Search

In this paper, we present a new method for isolating real roots of a bivariate nonlinear system. It is a subdivision method based on analyzing the local geometrical properties of the given system. We propose the concept of opposite monotone system in a box and use it to determine the existence and the uniqueness of a simple real zero of the system in the box. We have implemented our method and the experiments show the effectivity and efficiency of our approach, especially for high degree sparse polynomial systems. This paper is an extended version of the ISSAC'19 conference paper (Cheng and Wen (2019)). [ABSTRACT FROM AUTHOR]

Let E be an Anderson A -module over C ∞. The period lattice of E is related to its module of special functions by means of a non-canonical isomorphism introduced by the authors in [GM21]. In this paper, we explain how a modification of the inverse map is canonical by interpreting it as a residue morphism along the characteristic graph. This phenomenon has already been observed in various situations. The main innovation of this text is that of costability (costable admissible opens, costable site, etc.) which provides a convenient framework to develop the notion of sheaves of E (C ∞) -valued meromorphic functions on the rigid analytic plane. [ABSTRACT FROM AUTHOR]

Conway's real closed field No of surreal numbers is a sweeping generalization of the real numbers and the ordinals to which a number of elementary functions such as log and exponentiation have been shown to extend. The problems of identifying significant classes of functions that can be so extended and of defining integration for them have proven to be formidable. In this paper we address this and related unresolved issues by showing that extensions to No , and thereby integrals, exist for most functions arising in practical applications. In particular, we show they exist for a large subclass of the resurgent functions , a subclass that contains the functions that at ∞ are semi-algebraic, semi-analytic, analytic, meromorphic, and Borel summable as well as solutions to nonresonant linear and nonlinear meromorphic systems of ODEs or of difference equations. By suitable changes of variables we deal with arbitrarily located singular points. We further establish a sufficient condition for the theory to carry over to ordered exponential subfields of No more generally and illustrate the result with structures familiar from the surreal literature. The extensions of functions and integrals that concern us are constructive in nature, which permits us to work in NBG less the Axiom of Choice (for both sets and proper classes). Following the completion of the positive portion of the paper, it is shown that the existence of such constructive extensions and integrals of substantially more general types of functions (e.g. smooth functions) is obstructed by considerations from the foundations of mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we attempt to resolve the singularities of the zero variety of a C ∞ function of two variables as much as possible by using ordinary blowings up. As a result, we formulate an algorithm to locally express the zero variety in the "almost" normal crossings form, which is close to the normal crossings form but may include flat functions. As an application, we investigate analytic continuation of local zeta functions associated with C ∞ functions of two variables. As is well known, the desingularization theorem of Hironaka implies that the local zeta functions associated with real analytic functions admit the meromorphic continuation to the whole complex plane. On the other hand, it is recently observed that the local zeta function associated with a specific (non-real analytic) C ∞ function has a singularity different from the pole. From this observation, the following questions are naturally raised in the C ∞ case: how wide the meromorphically extendible region can be and what kinds of information essentially determine this region? This paper shows that this region can be described in terms of some kind of multiplicity of the zero variety of each C ∞ function. By using our blowings up algorithm, it suffices to investigate local zeta functions in the almost normal crossings case. This case can be effectively analyzed by using real analysis methods; in particular, a van der Corput-type lemma plays a crucial role in the determination of the above region. [ABSTRACT FROM AUTHOR]

For λ ≥ 0 , a C 2 function f defined on the unit disk D is said to be λ -analytic if D z ̄ f = 0 , where D z ̄ is the (complex) Dunkl operator given by D z ̄ f = ∂ z ̄ f − λ (f (z) − f (z ̄)) / (z − z ̄). The aim of the paper is to study several problems on the associated Bergman spaces A λ p (D) and Hardy spaces H λ p (D) for p ≥ 2 λ / (2 λ + 1) , such as boundedness of the Bergman projection, growth of functions, density, completeness, and the dual spaces of A λ p (D) and H λ p (D) , and characterization and interpolation of A λ p (D). [ABSTRACT FROM AUTHOR]

In this note, we provide a corrected proof of Lemma 3.2 in [4]. We also point out some improvements to Remark 5.5 in the same paper. [ABSTRACT FROM AUTHOR]

