Analytic Functions Of Complex Research Articles

Lê numbers and Newton diagram

We give an algorithm to compute the Lê numbers of (the germ of) a Newton non-degenerate complex analytic function f:(Cn,0)→(C,0) in terms of certain invariants attached to the Newton diagram of the function f+z1α1+⋯+zdαd, where d is the dimension of the critical locus of f and α1,…,αd are sufficiently large integers. This is a version for non-isolated singularities of a famous theorem of A.G. Kouchnirenko. As a corollary, we obtain that Newton non-degenerate functions with the same Newton diagram have the same Lê numbers.

Three-webs

Local differential-geometric properties of three-webs formed on a -dimensional manifold by foliations of codimension , and , respectively, are considered. In particular, three-webs defined by complex analytic functions of complex arguments belong to this class of webs. The structure equations of a three-web in an adapted co-frame (in particular, in a natural co-frame) are deduced; the canonical connection on the manifold of a web is introduced; formulae are obtained to calculate (in a natural co-basis) the components of the first structure tensor of a three-web in terms of the derivatives of the function of this web. Three special classes of three-webs are considered in detail: regular and group three-webs and also three-webs generated by holomorphic functions. Bibliography: 17 titles.

The dimension of the moduli spaces of curves defined by topologically non quasi-homogeneous functions

We consider a topological class of a germ of complex analytic function in two variables which does not belong to its jacobian ideal. Such a function is not quasi homogeneous. The 0-level of such a function defines a germ of analytic curve. Proceeding similarly to the homogeneous case Genzmer and Paul (2011) and the quasi homogeneous case Genzmer and Paul (2016), we describe an algorithm which computes the dimension of the generic strata of the local moduli space of curves.

Uniform parameterization of subanalytic sets and diophantine applications

We prove new parameterization theorems for sets definable in the structure $\mathbb{R}_{an}$ (i.e. for globally subanalytic sets) which are uniform for definable families of such sets. We treat both $C^r$-parameterization and (mild) analytic parameterization. In the former case we establish a polynomial (in $r$) bound (depending only on the given family) for the number of parameterizing functions. However, since uniformity is impossible in the latter case (as was shown by Yomdin via a very simple family of algebraic sets), we introduce a new notion, analytic quasi-parameterization (where many-valued complex analytic functions are used), which allows us to recover a uniform result. We then give some diophantine applications motivated by the question as to whether the $H^{o(1)}$ bound in the Pila-Wilkie counting theorem can be improved, at least for certain reducts of $\mathbb{R}_{an}$. Both parameterization results are shown to give uniform $(\log H)^{O(1)}$ bounds for the number of rational points of height at most $H$ on $\mathbb{R}_{an}$-definable Pfaffian surfaces. The quasi-parameterization technique produces the sharper result, but the uniform $C^r$-parametrization theorem has the advantage of also applying to $\mathbb{R}_{an}^{pow}$-definable families.

On the growth behaviour of Hironaka quotients

We consider a finite analytic morphism φ = (f, g) : (X, p) −→ (C 2 , 0) where (X, p) is a complex analytic normal surface germ and f and g are complex analytic function germs. Let π : (Y, E Y) → (X, p) be a good resolution of φ with exceptional divisor E Y = π −1 (p). We denote G(Y) the dual graph of the resolution π. We study the behaviour of the Hironaka quotients of (f, g) associated to the vertices of G(Y). We show that there exists maximal oriented arcs in G(Y) along which the Hironaka quotients of (f, g) strictly increase and they are constant on the connected components of the closure of the complement of the union of the maximal oriented arcs.

Resolution of singularities and geometric proofs of the Łojasiewicz inequalities

The Lojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). In this article, we first give an elementary geometric, coordinate-based proof of the Lojasiewicz inequalities in the special case where the function is $C^1$ with simple normal crossings. We then prove, partly following Bierstone and Milman (1997) and using resolution of singularities for real analytic varieties, that the gradient inequality for an arbitrary real or complex analytic function follows from the special case where it has simple normal crossings. In addition, we prove the Lojasiewicz inequalities when a function is $C^N$ and generalized Morse-Bott of order $N \geq 3$; we gave an elementary proof of the Lojasiewicz inequalities when a function is $C^2$ and Morse-Bott in arXiv:1708.09775v4 (finite-dimensional case) and arXiv:1706.09349 (infinite-dimensional case).

Certain basic information related to the Tremblay operator and some applications in connection therewith

AbstractIn this scientific note, an operator, which is the well-known Tremblay operator in the literature, is first introduced and some of its applications to certain analytic complex functions, which are normalized and analytic in the open unit disk, are then determined. In addition, certain special results of the related applications are also emphasized.

