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

We define L-functions for the class of real-analytic modular forms recently introduced by F. Brown. We establish their main properties and construct the analogue of period polynomial in cases of special interest, including those of modular iterated integrals of length one.

We propose new and original mathematical connections between Hamilton–Jacobi (HJ) partial differential equations (PDEs) with initial data and neural network architectures. Specifically, we prove that some classes of neural networks correspond to representation formulas of HJ PDE solutions whose Hamiltonians and initial data are obtained from the parameters of the neural networks. These results do not

We define the Euler function of a global field and recover the fundamental properties of the classical arithmetical function. In addition, we prove the holomorphicity of the associated zeta function. As an application, we recover analogs of the mean value theorems of Mertens and Erdős–Dressler–Bateman. The exposition is aimed at non-experts in arithmetic statistics, with the intention of providing

Predicting complex nonlinear turbulent dynamical systems using partial observations is an important topic. Despite the simplicity of the forecast based on the ensemble mean time series, several critical shortcomings in the ensemble mean forecast and using path-wise measurements to quantify the prediction error are illustrated in this article. Then, a new ensemble method is developed for improving the

We present two new sharp regularity results (regularizing effect and propagation of regularity) for viscosity solutions of uniformly convex homogeneous (space independent) Hamilton–Jacobi equations. The estimates do not depend on the convexity constants of the Hamiltonians. The sharp propagation of regularity result holds in dimension larger than one without additional smoothness assumptions on the

In this paper, we consider modeling missing dynamics with a nonparametric non-Markovian model, constructed using the theory of kernel embedding of conditional distributions on appropriate reproducing kernel Hilbert spaces (RKHS), equipped with orthonormal basis functions. Depending on the choice of the basis functions, the resulting closure model from this nonparametric modeling formulation is in the

We study the photonic band structure of a metal–dielectric periodic structure. The metallic component is described by the Drude model; therefore, the electric permittivity is frequency dependent, i.e., dispersive. Rather than solving a nonlinear eigenvalue problem for the band structure of the material, we follow a time-dependent formulation described in Raman and Fan (Phys Rev Lett 104:087401, 2010)

Geophysical turbulence has a wide range of spatiotemporal scales that requires a multiscale prediction model for efficient and fast simulations. Stochastic parameterization is a class of multiscale methods that approximates the large-scale behaviors of the turbulent system without relying on scale separation. In the stochastic parameterization of unresolved subgrid-scale dynamics, there are several

We discuss how to combine exponential time differencing technique with multi-step method to develop higher order in time linear numerical scheme that are energy stable for certain gradient flows with the aid of a generalized viscous damping term. As an example, a stabilized third order in time accurate linear exponential time differencing (ETD) scheme for the epitaxial thin film growth model without

This article is an overview of the vanishing cycles method in number theory over function fields. We first explain how this works in detail in a toy example and then give three examples which are relevant to current research. The focus will be a general explanation of which sorts of problems this method can be applied to.

In this paper, the right-sided quaternion Fourier transform (right-sided QFT), which is a non-trivial generalization of the real and complex Fourier transform, is studied on spaces of test functions and distributions. The continuous quaternion wavelet transform of periodic functions is also defined and its quaternion Fourier representation form is established. The Plancherel and inversion formulas

We prove that Galois cohomology satisfies several surprisingly strong versions of Koszul properties, under a well known conjecture, in the finitely generated case. In fact, these versions of Koszulity hold for all finitely generated maximal pro-p quotients of absolute Galois groups which are currently understood. We point out several of these unconditional results which follow from our work. We show

We construct certain operations on stable moduli spaces and use them to compare cohomology of moduli spaces of closed manifolds with tangential structure. We obtain isomorphisms in a stable range provided the p-adic valuation of the Euler characteristics agree, for all primes p not invertible in the coefficients for cohomology.