In this paper, we present ergodicity criteria for 1-Lipschitz functions on Z p , in terms of the van der Put coefficients as well as the inherent data associated with the function. These criteria are applied to provide sufficient conditions for ergodicity of the 1-Lipschitz p -adic functions with special features, such as everywhere/uniform differentiability with respect to the Mahler expansion. In particular, the ergodicity criteria are obtained for certain 1-Lipschitz functions on Z 2 and Z 3 , which are known as B -functions, in terms of the Mahler and van der Put expansions. These functions are locally analytic functions of order 1 (and therefore contain polynomials). For arbitrary primes p ≥ 5 , an ergodicity criterion of B -functions on Z p is introduced, which leads to an efficient and practical method of constructing ergodic polynomials on Z p that realize a given unicyclic permutation modulo p. Thus, a complete description of ergodic polynomials modulo p μ , which are reduced from all ergodic B -functions on Z p , is provided where μ = μ (p) = 3 for p ∈ { 2 , 3 } and μ = 2 for p ≥ 5. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the global well-posedness of solutions to magnetohydrodynamics (MHD) boundary layer equations in analytic function spaces. When the initial data is a small perturbation around a selected profile, and such a profile is governed by an one dimensional heat equation with a source term, we establish the global in time existence and uniqueness of analytic solutions to the two dimensional (2D) MHD boundary layer equations. It is noted that the far-field state of velocity is not required to be small initially, but decays to zero as time tends to infinity with suitable decay rates. The whole analysis is divided into two parts: When the initial far-field states are not small, we reformulate the original problem into a small perturbation problem by extracting a suitable background profile which is governed by a heat equation, then we prove a long-time existence of solutions, and the lower bound of lifespan of solutions is given in term of the small parameter of initial perturbation; After establishing the long-time existence, combining the decay properties of far-field state and the properties of solutions to the heat equation, it is shown that both the solution obtained in the first part and the background profile indeed satisfy some smallness requirements when time goes to the lower bound of lifespan of solutions. Then we can show the global existence of solutions after this moment. [ABSTRACT FROM AUTHOR]

In this paper we prove uniqueness in the inverse boundary value problem for quasilinear elliptic equations whose linear part is the Laplacian and nonlinear part is the divergence of a function analytic in the gradient of the solution. The main novelty in terms of the result is that the coefficients of the nonlinearity are allowed to be "anisotropic". As in previous works, the proof reduces to an integral identity involving the tensor product of the gradients of 3 or more harmonic functions. Employing a construction method using Gaussian quasi-modes, we obtain a convenient family of harmonic functions to plug into the integral identity and establish our result. [ABSTRACT FROM AUTHOR]

The potential-energy and quadrupole-moment functions of the H 2 ground electronic state are well known in literature (Komasa et al., 2019; Wolniewicz et al., 1998), and the line list of the vibrational–rotational transitions was calculated (Roueff et al., 2019). In this paper, we analyze the calculated intensities in order to learn how the intensities will change when analytic quadrupole-moment functions fitted to the ab initio and experimental data are used instead of spline-interpolated functions. We found that the use of splines does not deteriorate the intensities and does not lead to nonphysical saturation, as in heavier molecules, owing to the high precision of the ab initio data and the high density of the grid. The accuracy of the calculated intensities is estimated up to high overtones. Extraction of new spectroscopic information from the observational data that supplements the laboratory measurements is performed. The laboratory and observational data do not help increase the quality of the analytic functions. Numerous anomalies resulting from the destructive interference are identified in the calculated line lists, some of them being situated within the recently observed spectral regions, 1.5– 2. 5 μ m. The intensities of these anomalies can be sensitive to the form of the molecular functions as well as to the proton-to-electron mass ratio. In this connection, the similar Le Roy anomalies (Brown and LeRoy, 1973; Le Roy and Vrscay, 1975) also arising due to the destructive interference in the Lyman and Werner systems are discussed. [Display omitted] • The use of splines does not deteriorate the intensities. • New spectroscopic information from the observational data. • Numerous anomalies due to destructive interference are identified in the calculated line lists. • Similar Le Roy's anomalies in the Lyman and Werner systems are discussed. • The anomalies can be sensitive to the proton-to-electron mass ratio. [ABSTRACT FROM AUTHOR]

This is a sequel to the paper [4] , in which we introduced Drinfeld modular polynomials of higher rank, using an analytic construction. These polynomials relate the isomorphism invariants of Drinfeld F q [ T ] -modules of rank r ≥ 2 linked by isogenies of a specified type. In the current paper, we give an algebraic construction of greater generality, and prove a generalization of the Kronecker congruences relations, which describe what happens when modular polynomials associated to P -isogenies are reduced modulo a prime P . We also correct an error in [4] . [ABSTRACT FROM AUTHOR]

