Arbitrary Analytic Function Research Articles

Global Mapping Properties of Some Functions of Class S

The Lemma of Schwarz is one of the most surprising results in complex analysis in the sense that some very weak conditions on an analytic function in the unit disk |z| < 1 imply a very strict behavior of that function in the respective disk. What about the behavior of the function outside the unit disk? This is the question we deal with in this paper. The theory we presented in some previous publications was about univalent functions, not necessarily in the unit disk, but in the most general setting, namely in the fundamental domains of arbitrary analytic functions. Naturally, connections can be expected between the two fields of complex analysis. The purpose of this paper is to explore these connections and take advantage of the well established theory of univalent functions in order to advance the theory of fundamental domains.

Poincaré sphere trajectory encoding metasurfaces based on generalized Malus’ law

As a fundamental property of light, polarization serves as an excellent information encoding carrier, playing significant roles in many optical applications, including liquid crystal displays, polarization imaging, optical computation and encryption. However, conventional polarization information encoding schemes based on Malus’ law usually consider 1D polarization projections on a linear basis, implying that their encoding flexibility is largely limited. Here, we propose a Poincaré sphere (PS) trajectory encoding approach with metasurfaces that leverages a generalized form of Malus’ law governing universal 2D projections between arbitrary elliptical polarization pairs spanning the entire PS. Arbitrary polarization encodings are realized by engineering PS trajectories governed by either arbitrary analytic functions or aligned modulation grids of interest, leading to versatile polarization image transformation functionalities, including histogram stretching, thresholding and image encryption within non-orthogonal PS loci. Our work significantly expands the encoding dimensionality of polarization information, unveiling new opportunities for metasurfaces in polarization optics for both quantum and classical regimes.

Conformal Self Mappings of the Fundamental Domains of Analytic Functions and Computer Experimentation

Conformal self mappings of a given domain of the complex plane can be obtained by using the Riemann Mapping Theorem in the following way. Two different conformal mappings φ and ψ of that domain onto one of the standard domains: the unit disc, the complex plane or the Riemann sphere are taken and then ψ −1 ◦ φ is what we are looking for. Yet, this is just a theoretical construction, since the Riemann Mapping Theorem does not offer any concrete expression of those functions. The Möbius transformations are concrete, but they can be used only for particular circular domains. We are proving in this paper that conformal self mappings of any fundamental domain of an arbitrary analytic function can be obtained via Möbius transformations as long as we allow that domain to have slits. Moreover, those mappings enjoy group properties. This is a totally new topic. Although fundamental domains of some elementary functions are well known, the existence of such domains for arbitrary analytic functions has been proved only in our previous publications mentioned in the References section. No other publication exists on this topic and the reference list is complete. We deal here with conformal self mappings of fundamental domains in its whole generality and present sustaining illustrations. Those related to the case of Dirichlet functions represent a real achievement. Computer experimentation with these mappings are made for the most familiar analytic functions.

On exact solutions of the equation of continuity

The continuum of exact solutions of the two-dimensional continuity equation for stationary, planeparallel fluid flow is found in the article. Only specific solutions of this equation corresponding to a vortex-free fluid flow are described in the classical literature. At the same time, components of an arbitrary analytical function of a complex variable are used, playing the role of a velocity potential and a current function. In this paper we give a generalization of this method, which also gives vortex-free solutions to the continuity equation.

On bivariate fractional calculus with general univariate analytic kernels

We introduce a general bivariate fractional calculus, defined using a kernel based on an arbitrary univariate analytic function with an appropriate bivariate substitution. Various properties of the introduced general operators are established, including a series formula, function space mappings, and Fourier and Laplace transforms. A major result of this paper is a fractional Leibniz rule for the new operators, the derivation of which involves correcting a minor error in one of the classic textbooks on fractional calculus. We also solve some fractional differential equations using transform methods, revealing an interesting connection between bivariate type Mittag-Leffler functions.

Наилучшее приближение и значения поперечников некоторых классов аналитических функций в весовом пространстве Бергмана B_(2,γ)

In the paper, exact inequalities are found for the best approximation of an arbitrary analytic function f in the unit circle by algebraic complex polynomials in terms of the modulus of continuity of the m th order of the r th order derivative f^((r)) in the weighted Bergman space B_(2,γ). Also using the modulus of continuity of the m-th order of the derivative f(r), we introduce a class of functions W_m^((r) ) (h,Φ) analytic in the unit circle and defined by a given majorant Φ, h∈(0,π⁄n], n>r, monotonically increasing on the positive semiaxis. Under certain conditions on the majorant Φ, for the introduced class of functions, the exact values of some known n-widths are calculated. We use methods for solving extremal problems in normed spaces of functions analytic in a circle, as well as the method for estimating from below the n-widths of functional classes in various Banach spaces developed by V.M. Tikhomirov. The results presented in this paper are a continuation and generalization of some earlier results on the best approximations and values of widths in the weighted Bergman space B_(2,γ).

