Integral Representation In Terms Research Articles

Statistical mechanics of learning via reverberation in bidirectional associative memories

We study bi-directional associative neural networks that, exposed to noisy examples of an extensive number of random archetypes, learn the latter (with or without the presence of a teacher) when the supplied information is enough: in this setting, learning is heteroassociative – involving couples of patterns – and it is achieved by reverberating the information depicted from the examples through the layers of the network. By adapting Guerra’s interpolation technique, we provide a full statistical mechanical picture of supervised and unsupervised learning processes (at the replica symmetric level of description) obtaining analytically phase diagrams, thresholds for learning, a picture of the ground-state in plain agreement with Monte Carlo simulations and signal-to-noise outcomes. In the large dataset limit, the Kosko storage prescription as well as its statistical mechanical picture provided by Kurchan, Peliti, and Saber in the eighties is fully recovered. Computational advantages in dealing with information reverberation, rather than storage, are discussed for natural test cases. In particular, we show how this network admits an integral representation in terms of two coupled restricted Boltzmann machines, whose hidden layers are entirely built of by grand-mother neurons, to prove that by coupling solely these grand-mother neurons we can correlate the patterns they are related to: it is thus possible to recover Pavlov’s Classical Conditioning by adding just one synapse among the correct grand-mother neurons (hence saving an extensive number of these links for further information storage w.r.t. the classical auto-associative setting). Likewise, it is enough to equip the BAM’s synaptic coupling with its transpose matrix (and alternate among these two, e.g., by introducing a clock) to let these networks be able count sequences of patterns (still preserving Detailed Balance, at difference w.r.t. the auto-associative case).

Low Energy Analysis of Phase Shifts and Cross-Section for n – 3He System

"Considering the Hulth´en potential as the nuclear part of interaction regular solution is derived by exploiting the Frobenius method. The Jost function is constructed by utilizing its integral representation in terms of regular solution. The scattering phase shifts and cross-sections are calculated for the n−3He system from the on-shell Jost function. The results obtained are in close conformity with the previous works that exist in the literature."

Nonnegative solutions of the heat equation in a cylindrical domain and Widder's theorem

It is shown every nonnegative solution of the heat equation in a bounded cylindrical domain has an integral representation in terms of a trace triple consisting of a bottom trace, a corner trace and a lateral trace on its parabolic boundary. Conversely this trace triple uniquely determines the solution.

Analytical solution for nuclear plus atomic Hulthén potential — A perturbative approach

In this work, we construct the analytical solution of the Schrödinger equation for a combined nuclear plus atomic Hulthén potential with different range parameters, using the Frobenius method. The atomic Hulthén potential acts as the screened Coulomb potential to represent the very short-range electromagnetic interaction. It is intended to emphasize how the impact of the combined potential in these situations is routinely examined within the context of nuclear physics. The Jost function is calculated from its integral representation in terms of the regular solution. From the phase of the Jost function, the scattering phase shifts for different partial waves, and further differential scattering cross-section and total cross-sections are calculated for alpha-proton and proton–proton systems. On comparing with the existing experimental data, we conclude that the results obtained are in close conformity with the previous works that exist in the literature.

Asymptotics of the deformed higher order Airy-kernel determinants and applications

We study the one-parameter family of Fredholm determinants , , where stands for the integral operator acting on with the higher order Airy kernel. This family of determinants represents a new universal class of distributions which is a higher order analogue of the classical Tracy–Widom distribution. Each of the determinants admits an integral representation in terms of a special real solution to the nth member of the Painlevé II hierarchy. Using the Riemann–Hilbert approach, we establish asymptotics of the determinants and the associated higher order Painlevé II transcendents as for and , respectively. In the case of , we are able to calculate the constant term in the asymptotic expansion of the determinants, while for , the relevant asymptotics exhibit singular behaviours. Applications of our results are also discussed, which particularly include asymptotic statistical properties of the counting function for the random point process defined by the higher order Airy kernel.