Let ϕ be an analytic self-map of the n -ball, having 0 as the attracting fixed point and having full-rank near 0. We consider the generalized Schröder's equation, F ∘ ϕ = ϕ ′ ( 0 ) k F with k a positive integer and prove there is always a solution F with linearly independent component functions, but that such an F cannot have full rank except possibly when k = 1 . Furthermore, when k = 1 (Schröder's equation), necessary and sufficient conditions on ϕ are given to ensure F has full rank near 0 without the added assumption of diagonalizability as needed in the 2003 Cowen/MacCluer paper. In response to Enoch's 2007 paper, it is proven that any formal power series solution indeed represents an analytic function on the whole unit ball. How exactly resonance can lead to an obstruction of a full rank solution is discussed as well as some consequences of having solutions to Schröder's equation. [ABSTRACT FROM AUTHOR]

A family of Gaussian analytic functions (GAFs) has recently been linked to the Gabor transform of Gaussian white noises [4]. This answered pioneering work by Flandrin [10] , who observed that the zeros of the Gabor transform of white noise had a regular distribution and proposed filtering algorithms based on the zeros of a spectrogram. In this paper, we study in a systematic way the link between GAFs and a class of time-frequency transforms of Gaussian white noises. Our main observation is a correspondence between pairs (transform, GAF) and generating functions for classical orthogonal polynomials. This covers some classical time-frequency transforms, such as the Gabor transform and the Daubechies-Paul wavelet transform. It also unveils new windowed discrete Fourier transforms, which map white noises to fundamental GAFs. Moreover, we discuss subtleties in defining a white noise and its transform on infinite dimensional Hilbert spaces and its finite dimensional approximations. [ABSTRACT FROM AUTHOR]

We show a general method allowing the solution calculation, in the form of a power series, for a very large class of nonlinear Ordinary Differential Equations (ODEs), namely the real analytic σπ -ODEs (and, more in general, the real analytic σπ - reducible ODEs) in many indeterminates, characterized by an ODE-function given by generalized polynomials of the indeterminates and their derivatives, i.e. functions formally polynomial with exponents, though the exponent can be any real number, and whose coefficients are analytic time functions. The solution method consists in reducing the ODE to a certain canonical homogeneous quadratic ODE, named driver-type Riccati equation with a larger number of indeterminates, whose solutions include, as sub-solutions, the original solutions, and for which a recursive formula is shown, giving all coefficients of the solution power series directly from the ODE parameters. The reduction method is named exact quadratization and was formerly introduced in another our article, where we considered explicit ODEs only. In the present paper, which is self-contained to a large extent, we review and complete the theory of exact quadratization by solving issues, such as for instance the existence of a piecewise-quadratization, that had remained open, and also extend it to the more general case of an implicitly defined ODE. Finally, we argue that the result can be seen as a partial solution of a differential version of the 22nd Hilbert's problem. [ABSTRACT FROM AUTHOR]

We study metric properties of sets which are characterized by their approximation properties by smooth curves at scales of positive density and their relationships with known problems in complex dynamics and analysis. Three direct applications towards removability for bounded analytic functions, optimal bounds for Traveling Salesman problem, and Hausdorff dimension of unfolding attractors are discussed in the paper. The main technical ingredient is a construction, for every continuum K , of a finite Radon measure of bounded Menger curvature such that on every ball B (x , r) , x ∈ K , the measure is bounded by a universal constant multiple of r exp ⁡ (− g (x , r)) , where g (x , r) ≥ 0 is an explicit function. Theorem A is the main analytical result of the paper and provides a precise estimate on the growth of the function g (x , r) in terms of Jones' β numbers. Its dimension theory counterpart is stated as Theorems 1 and 2 and yields sharp estimates of the Hausdorff dimension of compacta in terms of asymptotic properties of log ⁡ g (x , r) / log ⁡ r for all x ∈ K except for a set of finite linear measure. [ABSTRACT FROM AUTHOR]

In a previous paper [31] we established the complex analytic manifold theory of the BMO-Teichmüller space. In this paper we identify the function space which is the tangent space to the BMO-Teichmüller space. [ABSTRACT FROM AUTHOR]