Interesting identities involving weighted representations of integers as sums of arbitrarily many squares

We consider the number of ways to write an integer as a sum of squares, a problem with a long history going back at least to Fermat. The previous studies in this area generally fix the number of squares which may occur and then either use algebraic techniques or connect these to coefficients of certain complex analytic functions with many symmetries known as modular forms, from which one may use techniques in complex and real analysis to study these numbers. In this paper, we consider sums with arbitrarily many squares, but give a certain natural weighting to each representation. Although there are a very large number of such representations of each integer, we see that the weighting induces massive cancellation, and we furthermore prove that these weighted sums are again coefficients of modular forms, giving precise formulas for them in terms of sums of divisors of the integer being represented.

An efficient adaptive forward–backward selection method for sparse polynomial chaos expansion

As an efficient uncertainty quantification (UQ) methodology for moment propagation and probability analysis of quantities of interest, polynomial chaos (PC) expansions have received broad and sustained attentions. However, the exponentially increasing cost of building PC representations with increasing dimension of uncertainty, i.e., the curse of dimensionality, seriously restricts the practical application of PC at the industrial level. Some efficient strategies applying adaptive basis selection algorithm for sparse optimization (or l1-minimization) of PC show great potential compared to the classical full PC. However, these strategies mainly focus on forward selection algorithms, which are incapable of correcting any error made by these algorithms. Hence, this paper develops a novel adaptive forward–backward selection (AFBS) algorithm for reconstructing sparse PC. The proposed algorithm by a reasonable combination of forward selection and adaptively backward elimination technique can efficiently correct mistakes made by earlier forward selection steps, which retains the most significant PC terms and discards redundant or insignificant ones. The accuracy of built PC metamodel is checked by a cross-validation procedure. As a consequence, the most significant PC terms are detected sequentially and corresponding PC coefficients are accurately recovered. It largely enhances the sparsity of PC and improves the prediction accuracy compared to the popular forward selection algorithms, e.g., least angle regressions (LARs). To validate the efficiency of the proposed algorithm, a complex analytical function with Gaussian distribution inputs and two challenging aerodynamic applications including a sonic boom propagation analysis considering atmospheric uncertainty and a natural-laminar-flow (NLF) airfoil computation under both geometrical and operational uncertainties are elaborated. With an in-depth comparison with some popular PC reconstruction methodologies, the performance of the devised AFBS method is assessed comprehensively. The results demonstrate that the proposed AFBS method can efficiently identify the significant PC contributions describing the problems, and reproduce sparser PC metamodel and more accurate UQ results than those classical full PC and LARs methods.

Solution for bending of a polygonal plate with cracks by the linear conjugate method

Stress–strain state of a polygonal multiply connected plate under bending is considered. The plate under the action of different force factors (bending moment M and uniformly distributed load of intensity q) is also carried out. Similar problems for a simple configuration plates were solved by different authors. But the bending problem for a plate with different length through cracks is considered first. Using the theory of a complex variable function and analytic functions (Kolosov-Muskheleshvili potential) the problem is solved by the linear conjugate method. Such plates (slabs) are widely used in shipbuilding (hull-deck of ships), in construction (interfloor slabs, balcony flooring slabs) in mechanical engineering, aircraft building, etc. Therewith, at first boundary condition for an entire plate (without cracks) under the action of the applied loads, (M and q) and then a problem on bending of a plate with cracks under the action of loads applied to crack faces and that are determined from the problem on entire plate (using the uniqueness of displacements $$v+iv$$ and uniqueness of the flexure w), are used. The obtained elliptic integrals F(x) and E(x) are calculated by the methods known from references. As a result for determining integral constants $$C_1 ,C_2 ,\ldots ,C_j $$ , we get a system of equations for the considered cases of applied loads. Some numerical examples are solved.

Fréchet Derivative and Analytic Functional Calculus

We calculate Frechet derivatives of the mapping $$a\mapsto f(a)$$, where a is a member of a complex unital Banach algebra and f is a complex analytic function in a neighborhood of the spectrum of a. Thus, the connection between the Frechet derivative and the analytic functional calculus is established.

Global Optimization for Quasi-Noncyclic Relatively Nonexpansive Mappings with Application to Analytic Complex Functions

The purpose of this article is to resolve a global optimization problem for quasi-noncyclic relatively nonexpansive mappings by giving an algorithm that determines an optimal approximate solution of the following minimization problem, min x ∈ A d ( x , T x ) , min y ∈ B d ( y , T y ) and min ( x , y ) ∈ A × B d ( x , y ) ; also, we provide some illustrative examples to support our results. As an application, the existence of a solution of the analytic complex function is discussed.

