We consider the probability that no points lie on g large intervals in the bulk of the Airy point process. We make a conjecture for all the terms in the asymptotics up to and including the oscillations of order 1, and we prove this conjecture for g=1.

The asymptotic spectrum of graphs, introduced by Zuiddam (Combinatorica, 2019), is the space of graph parameters that are additive under disjoint union, multiplicative under the strong product, normalized and monotone under homomorphisms between the complements. He used it to obtain a dual characterization of the Shannon capacity of graphs as the minimum of the evaluation function over the asymptotic

Gerards and Seymour conjectured that every graph with no odd Kt minor is (t − 1)-colorable. This is a strengthening of the famous Hadwiger’s Conjecture. Geelen et al. proved that every graph with no odd Kt minor is \(O(t\sqrt {\log t} )\)-colorable. Using the methods the present authors and Postle recently developed for coloring graphs with no Kt minor, we make the first improvement on this bound by

We show that the maximum cardinality of an equiangular line system in 14 and 16 dimensions is 28 and 40, respectively, thereby solving a longstanding open problem. We also improve the upper bounds on the cardinality of equiangular line systems in 19 and 20 dimensions to 74 and 94, respectively.

The main result of this paper is that, if Γ is a finite connected 4-valent vertex- and edge-transitive graph, then either Γ is part of a well-understood family of graphs, or every non-identity automorphism of Γ fixes at most 1/3 of the vertices. As a corollary, we get a similar result for 3-valent vertex-transitive graphs.

Pokrovskiy conjectured that there is a function f: ℕ → ℕ such that any 2k-strongly-connected tournament with minimum out and in-degree at least f(k) is k-linked. In this paper, we show that any (2k + 1)-strongly-connected tournament with minimum out-degree at least some polynomial in k is k-linked, thus resolving the conjecture up to the additive factor of 1 in the connectivity bound, but without the

Besides various asymptotic results on the concept of sum-product bases in the set of non-negative integers ℕ, we investigate by probabilistic arguments the existence of thin sets A, A′ of non-negative integers such that AA + A = ℕ and A′A′ + A′A′ = ℕ.

Let f be a smooth real function with strictly monotone first k derivatives. We show that for a finite set A, with ∣A + A∣ ≤K∣A∣, $$\left| {{2^k}f(A) - ({2^k} - 1)f(A)} \right|{ \gg _k}\,{\left| A \right|^{k + 1 - o(1)}}/{K^{{O_k}(1)}}.$$ We deduce several new sum-product type implications, e.g. that A+A being small implies unbounded growth for a many enough times iterated product set A ⋯ A.

We describe a family of graphs with queue-number at most 4 but unbounded stack-number. This resolves open problems of Heath, Leighton and Rosenberg (1992) and Blankenship and Oporowski (1999).

Let G be a finite group and let h be a positive integer. A BH(G, h) matrix is a G-invariant ∣G∣ × ∣G∣ matrix H whose entries are complex hth roots of unity such that H H* = ∣G∣I∣G∣, where H* denotes the complex conjugate transpose of H, and I∣G∣ denotes the identity matrix of order ∣G∣. In this paper, we give three new constructions of BH(G, h) matrices. The first construction is the first known family

Given a graph H, the extremal number ex(n, H) is the largest number of edges in an H-free graph on n vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing \(K_{s,t}^\prime \) for the subdivision of the bipartite graph Ks,t, we show that \({\rm{ex}}(n,K_{s,t}^\prime ) = O({n^{3/2 - {1 \over {2s}}}})\). This proves a conjecture of Kang, Kim

Given a graph H, the extremal number ex(n, H) is the largest number of edges in an H-free graph on n vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing \(K_{s,t}^\prime \) for the subdivision of the bipartite graph Ks,t, we show that \({\rm{ex}}(n,K_{s,t}^\prime) = O({n^{3/2 - {1 \over {2s}}}})\). This proves a conjecture of Kang, Kim

We find the asymptotic spectral distribution of random Kummer matrix. Then we formulate and prove a free analogue of HV independence property, which is known for classical Kummer and Gamma random variables and for Kummer and Wishart matrices. We also prove a related characterization of free-Kummer and free-Poisson (Marchenko–Pastur) non-commutative random variables.

