Miracle-dz/UniFAR-FTUs huggingface dataset:
Bounded Operators to $\ell$-K\"othe Spaces

For Fr{\'e}chet spaces E and F we write (E,F) \in {B} if every continuous linear operator from E to F is bounded. Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We obtain a necessary and sufficient condition for (E,F) \in {B} when F = \lambda^{l}(B). We say that a triple (E,F,G) has the bounded factorization property and write (E,F,G) \in {BF} if each continuous linear operator T : E \longrightarrow G that factors over F is bounded. We extend some results in \cite{Ter03} to l-K\"{o}the spaces and obtain a sufficient condition for (E,\lambda^{l_1}(A) \hat{otimes}_{pi} \lambda^{l_2}(B), \lambda^{l_3}(C)) \in {BF} when \lambda^{l_1}(A) and \lambda^{l_2}(B) are nuclear.
Imaging ocean waves by synthetic aperture radars with long integration times

Previously developed theories for the imaging mechanism of ocean waves by synthetic aperture radars (SAR's) apply only for short integration times T (typically, T s). To what extent long integration times modify these theories is investigated. First, an analytical expression describing the effect of systematic wave motions on radar imagery valid for an arbitrary integration time is derived. Numerical evaluation of the expression shows only small deviations from the previous expression valid for short integration times. Thus the restriction of the previous theory, ( \hat{\omega}T/2) \ll 1 , where \hat{\omega} is the radian frequency of the long waves, can be relaxed to (\hat{\omega}T/2) \lsim 1 . Second, the influence of the coherence time of the scene, randomness of wave field, on the imaging mechanism is investigated. It is argued that the scene coherence time is usually larger than the SAR coherent integration time T , implying that the azimuthal "image smear" for the case of ocean wave imaging is usually due to systematic wave motion rather than the scene coherence time.
Umbilic Points on the Finite and Infinite Parts of Certain Algebraic Surfaces

Abstract:The global qualitative behaviour of fields of principal directions for the graph of a real-valued polynomial function f on the plane is studied. We determine and analyse the projective extension of these fields and show that they are defined by an analytic quadratic form on the whole unit 2-sphere. We prove that every umbilic point at infinity of this extension has a Poincaré-Hopf index equal to 1/2, and the topological type of a Lemon when the degree of f is 2 and the topological type of a Monstar for higher degrees. As a consequence, we prove a Poincaré-Hopf type formula for the graph of f such that, if all umbilics are isolated, the sum of all indices of the principal directions at umbilic points depends only upon the number of real linear factors of the homogeneous part of highest degree of f. A similar analysis is carried out in the case of f being a homogeneous polynomial.
Generalized prolate spheroidal functions: algorithms and analysis

Generalized prolate spheroidal functions (GPSFs) arise naturally in the study of bandlimited functions as the eigenfunctions of a certain truncated Fourier transform. In one dimension, the theory of GPSFs (typically referred to as prolate spheroidal wave functions) has a long history and is fairly complete. Furthermore, more recent work has led to the development of numerical algorithms for their computation and use in applications. In this paper we consider the more general problem, extending the one dimensional analysis and algorithms to the case of arbitrary dimension. Specifically, we introduce algorithms for efficient evaluation of GPSFs and their corresponding eigenvalues, quadrature rules for bandlimited functions, formulae for interpolation via GPSF expansion, and various analytical properties of GPSFs. We illustrate the numerical and analytical results with several numerical examples.
Attractors for Delayed, Nonrotational von Karman Plates with Applications to Flow-Structure Interactions Without any Damping

This paper is devoted to a long-time behavior analysis of flow-structure interactions at subsonic and supersonic velocities. An intrinsic component of that analysis is the study of attractors corresponding to von Karman plate equations with delayed terms and without rotational terms. The presence of delay terms in the dynamical system leads to a loss of gradient structure, while the absence of rotational terms in von Karman plates leads to the loss of compactness of the orbits. Both of these features make the analysis of long-time behavior rather subtle, rendering the established tools in the theory of PDE and dynamical systems not applicable. We develop methodology that is capable of addressing this class of problems.
An efficient linear programming method for Optimal Transportation