Painlevé IV, Chazy II, and asymptotics for recurrence coefficients of semi‐classical Laguerre polynomials and their Hankel determinants

This paper studies the monic semi‐classical Laguerre polynomials based on previous work by Boelen and Van Assche, Filipuk et al., and Clarkson and Jordaan. Filipuk et al. proved that the diagonal recurrence coefficient satisfies the fourth Painlevé equation. In this paper, we show that the off‐diagonal recurrence coefficient fulfills the first member of Chazy II system. We also prove that the sub‐leading coefficient of the monic semi‐classical Laguerre polynomials satisfies both the continuous and discrete Jimbo–Miwa–Okamoto ‐form of Painlevé IV. By using Dyson's Coulomb fluid approach together with the discrete system for and , we obtain the large asymptotic expansions of the recurrence coefficients and the sub‐leading coefficient. The large asymptotics of the associated Hankel determinant (including the constant term) is derived from its integral representation in terms of the sub‐leading coefficient.

Analytic continuation and physical content of the gluon propagator

The analytic continuation of the gluon propagator is revised in the light of recent findings on the possible existence of complex conjugated poles. The contribution of the anomalous pole must be added when Wick rotating, leading to an effective Minkowskian propagator which is not given by the trivial analytic continuation of the Euclidean function. The effective propagator has an integral representation in terms of a spectral function which is naturally related to a set of elementary (complex) eigenvalues of the Hamiltonian, thus generalizing the usual K\"all\'en-Lehmann description. A simple toy model shows how the elementary eigenvalues might be related to actual physical quasiparticles of the nonperturbative vacuum.

Íntima relação entre o fator integrante e a função de Green para equações diferenciais de primeira ordem

For first-order ordinary differential equations, it is shown by a very simple and straightforward approach how the integral representation of the particular solution with the integrating factor as part of the integrand can be manipulated in favor of an integral representation in terms of the Green function.

Non-Zero Order of an Extended Temme Integral

A new three-dimensional integral containing f(x,y,z)Iv(xα) is derived where Iv(xα) is the Modified Bessel Function of the first kind and the integral is taken over the infinite cubic space 0<x<∞,0<y<∞,0<z<∞. The integral is not easily evaluated for complex ranges of the parameters. A representation in terms of the Hurwitz–Lerch zeta function, polylogarithm function and Riemann zeta functions are evaluated. This representation yields triple integral representations in terms of fundamental constants that can be derived. Almost all Lerch functions have an asymmetrical zero distribution.

Periodic homogenization in the context of structured deformations

An energy for first-order structured deformations in the context of periodic homogenization is obtained. This energy, defined in principle by relaxation of an initial energy of integral-type featuring contributions of bulk and interfacial terms, is proved to possess an integral representation in terms of relaxed bulk and interfacial energy densities. These energy densities, in turn, are obtained via asymptotic cell formulae defined by suitably averaging, over larger and larger cubes, the bulk and surface contributions of the initial energy. The integral representation theorem, the main result of this paper, is obtained by mixing blow-up techniques, typical in the context of structured deformations, with the averaging process underlying the theory of homogenization.

Asymptotic relations for semi-classical Laguerre orthogonal polynomials and the associated Hankel determinants

We study recurrence coefficients of semi-classical Laguerre orthogonal polynomials and the associated Hankel determinant generated by a semi-classical Laguerre weight w(x,t)=xαe−x−tx2,x∈(0,∞),α>0,t≥0. If t = 0, it is reduced to the classical Laguerre weight. For t > 0, this weight tends to zero faster than the classical Laguerre weight as x → ∞. In the finite n-dimensional case, we obtain two auxiliary quantities Rn(t) and rn(t) by using the Ladder operator approach. We show that the Hankel determinant has an integral representation in terms of Rn(t), where the quantity Rn(t) is closely related to a second-order nonlinear differential equation. Furthermore, we derive a second-order nonlinear differential equation and also a second-order differential equation for the auxiliary quantity σn(t)=−∑j=0n−1Rj(t), which is also related to the logarithmic derivative of the Hankel determinant. In the infinite n-dimensional case, we consider the asymptotic behaviors of the recurrence coefficients and the scaled Laguerre orthogonal polynomials by using the Coulomb fluid method.

