Analytic Functions Of Complex Research Articles

Symplectic transversality and the Pego-Weinstein theory

This paper studies the linear stability problem for solitary wave solutions of Hamiltonian PDEs. The linear stability problem is formulated in terms of the Evans function, a complex analytic function denoted by D(λ), where λ is the spectral parameter. The main result is the introduction of a new factor, denoted Π, in the Pego & Weinstein (1992) derivative formulaD″(0)=χΠdIdc, where I is the momentum of the solitary wave and c is the speed. Moreover this factor turns out to be related to transversality of the solitary wave, modeled as a homoclinic orbit: the homoclinic orbit is transversely constructed if and only if Π≠0. The sign of Π is a symplectic invariant, an intrinsic property of the solitary wave, and is a key new factor affecting the linear stability. The factor χ was already introduced by Bridges & Derks (1999) and is based on the asymptotics of the solitary wave. A supporting result is the introduction of a new abstract class of Hamiltonian PDEs built on a nonlinear Dirac-type equation, which model a wide range of PDEs in applications. Examples where the theory applies, other than Dirac operators, are the coupled mode equation in fluid mechanics and optics, the massive Thirring model, and coupled nonlinear wave equations. A calculation of D″(0) for solitary wave solutions of the latter class is included to illustrate the theory.

Composition of entire and analytic functions in the unit ball

In this paper, we investigate a composition of entire function of several complex variables and analytic function in the unit ball. We modified early known results with conditions providing equivalence of boundedness of $L$-index in a direction for such a composition and boundedness of $l$-index of initial function of one variable, where the continuous function $L:\mathbb{B}^n\to \mathbb{R}_+$ is constructed by the continuous function $l: \mathbb{C}^m\to \mathbb{R}_+.$ Taking into account new ideas from recent results on composition of entire functions, we remove a condition that a directional derivative of the inner function $\Phi$ in the composition does not equal to zero. Instead of the condition we construct a greater function $L(z)$ for which $F(z)=f(\underbrace{\Phi(z),\ldots,\Phi(z)}_{m\text{ times}})$ has bounded $L$-index in a direction, where $f\colon \mathbb{C}^m\to \mathbb{C}$ is an entire function of bounded $l$-index in the direction $(1,\ldots,1)$, $\Phi\colon \mathbb{B}^n\to \mathbb{C}$ is an analytic function in the unit ball.
 We weaken the condition $|\partial_{\mathbf{b}}^k\Phi(z)|\le K|\partial_{\mathbf{b}}\Phi(z)|^k$ for all $z\in\mathbb{B}^n$, where $K\geq 1$ is a constant, $\mathbf{b}\in\mathbb{C}^n\setminus\{0\}$ is a given direction and $${\partial_{\mathbf{b}} F(z)}:=\sum\limits_{j=1}^{n}\!\frac{\partial F(z)}{\partial z_{j}}{b_{j}}, \ \partial_{\mathbf{b}}^k F(z):=\partial_{\mathbf{b}}\big(\partial_{\mathbf{b}}^{k-1} F(z)\big).$$ It is replaced by the condition $|\partial_{\mathbf{b}}^k\Phi(z)|\le K(l(\Phi(z)))^{1/(N_{\mathbf{1}}(f,l)+1)}|\partial_{\mathbf{b}}\Phi(z)|^k$, where $N_{\mathbf{1}}(f,l)$ is the $l$-index of the function $f$ in the direction $\mathbf{1}=(1,\ldots,1).$ The described result is an improvement of previous one. It is also a new result for the one-dimensional case $n=1,$ $m=1$, i.e. for an analytic function $\Phi$ in the unit disc and for an entire function $f: \mathbb{C}\to\mathbb{C}$ of bounded $l$-index.

Book Review: Singularities of mappings: The local behaviour of smooth and complex analytic mappings

Book Review: Singularities of mappings: The local behaviour of smooth and complex analytic mappings

Renormalization of determinant lines in quantum field theory

On a compact manifold $M$, we consider the affine space $A$ of non self-adjoint perturbations of some invertible elliptic operator acting on sections of some Hermitian bundle, by some differential operator of lower order. We construct and classify all complex analytic functions on the Fr\'echet space $A$ vanishing exactly over non invertible elements, having minimal order and which are obtained by local renormalizations, a concept coming from quantum field theory, called renormalized determinants. The additive group of local polynomial functionals of finite degrees acts freely and transitively on the space of renormalized determinants. We provide different representations of the renormalized determinants in terms of spectral zeta determinants, Gaussian Free Fields, infinite product and renormalized Feynman amplitudes in perturbation theory in position space \`a la Epstein-Glaser. Specializing to the case of Dirac operators coupled to vector potentials and reformulating our results in terms of determinant line bundles, we prove our renormalized determinants define some complex analytic trivializations of some holomorphic line bundle over $A$ relating our results to a conjectural picture from some unpublished notes by Quillen [52] from April 1989.