An efficient method for computing solutions to the Optimal Transportation (OT) problem with a wide class of cost functions is presented. The standard linear programming (LP) discretization of the continuous problem becomes intractible for moderate grid sizes. A grid refinement method results in a linear cost algorithm. Weak convergence of solutions is stablished. Barycentric projection of transference plans is used to improve the accuracy of solutions. The method is applied to more general problems, including partial optimal transportation, and barycenter problems. Computational examples validate the accuracy and efficiency of the method. Optimal maps between nonconvex domains, partial OT free boundaries, and high accuracy barycenters are presented.
Mathematics and turbulence: where do we stand?

This contribution covers the topics presented by the authors at the “Fundamental Problems of Turbulence, 50 Years after the Marseille Conference 1961” meeting that took place in Marseille in 2011. It focuses on some of the mathematical approaches to fluid dynamics and turbulence. This contribution does not pretend to cover or answer, as the reader may discover, the fundamental questions in turbulence, however, it aims toward presenting some of the most recent advances in attacking these questions using rigorous mathematical tools. Moreover, we consider that the proofs of the mathematical statements (concerning, for instance, finite time regularity, weak solutions and vanishing viscosity) may contain information as relevant, to the understanding of the underlying problem, as the statements themselves.
On the τ-functions of the reduced Ostrovsky equation and the A(2)2 two-dimensional Toda system

The reciprocal link between the reduced Ostrovsky equation and the A(2)2 two-dimensional Toda (2D-Toda) system is used to construct the N-soliton solution of the reduced Ostrovsky equation. The N-soliton solution of the reduced Ostrovsky equation is presented in the form of pfaffian through a hodograph (reciprocal) transformation. The bilinear equations and the τ-function of the reduced Ostrovsky equation are obtained from the period 3-reduction of the B∞ or C∞ 2D-Toda system, i.e. the A(2)2 2D-Toda system. One of the τ-functions of the A(2)2 2D-Toda system becomes the square of a pfaffian which also becomes a solution of the reduced Ostrovsky equation. There is another bilinear equation which is a member of the 3-reduced extended BKP hierarchy. Using this bilinear equation, we can also construct the same pfaffian solution.
Idempotent states and the inner linearity property

We find an analytic formulation of the notion of Hopf image, in terms of the associated idempotent state. More precisely, if $\pi:A\to M_n(\mathbb C)$ is a finite dimensional representation of a Hopf $C^*$-algebra, we prove that the idempotent state associated to its Hopf image $A'$ must be the convolution Ces\`aro limit of the linear functional $\phi=tr\circ\pi$. We discuss then some consequences of this result, notably to inner linearity questions.
Recent tests of exponentiality against IFR alternatives: a survey

Testing of various classes of life distributions has been a subject of investigation for more than four decades. In this study, we restrict ourselves to the problem of testing exponentiality (which essentially means no aging) against positive aging, which is captured by the class of increasing failure rate alternatives. Recent tests are discussed and compared. The empirical size of the tests is obtained by simulation. Power computations, using simulations, are done for each test procedure. These comparisons are done both for small and large sample sizes. Suggestions are made regarding the choice of the test when a particular alternative is suspected.
Large deviations for randomly connected neural networks: I. Spatially extended systems

Abstract In a series of two papers, we investigate the large deviations and asymptotic behavior of stochastic models of brain neural networks with random interaction coefficients. In this first paper, we take into account the spatial structure of the brain and consider first the presence of interaction delays that depend on the distance between cells and then the Gaussian random interaction amplitude with a mean and variance that depend on the position of the neurons and scale as the inverse of the network size. We show that the empirical measure satisfies a large deviations principle with a good rate function reaching its minimum at a unique spatially extended probability measure. This result implies an averaged convergence of the empirical measure and a propagation of chaos. The limit is characterized through a complex non-Markovian implicit equation in which the network interaction term is replaced by a nonlocal Gaussian process with a mean and covariance that depend on the statistics of the solution over the whole neural field.
Extended eigenvarieties for overconvergent cohomology