In a recent paper, Flandrin [16] proposed filtering based on the zeros of a spectrogram with Gaussian window. His results are based on empirical observations on the distribution of the zeros of the spectrogram of white Gaussian noise. These zeros tend to be uniformly spread over the time–frequency plane, and not to clutter. Our contributions are threefold: we rigorously define the zeros of the spectrogram of continuous white Gaussian noise, we explicitly characterize their statistical distribution, and we investigate the computational and statistical underpinnings of the practical implementation of signal detection based on the statistics of the zeros of the spectrogram. The crux of our analysis is that the zeros of the spectrogram of white Gaussian noise correspond to the zeros of a Gaussian analytic function, a topic of recent independent mathematical interest [24]. [ABSTRACT FROM AUTHOR]

In this paper, we study a question proposed by Marques and Moreira about transcendental entire functions mapping Q into itself. In particular, we prove that there is no a transcendental entire function f such that f (Q) ⊆ Q and the denominator of f (p / q) is O (q) , for all rational number p / q , with q sufficiently large. [ABSTRACT FROM AUTHOR]

By the Voronin theorem, the set of shifts of the Riemann zeta-function ζ (s + i τ) , s = σ + i t , τ ∈ R , that approximate any given non-vanishing analytic function defined on { s ∈ C : 1 2 < σ < 1 } has a positive lower density. In the paper, it is proved that the same property of the above shifts remains valid in the intervals [ T , T + H ] with T 1 / 3 (log ⁡ T) 26 / 15 ⩽ H ⩽ T. [ABSTRACT FROM AUTHOR]

• Flexible ring on the two-dimensional elastic foundation; • updated analytical stiffness of the curved sidewall beam; • grounding contact with with the interative numerative method. This paper presents a complete tire model for heavy-load radial tires with a large section ratio that is based on a flexible carcass ring model, combined with a sidewall element with two-dimensional stiffness. It also establishes a two-dimensional analytical model of sidewall stiffness for heavy-duty tires based on preloaded beam deformation theory. The flexible ring model is used to examine the ground-contact modeling of heavy load tires and their stiffness characteristics are analyzed. Experimental and theoretical results show that flexible ring tire models based on nonlinear elastic foundations can accurately characterize the grounding mark of tires and their stiffness characteristics, with there being a good fit between the theoretical and experimental results. The analytic two-dimensional function for curved sidewalls developed in the paper can also be used to enrich other tire models, such as FTire and CDTire. [ABSTRACT FROM AUTHOR]

This work is the first, and main, of a series of papers in progress dedicated to Nijenhuis operators, i.e., fields of endomorphisms with vanishing Nijenhuis torsion. It serves as an introduction to Nijenhuis Geometry that should be understood in much wider context than before: from local description at generic points to singularities and global analysis. The goal of the present paper is to introduce terminology, develop new important techniques (e.g., analytic functions of Nijenhuis operators, splitting theorem and linearisation), summarise and generalise basic facts (some of which are already known but we give new self-contained proofs), and more importantly, to demonstrate that the research programme proposed in the paper is realistic by proving a series of new, not at all obvious, results. [ABSTRACT FROM AUTHOR]

Abstract: The classical -logarithm is a multivalued analytic function defined inductively: In this paper we give a simple explicit construction of the Grassmannian -logarithm, which is a multivalued analytic function on the quotient of the Grassmannian of -dimensional subspaces in in generic position to the coordinate hyperplanes by the natural action of the torus . The classical -logarithm appears at a certain one dimensional boundary stratum. We study Tate iterated integrals, which are homotopy invariant integrals of 1-forms where are rational functions. We give a simple explicit formula for the Tate iterated integral which describes the Grassmannian -logarithm. Another example is the Tate iterated integrals for the multiple polylogarithms on the moduli spaces , calculated in Section 4.4 of Goncharov (2005) [13] using the combinatorics of plane trivalent trees decorated by the arguments of the multiple polylogarithms. Variations of mixed Hodge–Tate structures on are described by a Hopf algebra . We upgrade Tate iterated integrals on a (rational) complex variety to elements of . The coproducts of these elements are very interesting invariants of the iterated integrals. In general their calculation is a non-trivial problem. We show however, that working modulo the ideal of generated by constant variations, there is a simple way to calculate the coproduct. It is a pleasure to dedicate this paper to Andrey Suslin, whose works Suslin (1984) [21] and Suslin (1991) [22] played an essential role in the development of the story. [Copyright &y& Elsevier]