Semi-classical Jacobi polynomials, Hankel determinants and asymptotics

We study orthogonal polynomials and Hankel determinants generated by a symmetric semi-classical Jacobi weight. By using the ladder operator technique, we derive the second-order nonlinear difference equations satisfied by the recurrence coefficient $\beta_n(t)$ and the sub-leading coefficient $\mathrm{p}(n,t)$ of the monic orthogonal polynomials. This enables us to obtain the large $n$ asymptotics of $\beta_n(t)$ and $\mathrm{p}(n,t)$ based on the result of Kuijlaars et al. [Adv. Math. \textbf{188} (2004) 337-398]. In addition, we show the second-order differential equation satisfied by the orthogonal polynomials, with all the coefficients expressed in terms of $\beta_n(t)$. From the $t$ evolution of the auxiliary quantities, we prove that $\beta_n(t)$ satisfies a second-order differential equation and $R_n(t)=2n+1+2\alpha-2t(\beta_n(t)+\beta_{n+1}(t))$ satisfies a particular Painlev\'{e} V equation under a simple transformation. Furthermore, we show that the logarithmic derivative of the associated Hankel determinant satisfies both the second-order differential and difference equations. The large $n$ asymptotics of the Hankel determinant is derived from its integral representation in terms of $\beta_n(t)$ and $\mathrm{p}(n,t)$.

An advanced boundary integral equation method for wave propagation analysis in a layered piezoelectric phononic crystal with a crack or an electrode

The paper presents an extended and advanced boundary integral equation method (BIEM) for simulating the elastic wave excitation and propagation in a layered piezoelectric phononic crystal with a strip-like crack or electrode. The method is based on the integral representation in terms of the Fourier transform of the Green's matrix for the whole piezoelectric laminate. A novel algorithm for constructing the Green's matrix, which allows taking into account the periodicity of the structure, is proposed. The obtained boundary integral equations are solved numerically using the Bubnov-Galerkin method with Chebyshev polynomials of the first and the second kind for a crack and an electrode respectively. The present method is compared with the standard finite element method (FEM), and the efficiency and convergence of the present method are also demonstrated by several representative numerical examples. The main advantages of the present extended and advanced BIEM lie in its strong capability of simulating unbounded piezoelectric laminates with a large number of layers (periodically or non-periodically arranged) as well as its high accuracy and efficiency compared with the FEM. The latter allows us to apply the method for a comprehensive parametric analysis and for the complex cases of multiple cracks and/or electrodes as well as for periodic arrays of cracks and/or electrodes.

CDF of non-central [formula omitted] distribution revisited. Incomplete hypergeometric type functions approach

The cumulative distribution function of the non-central chi-square distribution χν′2(λ),ν∈R+ possesses an integral representation in terms of a generalized Marcum Q-function. Regarding some already known-results, here we derive a simpler form of the cumulative distribution function for ν=2n∈N degrees of freedom. Also, we express these representations in terms of an incomplete Fox–Wright function pΨq(γ) and the generalized incomplete hypergeometric functions concerning the important special cases such as 1Γ1,2Γ1 and 2γ1. New identities are established between 1Γ1 and 2Γ1 as well.

A common parametrization for finite mode Gaussian states, their symmetries, and associated contractions with some applications

Let Γ(H) be the boson Fock space over a finite dimensional Hilbert space H. It is shown that every Gaussian symmetry admits a Klauder–Bargmann integral representation in terms of coherent states. Furthermore, Gaussian states, Gaussian symmetries, and second quantization contractions belong to a weakly closed self-adjoint semigroup E2(H) of bounded operators in Γ(H). This yields a common parametrization for these operators. It is shown that the new parametrization for Gaussian states is a fruitful alternative to the customary parametrization by position–momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries: (i) every Gaussian state ρ admits a factorization ρ=Z1†Z1, where Z1 is an element of E2(H) and has the form Z1=cΓ(P)exp∑r=1nλrar+∑r,s=1nαrsaras on the dense linear manifold generated by all exponential vectors, where c is a positive scalar, Γ(P) is the second quantization of a positive contractive operator P in H, ar, 1 ≤ r ≤ n, are the annihilation operators corresponding to the n different modes in Γ(H), λr∈C, and [αrs] is a symmetric matrix in Mn(C); (ii) an explicit particle basis expansion of an arbitrary mean zero pure Gaussian state vector along with a density matrix formula for a general Gaussian state in terms of its E2(H)-parameters; (iii) a class of examples of pure n-mode Gaussian states that are completely entangled; (iv) tomography of an unknown Gaussian state in Γ(Cn) by the estimation of its E2(Cn) parameters using O(n2) measurements with a finite number of outcomes.