On solutions of PDEs by using algebras

The components of complex analytic functions define solutions for the Laplace's equation, and in a simply connected domain, each solution of this equation is the first component of a complex analytic function. In this paper, we generalize this result; for each PDE of the form , and for each affine planar vector field φ, we give an algebra with unit e = e1, with respect to which the components of all functions of the form are all the solutions for this PDE, where is differentiable in the sense of Lorch with respect to . Solutions are also constructed for the following equations: , ‐order PDEs, and ‐order PDEs; among these are the bi‐harmonic, the bi‐wave, and the bi‐telegraph equations.

A new approach to solve the anti-plane crack problems by the method of fundamental solutions

This paper expands the method of fundamental solutions (MFS) to solve the anti-plane crack problems based on the traditional fundamental solution by the conformal mapping technique without using the domain decomposition technique and/or the crack Green's function. Based on the complex analysis and the conformal mapping technique, the approximate solution of the complex analytic function is derived for the MFS. The modified boundary conditions are deduced to construct the interpolation equations of the MFS. The conformal mapping technique naturally introduces singularities of the stress fields and the stress intensity factors (SIFs) for the anti-plane crack problems without using the crack Green's function and the domain-decomposition technique. The obtained numerical examples illustrate the validity and accuracy for the anti-plane crack problems. The proposed method is an expansion of the traditional MFS, and can be applied for many anti-plane elastic problems with the simple concept and the easy numerical implementation.

Restriction estimates to complex hypersurfaces

The restriction problem is relatively well understood for hypersurfaces and recent progresses have been made by means of bilinear and multilinear approaches and, more recently, by means of polynomial partitioning combined with those approaches. However, for surfaces with codimension bigger than 1, bilinear and multilinear generalizations of restriction estimates are more involved and effectiveness of these multilinear estimates is not so well understood yet. Regarding the restriction problem for surfaces with codimensions bigger than 1, the current state of the art is still at the level of the TT⁎ method which is known to be useful for obtaining Lq–L2 estimates. In this paper, we consider a special type of codimension 2 surfaces, given by the graphs of complex analytic functions, and attempt to make progress beyond the L2 restriction estimates.

Spectrally Simple Zeros of Zeon Polynomials

Combinatorial properties of zeons have been applied to graph enumeration problems, graph colorings, routing problems in communication networks, partition-dependent stochastic integrals, and Boolean satisfiability. Power series of elementary zeon functions are naturally reduced to finite sums by virtue of the nilpotent properties of zeons. Further, the zeon extension of any analytic complex function has zeon polynomial representations on associated equivalence classes of zeons. In this paper, zeros of polynomials over complex zeons are considered. Existing results for real zeon polynomials are extended to the complex case and new results are established. In particular, a fundamental theorem of zeon algebra is established for spectrally simple zeros of complex zeon polynomials, and an algorithm is presented that allows one to find spectrally simple zeros when they exist. As an application, inverses of zeon extensions of analytic functions are computed using polynomial methods.

A basis of analytic functionals for CFTs in general dimension

We develop an analytic approach to the four-point crossing equation in CFT, for general spacetime dimension. In a unitary CFT, the crossing equation (for, say, the s- and t-channel expansions) can be thought of as a vector equation in an infinite-dimensional space of complex analytic functions in two variables, which satisfy a boundedness condition at infinity. We identify a useful basis for this space of functions, consisting of the set of s- and t-channel conformal blocks of double-twist operators in mean field theory. We describe two independent algorithms to construct the dual basis of linear functionals, and work out explicitly many examples. Our basis of functionals appears to be closely related to the CFT dispersion relation recently derived by Carmi and Caron-Huot.

Hilbert’s Tenth Problem for Complex Meromorphic Functions in Several Variables

Abstract We prove that Hilbert’s Tenth Problem for meromorphic functions and for entire functions in several variables is unsolvable over the language of rings, together with constant symbols for two of the variables and a predicate for a place. This is the first result in the literature, which proves unsolvability of a diophantine problem for the ring of complex analytic functions in a number of variables. The proof rests upon an interplay between analytic geometry, analysis, number theory, and logic.

Two Points Taylor’s Type Representations for Analytic Complex Functions with Integral Remainders

Abstract In this paper we establish some two point weighted Taylor’s expansions for analytic functions f : D ⊆ ℂ→ ℂ defined on a convex domain D. Some error bounds for these expansions are also provided. Examples for the complex logarithm and the complex exponential are also given.