Bm$(1-xy,y-x)$-expansion formula and its applications

In this paper, we first show a $(1-xy,y-x)$-expansion formula for an arbitrary analytic function $F(x)$ with respect to the basis <disp-formula><tex-math><![CDATA[$$\bigg\{\prod_{k=0}^{n-1}\frac{x-b_k}{1-x_kx}\,\bigg|\, n\geq 0\bigg\},$$]]></tex-math></disp-formula> which is mainly based on the $(f,g)$-inversion formula. Subsequently, by specializing $F(x)$, $x_n$ and $b_n$, we not only find the corresponding proofs of many classical results such as the Rogers-Fine identity, Andrews' four parametric reciprocal theorem, and Ramanujan's ${~}_1\psi_1$ summation formula, but also construct a lot of $q$-series transformations and summation formulas, including a generalization of Andrews' WP Bailey lemma.

Microstructure and lattice defects in highly deformed metals by X-ray diffraction whole powder pattern modelling

AbstractBasic principles and recent developments of the Whole Powder Pattern Modelling (WPPM) approach are briefly reviewed and illustrated by a typical case study: a highly deformed metal powder (AISI 434) produced by high energy ball milling. Microstructural parameters, like crystalline domain size distribution, dislocation density, and arrangement parameters, are directly obtained by modelling the experimental pattern, without using arbitrary analytical profile functions.

Roots of random functions: A framework for local universality

:We investigate the local distribution of roots of random functions of the form $F_n(z)=\sum_{i=1}^n\xi_i\phi_i(z)$, where $\xi_i$ are independent random variables and $\phi_i(z)$ are arbitrary analytic functions. Starting with the fundamental works of Kac and Littlewood-Offord in the 1940s, random functions of this type have been studied extensively in many fields of mathematics.We develop a robust framework to solve the problem by reducing, via universality theorems, the calculation of the distribution of the roots and the interaction between them to the case where $\xi_i$ are Gaussian. In this special case, one can use the Kac-Rice formula and various other tools to obtain precise answers.Our framework has a wide range of applications, which include the most popular models of random functions, such as random trigonometric polynomials and all basic classes of random algebraic polynomials (Kac, Weyl, and elliptic). Each of these ensembles has been studied heavily by deep and diverse methods. Our method, for the first time, provides a unified treatment for all of them.Among the applications, we derive the first local universality result for random trigonometric polynomials with arbitrary coefficients. When restricted to the study of real roots, this result extends several recent results, proved for less general ensembles. For random algebraic polynomials, we strengthen several recent results of Tao and the second author, with significantly simpler proofs. As a corollary, we sharpen a classical result of Erd\"{o}s and Offord on real roots of Kac polynomials, providing an optimal error estimate. Another application is a refinement of a recent result of Flasche and Kabluchko on the roots of random Taylor series.

The polynomial Hermite-Padé -system for meromorphic functions on a compact Riemann surface

Given a tuple of germs of arbitrary analytic functions at a fixed point, we introduce the polynomial Hermite-Padé -system, which includes the Hermite-Padé polynomials of types I and II. In the generic case we find the weak asymptotics of the polynomials of the Hermite-Padé -system constructed from the tuple of germs of functions that are meromorphic on an -sheeted compact Riemann surface . We show that if for some meromorphic function on , then with the help of the ratios of polynomials of the Hermite-Padé -system we recover the values of on all sheets of the Nuttall partition of , apart from the last sheet. Bibliography: 18 titles.

Optimal measurement of field properties with quantum sensor networks.

We consider a quantum sensor network of qubit sensors coupled to a field f ( x ; θ ) analytically parameterized by the vector of parameters θ . The qubit sensors are fixed at positions x 1, …, x d . While the functional form of f ( x ; θ ) is known, the parameters θ are not. We derive saturable bounds on the precision of measuring an arbitrary analytic function q( θ ) of these parameters and construct the optimal protocols that achieve these bounds. Our results are obtained from a combination of techniques from quantum information theory and duality theorems for linear programming. They can be applied to many problems, including optimal placement of quantum sensors, field interpolation, and the measurement of functionals of parametrized fields.