Abstract: In this paper, we study the Gevrey regularity of spatially homogeneous Boltzmann equation without angular cutoff. We prove the propagation of Gevrey regularity for solutions with the Maxwellian decay to the Cauchy problem of spatially homogeneous Boltzmann equation. The idea we use here is based on the framework of Morimoto–Ukaiʼs recent paper (see [Y. Morimoto, S. Ukai, Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff, J. Pseudo-Differ. Oper. Appl. 1 (2010) 139–159]), but we extend the range of the index γ satisfying , and in this case we consider the kinetic factor in the form of instead of as Morimoto and Ukai did before. [Copyright &y& Elsevier]

Abstract: This paper deals with a generating function of the Jacobi polynomials that satisfies the following properties (I) and (II). (I) The generating function is the kernel of an integral operator that is unitary. (II) The image of the unitary operator is a reproducing kernel Hilbert space of analytic functions and the reproducing kernel is given as a special value of the generating function above. A generating function that satisfies (I) is given in Watanabe (1998) . The purpose of this paper is to give a generating function that satisfies (I) and (II). From a group theoretical point of view, a similar construction for zonal spherical functions is given in Watanabe (2006) . [Copyright &y& Elsevier]

Abstract: Analytic signals of finite energy in signal analysis are identical with non-tangential boundary limits of functions in the related Hardy spaces. With this identification this paper studies a subclass of the analytic signals that, with the amplitude-phase representation , , satisfy the relation a.e., signals in this subclass are called mono-components, and, in that case, the phase derivative is called the analytic instantaneous frequency of s. This paper proves that when , where is real-valued, band-limited with minimal bandwidth B and is real-valued, as the restriction on the real line of some entire function, then s is an analytic signal if and only if is a linear function, and with there holds . In the case s is a mono-component. This generalizes the corresponding result obtained by Xia and Cohen in 1999 in which is assumed to be a real-valued polynomial. [Copyright &y& Elsevier]

Abstract: In a recent paper (Buium et al., 2011 ), Buium et al. proved that f is a locally analytic function from the p-adic integers, to itself if and only if it is written as a restricted power series over with finitely many iterates of the Fermat quotient operator. In this paper we establish the function field analogs of their result. In addition, we deduce Wagnerʼs result, which is the function field analog of Mahlerʼs result for continuous functions on . [Copyright &y& Elsevier]

Abstract: The goal of this paper is to study operators of the form, where γ is a real analytic function defined on a neighborhood of the origin in , satisfying , ψ is a cutoff function supported near , and K is a “multi-parameter singular kernel” supported near . A main example is when K is a “product kernel.” We also study maximal operators of the form, We show that is bounded on (). We give conditions on γ under which T is bounded on (); these conditions hold automatically when K is a Calderón–Zygmund kernel. This is the final paper in a three part series. The first two papers consider the more general case when γ is . [Copyright &y& Elsevier]

Abstract This paper is concerned with the derivative nonlinear Schrödinger equation with periodic boundary conditions i u t + u x x + i (f (x , u , u ¯)) x = 0 , x ∈ T : = R / 2 π Z , where f is an analytic function of the form f (x , u , u ¯) = μ | u | 2 u + f ≥ 4 (x , u , u ¯) , 0 ≠ μ ∈ R , and f ≥ 4 (x , u , u ¯) denotes terms of order at least four in u , u ¯. We show the above equation possesses Cantor families of smooth quasi-periodic solutions of small amplitude. The proof is based on an infinite dimensional KAM theorem for unbounded perturbation vector fields. [ABSTRACT FROM AUTHOR]

Abstract Inverse scattering has been an active research area for the past thirty years. While very successful in many cases, progress has lagged when only limited-aperture measurement is available. In this paper, we perform some elementary study to recover data that can not be measured directly. In particular, we aim at recovering the full-aperture far field data from limited-aperture measurement. Due to the reciprocity relation, the multi-static response matrix (MSR) has a symmetric structure. Using the Green's formula and single layer potential, we propose two schemes to recover full-aperture MSR. The recovered data is tested by a recently proposed direct sampling method and the factorization method. Numerical results show that it is possible to, at least, partially recover the missing data and consequently improve the reconstruction of the scatterer. Highlights • Recovery of the full-aperture data from limited-aperture data in inverse scattering. • A novel approach to seek an analytic function in a suitable space governing the scattering problem. • Two schemes to recover the full-aperture data using the Green's formula and the single layer potential. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces H s , s > 3 / 2 for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Moreover, it is shown that the solution map for the generalized Camassa–Holm equation is Hölder continuous in H r -topology. Finally, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems (Helton et al. (2009) , de Oliviera et al. (2009) ). In the earlier paper (Helton et al. (2009) ) we characterized NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call “NC ball maps”. In this paper we turn to a more general dimension-free ball , called a “pencil ball”, associated with a homogeneous linear pencil For , define and let We study the generalization of NC ball maps to these pencil balls , and call them “pencil ball maps”. We show that every has a minimal dimensional (in a certain sense) defining pencil . Up to normalization, a pencil ball map is the direct sum of with an NC analytic map of the pencil ball into the ball. That is, pencil ball maps are simple, in contrast to the classical result of D''Angelo (1993) showing there is a great variety of such analytic maps from to when . To prove our main theorem, this paper uses the results of our previous paper (Helton et al. (2009) ) plus entirely different techniques, namely, those of completely contractive maps. What we do here is a small piece of the bigger puzzle of understanding how Linear Matrix Inequalities (LMIs) behave with respect to noncommutative change of variables. [Copyright &y& Elsevier]