We characterize the limiting fluctuations of traces of several independent Wigner matrices and deterministic matrices under mild conditions. A CLT holds but in general the families are not asymptotically free of second-order and the limiting covariance depends the limiting ∗-distribution of the deterministic matrices and their transposes and Hadamard products.

Assuming a covariance structure with blocked compound symmetry, it was showed that unbiased estimators for the covariance matrices are optimal under normality. In this paper, we derive the asymptotic distribution of the correlation matrix using unbiased estimators and discuss its use in hypothesis testing. The accuracy of the result is investigated through numerical simulation and the method is applied

We establish a moderate deviation principle for linear eigenvalue statistics of β-ensembles in the one-cut regime with a real-analytic potential. The main ingredient is to obtain uniform estimates for the correlators of a family of perturbations of β-ensembles using the loop equations.

We prove a central limit theorem (CLT) for the product of a class of random singular matrices related to a random Hill’s equation studied by Adams–Bloch–Lagarias. The CLT features an explicit formula for the variance in terms of the distribution of the matrix entries and this allows for exact calculation in some examples. Our proof relies on a novel connection to the theory of m-dependent sequences

Generalizing the $\star$‑graphs of Müller–Stach and Westrich, we describe a class of graphs whose associated graph hypersurface is equipped with a non-trivial torus action. For such graphs, we show that the Euler characteristic of the corresponding projective graph hypersurface complement is zero. In contrast, we also show that the Euler characteristic in question can take any integer value for a suitable

We study the local optimality of periodic point sets in $\mathbb{R}^n$ for energy minimization in the Gaussian core model, that is, for radial pair potential functions $f_c(r) = e^{-cr}$ with $c \gt 0$. By considering suitable parameter spaces for $m$-periodic sets, we can locally rigorously analyze the energy of point sets, within the family of periodic sets having the same point density. We derive

We investigate the Jacobi forms for the root system $E_8$ invariant under the Weyl group. This type of Jacobi forms has significance in Frobenius manifolds, Gromov–Witten theory and string theory. In 1992, Wirthmüller proved that the space of Jacobi forms for any irreducible root system not of type $E_8$ is a polynomial algebra. But very little has been known about the case of $E_8$. In this paper

We define one-parameter “massive” deformations of Maass forms and Jacobi forms. This is inspired by descriptions of plane gravitational waves in string theory. Examples include massive Green’s functions (that we write in terms of Kronecker–Eisenstein series) and massive modular graph functions.

We show that the coefficients of rational $2$-functions are contained in an abelian number field. More precisely, we show that the poles of such functions are poles of order one and given by roots of unity and rational residue.

In this paper, we investigate a relation between the Givental group of rank one and the Heisenberg–Virasoro symmetry group of the KP hierarchy. We prove, that only a two-parameter family of the Givental operators can be identified with elements of the Heisenberg–Virasoro symmetry group. This family describes triple Hodge integrals satisfying the Calabi–Yau condition. Using the identification of the

Noncommutative near-group fusion categories were completely classified in the previous work of the first author by using an operator algebraic method (and hence under the assumption of unitarity), and they were shown to be group theoretical though the corresponding pointed categories were not identified. In this note we give a purely algebraic construction of the noncommutative near-group fusion categories

We show that the shape of a complex cubic field lies on the geodesic of the modular surface defined by the field’s trace-zero form. We also prove a general such statement for all orders in étale Q-algebras. Applying a method of Manjul Bhargava and Piper H to results of Bhargava and Ariel Shnidman, we prove that the shapes lying on a fixed geodesic become equidistributed with respect to the hyperbolic

We completely classify all subbundles of a given vector bundle on the Fargues–Fontaine curve, where by a subbundle we mean a locally free subsheaf. Our classification is given in terms of a simple and explicit condition on Harder–Narasimhan polygons. Our proof is inspired by the proof of the main theorem in our previous work (Hong 2019), but also involves a number of nontrivial adjustments.

