American Institute of Mathematics

We give a new heuristic for all of the main terms in the integral moments of various families of primitive L-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined by the appropriate group averages. This lends support to the idea that arithmetical L-functions have a spectral interpretation, and that their value distributions can be modelled using Random Matrix Theory. Numerical examples show good agreement with our conjectures. 2000 Mathematics Subject Classification 11M26, 15A52.

In tasks involving the interpretation of medical images, suitably trained machine-learning models often exceed the performance of medical experts. Yet such a high-level of performance typically requires that the models be trained with relevant datasets that have been painstakingly annotated by experts. Here we show that a self-supervised model trained on chest X-ray images that lack explicit annotations performs pathology-classification tasks with accuracies comparable to those of radiologists. On an external validation dataset of chest X-rays, the self-supervised model outperformed a fully supervised model in the detection of three pathologies (out of eight), and the performance generalized to pathologies that were not explicitly annotated for model training, to multiple image-interpretation tasks and to datasets from multiple institutions.

Prolyl endopeptidases have potential for treating coeliac sprue, a disease of the intestine caused by proteolytically resistant peptides from proline-rich prolamins of wheat, barley and rye. We compared the properties of three similar bacterial prolyl endopeptidases, including the known enzymes from Flavobacterium meningosepticum (FM) and Sphingomonas capsulate (SC) and a novel enzyme from Myxococcus xanthus (MX). These enzymes were interrogated with reference chromogenic substrates, as well as two related gluten peptides (PQPQLPYPQPQLP and LQLQPFPQPQLPYPQPQLPYPQPQLPYPQPQPF), believed to play a key role in coeliac sprue pathogenesis. In vitro and in vivo studies were conducted to evaluate the activity, specificity and acid/protease stability of the enzymes. All peptidases were relatively resistant to acid, pancreatic proteases and membrane peptidases of the small intestinal mucosa. Although their activities against reference substrates were similar, the enzymes exhibited substantial differences with respect to chain length and subsite specificity. SC hydrolysed PQPQLPYPQPQLP well, but had negligible activity against LQLQPFPQPQLPYPQPQLPYPQPQLPYPQPQPF. In contrast, the FM and MX peptidases cleaved both substrates, although the FM enzyme acted more rapidly on LQLQPFPQPQLPYPQPQLPYPQPQLPYPQPQPF than MX. Whereas the FM enzyme showed a preference for Pro-Gln bonds, SC cleaved both Pro-Gln and Pro-Tyr bonds with comparable efficiency, and MX had a modest preference for Pro-(Tyr/Phe) sites over Pro-Gln sites. While a more comprehensive understanding of sequence and chain-length specificity may be needed to assess the relative utility of alternative prolyl endopeptidases for treating coeliac sprue, our present work has illustrated the diverse nature of this class of enzymes from the standpoint of proteolysing complex substrates such as gluten.

In upcoming papers by Conrey, Farmer and Zirnbauer there appear conjectural formulas for averages, over a family, of ratios of products of shifted L-functions. In this paper we will present various applications of these ratios conjectures to a wide variety of problems that are of interest in number theory, such as lower order terms in the zero statistics of L-functions, mollified moments of L-functions and discrete averages over zeros of the Riemann zeta function. In particular, using the ratios conjectures we easily derive the answers to a number of notoriously difficult computations.

Abstract Tree‐width, and variants that restrict the allowable tree decompositions, play an important role in the study of graph algorithms and have application to computer science. The zero forcing number is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by a graph. We establish relationships between these parameters, including several Colin de Verdière type parameters, and introduce numerous variations, including the minor monotone floors and ceilings of some of these parameters. This leads to new graph parameters and to new characterizations of existing graph parameters. In particular, tree‐width, largeur d'arborescence, path‐width, and proper path‐width are each characterized in terms of a minor monotone floor of a certain zero forcing parameter defined by a color change rule.

