Arbitrary Analytic Function Research Articles

More general sudden singularities

We present a general form for the solution of an expanding general-relativistic Friedmann universe that encounters a singularity at finite future time. The singularity occurs in the material pressure and acceleration whilst the scale factor, expansion rate and material density remain finite and the strong energy condition holds. We also show that the same phenomenon occurs, but under different conditions, for Friedmann universes in gravity theories arising from the variation of an action that is an arbitrary analytic function of the scalar curvature.

Singularities and enriched cycles

We introduce graded, enriched characteristic cycles as a method for encoding Morse modules of strata with respect to a constructible complex of sheaves. Using this new device, we obtain results for arbitrary complex analytic functions on arbitrarily singular complex analytic spaces.

Localizing attractors via a generalized La Salle principle

We are concerned with plane differential systems of the form x ̇ = P(x,y), y ̇ = Q(x,y) , with P, Q analytic. We propose a formal-numeric method to localize the attractors and the repellers of the system. Such a method consists of looking for a power series solution to a PDE of the type P ∂V ∂x + Q ∂V ∂y = μ(V) , with μ is an arbitrary analytic function. When μ( V) = ρ( V − V 2), ρ > 0, the attractors are contained in the set V = 1, the repellers in the set V = 0.

Squeezing of composite piezoceramic plate under the loading of regular normal pressure

The electroelastic boundary value problem for thin composite plate of arbitrary configuration is studied under the action of regular normal pressure. The plane deformation parallel to the direction of the piezoceramic polarization is considered. The representation of the electroelastic magnitudes through three arbitrary analytic functions with complex variable is used. Using the fundamental solution for unbounded composite plate, the integral representations of complex potentials are constructed. The mechanical and electric boundary conditions at the interface of the composite plate are satisfied. Then, the corresponding boundary value problem is reduced to a system of singular integral equations of second order with fixed singularity. The solution of the problem is obtained. Numerical results using the algorithm of the quadrature method are shown.

Periodic solutions of a system of complex ODEs

We consider the real evolution of the system of ODEs z ̈ n+z n= ∑ m=1, m≠n N g nm(z n−z m) −3,z n≡z n(t), z ̇ n≡dz n(t)/dt,n=1,…,N, in C N , namely we assume the N dependent variables z n , as well as the N( N−1) (arbitrary!) “coupling constants” g nm , to be complex numbers, while the independent variable t (“time”) is real. In this context we prove that there exists, in the phase space of the initial data z n(0), z ̇ n(0) , an open domain having infinite measure, such that all trajectories emerging from it are completely periodic with period 2 π, z n ( t+2 π)= z n ( t). However, completely real initial data z n(0), z ̇ n(0) are generally not included in this domain. There also hold analogous results for more general systems of ODEs, such as that which obtains by replacing the right-hand sides of the above equations of motion with arbitrary analytic functions F n( z) satisfying the scaling property F n(λ z)=λ −3F n( z) .

Optimizing the FESR extraction of ms from hadronic τ decay data

As is well known, the difference between ud and us vector and/or axial vector correlators, whose spectral functions are measurable in hadronic τ decay, allows one in principle, through the use of QCD sum rules, to determine the strange quark mass, m s . We show that, by studying the behavior of the relevant correlator difference in the complex q 2-plane, the freedom to choose arbitrary analytic weight functions present in the finite energy sum rule (FESR) implementation of this approach may be exploited to construct FESR's which (1) significantly reduce theoretical uncertainties and (2) remove problems associated with both the poor convergence of the OPE representation of the longitudinal part of the us vector and axial vector correlators and the large statistical errors in the us spectral data above the K* region. The result of this optimization is an extraction, based on present data, m s (2 GeV) = 115.1 ± 13.6 ± 11.8 ± 9.7, where the first error is statistical, the second due to that on V us , and the third theoretical. We show also that, for the new weights constructed here, analyses with errors in the us data in the K* region reduced by a factor of 2 would produce a determination of m s with the statistical error reduced significantly below that associated with the uncertainty in V us .

A subdivision scheme for Poisson curves and surfaces

The de Casteljau evaluation algorithm applied to a finite sequence of control points defines a Bézier curve. This evaluation procedure also generates a subdivision algorithm and the limit of the subdivision process is this same Bézier curve. Extending the de Casteljau subdivision algorithm to an infinite sequence of control points defines a new family of curves. Here, limits of this stationary non-uniform subdivision process are shown to be equivalent to curves whose control points are the original data points and whose blending functions are given by the Poisson distribution. Thus this approach generalizes standard subdivision techniques from polynomials to arbitrary analytic functions. Extensions of this new subdivision scheme from curves to tensor product surfaces are also discussed.

