Integer Index Research Articles

Orthogonality from disjoint support in reproducing kernel Hilbert spaces

We investigate reproducing kernel Hilbert spaces (RKHS) where two functions are orthogonal whenever they have disjoint support. Necessary and sufficient conditions in terms of feature maps for the reproducing kernel are established. We also present concrete examples of finite dimensional RKHS and RKHS with a translation invariant reproducing kernel. In particular, it is shown that a Sobolev space has the orthogonality from disjoint support property if and only if it is of integer index.

Entropy of Patterns of i.i.d. Sequences—Part I: General Bounds

Tight bounds on the block entropy of patterns of sequences generated by independent and identically distributed (i.i.d.) sources are derived. A pattern of a sequence is a sequence of integer indices with each index representing the order of first occurrence of the respective symbol in the original sequence. Since a pattern is the result of data processing on the original sequence, its entropy cannot be larger. Bounds derived here describe the pattern entropy as function of the original i.i.d. source entropy, the alphabet size, the symbol probabilities, and their arrangement in the probability space. Matching upper and lower bounds derived provide a useful tool for very accurate approximations of pattern block entropies for various distributions, and for assessing the decrease of the pattern entropy from that of the original i.i.d. sequence.

Optimal structural design family by genetic search and ant colony approach

PurposeGenetic Algorithm, as a generalized constructive search method, has already been applied to various fields of optimization problems using different encoding schemes. In conventional GAs, the optimum solution is usually announced as the fittest feasible individual achieved in a limited number of generations. In this paper, such a pseudo‐optimum is extended to a neighborhood structure, known as optimal design family.Design/methodology/approachIn this paper, the constructive feature of genetic search is combined with trail update strategy of ant colony approach in a discrete manner, in order to sample more competitive individuals from various subspaces of the search space as a dynamic‐memory of updating design family.FindingsThe proposed method is applied to structural layout and size optimization utilizing an efficient integer index encoding and its appropriate genetic operators. Different applications of the proposed method are illustrated using three truss and frame examples. In the first example, topological classes are identified during layout optimization. In the second example, an objective function containing the stress response, displacement response, and the weight of the structure is considered to solve the optimal design of non‐braced frames. This approach allows the selection of less sensitive designs among the family of solutions. The third example is selected for eigenvalue maximization with minimal number of bracings and structural weight for braced frames.Originality/valueIn this paper, a pseudo‐optimum is extended to a neighborhood structure, known as optimal design family.

UNSYMMETRICAL AND SYMMETRICAL ONE-RANGE ADDITION THEOREMS FOR SLATER TYPE ORBITALS AND COULOMB–YUKAWA-LIKE CORRELATED INTERACTION POTENTIALS OF INTEGER AND NONINTEGER INDICES

Using one-center expansion relations for the Slater type orbitals (STOs) of noninteger principal quantum numbers in terms of integer nSTOs derived in this study with the help of ψa-exponential type orbitals (ψa-ETOs, a = 1, 0, -1, -2,…), the general formulas through the integer nSTOs are established for the unsymmetrical and symmetrical one-range addition theorems for STOs and Coulomb–Yukawa-like correlated interaction potentials (CIPs) with integer and noninteger indices. The final results are especially useful for the computations of arbitrary multicenter multielectron integrals that arise in the Hartree–Fock–Roothaan (HFR) approximation and also in the correlated methods based upon the use of STOs as basis functions.

A high-speed, low-resource ASR back-end based on custom arithmetic

With the skyrocketing popularity of mobile devices, new processing methods tailored to a specific application have become necessary for low-resource systems. This work presents a high-speed, low-resource speech recognition system using custom arithmetic units, where all system variables are represented by integer indices and all arithmetic operations are replaced by hardware-based table lookups. To this end, several reordering and rescaling techniques, including two accumulation structures for Gaussian evaluation and a novel method for the normalization of Viterbi search scores, are proposed to ensure low entropy for all variables. Furthermore, a discriminatively inspired distortion measure is investigated for scalar quantization of forward probabilities to maximize the recognition rate. Finally, heuristic algorithms are explored to optimize system-wide resource allocation. Our best bit-width allocation scheme only requires 59 kB of ROMs to hold the lookup tables, and its recognition performance with various vocabulary sizes in both clean and noisy conditions is nearly as good as that of a system using a 32-bit floating-point unit. Simulations on various architectures show that, on most modern processor designs, we can expect a cycle-count speedup of at least three times over systems with floating-point units. Additionally, the memory bandwidth is reduced by over 70% and the offline storage for model parameters is reduced by 80%.

