Artin’s braid group \(B_n\) is generated by \(\sigma _1, \ldots , \sigma _{n-1}\) subject to the relations $$\begin{aligned} \sigma _i \sigma _{i+1} \sigma _i = \sigma _{i+1} \sigma _i \sigma _{i+1}, \quad \sigma _i\sigma _j = \sigma _j \sigma _i \text { if } |i-j|>1. \end{aligned}$$ For complex parameters \(q_1,q_2\) such that \(q_1q_2 \ne 0\), the group \(B_n\) acts on the vector space \(\mathbf

The classical criterion of Jensen for the Riemann hypothesis is that all of the associated Jensen polynomials have only real zeros. We find a new version of this criterion, using linear combinations of Hermite polynomials, and show that this condition holds in many cases. Detailed asymptotic expansions are given for the required Taylor coefficients of the xi function at 1/2 as well as related quantities

Every tropical ideal in the sense of Maclagan–Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove

Rigid meromorphic cocycles were introduced by Darmon and Vonk as a conjectural p-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of \({{\,\mathrm{SL}\,}}_2(\mathbb {Z}[1/p])\) which can be evaluated at real quadratic irrationalities, and the values thus obtained are conjectured to lie in algebraic extensions of the base field. In this

The u-plane integral is the contribution of the Coulomb branch to correlation functions of \({\mathcal {N}}=2\) gauge theory on a compact four-manifold. We consider the u-plane integral for correlators of point and surface observables of topologically twisted theories with gauge group \(\mathrm{SU}(2)\), for an arbitrary four-manifold with \((b_1,b_2^+)=(0,1)\). The u-plane contribution equals the

We consider a class of dissipative stochastic differential equations (SDE’s) with time-periodic coefficients in finite dimension, and the response of time-asymptotic probability measures induced by such SDE’s to sufficiently regular, small perturbations of the underlying dynamics. Understanding such a response provides a systematic way to study changes of statistical observables in response to perturbations

A generalized modular relation of the form \(F(z, w, \alpha )=F(z, iw,\beta )\), where \(\alpha \beta =1\) and \(i=\sqrt{-1}\), is obtained in the course of evaluating an integral involving the Riemann \(\Xi \)-function. This modular relation involves a surprising new generalization of the Hurwitz zeta function \(\zeta (s, a)\), which we denote by \(\zeta _w(s, a)\). We show that \(\zeta _w(s, a)\)

The modified transmission eigenvalue problem arises in inverse scattering theory for inhomogeneous media, by embedding the relevant medium into an other, artificially introduced inhomogeneous medium. In the present work, we examine the case where the artificial medium is characterised as a metamaterial, i.e. having a negative valued refractive index. Our aim is to construct an appropriate spectral

It is a classical derivation that the Wigner equation, derived from the Schrödinger equation that contains the quantum information, converges to the Liouville equation when the rescaled Planck constant \(\varepsilon \rightarrow 0\). Since the latter presents the Newton’s second law, the process is typically termed the (semi-)classical limit. In this paper, we study the classical limit of an inverse

For a given bundle \(\xi :E \rightarrow M\) over a manifold, configuration-section spaces parametrise finite subsets \(z \subseteq M\) equipped with a section of \(\xi \) defined on \(M \smallsetminus z\), with prescribed “charge” in a neighbourhood of the points z. These spaces may be interpreted physically as spaces of fields that are permitted to be singular at finitely many points, with constrained

We consider a class of one-dimensional mixed stochastic delay differential equations driven by Brownian motions and fractional Brownian motions with Hurst parameter \(H\in (\frac{1}{2},1).\) A existence and uniqueness result is proved by using a contraction principle and some priori estimations. Some previous works are generalized and improved partially.

We construct secondary cup and cap products on coarse (co-)homology theories from given cross and slant products. They are defined for coarse spaces relative to weak generalized controlled deformation retracts. On ordinary coarse cohomology, our secondary cup product agrees with a secondary product defined by Roe. For coarsifications of topological coarse (co-)homology theories, our secondary cup and

We introduce an analog of the Dirichlet-to-Neumann map for the Maxwell equation in a bounded domain. We show that it can be approximated by a pseudodifferential operator on the boundary with a matrix-valued symbol and we compute the principal symbol. As an application, we obtain a parabolic region free of the transmission eigenvalues associated with the Maxwell equation.