The rational Bernstein bases and the multirational blossoms

Rational Bezier curves are typically defined as projections of polynomial Bezier curves of one higher dimension. Here we introduce an alternative notion of rational Bezier curves defined in terms of rational Bernstein blending functions. These negative degree Bernstein bases share many properties with their polynomial kin: they are linearly independent, form a partition of unity, satisfy Descartes' Law of Signs, obey the standard recurrence relation, and have simple two term formulas for differentiation and degree elevation. But whereas the Bernstein bases of positive degree can represent only polynomial functions, Bernstein bases of negative degree can exactly represent arbitrary functions analytic in a neighborhood of zero. Moreover, whereas the Bernstein bases of positive degree can uniformly approximate arbitrary continuous functions on a compact interval, the Bernstein bases of negative degree can uniformly approximate all continuous functions that vanish at minus infinity. Nevertheless, despite these differences, most of the standard algorithms for Bezier curves, such as reparametrization, trimming, and conversion to and from monomial form extend quite naturally to their rational Bezier brethren. Bezier curves are closely associated with blossoming. The blossom of a degree n polynomial P( x) is the unique, symmetric, multiaffine function p( u 1,…, u n ) that reduces to P( x) along the diagonal. The kth Bezier control point of P( x) is given by the blossom value p(1,…,1,0,…,0), where the blossom parameters consist of k ones and ( n− k) zeros. Thus the blossom provides the dual functionals for standard Bezier curves. Here we introduce a new kind of blossom associated with rational Bernstein bases and with arbitrary analytic functions. The degree − n blossom of an analytic function F( x) is a function in two sets of parameters ( u 1,…, u k ) and ( v 1,…, v k+ n ). This new blossom of F( x) is the unique function f( u 1,…, u k / v 1,…, v k+ n ) that is bisymmetric, multiaffine in the u parameters, satisfies a cancellation property, and reduces to F( x) along the diagonal. The kth Bezier control point of an analytic function F( x) relative to the degree − n Bernstein basis is given by the blossom value f(1,…,1/0,…,0), where the blossom parameters consist of k ones and ( n+ k) zeros. The purpose of this paper is to explore some of the properties of these new rational Bezier curves and their associated multirational blossoms.

Decision Trees: Old and New Results

In this paper, we prove two general lower bounds for algebraic decision trees which test membership in a set S ⊆ R n which is defined by linear inequalities. Let rank( S ) be the maximal dimension of a linear sub- space contained in the closure of S (in Euclidean topology). First we show that any decision tree for S which uses products of linear functions (we call such functions mlf- functions ) must have depth at least n −rank( S ). This solves an open question raised by A. C. Yao and can be used to show that mlf-functions are not really more powerful than simple comparisons between the input variables when computing the largest k out of n elements. Yao proved this result in the special case when products of at most two linear functions are allowed. Our proof also shows that any decision tree for this problem must have exponential size. Using the same methods, we can give an alternative proof of Rabin's theorem, namely that the depth of any decision tree for S using arbitrary analytic functions is at least n −rank( S ).

Exact solutions of the Dirac equation with harmonic pseudoscalar, scalar or electric potential

A new class of exact solutions is obtained in explicit form for the Dirac equation with a pseudoscalar or scalar, or electric static potential. The potential may be an arbitrary harmonic function the gradient squared of which is a constant. It is shown that the set of such potentials is sufficiently ample. The simplest example is the linear function . Another example is the function , where is an arbitrary analytic function depending on the complex variable . The solutions are obtained using the technique of biquaternionic projection operators which itself is interesting since its possible applications are not limited to the situation considered in this paper.

Seiberg-Witten monopole equations and Riemann surfaces

The twice dimensionally reduced Seiberg-Witten monopole equations admit solutions depending on two real parameters ( b, c) and an arbitrary analytic function f ( z) determining a solution of Liouville's equation. The U(1) and manifold curvature 2-forms Fand R 2 1 are invariant under fractional SL(2, R) transformations of f ( z). When b = 1 2 and c= 0 and f( z) is the Fuchsian function uniformizing an algebraic function whose Riemann surface has genus p ≥ 2, the solutions, now completely SL (2, R) invariant, are the same surfaces accompanied by a U(1) bundle of c 1 = ±( p − 1) and a 1-component constant spinor.

General minimal surface solution for gravitational instantons

We construct the general instanton metric obtained from Weierstrass' general local solution for minimal surfaces using the correspondence between minimal surfaces in three-dimensional Euclidean space and gravitational instantons admitting two Killing vectors. The resulting metric contains one arbitrary analytic function and we show that it can be transformed to the Gibbons-Hawking form of an instanton metric that was reported earlier.