Towards a classification of icosahedral viruses in terms of indexed polyhedra

The standard Caspar & Klug classification of icosahedral viruses by means of triangulation numbers and the more recent novel characterization of Twarock leading to a Penrose-like tessellation of the capsid of viruses not obeying the Caspar-Klug rules can be obtained as a special case in a new approach to the morphology of icosahedral viruses. Considered are polyhedra with icosahedral symmetry and rational indices. The law of rational indices, fundamental for crystals, implies vertices at points of a lattice (here icosahedral). In the present approach, in addition to the rotations of the icosahedral group 235, crystallographic scalings play an important rôle. Crystallographic means that the scalings leave the icosahedral lattice invariant or transform it to a sublattice (or to a superlattice). The combination of the rotations with these scalings (linear, planar and radial) permits edge, face and vertex decoration of the polyhedra. In the last case, satellite polyhedra are attached to the vertices of a central polyhedron, the whole being generated by the icosahedral group from a finite set of points with integer indices. Three viruses with a polyhedral enclosing form given by an icosahedron, a dodecahedron and a triacontahedron, respectively, are presented as illustration. Their cores share the same polyhedron as the capsid, both being in a crystallographic scaling relation.

Azimuthons: Spatially Modulated Vortex Solitons

We introduce a novel class of spatially localized self-trapped ringlike singular optical beams in nonlinear media, the so-called azimuthons, which appear due to a continuous azimuthal deformation of vortex solitons. We demonstrate that the azimuthons are characterized by two independent integer indices, the topological charge m and the number N of the intensity peaks along the ring. Each soliton family includes azimuthons with negative, positive, and zero angular velocity.

The generalized M sets from a class complex mapping system

The method constructing the generalized Mandelbrot sets (the generalized M sets) for positive integer index number generated from the composition of two simple complex mapping which was put forward by Shirriff was expanded. Based on a class simple complex mapping system expanded by the author, a series of the generalized M sets for real index number were constructed. Using the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, the structural characteristics and evolutions of the generalized M sets were studied. The results show that the generalized M sets for integer index number have symmetry and fractal feature; while the generalized M sets for decimal index number have discontinuity and collapse, and their evolutions depend on the choice of the principal range of the phase angle. And the author sets forth the physical meaning for the generalized M sets.

R modes of slowly pulsating B stars

Slowly pulsating B (SPB) stars are g-mode pulsators in main-sequence stages with mass ranging from M∼ 3 to ∼8 M⊙. In this paper, we examine pulsational stability of low-m r modes in SPB stars by calculating fully non-adiabatic oscillations of uniformly rotating stars, where m is an integer representing the azimuthal wavenumber around the rotation axis. r modes are rotationally induced, non-axisymmetric, oscillation modes, whose oscillation frequency strongly depends on the rotation frequency Ω of the star. They are conveniently classified by using two integer indices m and l′≥ ∼m∼ that define the asymptotic oscillation frequency 2mΩ/[l′(l′+ 1)] in the limit of Ω→ 0. We confirm Savonije's and Townsend's finding that low-m, high-radial order, odd r modes with l′=m in SPB stars are excited by the same iron opacity bump mechanism that excites low-frequency g modes of the variables, when the rotation frequency Ω is sufficiently high. No even r modes with low m are found to be pulsationally unstable. Since the surface pattern of the temperature perturbation of odd modes is antisymmetric about the equator of the star, observed photometric amplitudes caused by the unstable odd r modes with l′=m are strongly dependent on the inclination angle between the axis of rotation and the line of sight. Applying the wave-meanflow interaction formalism to non-adiabatic r modes in rapidly rotating SPB models, we find that because of the r φ component of the Reynolds stress and the radial transport of the eddy fluctuation of density in the rotating star, the surface rotation is accelerated by the forcing due to the low l′=m unstable r modes. We suggest that the amount of angular momentum redistribution in the surface region of the stars can be comparable to that needed to sustain decretion discs found in Be systems. We make a brief comparison between non-adiabatic r-mode calculations done with and without the traditional approximation. In the traditional approximation, the local horizontal component of the rotation vector Ω is ignored in the momentum conservation equation, which makes it possible to represent the angular dependence of an oscillation mode using a single Hough function. We find that the oscillation frequencies of low-m r modes computed with and without the traditional approximation are qualitatively in good agreement. We also find that the pulsational instability of r modes in the traditional approximation appears weaker than that without the approximation.

