Single Analytic Function Research Articles

Adelic amplitudes and intricacies of infinite products

For every prime number $p$ it is possible to define a $p$-adic version of the Veneziano amplitude and its higher-point generalizations. Multiplying together the real amplitude with all its $p$-adic counterparts yields the adelic amplitude. At four points it has been argued that the adelic amplitude, after regulating the product that defines it, equals one. For the adelic 5-point amplitude, there exist kinematic regimes where no regularization is needed. This paper demonstrates that in special cases within this regime, the adelic product can be explicitly evaluated in terms of ratios of the Riemann zeta function, and observes that the 5-point adelic amplitude is not given by a single analytic function. Motivated by this fact to study new regularization procedures for the 4-point amplitude, an alternative formalism is presented, resulting in non-constant amplitudes that are piecewise analytic in the three scattering channels, including one non-constant adelic amplitude previously suggested in the literature. Decomposing the residues of these amplitudes into weighted sums of Gegenbauer polynomials, numerical evidence indicates that in special ranges of spacetime dimensions all the coefficients are positive, as required by unitarity.

Screw dislocations in piezoelectric laminates with four or more phases

We present an analytical solution to the problem of a screw dislocation in a four-phase piezoelectric laminate composed of two piezoelectric layers of equal thickness sandwiched between two semi-infinite piezoelectric media. A new version of the complex variable formulation is proposed such that the 2 × 2 real symmetric matrix appearing in the formulation becomes dimensionless. Using analytic continuation, the original boundary value problem is reduced to the identification of a single 2D analytic vector function which is completely determined following rigorous solution of the resulting linear recurrence relations in matrix form. An explicit expression for the image force acting on the piezoelectric screw dislocation is obtained once the single 2 × 2 real matrix function is identified. We also discuss the solution for a screw dislocation in an N -phase piezoelectric laminate composed of N − 2 piezoelectric layers of equal thickness sandwiched between two semi-infinite piezoelectric media.

Operators and multipliers on weighted Bergman spaces

In 1992, Blasco considered a weighted Bergman space using Dini-weight function. He proved that a linear operator T on weighted Bergman space to any Banach space X is bounded if and only if certain one parameter fractional derivative of a single X-valued analytic function satisfy some growth condition. In this paper, we use a three parameters fractional derivative of a single X-valued analytic function and provide an equivalent condition. Our technique uses the Gaussian hypergeometric functions. Furthermore we supply some conditions on the parameters a, b and c under which the Gaussian hypergeometric functions F(a, b; c; z) are Dini-weight.

A circular inclusion with inhomogeneous sliding imperfect interface in harmonic materials

In the following study we rigorously analyze the problem of a circular inclusion with inhomogeneous imperfect sliding interface in finite deformation of harmonic materials. The work begins by defining the inhomogeneous sliding boundary conditions characterized by two interface parameters corresponding to the normal and tangential coordinate directions (with respect to the interface boundary curve), respectively. Then, through the process of analytic continuation the problem is eventually reduced to the determination of a single analytic function given by an ordinary differential equation with variable coefficients. A specific example is selected to illustrate the method. The effects of the circumferential variation of the interface parameter on the mean stress at the interface and the average mean stress in the inclusion are discussed.

A circular inhomogeneity incorporating surface/interface strain gradient elasticity

We derive a rigorous solution to the anti-plane shear problem of a circular elastic inhomogeneity with an imperfect interface described by surface strain gradient elasticity. The loadings considered include a uniform remote stress field and a screw dislocation applied either in the matrix or in the inhomogeneity. By using analytical continuation, the problem is reduced to a third-order differential equation for a single analytic function. The analysis indicates that in the absence of the screw dislocation, the internal stress field is uniform and is reliant on two size-dependent parameters arising from surface strain gradient elasticity. The size-dependent image force acting on the screw dislocation is derived. The conditions for the existence of equilibrium positions for the dislocation in the matrix or in the inhomogeneity are obtained. Several numerical examples are presented to illustrate the solution. As an application of the solution, the stabilities of a misfit screw dislocation dipole in the matrix and a screw dislocation pair in the inhomogeneity are also discussed.

A circular inclusion with inhomogeneous non-slip imperfect interface in harmonic materials.