We investigate cohomological support varieties for finite-dimensional Lie superalgebras defined over fields of odd characteristic. Verifying a conjecture from our previous work, we show the support variety of a finite-dimensional supermodule can be realized as an explicit subset of the odd nullcone of the underlying Lie superalgebra. We also show the support variety of a finite-dimensional supermodule

Let F be a finite extension of ℚp. Let W(k) denote the Witt vectors of an algebraically closed field k of characteristic ℓ different from p, and let 𝒵 be the spherical Hecke algebra for GLn(F) over W(k). Given a Hecke character λ : 𝒵→ R, where R is an arbitrary W(k)-algebra, we introduce the universal unramified module ℳλ,R. We show ℳλ,R embeds in its Whittaker space and is flat over R, resolving

We consider a special theta lift 𝜃(f) from cuspidal Siegel modular forms f on Sp4 to “modular forms” 𝜃(f) on SO(4,4) in the sense of our prior work (Pollack 2020a). This lift can be considered an analogue of the Saito–Kurokawa lift, where now the image of the lift is representations of SO(4,4) that are quaternionic at infinity. We relate the Fourier coefficients of 𝜃(f) to those of f, and in particular

The aim of this paper is to study some modular contractions of the moduli space of stable pointed curves M¯g,n. These new moduli spaces, which are modular compactifications of Mg,n, are related to the minimal model program for M¯g,n and have been introduced by Codogni et al. (2018). We interpret them as log canonical models of adjoint divisors and we then describe the Shokurov decomposition of a region

We consider some additive properties of integers with restricted digit expansions. Let b ≥ 3 be an integer and Bb be the set of integers whose base b expansions have only digits {0,1}. Let a,b,c be three integers greater than 2. We give some estimates on the size of (Ba + Bb) ∩ Bc. In particular, under mild conditions, (Ba + Bb) ∩ Bc is a very thin set in the sense that for each 𝜖 > 0, as N →∞, #((Ba

Let p and q be multiplicatively independent natural numbers, and K the field ℂ(x1∕s∣s = 1,2,3…). Let p and q act on K as the Mahler operators x↦xp and x↦xq. Schäfke and Singer (2019) showed that a finite-dimensional vector space over K, carrying commuting structures of a p-Mahler module and a q-Mahler module, is obtained via base change from a similar object over ℂ. As a corollary, they gave a new

We prove for every graph H there exists ɛ > 0 such that, for every graph G with |G|≥2, if no induced subgraph of G is a subdivision of H, then either some vertex of G has at least ɛ|G| neighbours, or there are two disjoint sets A, B ⊆ V(G) with |A|,|B|≥ɛ|G| such that no edge joins A and B. It follows that for every graph H, there exists c>0 such that for every graph G, if no induced subgraph of G or

We consider the singular linear statistic of the Laguerre unitary ensemble (LUE) consisting of the sum of the reciprocal of the eigenvalues. It is observed that the exponential generating function for this statistic can be written as a Toeplitz determinant with entries given in terms of particular K Bessel functions. Earlier studies have identified the same determinant, but with the K Bessel functions

Complex Wishart matrices are a class of random matrices with numerous emerging applications. In particular, the statistical characterization of such class of random matrices is essential for solving problems in various fields, including statistics, finance, physics and engineering. This paper establishes a new way to solve such problems based on the statistical moments and correlation of the minors