Generalized index for Hamiltonian systems with applications

Some classes of linear Hamiltonian equations with periodic coefficients (nondegenerate, strongly stable, completely unstable) are determined by the disposition of the Floquet multipliers. Periodic solutions of nonlinear equations (e.g. elliptic or hyperbolic solutions) are also defined by the multipliers of the corresponding variational equation. In this paper, we consider a general set M of linear Hamiltonian equations with multipliers satisfying some arbitrary conditions and a specific condition on the multiplier lying at some point of the unit circle (all known sets admit such a definition). We show that the set M consists of a finite number of subsets Mi which comprise a countable number of domains within which any two Hamiltonians can be continuously deformed into each other. The corresponding integer index q is expressed through the eigenvalues of some self-adjoint problem. It is shown that this index (and, therefore, the known indices relating to specific sets) increases on increasing the Hamiltonian. Using the obtained results, some known and new sets are studied from the unified point of view. It is shown that for the sets of nondegenerate and completely unstable equations, the domains are directionally convex; for strongly stable equations, necessary and sufficient conditions for directional convexity are found. The results are applied to problems of existence and stability of periodic solutions of nonlinear Hamiltonian equations.

A game based on spectral graph theory

We present a mathematical game for two players based on spectral graph theory. We solve some cases and discuss a general strategy for unsolved ones. In addition, we present some data on graphs with integer index.

Effect of suprathermal particles on the quiet Sun radio emission

The bremsstrahlung emissivity and absorption coefficient in the radiofrequency range are derived under the assumption that the electron population is not purely thermal, but presents a tail of high energy particles. This population is approximated by a two-component Maxwellian distribution and by the kappa-functions of different (integer) index. It is shown that, if the temperature ratio of the two Maxwellians is larger than 10, the absorption coefficient and the effective temperature (the quantities entering the radio transfer equation) depend only on the fraction R of particles in the highest temperature Maxwellian. In the case of kappa-functions the above quantities depend on the index n of the functions. The microwave radio spectrum is computed for different values of R and for $3 \leq n \leq 6$, finding, in all cases, brightness temperatures lower than those computed with a pure thermal distribution. This could explain some inconsistencies found between radio and EUV observations.

Numerical Representations of the Incomplete Gamma Function of Complex-Valued Argument

Various approaches to the numerical representation of the incomplete Gamma function γ(m+1/2,z) for complex arguments z and non-negative small integer indices m are compared with respect to numerical fitness (accuracy and speed). We consider power series, Laurent series, classical numerical methods of sampling the basic integral representation, and others not yet covered by the literature. The most suitable scheme is the construction of Taylor expansions around nodes of a regular, fixed grid in the z-plane, which stores a static matrix of higher derivatives. This is the obvious extension to a procedure that is in common use for real-valued z.

Minima of initial segments of infinite sequences of reals

AbstractSuppose that 〈xk〉k∈ℕ is a countable sequence of real numbers. Working in the usual subsystems for reverse mathematics, RCA0 suffices to prove the existence of a sequence of reals 〈uk〉k∈ℕ such that for each k, uk is the minimum of {x0, x1, …, xk}. However, if we wish to prove the existence of a sequence of integer indices of minima of initial segments of 〈xk〉k∈ℕ, the stronger subsystem WKL0 is required. Following the presentation of these reverse mathematics results, we will derive computability theoretic corollaries and use them to illustrate a distinction between computable analysis and constructive analysis. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Interplay of topology and quantization: topological energy quantization in a cavity

The interplay between quantization and topology is investigated in the frame of a topological model of electromagnetism proposed by the author. In that model, the energy of electromagnetic radiation in a cubic cavity is E=(d/4)ℏω where d is a topological integer index equal to the degree of a map between two orbifolds.

