Antisymmetric Functions Research Articles

A note on the exchangeability condition in Stein’s method

We show by a surprisingly simple argument that the exchangeability condition, which is key to the exchangeable pair approach in Stein’s method for distributional approximation, can be omitted in many standard settings. This is achieved by replacing the usual antisymmetric function by a simpler one, for which only equality in distribution is required. In the case of normal approximation we also slightly improve the constants appearing in the previous results. For the Poisson approximation, a different antisymmetric function is used, and additional error terms are needed if the bound is to be extended beyond the exchangeable setting.

Jensen's Inequality for Convex-Concave Antisymmetric Functions and Applications

The weighted Jensen inequality for convex-concave antisymmetric functions is proved and some applications are given.

Simple proofs of open problems about the structure of involutions in the Riordan group

We prove that if D = ( g ( x ) , f ( x ) ) is an element of order 2 in the Riordan group then g ( x ) = ± exp [ Φ ( x , xf ( x ) ] for some antisymmetric function Φ ( x , z ) . Also we prove that every element of order 2 in the Riordan group can be written as BMB - 1 for some element B and M = ( 1 , - 1 ) in the Riordan group. These proofs provide solutions to two open problems presented by L. Shapiro [L.W. Shapiro, Some open questions about random walks, involutions, limiting distributions and generating functions, Adv. Appl. Math. 27 (2001) 585–596].

(Anti)symmetric multivariate exponential functions and corresponding Fourier transforms

We define and study symmetrized and antisymmetrized multivariate exponential functions. They are defined as determinants and antideterminants of matrices whose entries are exponential functions of one variable. These functions are eigenfunctions of the Laplace operator on the corresponding fundamental domains satisfying certain boundary conditions. To symmetric and antisymmetric multivariate exponential functions there correspond Fourier transforms. There are three types of such Fourier transforms: expansions into the corresponding Fourier series, integral Fourier transforms and multivariate finite Fourier transforms. Eigenfunctions of the integral Fourier transforms are found.

Magnetic structure factor and pair correlation function for a semiconductor quantumwire

In this paper we report the spin correlation and its effects on the many-body properties ofa quantum wire whose width is given practical consideration. The single-poleapproximation, which is of much relevance in studying the many-body properties of aninteracting one-dimensional electron gas (1DEG) because of the severe constraintsimposed on the phase space of plasma–hole excitations, has been used to obtainanalytical results. The local-field corrections are incorporated within the Hubbardapproximation (HA). We find the results obtained for low electron densities under thesingle-pole approximation on the paramagnon dispersion relation, magnetic structurefactor and the symmetric and anti-symmetric pair correlation function to be ingood quantitative agreement with those obtained under heavily computationalself-consistent field calculations. Unlike prior reported work on the magnetic structurefactor and pair correlation function, our calculation is applicable to quantum wiresof width less than 10 nm, where interesting quantum size effects are observed.

Antisymmetric Orbit Functions

In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space $E_n$ are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. These functions are closely related to irreducible characters of a compact semisimple Lie group $G$ of rank $n$. Up to a sign, values of antisymmetric orbit functions are repeated on copies of the fundamental domain $F$ of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space $E_n$. Antisymmetric orbit functions are solutions of the corresponding Laplace equation in $E_n$, vanishing on the boundary of the fundamental domain $F$. Antisymmetric orbit functions determine a so-called antisymmetrized Fourier transform which is closely related to expansions of central functions in characters of irreducible representations of the group $G$. They also determine a transform on a finite set of points of $F$ (the discrete antisymmetric orbit function transform). Symmetric and antisymmetric multivariate exponential, sine and cosine discrete transforms are given.

Sparse Grids, Adaptivity, and Symmetry

Sparse grid methods represent a powerful and efficient technique for the representation and approximation of functions and particularly the solutions of partial differential equations in moderately high space dimensions. To extend the approach to truly high-dimensional problems as they arise in quantum chemistry, an additional property has to be brought into play, the symmetry or antisymmetry of the functions sought there.In the present article, an adaptive sparse grid refinement scheme is developed that takes full advantage of such symmetry properties and for which the amount of work and storage remains strictly proportional to the number of degrees of freedom. To overcome the problems with the approximation of the inherently complex antisymmetric functions, augmented sparse grid spaces are proposed.