In this work, a rigorous study is presented for the problem associated with a circular inclusion embedded in an infinite matrix in finite plane elastostatics where both the inclusion and matrix are comprised a harmonic material. The inclusion/matrix boundary is treated as a circumferentially inhomogeneous imperfect interface that is described by a linear spring-type imperfect interface model where in the tangential direction, the interface parameter is infinite in magnitude and in the normal direction, the interface parameter is finite in magnitude (the so-called non-slip interface condition). Through the repeated use of the technique of analytic continuation, the boundary value problem for four analytic functions is reduced to solve a single first-order linear ordinary differential equation with variable coefficients for a single analytic function defined within the inclusion. The unknown coefficients of said function are then found via various analyticity requirements. The method is illustrated, using a specific example of a particular class of inhomogeneous non-slip imperfect interface. The results from these calculations are then contrasted with the results from the homogeneous imperfect interface. These comparisons indicate that the circumferential variation of interface damage has a pronounced effect on the average boundary stress.

On Hermitian separability of the next-to-leading order BFKL kernel for the adjoint representation of the gauge group in the planar $$N=4$$ N = 4 SYM

We analyze a modification of the BFKL kernel for the adjoint representation of the colour group in the maximally supersymmetric (N=4) Yang-Mills theory in the limit of a large number of colours, related to the modification of the eigenvalues of the kernel suggested by S. Bondarenko and A. Prygarin in order to reach the Hermitian separability of the eigenvalues. We restore the modified kernel in the momentum space. It turns out that the modification is related only to the real part of the kernel and that the correction to the kernel can not be presented by a single analytic function in the entire momentum region, which contradicts the known properties of the kernel.

4D hyperspherical harmonic (HyperSPHARM) representation of surface anatomy: a holistic treatment of multiple disconnected anatomical structures.

Image-based parcellation of the brain often leads to multiple disconnected anatomical structures, which pose significant challenges for analyses of morphological shapes. Existing shape models, such as the widely used spherical harmonic (SPHARM) representation, assume topological invariance, so are unable to simultaneously parameterize multiple disjoint structures. In such a situation, SPHARM has to be applied separately to each individual structure. We present a novel surface parameterization technique using 4D hyperspherical harmonics in representing multiple disjoint objects as a single analytic function, terming it HyperSPHARM. The underlying idea behind HyperSPHARM is to stereographically project an entire collection of disjoint 3D objects onto the 4D hypersphere and subsequently simultaneously parameterize them with the 4D hyperspherical harmonics. Hence, HyperSPHARM allows for a holistic treatment of multiple disjoint objects, unlike SPHARM. In an imaging dataset of healthy adult human brains, we apply HyperSPHARM to the hippocampi and amygdalae. The HyperSPHARM representations are employed as a data smoothing technique, while the HyperSPHARM coefficients are utilized in a support vector machine setting for object classification. HyperSPHARM yields nearly identical results as SPHARM, as will be shown in the paper. Its key advantage over SPHARM lies computationally; HyperSPHARM possess greater computational efficiency than SPHARM because it can parameterize multiple disjoint structures using much fewer basis functions and stereographic projection obviates SPHARM's burdensome surface flattening. In addition, HyperSPHARM can handle any type of topology, unlike SPHARM, whose analysis is confined to topologically invariant structures.

Measuring social inequality with quantitative methodology: Analytical estimates and empirical data analysis by Gini and [formula omitted] indices

Social inequality manifested across different strata of human existence can be quantified in several ways. Here we compute non-entropic measures of inequality such as Lorenz curve, Gini index and the recently introduced $k$ index analytically from known distribution functions. We characterize the distribution functions of different quantities such as votes, journal citations, city size, etc. with suitable fits, compute their inequality measures and compare with the analytical results. A single analytic function is often not sufficient to fit the entire range of the probability distribution of the empirical data, and fit better to two distinct functions with a single crossover point. Here we provide general formulas to calculate these inequality measures for the above cases. We attempt to specify the crossover point by minimizing the gap between empirical and analytical evaluations of measures. Regarding the $k$ index as an `extra dimension', both the lower and upper bounds of the Gini index are obtained as a function of the $k$ index. This type of inequality relations among inequality indices might help us to check the validity of empirical and analytical evaluations of those indices.

