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

False theta functions closely resemble ordinary theta functions; however, they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among other things gives an efficient way to compute their obstruction to modularity. This has potential applications for a variety of contexts where false and partial theta

In this paper, we consider the iterative method of subspace corrections with random ordering. We prove identities for the expected convergence rate and use these results to provide sharp estimates for the expected error reduction per iteration. We also study the fault-tolerant features of the randomized successive subspace correction method by rejecting corrections when faults occur and show that the

In 1979, in the course of the proof of the irrationality of \(\zeta (2)\) Apéry introduced numbers \(b_n\) that are, surprisingly, integral solutions of the recursive relation$$\begin{aligned} (n+1)^2 u_{n+1} - (11n^2+11n+3)u_n-n^2u_{n-1} = 0. \end{aligned}$$Indeed, \(b_n\) can be expressed as \(b_n= \sum _{k=0}^n {n \atopwithdelims ()k}^2{n+k \atopwithdelims ()k}\). Zagier performed a computer search

Recently, Ono et al. answered problems of Manin by defining zeta-polynomials \(Z_f(s)\) for even weight newforms \(f\in S_k(\varGamma _0(N)\); these polynomials can be defined by applying the “Rodriguez-Villegas transform” to the period polynomial of f. It is known that these zeta-polynomials satisfy a functional equation \(Z_f(s) = \pm \, Z_f(1-s)\) and they have a conjectural arithmetic-geometric

In order to better understand the structure of classical rings of invariants for binary forms, Dixmier proposed, as a conjectural homogeneous system of parameters, an explicit collection of invariants previously studied by Hilbert. We generalize Dixmier’s collection and show that a particular subfamily is algebraically independent. Our proof relies on showing certain alternating sums of products of

We establish three infinite families of quantum Jacobi forms, arising in the diverse areas of number theory, topology, and mathematical physics, and unified by partial Jacobi theta functions.

We characterize Zagier’s generating series of extended period polynomials of normalized Hecke eigenforms for \({{\,\mathrm{PSL}\,}}_2(\mathbb {Z})\) in terms of the period relations and existence of a suitable factorization. For this, we prove a characterization of the Kronecker function as the “fundamental solution” of the Fay identity.

We use the method of Shintani, as developed by Taniguchi and Thorne, to prove the quantitative equidistribution of the shape of cubic fields when the fields are ordered by discriminant.

The classical Maass lift is a map from holomorphic Jacobi forms to holomorphic scalar-valued Siegel modular forms. Automorphic representation theory predicts a non-holomorphic and vector-valued analogue for Hecke eigenforms. This paper is the first part of a series of papers. In this series of papers, we provide an explicit construction of the non-holomorphic Maass lift that is linear and also applies

In recent years, deep learning has led to impressive results in many fields. In this paper, we introduce a multiscale artificial neural network for high-dimensional nonlinear maps based on the idea of hierarchical nested bases in the fast multipole method and the \(\mathcal {H}^2\)-matrices. This approach allows us to efficiently approximate discretized nonlinear maps arising from partial differential

We introduce and study higher depth quantum modular forms. We construct two families of examples coming from rank two false theta functions, whose “companions” in the lower half-plane can be also realized both as double Eichler integrals and as non-holomorphic theta series having values of “double error” functions as coefficients. In particular, we prove that the false theta functions of \(\mathfrak

Many physical equations have the form \(\mathbf{J }(\mathbf{x })=\mathbf{L }(\mathbf{x })\mathbf{E }(\mathbf{x })-\mathbf{h }(\mathbf{x })\) with source \(\mathbf{h }(\mathbf{x })\) and fields \(\mathbf{E }\) and \(\mathbf{J }\) satisfying differential constraints, symbolized by \(\mathbf{E }\in \mathcal E\), \(\mathbf{J }\in \mathcal J\) where \(\mathcal E\), \(\mathcal J\) are orthogonal spaces.

We study the distribution of the sequence of elements of the discrete dynamical system generated by the Möbius transformation \(x \mapsto (ax + b)/(cx + d)\) over a finite field of p elements. Motivated by a recent conjecture of P. Sarnak, we obtain nontrivial estimates of exponential sums with such sequences that imply that trajectories of this dynamical system are disjoined with the Möbius function

In the present paper, we investigate Taelman L-values corresponding to Drinfeld modules over Tate algebras of arbitrary rank. Using our results, we also introduce an L-series converging in Tate algebras which can be seen as a generalization of Pellarin L-series.

Multivariable Appell functions show up in the work of Kac and Wakimoto in the computation of character formulas for certain \(s \ell (m,1)^\wedge \) modules. Bringmann and Ono showed that the character formulas for the \(s \ell (m,1)^\wedge \) modules \(L(\varLambda _{(s)})\), where \(L(\varLambda _{(s)})\) is the irreducible \(s \ell (m,1)^\wedge \) module with the highest weight \(\varLambda _{(s)}\)

Quantized deep neural networks (QDNNs) are attractive due to their much lower memory storage and faster inference speed than their regular full-precision counterparts. To maintain the same performance level especially at low bit-widths, QDNNs must be retrained. Their training involves piecewise constant activation functions and discrete weights; hence, mathematical challenges arise. We introduce the

In this survey paper, we study Manin’s conjecture from a geometric perspective. The focus of the paper is the recent conjectural description of the exceptional set in Manin’s conjecture due to Lehmann–Sengupta–Tanimoto. After giving an extensive background, we give a precise description of this set and compute it in many examples.

Recently, a new singularity formation scenario for the 3D axi-symmetric Euler equation and the 2D inviscid Boussinesq system has been proposed by Hou and Luo (Multiscale Model Simul 12(4):1722–1776, 2014, PNAS 111(36):12968–12973, 2014) based on extensive numerical simulations. As the first step to understand the scenario, models with simplified sign-definite Biot–Savart law and forcing have recently

We propose a new fourth-order compact time-splitting (\(S_\mathrm{4c}\)) Fourier pseudospectral method for the Dirac equation by splitting the Dirac equation into two parts together with using the double commutator between them to integrate the Dirac equation at each time interval. The method is explicit, fourth-order in time and spectral order in space. It is unconditionally stable and conserves the

Recent work linking deep neural networks and dynamical systems opened up new avenues to analyze deep learning. In particular, it is observed that new insights can be obtained by recasting deep learning as an optimal control problem on difference or differential equations. However, the mathematical aspects of such a formulation have not been systematically explored. This paper introduces the mathematical

We are interested in an inverse medium problem with internal data. This problem is originated from multi-waves imaging. We aim in the present work to study the well posedness of the inversion in terms of the boundary conditions. We precisely show that we have actually a stability estimate of Hölder type. For sake of simplicity, we limited our study to the class of Helmholtz equations \(\Delta + V\)

Using the locally compact abelian group \(\mathbb {T}\times \mathbb {Z}\), we assign a meromorphic function to each ideal triangulation of a 3-manifold with torus boundary components. The function is invariant under all 2–3 Pachner moves, and thus is a topological invariant of the underlying manifold. If the ideal triangulation has a strict angle structure, our meromorphic function can be expanded

We prove that Cohen’s Maass wave form and Li–Ngo–Rhoades’ Maass wave form are Hecke eigenforms with respect to certain Hecke operators. As a corollary, we find new identities of the pth coefficients of these Maass wave forms in terms of pth root of unity.