E-Orbit Functions

We review and further develop the theory of E-orbit functions. They are func- tions on the Euclidean spaceEn obtained from the multivariate exponential function by sym- metrization by means of an even part We of a Weyl group W, corresponding to a Coxeter- Dynkin diagram. Properties of such functions are described. They are closely related to symmetric and antisymmetric orbit functions which are received from exponential functions by symmetrization and antisymmetrization procedure by means of a Weyl group W. The E-orbit functions, determined by integral parameters, are invariant with respect to even part W aff e of the affine Weyl group corresponding to W. The E-orbit functions determine a symmetrized Fourier transform, where these functions serve as a kernel of the transform. They also determine a transform on a finite set of points of the fundamental domain F e of the group W aff e (the discrete E-orbit function transform).

Two-dimensional Raman spectra of atomic solids and liquids

We calculate third- and fifth-order Raman spectra of simple atoms interacting through a soft-core potential by means of molecular-dynamics (MD) simulations. The total polarizability of molecules is treated by the dipole-induced dipole model. Two- and three-body correlation functions of the polarizability at various temperatures are evaluated from equilibrium MD simulations based on a stability matrix formulation. To analyze the processes involved in the spectroscopic measurements, we divide the fifth-order response functions into symmetric and antisymmetric integrated response functions; the symmetric one is written as a simple three-body correlation function, while the antisymmetric one depends on a stability matrix. This analysis leads to a better understanding of the time scales and molecular motions that govern the two-dimensional (2D) signal. The 2D Raman spectra show novel differences between the solid and liquid phases, which are associated with the decay rates of coherent motions. On the other hand, these differences are not observed in the linear Raman spectra.

Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra

This paper studies the behavior under iteration of the maps T ab (x,y) = (F ab (x) − y, x) of the plane ℝ2, in which F ab (x) = ax if x ≥ 0 and bx if x < 0. These maps are area-preserving homeomorphisms of ℝ2 that map rays from the origin to rays from the origin. Orbits of the map correspond to solutions of the nonlinear difference equation x n+2 = 1/2(a − b)|x n+1|+1/2(a+b)x n+1 − x n . This difference equation can be rewritten in an eigenvalue form for a nonlinear difference operator of Schrödinger type − x n+2+2x n+1 − x n +V μ(x n+1)x n+1 = Ex n+1, in which μ = (1/2)(a − b) is fixed, and V μ(x) = μ(sgn(x)) is an antisymmetric step function potential, and the energy E = 2 − 1/2(a+b). We study the set Ω SB of parameter values where the map T ab has at least one bounded orbit, which correspond to l ∞-eigenfunctions of this difference operator. The paper shows that for transcendental μ the set Spec∞[μ] of energy values E having a bounded solution is a Cantor set. Numerical simulations suggest the possibility that these Cantor sets have positive (one-dimensional) measure for all real values of μ.

The Total Meridional Heat Flux and Its Oceanic and Atmospheric Partition

Abstract Atmospheric meridional heat transport is inferred as a residual from the Earth Radiation Budget Experiment (ERBE) data and in situ oceanic estimates. Reversing the conventional approach of computing the ocean as an atmospheric model residual is done to permit calculation of a preliminary uncertainty estimate for the atmospheric flux. The structure of the ERBE errors is itself an important uncertainty. Total energy transport is almost indistinguishable from a hemispherically antisymmetric analytic function, despite the great asymmetry of the oceanic heat fluxes. ERBE data appear sufficiently noisy so that a considerable range of atmospheric transports remains possible: the maximum atmospheric value lies between 3 and 5 PW in the Northern Hemisphere, at one standard deviation, although the values are sensitive to the noise assumptions made here. The Northern Hemisphere ocean and atmosphere carry comparable poleward heat fluxes to about 28°N where the oceanic flux drops rapidly, but does not actually vanish until the oceanic surface area goes to zero. Within the estimated error bars, there is a remarkable antisymmetry about the equator of the combined ocean and atmospheric transports, despite the marked oceanic transport asymmetry.

On the posterior median estimators of possibly sparse sequences

We adopt the Bayesian paradigm and discuss certain properties of posterior median estimators of possibly sparse sequences. The prior distribution considered is a mixture of an atom of probability at zero and a symmetric unimodal distribution, and the noise distribution is taken as another symmetric unimodal distribution. We derive an explicit form of the corresponding posterior median and show that it is an antisymmetric function and, under some conditions, a shrinkage and a thresholding rule. Furthermore we show that, as long as the tails of the nonzero part of the prior distribution are heavier than the tails of the noise distribution, the posterior median, under some constraints on the involved parameters, has the bounded shrinkage property, extending thus recent results to larger families of prior and noise distributions. Expressions of posterior distributions and posterior medians in particular cases of interest are obtained. The asymptotes of the derived posterior medians, which provide valuable information of how the corresponding estimators treat large coefficients, are also given. These results could be particularly useful for studying frequentist optimality properties and developing statistical techniques of the resulting posterior median estimators of possibly sparse sequences for a wider set of prior and noise distributions.