Abstract: Text: In this paper, we prove a generalization of Mertens'' theorem to Beurling primes, namely that , where γ is Euler''s constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit exists. We also show that this limit coincides with ; for ordinary primes this claim is called Meissel''s theorem. Finally, we will discuss a problem posed by Beurling, namely how small can be made for a Beurling prime number system , where P is the rational primes. We prove that for each there exists a Q such that and conjecture that this is the best possible bound. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=Kw3iNo3fAbk/. [Copyright &y& Elsevier]

Abstract There are various methods for fitting data to circles or ellipses in many different types of applied problems. However, the design of such studies is rarely discussed and for the few that do, model errors are commonly assumed to be homoscedastic and uncorrelated. This paper provides an analytic description of the D -optimal designs for estimating parameters in the bivariate Berman model on an arc when errors are correlated and heteroscedastic. We evaluate D -efficiencies and relative efficiencies of the commonly used equidistant sampling methods and show that such designs can be inefficient. [ABSTRACT FROM AUTHOR]

The Borel map j ∞ takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. In the literature, it is well known that the restriction of j ∞ to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space. In this paper, we are interested in studying how large the image of j ∞ is and we investigate the size and the structure of this image by using different approaches (Baire residuality, prevalence and lineability). We give an answer to this question in the very general setting of quasianalytic ultradifferentiable classes defined by weight matrices, which contains as particular cases the classes defined by a single weight sequence or by a weight function. [ABSTRACT FROM AUTHOR]

In this paper, we consider a one-dimensional transmission problem in a bounded domain with a delay in porous-elasticity. Using a semigroup theorem, under suitable assumption on the weight of the delay term, we establish the well-posedness of the system. Furthermore, using the method developed by Z. Liu and S. Zheng we show that the semigroup associated with the dissipative system is analytic and consequently exponentially stable. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the existence of exact analytic solutions to the Shabat equation where μ and q are complex parameters. This equation is the simplest self-similar reduction of so-called dressing chain for constructing and analyzing exactly solvable Schrödinger equation. It is so relevant to the study of the q-oscillator algebra in quantum mechanics. The main aim of this paper is to investigate whether the Shabat equation with has a non-trivial analytic solution under the Brjuno condition. [Copyright &y& Elsevier]

Abstract: In this paper, we establish a decomposition theorem for polyharmonic functions and consider its applications to some Dirichlet problems in the unit disc. By the decomposition, we get the unique solution of the Dirichlet problem for polyharmonic functions (PHD problem) and give a unified expression for a class of kernel functions associated with the solution in the case of the unit disc introduced by Begehr, Du and Wang. In addition, we also discuss some quasi-Dirichlet problems for homogeneous mixed-partial differential equations of higher order. It is worthy to note that the decomposition theorem in the present paper is a natural extension of the Goursat decomposition theorem for biharmonic functions. [Copyright &y& Elsevier]

Abstract: In a recent paper, we generalized Washburn’s analytical solution for capillary flow in a horizontally oriented tube by accounting for a dynamic contact angle. In this paper, we derive solutions for flow in inclined tubes that account for gravity. We again consider two general models for dynamic contact angle: the uncompensated Young force on the contact line depends on the capillary number in the form of (1) a power law with exponent , or (2) a polynomial. A dimensional analysis shows that, aside from the parameters for the model for the uncompensated Young force, the problem is defined through four nondimensional parameters: (1) the advancing equilibrium contact angle, (2) the initial contact angle, (3) a Bond number, and (4) nondimensional liquid pressure at the tube inlet relative to the constant gas pressure. For both contact angle models, we derive analytical solutions for the travel time of the gas–liquid interface as a function of interface velocity. The interface position as a function of travel time can be obtained through numerical integration. For the power law and (an approximation of Cox’s model for dynamic contact angle), we obtain an analytical solution for travel time as a function of interface position, as Washburn did for constant contact angle. Four different flow scenarios may occur: the interface moves (1) upward and approaches the height of capillary rise, (2) downward with the steady-state velocity, (3) downward while approaching the steady-state velocity from an initially higher velocity, or (4) downward while approaching the steady-state velocity from an initially smaller velocity. [Copyright &y& Elsevier]