A Review of Properties of Flow–Density Functions

Flow–density functions are often described as the fundamental relationship of traffic theory and are the basis of kinematic wave models or hydrodynamic models which are used to describe, predict or analyse traffic behaviour. We set out several different properties that it is generally agreed that flow–density functions should satisfy. We then take the many forms of flow–density functions that have been proposed or used over the past several decades, derive the properties of each one and consider whether, or to what extent, they satisfy each of the desirable properties. We find that few, if any, flow–density functions satisfy all of the desirable properties. The main reasons for this are that, in almost all cases, there are not enough independent parameters in the flow–density functions to capture all of the desirable properties and that, in almost all cases, the flow–density functions are specified as a single analytic function rather than a different function for each range of traffic behaviours, such as free-flow, congested and heavily congested traffic.

The van der Waals potential of the magnesium dimer

The ground state van der Waals potential of the magnesium dimer is described by the Tang-Toennies potential model, which requires five essential parameters. Among them, the three dispersion coefficients C(6), C(8), and C(10) are available from accurate ab initio calculations. The other two are the Born-Mayer parameters in A exp(-bR). In this paper, we show that A and b can be determined from the self-consistent Hartree-Fock calculations and the experimental dissociation energy D(0). The predicted well depth D(e) and equilibrium distance R(e) are in nearly perfect agreement with the experiment. In fact, the entire potential energy curve, which is given by a single analytic function, is in excellent agreement with the pointwise potential energies constructed from the spectroscopic measurements in the interval of 6a(0)-14a(0) and in good agreement with the experimental repulsive potential determined from Franck-Condon factors of the bound-free transitions for R less than 6a(0). The reduced potential of Mg(2) is analyzed in terms of its components, and the number of terms in the dispersion series necessary for convergence is investigated.

Complex replica zeros of ±J Ising spin glass at zero temperature

Zeros of the nth moment of the partition function [Zn] are investigated in a vanishing temperature limit β → ∞, n → 0 keeping y = βn ∼ O(1). In this limit, the moment parameterized by y characterizes the distribution of the ground-state energy. We numerically investigate the zeros for ±J Ising spin-glass models with tree and several other systems, which can be carried out with a feasible computational cost by a symbolic operation based on the Bethe–Peierls method. For several tree systems we find that the zeros tend to approach the real axis of y in the thermodynamic limit implying that the moment cannot be described by a single analytic function of y as the system size tends to infinity, which may be associated with breaking of the replica symmetry. However, examination of the analytical properties of the moment function and assessment of the spin-glass susceptibility indicate that the breaking of analyticity is relevant to neither one-step nor full replica symmetry breaking.

Green's function in plane anisotropic bimaterials with imperfect interface

The fundamental solutions or Green's functions for 2D or 3D anisotropic media with imperfect interface remain a challenging problem. In this paper, a general method is presented for the rigorous solution for the 2D Green's function in an anisotropic elastic bimaterial subject to a line force or a line dislocation. Most significant is the fact that the bonding along the bimaterial interface is considered to be homogeneous imperfect. Specifically, the tractions are continuous but the displacements are discontinuous and proportional, in terms of interface stiffness parameters, to their respective traction components. Using complex variable techniques, the basic boundary-value problem for two analytic vector functions is reduced to a coupled linear first-order differential equation for a single analytic vector function defined in the lower half space. The coupled linear differential equation for the single analytic vector function can be subsequently decoupled into three independent linear first-order differential equations for three newly defined analytic functions. Closed-form solutions for the 2D Green's function are derived in terms of the exponential integral. Unlike previous works which involve some sort of inverse transform method to obtain the physical quantities from the transform domain, the key feature of the present method is that the physical quantities can be readily calculated without the need to perform any inverse transform operations.

A Circular Inclusion with Circumferentially Inhomogeneous Non-Slip Interface in Plane Elasticity

A rigorous solution is presented for a problem associated with a circular inclusion embedded within an infinite matrix in plane elastostatics. The bonding at the inclusion-matrix interface is assumed to be imperfect. Specifically, the jump in the normal displacement is assumed to be proportional to the normal traction with the proportionality parameter taken to be circumferentially inhomogeneous. In addition, we assume that displacements in the tangential direction are continuous. This type of interface is generally referred to as an inhomogeneous non-slip interface. Using the principle of analytic continuation, the basic boundary-value problem for four analytic functions is reduced to a first-order differential equation for a single analytic function defined inside the circular inclusion. The resulting closed-form solutions include a finite number of unknown constants determined by analyticity requirements and certain other supplementary conditions. The method is illustrated using several specific examples of a particular class of inhomogeneous non-slip interface. The results from these calculations are compared with the corresponding results when the interface imperfections are homogeneous. These comparisons indicate that the circumferential variation of interface damage has a significant effect on even the average stresses induced within a circular inclusion.