Ultrafast molecular dynamics of liquid aromatic molecules and the mixtures with CCl4

The ultrafast molecular dynamics of liquid aromatic molecules, benzene, toluene, ethylbenzene, cumene, and 1,3-diphenylpropane, and the mixtures with CCl(4) have been investigated by means of femtosecond optical heterodyne-detected Raman-induced Kerr effect spectroscopy. The picosecond Kerr transients of benzene, toluene, ethylbenzene, and cumene and the mixtures with CCl(4) show a biexponential feature. 1,3-Diphenylpropane and the mixtures with CCl(4) show triexponential picosecond Kerr transients. The slow relaxation time constants of the aromatic molecules and the mixtures with CCl(4) are qualitatively described by the Stoke-Einstein-Debye hydrodynamic model. The ultrafast dynamics have been discussed based on the Kerr spectra in the frequency range of 0-800 cm(-1) obtained by the Fourier transform analysis of the Kerr transients. The line shapes of the low-frequency intermolecular spectra located at 0-180 cm(-1) frequency range have been analyzed by two Brownian oscillators ( approximately 11 cm(-1) and approximately 45 cm(-1) peaks) and an antisymmetric Gaussian function ( approximately 65 cm(-1) peak). The spectrum shape of 1,3-diphenylpropane is quite different from the spectrum shapes of the other aromatic molecules for the low magnitude of the low-frequency mode of 1,3-diphenylpropane and/or an intramolecular vibration. Although the concentration dependences of the low- and intermediate-frequency intermolecular modes (Brownian oscillators) do not show a significant trend, the width of high-frequency intermolecular mode (antisymmetric Gaussian) becomes narrower with the higher CCl(4) concentration for all the aromatics mixtures with CCl(4). The result indicates that the inhomogeneity of the intermolecular vibrational mode in aromatics/CCl(4) mixtures is decreasing with the lower concentration of aromatics. The intramolecular vibrational modes of the aromatic molecules observed in the Kerr spectra are also shown with the calculation results based on the density functional theory.

Algorithms for Numerical Analysis in High Dimensions

Nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension. The separated representation, introduced previously, allows many operations to be performed with scaling that is formally linear in the dimension. In this paper we further develop this representation by (i) discussing the variety of mechanisms that allow it to be surprisingly efficient; (ii) addressing the issue of conditioning; (iii) presenting algorithms for solving linear systems within this framework; and (iv) demonstrating methods for dealing with antisymmetric functions, as arise in the multiparticle Schrodinger equation in quantum mechanics. Numerical examples are given.

Electron-impact single and double ionization of helium

Electron-impact ionization cross sections for helium are calculated using time-dependent close-coupling theory. The total wave function for the three electron system is expanded in nine dimensions, where three dimensions are represented on a radial lattice and a coupled channels expansion is used to represent the other six dimensions. Collision cross sections are obtained by t{yields}{infinity} projection onto fully antisymmetric spatial and spin functions, with care as to orthogonality of different representations. Cross sections are also obtained using time-independent first- and second-order perturbative distorted-wave theory. Total cross sections are calculated at incident energies above the double ionization threshold for electron-impact single ionization leaving He{sup +} in the 1s, 2s, and 2p states and for electron-impact double ionization. Both the single ionization cross section, leaving He{sup +} in the 1s ground state, and the double ionization cross section are in excellent agreement with previous absolute experimental measurements.

Lattice calculations of the photoionization of Li.

Calculations are presented for the double photoionization (with excitation) and triple photoionization of the Li atom. The motion of all three electrons is treated equally by solving the time-dependent Schrödinger equation in nine dimensions. A radial lattice is used to represent three of the nine dimensions, while a coupled channel expansion is used to represent the other six dimensions. Probabilities for photoionization are obtained by t--> infinity projection onto fully antisymmetric spatial and spin functions. Double photoionization cross sections for lithium leaving the ion in the 1s, 2s, and 2p states are presented. Good agreement is found with the measurements of Huang et al. [Phys. Rev. A 59, 3397 (1999)]] for the total double photoionization cross section and with the measurements of Wehlitz et al. [Phys. Rev. Lett. 81, 1813 (1998)]] for the triple photoionization cross section.