Large-dimensional (LD) random matrix theory, RMT for short, which originates from the research field of quantum physics, has shown tremendous capability in providing deep insights into large-dimensional systems. With the fact that we have entered an unprecedented era full of massive amounts of data and large complex systems, RMT is expected to play more important roles in the analysis and design of

Additivity violation of minimum output entropy, which shows non-classical properties in quantum communication, had been proved in most cases for random quantum channels defined by Haar-distributed unitary matrices. In this paper, we investigate random completely positive maps made of Gaussian Unitary Ensembles and Ginibre Ensembles regarding this matter. Using semi-circular systems and circular systems

Following the work of Brown, we can canonically associate a family of motivic periods — called the motivic Feynman amplitude — to any convergent Feynman integral, viewed as a function of the kinematic variables. The motivic Galois theory of motivic Feynman amplitudes provides an organizing principle, as well as strong constraints, on the space of amplitudes in general, via Brown’s “small graphs principle”

Building on work of Kontsevich, we introduce a definition of the entropy of a finite probability distribution in which the ‘probabilities’ are integers modulo a prime $p$. The entropy, too, is an integer $\operatorname{mod} p$. Entropy $\operatorname{mod} p$ is shown to be uniquely characterized by a functional equation identical to the one that characterizes ordinary Shannon entropy. We also establish

Given a number field $K$ with a Hecke character $\chi$, for each place $\nu$ we study the free scalar field theory whose kinetic term is given by the regularized Vladimirov derivative associated to the local component of $\chi$. These theories appear in the study of $p$‑adic string theory and $p$‑adic AdS/CFT correspondence. We prove a formula for the regularized Vladimirov derivative in terms of the

We derive new functional equations for Nielsen polylogarithms. We show that, when viewed $\operatorname{modulo} \mathrm{Li}_5$ and products of lower weight functions, the weight $5$ Nielsen polylogarithm $S_{3,2}$ satisfies the dilogarithm five-term relation. We also give some functional equations and evaluations for Nielsen polylogarithms in weights up to $8$, and general families of identities in

We investigate the orthogonal polynomials associated with a singularly perturbed Pollaczek–Jacobi type weight wPJ2(x,t;α,β)=xα(1−x)βe−tx(1−x), where t∈[0,∞), α>0, β>0 and 0

Optimal portfolio selection problems are determined by the (unknown) parameters of the data generating process. If an investor wants to realize the position suggested by the optimal portfolios, he/she needs to estimate the unknown parameters and to account for the parameter uncertainty in the decision process. Most often, the parameters of interest are the population mean vector and the population

In this paper, a ridgelized Hotelling’s T2 test is developed for a hypothesis on a large-dimensional mean vector under certain moment conditions. It generalizes the main result of Chen et al. [A regularized Hotelling’s t2 test for pathway analysis in proteomic studies, J. Am. Stat. Assoc.106(496) (2011) 1345–1360.] by relaxing their Gaussian assumption. This is achieved by establishing an exact four-moment

Consider two random vectors C11/2x∈ℝp and C21/2y∈Rq, where the entries of x and y are i.i.d. random variables with mean zero and variance one, and C1 and C2 are respectively, p×p and q×q deterministic population covariance matrices. With n independent samples of (C11/2x,C21/2y), we study the sample correlation between these two vectors using canonical correlation analysis. Under the high-dimensional

Recently Navarro proposed a strengthening of the unsolved McKay conjecture using Galois automorphisms. We prove that the Navarro conjecture and its blockwise version hold for the alternating groups.

Let A be an abelian variety over a global function field K of characteristic p. We study the μ-invariant appearing in the Iwasawa theory of A over the unramified ℤp-extension of K. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate–Shafarevich group of A and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his

A Drinfeld module has a 𝔭-adic Tate module not only for every finite place 𝔭 of the coefficient ring but also for 𝔭 = ∞. This was discovered by J.-K. Yu in the form of a representation of the Weil group. Following an insight of Taelman we construct the ∞-adic Tate module by means of the theory of isocrystals. This applies more generally to pure A-motives and to pure F-isocrystals of p-adic cohomology

Let H be a weak Hopf algebra that is a finitely generated module over its affine center. We show that H has finite self-injective dimension and so the Brown–Goodearl conjecture holds in this special weak Hopf setting.