Unified analytic representation of physical sputtering yield

Abstract Generalized energy parameter η=η(e,δ) and normalized sputtering yield Y (η) , where e=E/ETF and δ=Eth/ETF, are introduced to achieve a unified representation of all available experimental and sputtering data at normal ion incidence. The sputtering data in the new Y (η) representation retain their original uncertainties. The Y (η) data can be fitted to a simple three-parameter analytic expression with an rms deviation of 32%, well within the uncertainties of original data. Both η and Y (η) have correct physical behavior in the threshold and high-energy regions. The available theoretical data produced by the TRIM.SP code can also be represented by the same single analytic function Y (η) with a similar accuracy.

Analytic Expressions to Represent the Hazard Ultraviolet Action Spectrum of ICNIRP & ACGIH

Relative spectral effectiveness values for hazard evaluation of ultraviolet radiation, as tabulated according to the action spectrum of the International Committe on Non-ionizing Radiation Protection (ICNIRP) and the American Conference of Governmental and Industrial Hygienists (ACGIH) can be substituted or interpolated in relevant spectral regions by one single analytic function, or by three simpler functions. The functions are useful for calculation of values of weighted spectral irradiance from 210-400 nm. The spectral weighting factors S(λ) follow the ICNIRP and ACGIH curve of tabulated values with enough accuracy for all practical purposes.

A Circular Inclusion With Circumferentially Inhomogeneous Sliding Interface in Plane Elastostatics

A general method is presented to obtain the rigorous solution for a circular inclusion embedded within an infinite matrix with a circumferentially inhomogeneous sliding interface in plane elastostatics. By virtue of analytic continuation, the basic boundary value problem for four analytic functions is reduced to a first-order differential equation for a single analytic function inside the circular inclusion. The finite form solution is obtained that includes a finite number of unknown constants determined by the analyticity of the solution and certain other auxiliary conditions. With this method, the exact values of the average stresses within the circular inclusion can be calculated without solving the full problem. Several specific examples are used to illustrate the method. The effects of the circumferential variation of the interface parameter on the mean stress at the interface and the average stresses within the inclusion are discussed.

A circular inclusion with circumferentially inhomogeneous interface in antiplane shear

A general method is developed for the rigorous solution of a problem associated with a circular inclusion embedded within an infinite matrix in antiplane shear. The bonding at the inclusion–matrix interface is assumed to be imperfect. Most significant is the fact that the imperfection in the interface is assumed to be circumferentially inhomogeneous. Using analytic continuation, the basic boundary–value problem for two analytic functions is reduced to a first‐order differential equation for a single analytic function and the closed‐form solution is obtained. The method is illustrated using several specific examples. The results from these examples are compared to the corresponding results when the imperfection in the interface is homogeneous. These comparisons illustrate how the circumferential variation of the parameter describing the imperfection has a pronounced effect on the stresses induced within the inclusion.

Bender-Wu branch points in the cubic oscillator

The analytic continuation of the resonances of the cubic anharmonic oscillator to complex values of the coupling constant is studied with semiclassical and numerical methods. Bender-Wu branch points, at which level crossing occurs, are calculated and labelled by a process of analytic continuation. The different resonances are the values that a single analytic function takes on different sheets of a Riemann surface whose topology is described.

Operators on weighted Bergman spaces (0<p≤1) and applications

We describe the boundedness of a linear operator from Bp(ρ) = {f : D → C analytic : (∫ D ρ(1 − |z|) (1 − |z|) |f(z)| dA(z) )1/p < ∞} , for 0 < p ≤ 1 under some conditions on the weight function ρ, into a general Banach space X by means of the growth conditions at the boundary of certain fractional derivatives of a single X-valued analytic function. This, in particular, allows us to characterize the dual of Bp(ρ) for 0 < p < 1 and to give a formulation of generalized Carleson measures in terms of the inclusion B1(ρ) ⊂ L(D,μ). We then apply the result to the study of multipliers, Hankel operators and composition operators acting on Bp(ρ) spaces. 1991 Math. Subject Class. : Primary 47B38, 47B35 Secondary 42A45