Abstract: The paper deals with the problem of ideals of : describe increasing functions such that for all bounded analytic functions in the unit disc the condition implies that τ belong to the ideal generated by , i.e. that there exist bounded analytic functions such that . It was proved earlier by the author that the function does not satisfy this condition. The strongest known positive result in this direction due to J. Pau states that the function works. However, there was always a suspicion that the critical exponent at is 1 and not 3/2. This suspicion turned out (at least partially) to be true, 3/2 indeed is not the critical exponent. The main result of the paper is that one can take for φ any function of form , where is a bounded non-increasing function satisfying . In particular any of the functions works. [Copyright &y& Elsevier]

Systems of analytic functions which are simultaneously orthogonal over each of two domains were apparently first studied in particular cases by Walsh and Szego¨, and in full generality by Bergman. In principle, these are very interesting objects, allowing application to analytic continuation that is not restricted (as Weierstrassian continuation via power series) either by circular geometry or considerations of locality. However, few explicit examples are known, and in general one does not know even gross qualitative features of such systems. The main contribution of the present paper is to prove qualitative results in a quite general situation.It is by now very well known that the phenomenon of “double orthogonality” is not restricted to Bergman spaces of analytic functions, nor even indeed has it any intrinsic relation to analyticity; its essence is an eigenvalue problem arising whenever one considers the operator of restriction on a Hilbert space of functions on some set, to a subset thereof, provided this restriction is injective and compact. However, in this paper only Hilbert spaces of analytic functions are considered, especially Bergman spaces. In the case of the Hardy spaces Fisher and Micchelli discovered remarkable qualitative features of doubly orthogonal systems, and we have shown how, based on the classical potential-theoretic notion of balayage, and its modern generalizations, one can deduce analogous results in the Bergman space set-up, but with restrictions imposed on the geometry of the considered domains and measures; these were not needed in the Fisher–Micchelli analysis, but are necessary here as shown by examples.From a more constructive point of view we study the Bergman restriction operator between the unit disk and a compactly contained quadrature domain and show that the representing kernel of this operator is rational and it is expressible (as an inversion followed by a logarithmic derivative) in terms of the polynomial equation of the boundary of the inner domain. [Copyright &y& Elsevier]

We study numerical integration for a weighted Korobov space of analytic periodic functions for which the Fourier coefficients decay exponentially fast. In particular, we are interested in how the error depends on the dimension d. Many recent papers deal with this problem or similar problems and provide matching necessary and sufficient conditions for various notions of tractability. In most cases even simple algorithms are known which allow to achieve these notions of tractability. However, there is a gap in the literature: while for the notion of exponential-weak tractability one knows matching necessary and sufficient conditions, so far no explicit algorithm has been known which yields the desired result. In this paper we close this gap and prove that Korobov lattice rules are suitable algorithms in order to achieve exponential-weak tractability for integration in weighted Korobov spaces of analytic periodic functions. [ABSTRACT FROM AUTHOR]

• IFGF provides an extremely simple new paradigm for operator acceleration. • IFGF is solely based on interpolation of a certain "Analytic Factor". • IFGF does not utilize FFT, translation operators or special functions expansions. • IFGF does not use high-dimensional linear-algebra factorizations or parabolic scaling. • IFGF does not suffer from the parallel "communication bottleneck". This paper presents a novel Interpolated Factored Green Function method (IFGF) for the accelerated evaluation of the integral operators in scattering theory and other areas. Like existing acceleration methods in these fields, the IFGF algorithm evaluates the action of Green function-based integral operators at a cost of O (N log ⁡ N) operations for an N -point surface mesh. The IFGF strategy, which leads to an extremely simple algorithm, capitalizes on slow variations inherent in a certain Green function analytic factor , which is analytic up to and including infinity, and which therefore allows for accelerated evaluation of fields produced by groups of sources on the basis of a recursive application of classical interpolation methods. Unlike other approaches, the IFGF method does not utilize the Fast Fourier Transform (FFT), and is thus better suited than other methods for efficient parallelization in distributed-memory computer systems. Only a serial implementation of the algorithm is considered in this paper, however, whose efficiency in terms of memory and speed is illustrated by means of a variety of numerical experiments—including a 43 min., single-core operator evaluation (on 10 GB of peak memory), with a relative error of 1.5 × 10 − 2 , for a problem of acoustic size of 512 wavelengths. [ABSTRACT FROM AUTHOR]