Recently, Andreatta, Iovita and Pilloni have constructed spaces of overconvergent modular forms in characteristic p, together with a natural extension of the Coleman-Mazur eigencurve over a compactified (adic) weight space. Similar ideas have also been used by Liu, Wan and Xiao to study the boundary of the eigencurve. This all goes back to an idea of Coleman. In this article, we construct natural extensions of eigenvarieties for arbitrary reductive groups G over a number field which are split at all places above p. If G is GL(2)/Q, then we obtain a new construction of the extended eigencurve of Andreatta-Iovita-Pilloni. If G is an inner form of GL(2) associated to a definite quaternion algebra, our work gives a new perspective on some of the results of Liu-Wan-Xiao. We build our extended eigenvarieties following Hansen's construction using overconvergent cohomology. One key ingredient is a definition of locally analytic distribution modules which permits coefficients of characteristic p (and mixed characteristic). When G is GL(n) over a totally real or CM number field, we also construct a family of Galois representations over the reduced extended eigenvariety.
Optimal error estimates and energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell's equations

This paper is concerned with the optimal error estimates and energy conservation properties of the alternating direction implicit finite-difference time-domain (ADI-FDTD) method which is a popular scheme for solving the 3D Maxwell equations. Precisely, for the case with a perfectly electric conducting (PEC) boundary condition we establish the optimal second-order error estimates in both space and time in the discrete $H^1$-norm for the ADI-FDTD scheme and prove the approximate divergence preserving property that if the divergence of the initial electric and magnetic fields are zero then the discrete $L^2$-norm of the discrete divergence of the ADI-FDTD solution is approximately zero with the second-order accuracy in both space and time. A key ingredient is two new discrete energy norms which are second-order in time perturbations of two new energy conservation laws for the Maxwell equations introduced in this paper. Furthermore, we prove that, in addition to two known discrete energy identities which are second-order in time perturbations of two known energy conservation laws, the ADI-FDTD scheme also satisfies two new discrete energy identities which are second-order in time perturbations of the two new energy conservation laws. This means that the ADI-FDTD scheme is unconditionally stable under the four discrete energy norms. Experimental results are presented which confirm the theoretical results.
Empirical Spectral Model of Surface Pressure Fluctuations

An empirical model of the surface pressure spectrum beneath a two-dimensional, zero-pressure-gradient boundary layer is presented that is based on the experimental surface pressure spectra measured by seven research groups. The measurements cover a large range of Reynolds number, 1.4 x 10 3 <Re θ < 2.34 x 10 4 . The model is a simple function of the ratio of the timescales of the outer to inner boundary layer. It incorporates the effect of Reynolds number through the timescale ratio and compares well to experimental data. It is proposed that the effect of Reynolds number is more aptly described as the effect of the range of relevant scales. Spectral features of the experimental data and the scaling behavior of the surface pressure spectrum are also discussed
Matricial model for the free multiplicative convolution

This paper investigates homomorphisms \`a la Bercovici-Pata between additive and multiplicative convolutions. We also consider their matricial versions which are associated with measures on the space of Hermitian matrices and on the unitary group. The previous results combined with a matricial model of Benaych-Georges and Cabanal-Duvillard allows us to define and study the large N limit of a new matricial model on the unitary group for free multiplicative L\'evy processes.
Rolling Manifolds: Intrinsic Formulation and Controllability