We develop a theory of enhanced diffusivity and skewness of the longitudinal distribution of a diffusing tracer advected by a periodic time-varying shear flow in a straight channel. Although applicable to any type of solute and fluid flow, we restrict the examples of our theory to the tracer advected by flows which are induced by a periodically oscillating wall in a Newtonian fluid between two infinite

The levels of the factors of a holomorphic eta quotient f are bounded above with respect to the weight and the level of f. Unfortunately, this bound remains implicit due to the ineffectiveness of Mersmann’s finiteness theorem. On the other hand, for checking whether f is irreducible, it is essential to know at least an explicit upper bound for the minimum \(m_f\) among the levels of the proper factors

This paper considers the enhanced symplectic “category” for purposes of quantizing quasi-Hamiltonian (G)-spaces, where (G) is a compact simple Lie group. Our starting point is the well-acknowledged analogy between the cotangent bundle \(T^*G\) in Hamiltonian geometry and the internally fused double \(D(G)=G\times G\) in quasi-Hamiltonian geometry. Guillemin and Sternberg consider the former, studying

In this paper, we consider the numerical approximation of the Steklov eigenvalue problem that arises in inverse acoustic scattering. The underlying scattering problem is for an inhomogeneous isotropic medium. These eigenvalues have been proposed to be used as a target signature since they can be recovered from the scattering data. A Galerkin method is studied where the basis functions are the Neumann

Particle swarm optimization (PSO) is a member of nature-inspired metaheuristic algorithms. Its formulation is simple and does not need the computation of derivatives. It and its many variants have been applied to many different types of optimization problems across several disciplines. There have been many attempts to study the convergence properties of PSO, but a rigorous and complete proof of its

Deforestation is a major threat to global environmental wellness, with illegal logging as one of the major causes. Recently, there has been increased effort to model environmental crime, with the goal of assisting law enforcement agencies in deterring these activities. We present a continuous model for illegal logging applicable to arbitrary domains. We model the practice of criminals under influence

We present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs). The purpose of this analysis is twofold: First, we prove that continuous nonlinear functionals, functional derivatives, and FDEs can be approximated uniformly on any compact subset of a real Banach space admitting a basis by high-dimensional

In this article, we find finite field analogues of certain identities satisfied by the classical \({_4}F_3\)-hypergeometric series and Appell series. As an application, we find new summation and product formulas satisfied by the Gaussian hypergeometric series. For example, we express a \({_4}F_3\)-Gaussian hypergeometric series as a sum of two \({_2}F_1\)-Gaussian hypergeometric series. We also find

Recent work of Kass–Wickelgren gives an enriched count of the 27 lines on a smooth cubic surface over arbitrary fields. Their approach using \(\mathbb {A}^1\)-enumerative geometry suggests that other classical enumerative problems should have similar enrichments, when the answer is computed as the degree of the Euler class of a relatively orientable vector bundle. Here, we consider the closely related

Vakil and Matchett-Wood (Discriminants in the Grothendieck ring of varieties, 2013. arXiv:1208.3166) made several conjectures on the topology of symmetric powers of geometrically irreducible varieties based on their computations on motivic zeta functions. Two of those conjectures are about subspaces of \(\text {Sym}^n(\mathbb {P}^1)\). In this note, we disprove one of them and prove a stronger form

We present a list of problems in arithmetic topology posed at the June 2019 PIMS/NSF workshop on “Arithmetic Topology.” Three problem sessions were hosted during the workshop in which participants proposed open questions to the audience and engaged in shared discussions from their own perspectives as working mathematicians across various fields of study. Participants were explicitly asked to provide

These are lecture notes from the conference Arithmetic Topology at the Pacific Institute of Mathematical Sciences on applications of Morel’s \({\mathbb {A}}^1\)-degree to questions in enumerative geometry. Additionally, we give a new dynamic interpretation of the \({\mathbb {A}}^1\)-Milnor number inspired by the first-named author’s enrichment of dynamic intersection numbers.

Given \(d\in {\mathbb {N}}\), we prove that any polarized Enriques surface (over any field k of characteristic \(p \ne 2\) or with a smooth K3 cover) of degree greater than \(12d^2\) contains at most 12 rational curves of degree at most d. For \(d>2\), we construct examples of Enriques surfaces of high degree that contain exactly 12 rational degree-d curves.

We define and prove basic properties of a lifting invariant of curves over an algebraically closed field k with a map to the projective line \({\mathbb {P}}^1_{k}\) that was introduced by Ellenberg, Venkatesh, and Westerland. In other work of Liu, Zureick-Brown, and the author, this lifting invariant, and its connection to components of Hurwitz schemes, is applied to help understand the average number

We generalize a theorem of Iwasawa on capitulation kernels of class groups over \({{\mathbb {Z}}}_p\)-extensions of number fields to all \({{\mathbb {Z}}}_p^d\)-extensions and discuss some of its applications.

The algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular

Let \(k \ge 2\) be an integer. Let q be a prime power such that \(q \equiv 1 ({\mathrm{mod}}\,\,k)\) if q is even, or, \(q \equiv 1 ({\mathrm{mod}}\,\,2k)\) if q is odd. The generalized Paley graph of order q, \(G_k(q)\), is the graph with vertex set \(\mathbb {F}_q\) where ab is an edge if and only if \({a-b}\) is a kth power residue. We provide a formula, in terms of finite field hypergeometric functions

Product identities in two variables x, q expand infinite products as infinite sums, which are linear combinations of theta functions; famous examples include Jacobi’s triple product identity, Watson’s quintuple identity, and Hirschhorn’s septuple identity. We view these series expansions as representations in canonical bases of certain vector spaces of quasiperiodic meromorphic functions (related to

We define generalised zeta functions associated with indefinite quadratic forms of signature \((g-1,1)\)—and more generally, to complex symmetric matrices whose imaginary part has signature \((g-1,1)\)—and we investigate their properties. These indefinite zeta functions are defined as Mellin transforms of indefinite theta functions in the sense of Zwegers, which are in turn generalised to the Siegel

The representation of clouds and associated processes of rain and snow formation remain one of the major uncertainties in climate and weather prediction models. In a companion paper (part I), we systematically derived a two-moment bulk cloud microphysics model for warm rain based on the kinetic coalescence equation (KCE) and the use of stochastic approximations to close the high order moment terms

In this work, we study the structure and cardinality of maximal sets of commuting and anticommuting Paulis in the setting of the abelian Pauli group. We provide necessary and sufficient conditions for anticommuting sets to be maximal and present an efficient algorithm for generating anticommuting sets of maximum size. As a theoretical tool, we introduce commutativity maps and study properties of maps

In the original article there are errors in Eqs. (2) and (4) and in the paragraph following Eq. (4).

Crystabelline representations are representations of the absolute Galois group \(\smash {G_{\mathbb {Q}_p}}\) over \(\smash {\smash {\overline{\mathbb {Q}}_p}}\) that become crystalline on \(\smash {G_{F}}\) for some abelian extension \(\smash {F/\mathbb {Q}_p}\). Their relation to modular forms is that the representation associated with a finite slope newform of level divisible by \(\smash {p^2}\)

We propose a mathematical methodology to two-moment parameterization of bulk warm cloud microphysics based on a stochastic representation of the high-order moment terms. Unlike previous bulk parameterizations, the stochastic closure approach proposed here does not assume any particular droplet size distribution, all parameters have physical meanings which are recoverable from data, and the resultant

We study the distribution of the sequence of elements of the discrete dynamical system generated by iterations of the Möbius map \(x \mapsto (ax + b)/(cx+d)\) over a finite field of p elements at the moments of time that correspond to prime numbers. In particular, we obtain nontrivial estimates of exponential sums with such sequences.

We classify functions \(f:(a,b)\rightarrow \mathbb {R}\) which satisfy the inequality $$\begin{aligned} {\text {tr}}f(A)+f(C)\ge {\text {tr}}f(B)+f(D) \end{aligned}$$ when \(A\le B\le C\) are self-adjoint matrices, \(D= A+C-B\), the so-called trace minmax functions. (Here \(A\le B\) if \(B-A\) is positive semidefinite, and f is evaluated via the functional calculus.) A function is trace minmax if and

The problem of data-driven identification of coherent observables of measure-preserving, ergodic dynamical systems is studied using kernel integral operator techniques. An approach is proposed whereby complex-valued observables with approximately cyclical behavior are constructed from a pair of eigenfunctions of integral operators built from delay-coordinate mapped data. It is shown that these observables

In this paper, we study a variation in a conjecture of Debarre on positivity of cotangent bundles of complete intersections. We establish the ampleness of Schur powers of cotangent bundles of generic complete intersections in projective manifolds, with high enough explicit codimension and multi-degrees. Our approach is naturally formulated in terms of flag bundles and allows one to reach the optimal

Asymptotic models have provided valuable insight into the atmosphere and its dynamics. Nevertheless, one shortcoming of the classic asymptotic models, such as the quasi-geostrophic (QG) equations, is that they describe a “dry” atmosphere and do not account for water vapor, clouds, and rainfall. Recently, precipitating QG (PQG) equations were derived in an asymptotic limit, starting from atmospheric

We establish a scale separation of Kolmogorov width type between subspaces of a given Banach space under the condition that a sequence of linear maps converges much faster on one of the subspaces. The general technique is then applied to show that reproducing kernel Hilbert spaces are poor \(L^{2}\)-approximators for the class of two-layer neural networks in high dimension, and that multi-layer networks

We prove effective forms of the Sato–Tate conjecture for holomorphic cuspidal newforms which improve on the author’s previous work (solo and joint with Lemke Oliver). We also prove an effective form of the joint Sato–Tate distribution for two twist-inequivalent newforms. Our results are unconditional because of recent work of Newton and Thorne.