Abstract. There is a growing body of evidence giving strong evidence that zeros of families of L-functions follow distribution laws of eigenvalues of random matrices. This philosophy is known as the random matrix model or the Katz-Sarnak philosophy. The random matrix model makes predictions for the average distribution of zeros near the central point for families of L-functions. We study the low-lying zeros for families of elliptic curve L-functions. For these L-functions there is special arithmetic interest in any zeros at the central point (by the conjecture of Birch and Swinnerton-Dyer and the impressive partial results towards resolving the conjecture). We calculate the density of the low-lying zeros for various families of elliptic curves. Our main foci are the family of all elliptic curves and a large family with positive rank. A main challenge has been to obtain results with test functions that are concentrated close to the origin since the central point is a location of great interest. An application is an improvement on the upper bound of the average rank of the family of all elliptic curves (conditional on the Generalized Riemann hypothesis (GRH)). The upper bound obtained is less than 2, which shows that a positive proportion of curves in the family have algebraic rank equal to analytic rank and finite Tate-Shafarevich group. We show that there is an extra contribution to the density of the low-lying zeros from the family with positive rank (presumably from the “extra ” zero at the central point). 1 2 1.

Following an idea of A. Berenstein, we define a commutor for the category of crystals of a finite-dimensional complex reductive Lie algebra. We show that this endows the category of crystals with the structure of a coboundary category. Similarly to the role of the braid group in braided categories, a group naturally acts on multiple tensor products in coboundary categories. We call this group the cactus group and identify it as the fundamental group of the moduli space of marked, real, genus-zero stable curves

We conjecture the true rate of growth of the maximum size of the Riemann zeta-function and other L-functions. We support our conjecture using arguments from random matrix theory, conjectures for moments of L-functions, and also by assuming a random model for the primes.

We study the dynamics of systems on networks from a linear algebraic perspective. The control theoretic concept of controllability describes the set of states that can be reached for these systems. Our main result says that controllability in the quantum sense, expressed by the Lie algebra rank condition, and controllability in the sense of linear systems, expressed by the controllability matrix rank condition, are equivalent conditions. We also investigate how the graph theoretic concept of a zero forcing set impacts the controllability property; if a set of vertices is a zero forcing set, the associated dynamical system is controllable. These results open up the possibility of further exploiting the analogy between networks, linear control systems theory, and quantum systems Lie algebraic theory. This study is motivated by several quantum systems currently under study, including continuous quantum walks modeling transport phenomena.

We determine the distribution of the sandpile group (or Jacobian) of the Erdős-Rényi random graph <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis n comma q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G(n,q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> goes to infinity. We prove the distribution converges to a specific distribution conjectured by Clancy, Leake, and Payne. This distribution is related to, but different from, the Cohen-Lenstra distribution. Our proof involves first finding the expected number of surjections from the sandpile group to any finite abelian group (the “moments” of a random variable valued in finite abelian groups). To achieve this, we show a universality result for the moments of cokernels of random symmetric integral matrices that is strong enough to handle dependence in the diagonal entries. The methods developed to prove this result include inverse Littlewood-Offord theorems over finite rings and new techniques for studying homomorphisms of finite abelian groups with not only precise structure but also approximate versions of that structure. We then show these moments determine a unique distribution despite their <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Superscript k squared"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">p^{k^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -size growth. In particular, our theorems imply universality of singularity probability and ranks mod <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for symmetric integral matrices.

In this paper we study the simplest deformation on a sequence of orthogonal polynomials, namely, replacing the original (or reference) weight $w_0(x)$ defined on an interval by $w_0(x)e^{-tx}.$ It is a well-known fact that under such a deformation the recurrence coefficients denoted as $\alpha_n$ and $\beta_n$ evolve in $t$ according to the Toda equations, giving rise to the time dependent orthogonal polynomials, using Sogo's terminology. The resulting "time-dependent" Jacobi polynomials satisfy a linear second order ode. We will show that the coefficients of this ode are intimately related to a particular Painlev\'e V. In addition, we show that the coefficient of $z^{n-1}$ of the monic orthogonal polynomials associated with the "time-dependent" Jacobi weight, satisfies, up to a translation in $t,$ the Jimbo-Miwa $\sigma$-form of the same $P_{V};$ while a recurrence coefficient $\alpha_n(t),$ is up to a translation in $t$ and a linear fractional transformation $P_{V}(\alpha^2/2,-\beta^2/2, 2n+1+\alpha+\beta,-1/2).$ These results are found from combining a pair of non-linear difference equations and a pair of Toda equations. This will in turn allow us to show that a certain Fredholm determinant related to a class of Toeplitz plus Hankel operators has a connection to a Painlev\'e equation.