A functionally-analytic method for modelling axial-symmetric flows of ideal fluid

Abstract We consider axial-symmetric stationary flows of the ideal incompressible fluid as an important case of potential solenoid fields. We use an integral expression of the Stokes flow function via the corresponding complex analytic function for solving a boundary value problem with respect to a steady streamline of the ideal incompressible fluid along an axial-symmetric body. We describe the solvability of the problem in terms of the singularities of the mentioned complex analytic function. The obtained results are illustrated by concrete examples of modelling of steady axial-symmetric flows.

Moduli spaces of a family of topologically non quasi-homogeneous functions

We consider a topological class of a germ of complex analytic function in two variables which does not belong to its jacobian ideal. Such a function is not quasi homogeneous. Each element f in this class induces a germ of foliation (df = 0). Proceeding similarly to the homogeneous case [2] and the quasi homogeneous case [3] treated by Genzmer and Paul, we describe the local moduli space of the foliations in this class and give analytic normal forms. We prove also the uniqueness of these normal forms.

Local topological algebraicity with algebraic coefficients of analytic sets or functions

We prove that any complex or real analytic set or function germ is topologically equivalent to a germ defined by polynomial equations whose coefficients are algebraic numbers.

Two Elementary Analytic Functions and Their Relationship with Hardy and Bergman Spaces

We prove that the function f(z)=1/(1-z) which is holomorphic in the open unit disc centered at the origin, is an element of a Hardy space H^p if and only if p 0}. Although the assertions claimed above refer to complex analytic functions, and the involved spaces are function spaces of analytic complex functions, the proofs from below are based on results and methods of real analysis

A nullstellensatz for linear partial differential equations with polynomial coefficients

In this paper an equation means a homogeneous linear partial differential equation in $n$ unknown functions of $m$ variables which has real or complex polynomial coefficients. The solution set consists of all $n$-tuples of real or complex analytic functions that satisfy the equation. For a given system of equations we would like to characterize its Weyl closure, i.e. the set of all equations that vanish on the solution set of the given system. It is well-known that in many special cases the Weyl closure is equal to $B_m(F)N \cap A_m(F)^n$ where $F$ is either the field of real or complex numbers, $A_m(F)$ (respectively $B_m(F)$) consists of all linear partial differential operators with coefficients in $F[x_1,\ldots,x_m]$ (respectively $F(x_1,\ldots,x_m)$) and $N$ is the submodule of $A_m(F)^n$ generated by the given system. Our main result is that this formula holds in general. In particular, we do not assume that the module $A_m(F)^n/N$ has finite rank which used to be a standard assumption. Our approach works also for the real case which was not possible with previous methods. Moreover, our proof is constructive as it depends only on the Riquier-Janet theory.

New Trace Formulas in Terms of Resonances for Three-Dimensional Schrödinger Operators

We consider the Schrodinger operator −Δ+V (x) in L2(R3) with a real shortrange (integrable) potential V. Using the associated Fredholm determinant, we present new trace formulas, in particular, on expressed in terms of resonances and eigenvalues only. We also derive expressions of the Dirichlet integral, and the scattering phase. The proof is based on a change of view the point for the above mentioned problems from that of operator theory to that of complex analytic (entire) function theory.

A High-Order Method for Evaluating Derivatives of Harmonic Functions in Planar Domains

We propose a high-order integral equation based method for evaluating interior and boundary derivatives of harmonic functions that are specified by their Dirichlet data in planar domains. The tangential derivative of the given Dirichlet data is used to form a complementary Neumann problem, whose solution is a harmonic conjugate of the function whose derivatives we seek. We use a high-order Nyström method to compute the Dirichlet trace of the harmonic conjugate on the domain boundary. The tangential derivative of this harmonic conjugate, effected via an FFT, is the normal derivative of the original function. Because the original and conjugate harmonic functions are the real and imaginary parts of a complex analytic function, we are able to use Cauchy's integral formulas to compute function values and derivatives inside the domain. Several numerical experiments, on smooth domains and domains with corners, illustrate the rapid convergence and high accuracy of the proposed approach.

Finding Zeros of Analytic Functions and Local Eigenvalue Analysis Using Contour Integral Method in Examples

A numerical method for computing zeros of analytic complex functions is presented. It relies on Cauchy's residue theorem and the method of Newton's identities, which translates the problem to finding zeros of a polynomial. In order to stabilize the numerical algorithm, formal orthogonal polynomials are employed. At the end the method is adapted to finding eigenvalues of a matrix pencil in a bounded domain in the complex plane. This work is based on a series of papers of Professor Sakurai and collaborators. Our aim is to make their work available by means of a systematic study of properly chosen examples.