Let L be a lattice of full rank in n-dimensional real space. A vector in L is called i-sparse if it has no more than i nonzero coordinates. We define the ith successive sparsity level of L, \(s_i(L)\), to be the minimal s so that L has s linearly independent i-sparse vectors, then \(s_i(L) \le n\) for each \(1 \le i \le n\). We investigate sufficient conditions for \(s_i(L)\) to be smaller than n and

We show the compactly supported motive of the moduli stack of degree n rational curves on the weighted projective stack \({\mathcal {P}}(a,b)\) is of mixed Tate type over any base field K with \(\hbox {char}(K) \not \mid a,b\) and has class \({\mathbb {L}}^{(a+b)n+1}-{\mathbb {L}}^{(a+b)n-1}\) in the Grothendieck ring of stacks. In particular, this improves upon the results of (Han and Park in Math

In this paper, a double-pivot simplex method is proposed. Two upper bounds of iteration numbers are derived. Applying one of the bounds to some special linear programming (LP) problems, such as LP with a totally unimodular matrix and Markov decision problem with a fixed discount rate, indicates that the double-pivot simplex method solves these problems in a strongly polynomial time. Applying the other

We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level \(\varGamma _0(N)\) and integral weight \(k\ge 2\) coincides with the field generated by the single-valued periods of a certain motive attached to \(\varGamma _0(N)\). This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres–Broadhurst

The geodesic flows are studied on real forms of complex semi-simple Lie groups with respect to a left-invariant (pseudo-)Riemannian metric of rigid body type. The Williamson types of the isolated relative equilibria on generic adjoint orbits are determined.

In this paper, we prove an explicit version of a function field analogue of a classical result of Odoni (Mathematika 22(1):71–80, 1975) about norms in number fields, in the case of a cyclic Galois extensions. In the particular case of a quadratic extension, we recover the result Bary-Soroker et al. (Finite Fields Appl 39:195–215, 2016) which deals with finding asymptotics for a function field version

We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing measures. Our exposition is complemented by a brief study of arising complexity questions.

Analytical formulas for effective drift, diffusivity, run times, and run lengths are derived for an intracellular transport system consisting of a cargo attached to two cooperative but not identical molecular motors (for example, kinesin-1 and kinesin-2) which can each attach and detach from a microtubule. The dynamics of the motor and cargo in each phase are governed by stochastic differential equations

This article is a survey of conjectures and results on reductive algebraic groups having good reduction at a suitable set of discrete valuations of the base field. Until recently, this subject has received relatively little attention, but now it appears to be developing into one of the central topics in the emerging arithmetic theory of (linear) algebraic groups over higher-dimensional fields. The

The equivalence of Benjamini–Schramm convergence and zeta-convergence, known for graphs, is proven for sequences of compact Riemann surfaces. A program is initialized, to extend this connection to arbitrary locally homogeneous spaces.

Closed-form evaluations of certain integrals of \(J_{0}(\xi )\), the Bessel function of the first kind, have been crucial in the studies on the electromagnetic field of alternating current in a circuit with two groundings, as can be seen from the works of Fock and Bursian, Schermann, etc. Koshliakov’s generalization of one such integral, which contains \(J_s(\xi )\) in the integrand, encompasses several

We study the space \({{\,\mathrm{Hol}\,}}_d(\mathbb {CP}^m,\mathbb {CP}^n)\) of degree d algebraic maps \(\mathbb {CP}^m \rightarrow \mathbb {CP}^n\), from the point of view of homological stability as discovered by Segal (Acta Math 143(1–2):39–72, 1979) and later explored by Mostovoy (Topol Appl 45(2):281–293, 2006), Cohen et al. (Acta Math 166:163–221, 1991), Farb and Wolfson (N Y J Math 22:801–821

In a beautiful paper, Deligne and Illusie (Invent Math 89(2):247–270, 1987) proved the degeneration of the Hodge-to-de Rham spectral sequence using positive characteristic methods. Kato (in: Igusa (ed) ALG analysis, geographic and numbers theory, Johns Hopkins University Press, Baltimore, 1989) generalized this result to logarithmic schemes. In this paper, we use the theory of twisted derived intersections

In this paper, we compute the distributions of various markings on smooth cubic surfaces defined over the finite field \(\mathbb {F}_q\), for example the distribution of pairs of points, ‘tritangents’ or ‘double sixes’. We also compute the (rational) cohomology of certain associated bundles and covers over complex numbers.

We study the nonlinear coupling mechanism and turbulent transition in magnetically confined plasma flows based on two representative limiting regime dynamics. The two-field flux-balanced Hasegawa–Wakatani (BHW) model is taken as a simplified approximation to the key physical processes in the energy-conserving nonlinear plasma flows. The limiting regimes separate the effects of finite non-adiabatic