S}(n)$ denote the number of solutions of the equation $n=s+s'$ with $s, s'\in S$ and $s \lt s'$. Let $r_{1}, r_{2}, m$ be integers with $0 \lt r_{1} \lt r_{2} \lt m$ and $2\mid r_{1}$. In this paper, we prove that there exist two sets C and D with $C\cup D=\mathbb{N}$ and $C\cap D=(r_{1}+m\mathbb{N})\cup (r_{2}+m\mathbb{N})$ such that $R_{C}(n)=R_{D}(n)$ for all $n\in\mathbb{N}$ if and only if there exists a positive integer l such that $r_{1}=2^{2l+1}-2, r_{2}=2^{2l+1}-1, m=2^{2l+2}-2$.

Let Γ be a finite graph and let $A(\Gamma)$ be the corresponding right-angled Artin group. From an arbitrary basis $\mathcal B$ of $H^1(A(\Gamma),\mathbb F)$ over an arbitrary field, we construct a natural graph $\Gamma_{\mathcal B}$ from the cup product, called the cohomology basis graph. We show that $\Gamma_{\mathcal B}$ always contains Γ as a subgraph. This provides an effective way to reconstruct the defining graph Γ from the cohomology of $A(\Gamma)$, to characterize the planarity of the defining graph from the algebra of $A(\Gamma)$ and to recover many other natural graph-theoretic invariants. We also investigate the behaviour of the cohomology basis graph under passage to elementary subminors and show that it is not well-behaved under edge contraction.

Recently, it is proven that positive harmonic functions defined in the unit disc or the upper half-plane in $\mathbb{C}$ are contractions in hyperbolic metrics [14]. Furthermore, the same result does not hold in higher dimensions as shown by given counterexamples [16]. In this paper, we shall show that positive (or bounded) harmonic functions defined in the unit ball in $\mathbb{R}^{n}$ are Lipschitz in hyperbolic metrics. The involved method in main results allows to establish essential improvements of Schwarz type inequalities for monogenic functions in Clifford analysis [24, 25] and octonionic analysis [21] in a unified approach.

In this paper, we show that for a large natural class of vertex operator algebras (VOAs) and their modules, the Zhu’s algebras and bimodules (and their g-twisted analogs) are Noetherian. These carry important information about the representation theory of the VOA, and its fusion rules, and the Noetherian property gives the potential for (non-commutative) algebro-geometric methods to be employed in their study.

The delta invariant interprets the criterion for the K-(poly)stability of log terminal Fano varieties. In this paper, we determine local delta invariants for all weak del Pezzo surfaces with the anti-canonical degree $\geq 5$.

We prove a large deviation principle for the slow-fast rough differential equations (RDEs) under the controlled rough path (RP) framework. The driver RPs are lifted from the mixed fractional Brownian motion (FBM) with Hurst parameter $H\in (1/3,1/2)$. Our approach is based on the continuity of the solution mapping and the variational framework for mixed FBM. By utilizing the variational representation, our problem is transformed into a qualitative property of the controlled system. In particular, the fast RDE coincides with Itô stochastic differential equation (SDE) almost surely, which possesses a unique invariant probability measure with frozen slow component. We then demonstrate the weak convergence of the controlled slow component by averaging with respect to the invariant measure of the fast equation and exploiting the continuity of the solution mapping.

This work is concerned with the exponential turnpike property for optimal control problems of particle systems and their mean-field limit. Under the assumption of the strict dissipativity of the cost function, exponential estimates for both optimal states and optimal control are proven. Moreover, we show that all the results for particle systems can be preserved under the limit in the case of infinitely many particles.