Color opponencies in the system of the uniform color scales of the Optical Society of America.

Many problems regarding color opponency are still unsolved. In this study the system of the uniform color scales of the Optical Society of America (OSA-UCS) is analyzed with the aim of obtaining answers to very general questions on color opponency. The perceptual color opponencies in the OSA-UCS system, represented by its coordinates (j, g), appear to work in a mutually interacting way. On the hypothesis that such an interaction is due to a linear mixing of a pair of independent opponent mechanisms with scales satisfying a proper Weber fraction, three chromatic opponency functions are derived, whose sum is equal to zero. These functions are the logarithms of the ratios of two tristimulus values in a proper reference frame (called the "main reference frame") and therefore are antisymmetric and zero-degree homogeneous functions of these tristimulus values. Any pair of these three functions is a set of two independent functions. A new formula for color opponency in the OSA-UCS system is derived in which the perceptual color opponencies (j, g) are written as products of the lightness by a proper linear mixing of any pair of the three chromatic opponency functions. All this is possible because the lattice of the OSA-UCS system is composed of geodesic lines.

Spectral decomposition of an elementary 3-fermion 2-body operator

The eigenvalues and eigenfunctions of an elementary 3-fermion 2-body operator 3 P 2 g ∧ I 1 A 3∑ 1≤ i≤ j≤3 P 2 g ( i, j) A 3 acting on a 3-particle antisymmetric finite-dimensional Hilbert space have been found. Here P g 2 denotes the projection operator onto a 2-particle antisymmetric function g 2, while A 3 denotes the 3-particle antisymmetrizing operator.

Collisionless reconnection of magnetic field lines in dusty plasmas

Stability of a thin current sheet in an inhomogeneous dusty plasma is investigated. A new mechanism for the collisionless reconnection of the magnetic field is found. It is associated with the linear instability of a surface shear-Alfvén mode which exists at the boundary between two plasma regions with different orientations of the sheared magnetic field. This new mode is possible only in dusty plasmas in which the electron and ion densities are inhomogeneous and different from each other. The dispersion properties of the surface mode are found to be strongly dependent on the spatial profile of the current sheet. The unstable linear mode is found in the case of a current sheet with steep edges. It is identified as the bending- (or kink-) mode, whose electrostatic and vector potentials, φ and Az, are symmetric and antisymmetric functions, respectively. In such a plasma configuration, the surface tearing mode (φ-antisymmetric, Az-symmetric) cannot propagate as an eigenmode. It arises only as a forced mode through the resonant coupling with the bending mode and the local convective cell. The increment of the bending/tearing instability is estimated from the second-order contribution of the resonance.

Spectral signature of magnetic flux tubes in sunspot penumbrae

We study the polarization of spectral lines in the penumbra by integrating the radiative transfer equation of polarized light for a three-dimensional model atmosphere of a sunspot. In this model, the Evershed flow is confined to magnetic flux tubes which are embedded in a static background atmosphere, in accordance with the moving tube model of Schlichenmaier et al. (1998a,b). The gradients and/or discontinuities in the fluid velocity and the magnetic field at the flux tube boundaries give rise to asymmetric Stokes profiles. We concentrate on the Stokes-V profiles and study the net circular polarization (NCP) of two photospheric spectral lines of neutral iron, Fe I 630.25 nm and Fe I 1564.8 nm. The dierent behavior of these two lines, which are exemplary for atomic spectral lines with a large Lande factor and significantly dierent wavelength, is based on the dierence in excitation potential of the corresponding atomic transitions and the fact that the wavelength dependence of the Doppler shift is linear, while that of the Zeeman splitting is quadratic. We find that the azimuthal variation of the NCP,N( ), is a predominantly antisymmetric function of with respect to the line connecting disk center and spot center (line-of-symmetry) for the infrared line of Fe I 1564.8 nm, while the variation is predominantly symmetric for Fe I 630.25 nm. We show that the antisymmetric variation is caused by anomalous dispersion (Faraday pulsation) and the discontinuity in the azimuthal angle of the magnetic field, which is due to the relative inclination between flux tube and background field. We then compute synthetic NCP maps of a sunspot and compare them with observational results. Finally, the center-to-limb variation of the NCP,N(), of these spectral lines is investigated. We show that the location of the zero-crossing point ofN() on the center side of the line-of-symmetry represents a diagnostic tool to determine the inclination angle of the Evershed flow: A vanishing NCP on the center-side of the line-of-symmetry is an indirect evidence of downflows in the penumbra.