When the dependence of the function on uncertain variables is non-monotonic in interval, the interval of function obtained by the classic interval extension based on the first order Taylor series will exhibit significant errors. In order to reduce theses errors, the improved format of the interval extension with the first order Taylor series is developed here considering the monotonicity of function. Two typical mathematic examples are given to illustrate this methodology. The vibration of a beam with lumped masses is studied to demonstrate the usefulness of this method in the practical application, and the necessary input data of which are only the function value at the central point of interval, sensitivity and deviation of function. The results of above examples show that the interval of function from the method developed by this paper is more accurate than the ones obtained by the classic method. [ABSTRACT FROM AUTHOR]

It is well known that the Hilbert matrix operator H is a bounded operator from the Bergman space A p into A p if and only if 2 < p < ∞ . In [5] it was shown that the norm of the Hilbert matrix operator H on the Bergman space A p is equal to π sin ⁡ 2 π p , when 4 ≤ p < ∞ , and it was also conjectured that ‖ H ‖ A p → A p = π sin ⁡ 2 π p , when 2 < p < 4 . In this paper we prove this conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we use the multivariate analytic techniques of Pemantle and Wilson to derive asymptotic formulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Then, we apply these results to a generating function encoding information about the stationary distributions of a graph coloring algorithm studied by Butler, Chung, Cummings, and Graham (2015). Historically, Flajolet and Odlyzko (1990) analyzed the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Gao and Richmond (1992) and Hwang (1996, 1998), in both cases by immediately reducing the multivariate case to the univariate case. Pemantle and Wilson (2013) outlined new multivariate analytic techniques and used them to analyze the coefficients of rational generating functions. These multivariate techniques are used here to analyze functions with algebraic singularities. [ABSTRACT FROM AUTHOR]

It has been known for some time that the Green's function of a planar domain can be defined in terms of the exit time of Brownian motion, and this definition has been extended to stopping times more general than exit times. In this paper, we extend the notion of conformal invariance of Green's function to analytic functions which are not injective, and use this extension to calculate the Green's function for a stopping time defined by the winding of Brownian motion. These considerations lead to a new proof of the Riemann mapping theorem. We also show how this invariance can be used to deduce several identities, including the standard infinite product representations of several trigonometric functions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an integrable two-component peakon system, which was proposed by Xia, Qiao and Zhou. We present a precise blow-up scenario and a new blow-up result for strong solutions to the system. Moreover, we prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a measure of the non-univalency of an analytic function, and we use it in order to find the best approximation of analytic function by a locally univalent normalized analytic function. [ABSTRACT FROM AUTHOR]

In this paper we estimate the norm and the essential norm of weighted composition operators from a large class of – non-necessarily reflexive – Banach spaces of analytic functions on the open unit disk into weighted type Banach spaces of analytic functions and Bloch type spaces. We also show the equivalence of compactness and weak compactness of weighted composition operators from these weighted type spaces into a class of Banach spaces of analytic functions, that includes a large family of conformally invariant spaces like BMOA and analytic Besov spaces. [ABSTRACT FROM AUTHOR]

Bohr's phenomenon, first introduced by Harald Bohr in 1914, deals with the largest radius r , 0 < r < 1 , such that the inequality ∑ k = 0 ∞ | a k | r k ≤ 1 holds whenever the inequality | ∑ k = 0 ∞ a k z k | ≤ 1 holds for all | z | < 1 . The exact value of this largest radius known as Bohr's radius , which is r b = 1 / 3 , was discovered long ago. In this paper, we first discuss Bohr's phenomenon for the classes of even and odd analytic functions and for alternating series. Then we discuss Bohr's phenomenon for the class of analytic functions from the unit disk into the wedge domain W α = { w : | arg ⁡ w | < π α / 2 } , 1 ≤ α ≤ 2 . In particular, we find Bohr's radius for this class. [ABSTRACT FROM AUTHOR]