The unitary extension principle for locally compact abelian groups with co-compact subgroups

The unitary extension principle by Ron and Shen is one of the cornerstones of wavelet frame theory; it leads to tight frames for L 2 ( R ) L^{2}(\mathbb {R}) and associated expansions of functions f ∈ L 2 ( R ) f\in L^{2}(\mathbb {R}) of similar type as those for orthonormal wavelet bases. In this paper, the unitary extension principle is extended to the setting of a locally compact abelian group, equipped with a collection of nested co-compact subgroups. Unlike all previously known generalizations of the unitary extension principle, the current one is taking place within the setting of continuous frames, which means that the resulting decompositions of functions in the underlying Hilbert space in general are given in terms of integral representations rather than discrete sums. The frame elements themselves appear by letting a collection of modulation operators act on a countable family of basic functions.

Painlevé V and the Hankel determinant for a singularly perturbed Jacobi weight

We study the Hankel determinant generated by a singularly perturbed Jacobi weightw(x,t):=(1−x2)αe−tx2,x∈[−1,1],α>0,t≥0. If t=0, it is reduced to the classical symmetric Jacobi weight. For t>0, the factor e−tx2 induces an infinitely strong zero at the origin. This Hankel determinant is related to the Wigner time-delay distribution in chaotic cavities.In the finite n dimensional case, we obtain two auxiliary quantities Rn(t) and rn(t) by using the ladder operator approach. We show that the Hankel determinant has an integral representation in terms of Rn(t), where Rn(t) is closely related to a particular Painlevé V transcendent. Furthermore, we derive a second-order nonlinear differential equation and also a second-order difference equation for the logarithmic derivative of the Hankel determinant. This quantity can be expressed in terms of the Jimbo-Miwa-Okamoto σ-function of a particular Painlevé V. Then we consider the asymptotics of the Hankel determinant under a suitable double scaling, i.e. n→∞ and t→0 such that s=2n2t is fixed. Based on previous results by using the Coulomb fluid method, we obtain the large s and small s asymptotic behaviors of the scaled Hankel determinant, including the constant term in the asymptotic expansion.

Uniform flow past a closed body at low Reynolds number employing a novel matching in a boundary element formulation

Consider a two dimensional steady low Reynolds number flow past a circular cylinder. A boundary integral representation that matches an outer Oseen flow and inner Stokes flow is given, and the matching error is shown to be smallest when the outer domain is as close as possible to the body. Also, it is shown that as the Greenâs function is approached, the oseenlet becomes the stokeslet to leading order and has the same order of magnitude error as the matching error. This means a novel boundary integral representation in terms of oseenlets is possible. To test this, a corresponding boundary element code is developed which uses point collocation weighting functions, linear shape functions, two-point Gaussian quadrature with analytic removal of the Greenâs function singularity for the integrations. The method is compared against various methods for the benchmark problem of flow past a circular cylinder. In particular, the drag coefficient is used for the comparison. The advantage of this method over existing ones is demonstrated and discussed particularly in the Reynolds number range Re=1∼4.

Age representation of Lévy walks: partial density waves, relaxation and first passage time statistics

Lévy walks (LWs) define a fundamental class of finite velocity stochastic processes that can be introduced as a special case of continuous time random walks. Alternatively, there is a hyperbolic representation of them in terms of partial probability density waves. Using the latter framework we explore the impact of aging on LWs, which can be viewed as a specific initial preparation of the particle ensemble with respect to an age distribution. We show that the hyperbolic age formulation is suitable for a simple integral representation in terms of linear Volterra equations for any initial preparation. On this basis relaxation properties, i.e. the convergence towards equilibrium of a generic thermodynamic function dependent on the spatial particle distribution, and first passage time statistics in bounded domains are studied by connecting the latter problem with solute release kinetics. We find that even normal diffusive LWs, where the long-term mean square displacement increases linearly with time, may display anomalous relaxation properties such as stretched exponential decay. We then discuss the impact of aging on the first passage time statistics of LWs by developing the corresponding Volterra integral representation. As a further natural generalization the concept of LWs with wearing is introduced to account for mobility losses.

Peters type polynomials and numbers and their generating functions: Approach with p‐adic integral method

The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p‐adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well‐known special numbers and polynomials are presented. By using p‐adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol‐type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol‐Bernoulli polynomials, the Apostol‐Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well‐known formulas. Finally, two open problems for interpolation functions for Apostol‐type Peters numbers and polynomials are revealed.