We study the inequities in the distribution of Frobenius elements in Galois extensions of the rational numbers with Galois groups that are either dihedral D2n or (generalized) quaternion ℍ2n of two-power order. In the spirit of recent work of Fiorilli and Jouve (2020), we study, under natural hypotheses, some families of such extensions, in a horizontal aspect, where the degree is fixed, and in a vertical

We prove that for any proper smooth formal scheme 𝔛 over 𝒪K, where 𝒪K is the ring of integers in a complete discretely valued nonarchimedean extension K of ℚp with perfect residue field k and ramification degree e, the i-th Breuil–Kisin cohomology group and its Hodge–Tate specialization admit nice decompositions when ie < p − 1. Thanks to the comparison theorems in the recent works of Bhatt, Morrow

We study the high-dimensional asymptotic regimes of correlated Wishart matrices d−1𝒴𝒴T, where 𝒴 is a n×d Gaussian random matrix with correlated and non-stationary entries. We prove that under different normalizations, two distinct regimes emerge as both n and d grow to infinity. The first regime is the one of central convergence, where the law of the properly renormalized Wishart matrices becomes

We develop a general method for computing integral points on modular curves, based on Baker’s inequality. As an illustration, we show that for 11 ≤ p < 101, the only integral points on the curve Xns+(p) are the CM points.

A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of covering systems with distinct moduli was initiated by Erdős in 1950, and over the following decades numerous problems were posed regarding their properties. One particularly notorious question, due to Erdős, asks whether there exist covering systems whose moduli are distinct and all

Let A be an abelian variety over a number field F and let p be a prime. Cohen–Lenstra–Delaunay-style heuristics predict that the Tate–Shafarevich group of (As) should contain an element of order p for a positive proportion of quadratic twists As of A. We give a general method to prove instances of this conjecture by exploiting independent isogenies of A. For each prime p, there is a large class of

We investigate in a statistical fashion the smallest height of a rational point on a Fano hypersurface defined over the field of rational numbers. Along the way, we establish an average version of Manin’s conjecture about the number of rational points of bounded height on Fano varieties for the complete family of Fano hypersurfaces of fixed degree and dimension.

We show, in the large q limit, that the average size of n-Selmer groups of elliptic curves of bounded height over 𝔽q(t) is the sum of the divisors of n. As a corollary, again in the large q limit, we deduce that 100% of elliptic curves of bounded height over 𝔽q(t) have rank 0 or 1.

We show that if a rational map is constant on each isomorphism class of unpolarized abelian varieties of a given dimension, then it is a constant map. Our results shed light on a question raised by Boneh et al. (J. Math. Cryptol. 14:1 (2020), 5–14) concerning a proposal for multiparty noninteractive key exchange.

Let k be a field of positive characteristic. We prove that the only linear relations between the Hodge numbers hi,j(X) = dimHj(X,ΩXi) that hold for every smooth proper variety X over k are the ones given by Serre duality. We also show that the only linear combinations of Hodge numbers that are birational invariants of X are given by the span of the hi,0(X) and the h0,j(X) (and their duals hi,n(X) and

It is known that if p > 37 is a prime number and E∕ℚ is an elliptic curve without complex multiplication, then the image of the mod p Galois representation ρ̄E,p : Gal(ℚ¯∕ℚ) → GL(E[p]) of E is either the whole of GL(E[p]), or is contained in the normaliser of a nonsplit Cartan subgroup of GL(E[p]). In this paper, we show that when p > 1.4 × 107, the image of ρ̄E,p is either GL(E[p]), or the full normaliser