In this paper, we consider two cases of rolling of one smooth connected complete Riemannian manifold $(M,g)$ onto another one $(\hM,\hg)$ of equal dimension $n\geq 2$. The rolling problem $(NS)$ corresponds to the situation where there is no relative spin (or twist) of one manifold with respect to the other one. As for the rolling problem $(R)$, there is no relative spin and also no relative slip. Since the manifolds are not assumed to be embedded into an Euclidean space, we provide an intrinsic description of the two constraints ''without spinning'' and ''without slipping'' in terms of the Levi-Civita connections $\nabla^{g}$ and $\nabla^{\hg}$. For that purpose, we recast the two rolling problems within the framework of geometric control and associate to each of them a distribution and a control system. We then investigate the relationships between the two control systems and we address for both of them the issue of complete controllability. For the rolling $(NS)$, the reachable set (from any point) can be described exactly in terms of the holonomy groups of $(M,g)$ and $(\hM,\hg)$ respectively, and thus we achieve a complete understanding of the controllability properties of the corresponding control system. As for the rolling $(R)$, the problem turns out to be more delicate. We first provide basic global properties for the reachable set and investigate the associated Lie bracket structure. In particular, we point out the role played by a curvature tensor defined on the state space, that we call the \emph{rolling curvature}. In the case where one of the manifolds is a space form (let say $(\hM,\hg)$), we show that it is enough to roll along loops of $(M,g)$ and the resulting orbits carry a structure of principal bundle which preserves the rolling $(R)$ distribution. In the zero curvature case, we deduce that the rolling $(R)$ is completely controllable if and only if the holonomy group of $(M,g)$ is equal to $SO(n)$. In the nonzero curvature case, we prove that the structure group of the principal bundle can be realized as the holonomy group of a connection on $TM\oplus \R$, that we call the rolling connection. We also show, in the case of positive (constant) curvature, that if the rolling connection is reducible, then $(M,g)$ admits, as Riemannian covering, the unit sphere with the metric induced from the Euclidean metric of $\R^{n+1}$. When the two manifolds are three-dimensional, we provide a complete local characterization of the reachable sets when the two manifolds are three-dimensional and, in particular, we identify necessary and sufficient conditions for the existence of a non open orbit. Besides the trivial case where the manifolds $(M,g)$ and $(\hM,\hg)$ are (locally) isometric, we show that (local) non controllability occurs if and only if $(M,g)$ and $(\hM,\hg)$ are either warped products or contact manifolds with additional restrictions that we precisely describe. Finally, we extend the two types of rolling to the case where the manifolds have different dimensions.
The Vertex Reinforced Jump Process and a Random Schr\"odinger operator on finite graphs

We introduce a new exponential family of probability distributions, which can be viewed as a multivariate generalization of the Inverse Gaussian distribution. Considered as the potential of a random Schr\"odinger operator, this exponential family is related to the random field that gives the mixing measure of the Vertex Reinforced Jump Process (VRJP), and hence to the mixing measure of the Edge Reinforced Random Walk (ERRW), the so-called magic formula. In particular, it yields by direct computation the value of the normalizing constants of these mixing measures, which solves a question raised by Diaconis. The results of this paper are instrumental in [Sabot-Zeng,2015], where several properties of the VRJP and the ERRW are proved, in particular a functional central limit theorem in transient regimes, and recurrence of the 2-dimensional ERRW.
Symmetry and resonant modes in platonic grating stacks

We study the flexural wave modes existing in finite stacks of gratings containing rigid, zero-radius pins. We group the modes into even and odd classes, and derive dispersion equations for each. We study the recently discovered elasto-dynamically inhibited transmission (EDIT) phenomenon, and relate it to the occurrence of trapped waves of even and odd symmetries being simultaneously resonant. We show how the EDIT interaction may be steered over a wide range of frequencies and angles, using a strategy in which the single-grating reflectance is kept high, so enabling the quality factors of the even and odd resonances to be kept large.
Physarum Dynamics and Optimal Transport for Basis Pursuit

We study the connections between Physarum Dynamics and Dynamic Monge Kantorovich (DMK) Optimal Transport algorithms for the solution of Basis Pursuit problems. We show the equivalence between these two models and unveil their dynamic character by showing existence and uniqueness of the solution for all times and constructing a Lyapunov functional with negative Lie-derivative that drives the large-time convergence. We propose a discretization of the equation by means of a combination of implicit time-stepping and Newton method yielding an efficient and robust method for the solution of general basis pursuit problems. Finally, we propose a simple modification to the dynamic equation that can be interpreted as a gradient flow system for the Lyapunov functional. Several numerical experiments run on literature benchmark problems are used to show the accuracy, efficiency, and robustness of the proposed method.
RTT realization of quantum affine superalgebras and tensor products