A Comparison of Norms: Bicomplex Root and Ratio Tests and an Extension Theorem

A new root and ratio test for bicomplex power series produces the bicomplex extension theorem, where any complex power series with positive radius of convergence R extends, via a simple change of the complex domain variable z to the bicomplex variable , to a bicomplex function analytic in a four real-dimensional ball with the same positive radius R. In this way, any complex-analytic function has a natural extension to a bicomplex-analytic function.

LONGITUDINAL SHIFT OF THE BINDER AND INCLUSIONS OF A COMPOSITE WEAKENED BY TWO-PERIODIC RIGHT LINEAR CRACKS

In this paper, we consider antiplane deformation for an isotropic elastic material consisting of an infinite system of parallel identical circular cylindrical fibers covered with a uniform cylindrical film uniformly covering the surface of each fiber and a bonding medium weakened by a doubly periodic system of rectilinear cracks. Each washer has a centrally located crack that is less than the diameter of the washer. The presented stresses and their displacements are expressed in terms of an analytical function. For the solution, the well-known position is used that the displacement in the case of an antiplane shear is a harmonic function. A known representation of the solution in each area is applied through the corresponding complex analytical function. Three analytic functions are represented by Laurent series. Satisfying the boundary condition on the contours of holes and crack faces, the problem is reduced to two infinite algebraic systems with respect to the sought coefficients and to two singular integral equations with a Cauchy-type kernel. Then the singular integral equation is reduced to a finite algebraic system of equations by the Multopp – Kalandia method. The procedure for calculating the stress intensity coefficients is given. The numerical implementation of the described method is given at IBM. The results of calculations of the critical load depending on the crack length and elastic geometric parameters of the perforated medium are presented. Keywords: isotropically elastic material, doubly periodic lattice, rectilinear cracks, stress intensity factor, mean stresses, critical load, circular hole, longitudinal shear.

Quantum calculus of Fibonacci divisors and infinite hierarchy of Bosonic–Fermionic Golden quantum oscillators

Starting from divisibility problem for Fibonacci numbers, we introduce Fibonacci divisors, related hierarchy of Golden derivatives in powers of the Golden Ratio and develop corresponding quantum calculus. By this calculus, the infinite hierarchy of Golden quantum oscillators with integer spectrum determined by Fibonacci divisors, the hierarchy of Golden coherent states and related Fock–Bargman representations in space of complex analytic functions are derived. It is shown that Fibonacci divisors with even and odd [Formula: see text] describe Golden deformed bosonic and fermionic quantum oscillators, correspondingly. By the set of translation operators we find the hierarchy of Golden binomials and related Golden analytic functions, conjugate to Fibonacci number [Formula: see text]. In the limit [Formula: see text], Golden analytic functions reduce to classical holomorphic functions and quantum calculus of Fibonacci divisors to the usual one. Several applications of the calculus to quantum deformation of bosonic and fermionic oscillator algebras, [Formula: see text]-matrices, geometry of hydrodynamic images and quantum computations are discussed.

The unbiased diffusion Monte Carlo: a versatile tool for two-electron systems confined in different geometries

Computational codes based on the diffusion Monte Carlo method can be used to determine the quantum state of two-electron systems confined by external potentials of various natures and geometries. In this work, we show how the application of this technique in its simplest form, that does not employ complex analytic guess functions, allows to obtain satisfactory results and, at the same time, to write programs that are readily adaptable from one type of confinement to another. This adaptability allows an easy exploration of the many possibilities in terms of both geometry and structure of the system. To illustrate these results, we present calculations in the case of two-electron hydrogen-based species (hbox {H}_{2} and hbox {H}_{3}^{+}) and two different types of confinement, nanotube-like and octahedral crystal field.Graphic abstract

非均匀复变解析函数零点的孤立性及惟一性

非均匀复变解析函数零点的孤立性及惟一性

On the complex q-Appell polynomials

The purpose of this article is to generalize the ring of \(q\)-Appell polynomials to the complex case. The formulas for \(q\)-Appell polynomials thus appear again, with similar names, in a purely symmetric way. Since these complex \(q\)-Appell polynomials are also \(q\)-complex analytic functions, we are able to give a first example of the \(q\)-Cauchy-Riemann equations. Similarly, in the spirit of Kim and Ryoo, we can define \(q\)-complex Bernoulli and Euler polynomials. Previously, in order to obtain the \(q\)-Appell polynomial, we would make a \(q\)-addition of the corresponding \(q\)-Appell number with \(x\). This is now replaced by a \(q\)-addition of the corresponding \(q\)-Appell number with two infinite function sequences \(C_{\nu,q}(x,y)\) and \(S_{\nu,q}(x,y)\) for the real and imaginary part of a new so-called \(q\)-complex number appearing in the generating function. Finally, we can prove \(q\)-analogues of the Cauchy-Riemann equations.

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.