We investigate the motion of a thin rigid body in Stokes flow and the corresponding slender body approximation used to model sedimenting fibers. In particular, we derive a rigorous error bound comparing a regularized version of the rigid slender body approximation to the classical PDE for rigid motion in the case of a closed loop with constant radius. Our main tool is the slender body PDE framework

We study the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard–Fuchs differential equations; we count points using Gauss sums and rewrite this in terms of finite-field hypergeometric sums; then we match up each differential equation to a factor of the zeta function, and we write this in terms

In this paper, we study the rarefaction wave case of the regularized Riemann problem proposed by Chu, Hong and Lee in SIMA MMS, 2020, for compressible Euler equations with a small parameter \(\nu \). The solutions \(\rho _\nu \) and \(v_\nu \) of such problems stand for the density and velocity of gas flow near vacuum, respectively. We show that as \(\nu \) approaches 0, the solutions \(\rho _\nu \)

By giving the definition of the sum of a series indexed by a set on which a group operates, we prove that the sum of the series that defines the Riemann zeta function, the Epstein zeta function and a few other series indexed by \({\mathbb {Z}}^k\) has an intrinsic meaning as a complex number, independent of the requirements of analytic continuation. The definition of the sum requires nothing more than

We consider the Dirichlet-to-Neumann operator (DNO) on nearly circular and nearly spherical domains in two and three dimensions, respectively. Treating such domains as perturbations of the ball, we prove the analyticity of the DNO with respect to the domain perturbation parameter. Consequently, the Steklov eigenvalues are also shown to be analytic in the domain perturbation parameter. To obtain these

D’Arcais-type polynomials encode growth and non-vanishing properties of the coefficients of powers of the Dedekind eta function. They also include associated Laguerre polynomials. We prove growth conditions and apply them to the representation theory of complex simple Lie algebras and to the theory of partitions, in the direction of the Nekrasov–Okounkov hook length formula. We generalize and extend

The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form \((\mu \star \Lambda _1^{\star k_1} \star \Lambda _2^{\star k_2} \star \cdots \star \Lambda _d^{\star k_d})\) is computed unconditionally by means of the autocorrelation of ratios of \(\zeta \) techniques from Conrey et al. (Proc Lond Math Soc (3) 91:33–104, 2005), Conrey et al

We prove an identity relating Mahler measures of a certain family of non-tempered polynomials to those of tempered polynomials. Evaluations of Mahler measures of some polynomials in the first family are also given in terms of special values of L-functions and logarithms.Finally, we prove Boyd’s conjectures for conductor 30 elliptic curves using our new identity, Brunault–Mellit–Zudilin’s formula and

This paper provides a rigorous convergence rate and complexity analysis for a recently introduced framework, called PDE acceleration, for solving problems in the calculus of variations and explores applications to obstacle problems. PDE acceleration grew out of a variational interpretation of momentum methods, such as Nesterov’s accelerated gradient method and Polyak’s heavy ball method, that views

We derive new gradient flows of divergence functions in the probability space embedded with a class of Riemannian metrics. The Riemannian metric tensor is built from the transported Hessian operator of an entropy function. The new gradient flow is a generalized Fokker–Planck equation and is associated with a stochastic differential equation that depends on the reference measure. Several examples of

The Cheeger constant is a measure of the edge expansion of a graph and as such plays a key role in combinatorics and theoretical computer science. In recent years, there is an interest in k-dimensional versions of the Cheeger constant that likewise provide quantitative measure of cohomological acyclicity of a complex in dimension k. In this paper, we study several aspects of the higher Cheeger constants

We investigate the invisibility via anomalous localized resonance of a general source in anisotropic media for electromagnetic waves. To this end, we first introduce the concept of doubly complementary media in the electromagnetic setting. These are media consisting of negative-index metamaterials in a shell and positive-index materials in its complement for which the shell is complementary to a part

For each \(t \in \mathbb {R}\), define the entire function$$\begin{aligned} H_t(z){:=}\,\int _0^\infty e^{tu^2} \varPhi (u) \cos (zu)\ \mathrm{d}u, \end{aligned}$$where \(\varPhi \) is the super-exponentially decaying function$$\begin{aligned} \varPhi (u){:=}\,\sum _{n=1}^\infty (2\pi ^2 n^4 e^{9u} - 3\pi n^2 e^{5u} ) \exp (-\pi n^2 e^{4u}). \end{aligned}$$This is essentially the heat flow evolution