We use the RTT realization of the quantum affine superalgebra associated with the Lie superalgebra $\mathfrak{gl}(M,N)$ to study its finite-dimensional representations and their tensor products. In the case $\mathfrak{gl}(1,1)$, the cyclicity condition of tensor products of finite-dimensional simple modules is determined completely in terms of zeros and poles of rational functions. This in turn induces cyclicity of some particular tensor products of Kirillov-Reshetikhin modules related to $\mathfrak{gl}(M,N)$.
Numerical Study on Heat Reduction of Various Counterflowing Jets over Highly Blunt Cone in Hypersonic Flow

In this paper, the effectiveness of counterflowing jets as heat-reduction devices for large-angle blunt cones flying at hypersonic Mach numbers is numerically simulated with various coolant jets. Different jet conditions have been chosen to investigate the effect of the counterflow jet on the surrounding flow field of nose cone. The compressible, unsteady, axisymmetric Navier-Stokes equations are solved with SST turbulence model for free stream Mach number of 5.75 at 0° angle of attack with and without gas injection. The coolant gas (air, Carbon Dioxide, and helium) is chosen to inject into the hypersonic flow at the nose of the model. The numerical results presented the surface heat reduction for different coolant jets. According to the investigation of various conditions of opposing jets, important phenomena of flow field and some effective jet conditions are found.
Development and validation of a tokamak skin effect transformer model

A lumped parameter, state space model for a tokamak transformer including the slow flux penetration in the plasma (skin effect transformer model) is presented. The model does not require detailed or explicit information about plasma profiles or geometry. Instead, this information is lumped in system variables, parameters and inputs. The model has an exact mathematical structure built from energy and flux conservation theorems, predicting the evolution and non-linear interaction of plasma current and internal inductance as functions of the primary coil currents, plasma resistance, non-inductive current drive and the loop voltage at a specific location inside the plasma (equilibrium loop voltage). Loop voltage profile in the plasma is substituted by a three-point discretization, and ordinary differential equations are used to predict the equilibrium loop voltage as a function of the boundary and resistive loop voltages. This provides a model for equilibrium loop voltage evolution, which is reminiscent of the skin effect. The order and parameters of this differential equation are determined empirically using system identification techniques. Fast plasma current modulation experiments with random binary signals have been conducted in the TCV tokamak to generate the required data for the analysis. Plasma current was modulated under ohmic conditions between 200 and 300 kA with 30 ms rise time, several times faster than its time constant L/R ≈ 200 ms. A second-order linear differential equation for equilibrium loop voltage is sufficient to describe the plasma current and internal inductance modulation with 70% and 38% fit parameters, respectively. The model explains the most salient features of the plasma current transients, such as the inverse correlation between plasma current ramp rates and internal inductance changes, without requiring detailed or explicit information about resistivity profiles. This proves that a lumped parameter modelling approach can be used to predict the time evolution of bulk plasma properties such as plasma inductance or current with reasonable accuracy; at least under ohmic conditions without external heating and current drive sources.
Convergence with probability one of stochastic approximation algorithms whose average is cooperative

We consider a stochastic approximation process where F : m m is a C 2 irreducible cooperative dissipative vector field, n n 0 is a sequence of positive numbers decreasing to 0 and Un n 0 a sequence of uniformly bounded m martingale differences. We show that under certain conditions on n and Un the sequence xn n 0 converges with probability one toward the equilibria set of the vector field F .
Random walks on randomly oriented lattices

Simple random walks on various types of partially horizontally oriented regular lattices are considered. The horizontal orientations of the lattices can be of various types (deterministic or random) and depending on the nature of the orientation the asymptotic behaviour of the random walk is shown to be recurrent or transient. In particular, for randomly horizontally oriented lattices the random walk is almost surely transient.