Analogues of the Kolosov–Muskhelishvili General Representation Formulas and Cauchy–Riemann Conditions in the Theory of Elastic Mixtures

Abstract Analogues of the well-known Kolosov–Muskhelishvili formulas of general representations are obtained for nonhomogeneous equations of statics in the case of the theory of elastic mixtures. It is shown that in this theory the displacement and stress vector components, as well as the stress tensor components, are represented through four arbitrary analytic functions. The usual Cauchy–Riemann conditions are generalized for homogeneous equations of statics in the theory of elastic mixtures.

A note on antiplane deformations of inhomogeneous elastic materials

This paper formally obtains the solution of the equations of antiplane inhomogeneous elasticity in terms of an arbitrary analytic function for the case when the shear modulus varies continuously with two Cartesian coordinates. The solution is used to solve a particular boundary value problem involving a crack in an inhomogeneous material.

The qs-Hermite Polynomial and the Representations of Heisenberg and Quantum Heisenberg Algebra

By introducing a pair of canonical conjugate two-parameter deformed operators Dqs, Xqs we can naturally obtain the form of qs-analogous Taylor series for an arbitrary analytic function, and explicitly construct the realizations of Heisenberg and two-parameter deformed quantum Heisenberg algebra by means of the operators Dqs and Xqs, and it is shown that the qs-analogous Hermite polynomials are the representations of Heisenberg and the quantum Heisenberg algebra.

Analogues of the Kolosov–Muskhelishvili General Representation Formulas and Cauchy–Riemann Conditions in the Theory of Elastic Mixtures

Analogues of the well-known Kolosov–Muskhelishvili formulas of general representations are obtained for nonhomogeneous equations of statics in the case of the theory of elastic mixtures. It is shown that in this theory the displacement and stress vector components, as well as the stress tensor components, are represented through four arbitrary analytic functions. The usual Cauchy–Riemann conditions are generalized for homogeneous equations of statics in the theory of elastic mixtures.

A method of solving plane initial and boundary-value problems of the dynamic theory of elasticity

A method for solving plane initial and boundary-value problems (1BVP) in the dynamic theory of elasticity (DTE), in which the solutions are represented in the form of a continuous planar superposition of arbitrary analytic functions which are complex plane waves of arbitrary shape is described. By representing the solutions in this form it is possible to use the Radon transformation, by means of which plant: lBVP of the DTE reduce to boundary-value problems of the theory of functions of a complex variable: in the simplest cases of those of Dirichlet or Keldysh-Sedov, and in more complicated cases to Riemann-Hilbert or Riemann ( matching) problems. The method is illustrated on some basic IBVP of the DTE in a half-plane. An analytic solution for the basic mixed problem of the DTE, in which there is no boundary condition that applies over the entire infinite interval of the half-plane boundary, is :found for the fast time.

Liouville vortex and φ4 kink solutions of the Seiberg–Witten equations

The Seiberg–Witten equations, when dimensionally reduced to R2, naturally yield the Liouville equation, whose solutions are parametrized by an arbitrary analytic function g(z). The magnetic flux Φ is the integral of a singular Kaehler form involving g(z); for an appropriate choice of g(z), N coaxial or separated vortex configurations with Φ=2πN/e are obtained when the integral is regularized. The regularized connection in the R1 case coincides with the kink solution of φ4 theory.

How generic are null spacetime singularities?

The spacetime singularities inside realistic black holes are sometimes thought to be spacelike and strong, since there is a generic class of solutions (BKL) to Einstein's equations with these properties. We show that null, weak singularities are also generic, in the following sense: there is a class of vacuum solutions containing null, weak singularities, depending on 8 arbitrary (up to some inequalities) analytic initial functions of 3 spatial coordinates. Since 8 arbitrary functions are needed (in the gauge used here) to span the generic solution, this class can be regarded as generic. \textcopyright{} 1996 The American Physical Society.

Conservation laws and canonical forms in the Stroh formalism of anisotropic elasticity

We find and classify all first-order conservation laws in the Stroh formalism. All possible non-semisimple degeneracies are considered. The laws are found to depend on three arbitrary analytic functions. In some instances, there is an “extra” law which is quadratic in ∇ u \nabla u . Separable and inseparable canonical forms for the stored energy function are given for each type of degeneracy and they are used to compute the conservation laws. The existence of a real Stroh eigenvector is found to be a necessary and sufficient condition for separability. The laws themselves are stated in terms of the Stroh eigenvectors.