and the class number problem by

Tropical polytopes are images of polytopes in an affine space over the Puiseux series field under the degree map. This viewpoint gives rise to a family of cellular resolutions of monomial ideals that generalize the hull complex of Bayer and Sturmfels [Bayer and Sturmfels 98], instances of which improve upon the hull resolution in the sense of being smaller. We also suggest a new definition of a face of a tropical polytope, which has nicer properties than previous definitions; we give examples and provide many conjectures and directions for further research in this area.

There is a growing body of work giving strong evidence that zeros of families of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -functions follow distribution laws of eigenvalues of random matrices. This philosophy is known as the random matrix model or the Katz-Sarnak philosophy. The random matrix model makes predictions for the average distribution of zeros near the central point for families of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -functions. We study these low-lying zeros for families of elliptic curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -functions. For these <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -functions there is special arithmetic interest in any zeros at the central point (by the conjecture of Birch and Swinnerton-Dyer and the impressive partial results towards resolving the conjecture). We calculate the density of the low-lying zeros for various families of elliptic curves. Our main foci are the family of all elliptic curves and a large family with positive rank. An important challenge has been to obtain results with test functions that are concentrated close to the origin since the central point is a location of great arithmetical interest. An application of our results is an improvement on the upper bound of the average rank of the family of all elliptic curves (conditional on the Generalized Riemann Hypothesis (GRH)). The upper bound obtained is less than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which shows that a positive proportion of curves in the family have algebraic rank equal to analytic rank and finite Tate-Shafarevich group. We show that there is an extra contribution to the density of the low-lying zeros from the family with positive rank (presumably from the “extra" zero at the central point).

We consider the limiting behavior of discriminants, by which we mean informally the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety X and linear systems on X. These are connected—we use the first to understand the second. We describe their classes in the Grothendieck ring of varieties, as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization is given in terms of motivic zeta values. Motivated by our results, we ask whether the symmetric powers of geometrically irreducible varieties stabilize in the Grothendieck ring (in an appropriate sense). Our results extend parallel results in both arithmetic and topology. We give a number of reasons for considering these questions, and we propose a number of new conjectures, both arithmetic and topological.

In this paper we study the characteristic or generating function of a certain discontinuous linear statistic of the Laguerre unitary ensembles and show that this is a particular fifth Painleve transcendent in the variable t, the position of the discontinuity. The proof of the ladder operators adapted to orthogonal polynomial with discontinuous weight announced sometime ago [13] is presented here, followed by the nonlinear difference equations satisfied by two auxiliary quantities and the derivation of the Painleve equation.

The minimum rank of a directed graph Γ is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i,j) is an arc in Γ and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i ≠ j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. Maximum nullity is equal to the difference between the order of the graph and minimum rank in either case. Definitions of various graph parameters used to bound symmetric maximum nullity, including path cover number and zero forcing number, are extended to digraphs, and additional parameters related to minimum rank are introduced. It is shown that for directed trees, maximum nullity, path cover number, and zero forcing number are equal, providing a method to compute minimum rank for directed trees. It is shown that the minimum rank problem for any given digraph or zero-nonzero pattern may be converted into a symmetric minimum rank problem.

We compute the fourth moment of Dirichlet L-functions at the central point for prime moduli, with a power savings in the error term.

We introduce the cotangent sum and prove that it satisfies ⁠, where g is a function which is analytic on the complex plane minus the negative real axis. The sum arises in connection with the Nyman–Beurling approach to the Riemann hypothesis.

Abstract. We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length T 1