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).

Polynomial BF -type action for general relativity and anti-self-dual gravity

We report a gravitational $BF$-type action principle propagating two (complex) degrees of freedom that, besides the gauge connection and the $B$ field, only employs an additional Lagrange multiplier. The action depends on two parameters and remarkably is polynomial in the $B$ field. For a particular choice of the involved parameters the action provides an alternative description of (complex) general relativity with a nonvanishing cosmological constant, whereas another choice corresponds to anti-self-dual gravity. Generic values of the parameters produce "close neighbors" of general relativity, although there is a peculiar choice of the parameters that leads to a Hamiltonian theory with two scalar constraints. Given the nontrivial form of the resulting scalar constraint for these models, we consider a more general setting where the scalar constraint is replaced with an arbitrary analytic function of some fundamental variables and show that the Poisson algebra involving this constraint together with the Gauss and vector constraints of the Ashtekar formalism closes, thus generating an infinite family of gravitational models that propagate the same number of degrees of freedom as general relativity.

On the analyticity of waveforms in complexified relatively undistorted progressing waves

We consider two classes of exact solutions of the wave equation, which play an important role in the linear theory of localized wave propagation. These are the complexified spherical wave, known as the “complex source wavefield”, and the complexified Bateman solution. Both involve an arbitrary analytic function of a complex argument, known as the waveform. We explore a possibility of considering non-analytical waveforms and establish that in the both cases the analyticity is unavoidable.

Existence and classification of three-dimensional Lorentzian manifolds with prescribed distinct Ricci eigenvalues

We prove that there exists at least one three-dimensional Lorentzian manifold with any prescribed distinct Ricci eigenvalues, which are given as arbitrary real analytic functions. Moreover, we prove that such Lorentzian manifolds depend locally on three arbitrary functions of two variables.

Generalization of Padé approximation from rational functions to arbitrary analytic functions — Theory

Generalization of Padé approximation from rational functions to arbitrary analytic functions — Theory

The unified group classification method for the generalized nonlinear wave equation and its partial difference schemes

This paper is concerned with the generalized nonlinear second-order equation and its discrete counterpart furnished with different lattices, respectively. By the unified symmetry analysis method, the complete group classifications of the equations are performed. In the sense of point symmetry, all of the vector fields of the continuous nonlinear equation are obtained. As its special cases, the vector fields of some other important nonlinear equations are provided. Then, we develop the unified group classification method for dealing with partial difference schemes (PΔS), all of the point symmetries of the difference schemes are presented with respect to the given arbitrary analytic function f(u).

Lie group analysis of a generalized Krichever-Novikov differential-difference equation

The symmetry algebra of the differential-difference equation \documentclass[12pt]{minimal}\begin{document}$\dot{u}_n = N/D$\end{document}u̇n=N/D with D = un+1 − un−1 and N = P(un)un+1un−1 + Q(un)(un+1 + un−1) + R(un), where P, Q, and R are arbitrary analytic functions is shown to have the dimension 1 ⩽ dimL ⩽ 5. When P, Q, and R are specific second order polynomials in un (depending on 6 constants) this is the integrable discretization of the Krichever–Novikov equation. We find 3 cases when the arbitrary functions are not polynomials and the symmetry algebra satisfies dimL = 2. These cases are shown not to be integrable. The symmetry algebras are used to reduce the equations to purely difference ones. The symmetry group is also used to impose periodicity un+N = un and thus to reduce the differential-difference equation to a system of N coupled ordinary three points difference equations.

On existence and representation of solutions for general degenerate Beltrami equations

We study the general degenerate Beltrami equations in the unit disk in Given an arbitrary analytic function with isolated singularities in we find criteria for the existence of a solution of the form where stands for a regular himeomorphism of onto itself.

Correlation Functions for Random Complex Zeroes: Strong Clustering and Local Universality

We prove strong clustering of k-point correlation functions of of Gaussian Entire Functions. In the course of the proof, we also obtain universal local bounds for k-point functions of of arbitrary nondegenerate Gaussian analytic functions. In the second part of the paper, we show that strong clustering yields the asymptotic normality of fluctuations of some linear statistics of of Gaussian Entire Functions, in particular, of the number of in measurable domains of large area. This complements our recent results from the paper Fluctuations in random complex zeroes (arXiv:1003.4251v1).