Boolean Scheme for Programming Trial Moves That Involve Molecule Insertion and Removal in Monte Carlo Simulation

A Boolean scheme is presented for programming trial moves that involve molecule insertion and removal in both grand canonical (GCMC) and Gibbs ensemble (GEMC) Monte Carlo simulation methods. The Boolean scheme makes use of logical data type arrays for keeping track of molecules being inserted and removed, instead of the integer index array technique of the traditional Nicholson's scheme. An exploratory evaluation of the relative computational performance of the two schemes is presented. For GCMC simulation of simple fluids, Nicholson's scheme appears to be more efficient than the Boolean scheme. In contrast, for GEMC simulation of vapor-liquid equilibria (VLE) in binary systems, the Boolean scheme seems to become more efficient as the complexity of the intermolecular potential increases. However, considering the limitations intrinsic to any evaluation of computational performance, the two schemes should be regarded as alternative tools to be tried for the particular application of interest to the user.

Generalized Hermite polynomials associated with functions of parabolic cylinder

Associated Hermite polynomials He ν n ( z) which generalize the usual (scaled) Hermite polynomials He n ( z) corresponding to ν=0 are introduced for the purpose to represent the raising and lowering of the indices of functions of the parabolic cylinder D ν ( z) in finite integer steps. Properties of these polynomials such as recursion relations, explicit representations and the differential equation are derived. The generation of the associated Hermite polynomials from the usual Hermite polynomials by differential operators representable by means of the confluent hypergeometric function is given. An application for the explicit calculation of the functions of the parabolic cylinder for negative integer indices is discussed. Other applications are visible for the investigation of the zeros of the functions of the parabolic cylinder.

Proof of the generalized Lieb-Wehrl conjecture for integer indices larger than one

Gnutzmann and Zyczkowski have proposed the Renyi-Wehrl entropy as a generalization of the Wehrl entropy, and conjectured that its minimum is obtained for coherent states. We prove this conjecture for the Renyi index q=2,3,... in the cases of compact semisimple Lie groups. A general formula for the minimum value is given.

Performance evaluation of generalized selection diversity systems over Nakagami‐m fading channels

AbstractThe generalized selection combining (GSC) scheme that adaptively combines a subset of M strongest paths out of L available diversity paths finds applications in several wideband receivers and broadband wireless communications. In this paper, exact closed‐form expressions for the moment generating function (MGF), the probability density function (PDF) and the cumulative density function (CDF) of the GSC(M, L) output signal‐to‐noise ratio (SNR) in independent and identically distributed (i.i.d) Nakagami‐m fading channels are derived while the fading index is a positive integer. These expressions hold for any M and L and provide a comprehensive framework for performance analysis including the derivation of closed‐form formulas for the average symbol error probability (ASEP) of a broad class of binary and M‐ary modulations, mean combined SNR and the outage probability of GSC(M, L) receiver structures. When the Nakagami‐m fading index is not an integer, the MGF of GSC(M, L) output SNR is derived as an (M − 1)‐fold infinite series. With this MGF, analytical expressions for both the outage probability and error rates can be readily obtained. An easily programmable recursive solution of the MGF of GSC(M, L) output SNR is also outlined for both the positive integer and noninteger fading severity index cases. Copyright © 2002 John Wiley & Sons, Ltd.

Form and folding of pentameric human serum Amyloid P component

Abstract The molecular prismatic forms which enclose the tertiary and the quaternary structure of the human serum Amyloid P component show remarkable ratios expressible in terms of the golden mean. A scale-rotation crystallographic point group, leaving these forms invariant, allows to explain these golden mean proportions and to assign integer indices to the vertices of the molecular forms. Cα -positions at form-confined folding points are identified and discussed on the basis of scale-rotations relating a number of corresponding ideal positions indexed by rational numbers.