Expansion Of Sum Research Articles

Analytic continuation of harmonic sums with purely imaginary indices near the integer values

We present a simple algebraic method for the analytic continuation of harmonic sums with integer real or purely imaginary indices near negative and positive integers. We provide a MATHEMATICA code for exact expansion of harmonic sums in a small parameter near these integers. As an application, we consider the analytic continuation of the anomalous dimension of twist-1 operators in the ABJM model, which contains nested harmonic sums with purely imaginary indices. We found that in the BFKL-like limit the result has the same single-logarithmic behavior as in [Formula: see text] SYM and QCD, however, we did not find a general expression for the “BFKL Pomeron” eigenvalue in this model. For the slope function, we found full agreement with the expansion of the known general result and give predictions for the first three perturbative terms in the expansion of the next-to-slope function. The proposed method of analytic continuation can also be used for other generalization of nested harmonic sums.

A Berry–Esseen theorem and Edgeworth expansions for uniformly elliptic inhomogeneous Markov chains

We prove a Berry–Esseen theorem and Edgeworth expansions for partial sums of the form \(\displaystyle S_N=\sum \nolimits _{n=1}^{N}f_n(X_n,X_{n+1})\), where \(\{X_n\}\) is a uniformly elliptic inhomogeneous Markov chain and \(\{f_n\}\) is a sequence of uniformly bounded functions. The Berry–Esseen theorem holds without additional assumptions, while expansions of order 1 hold when \(\{f_n\}\) is irreducible, which is an optimal condition. For higher order expansions, we then focus on two situations. The first is when the essential supremum of \(f_n\) is of order \(O(n^{-{\beta }})\) for some \({\beta }\in (0,1/2)\). In this case it turns out that expansions of any order \(r<\frac{1}{1-2{\beta }}\) hold, and this condition is optimal. The second case is uniformly elliptic chains on a compact Riemannian manifold. When \(f_n\) are uniformly Lipschitz continuous we show that \(S_N\) admits expansions of all orders. When \(f_n\) are uniformly Hölder continuous with some exponent \({\alpha }\in (0,1)\), we show that \(S_N\) admits expansions of all orders \(r<\frac{1+{\alpha }}{1-{\alpha }}\). For Hölder continues functions with \({\alpha }<1\) our results are new also for uniformly elliptic homogeneous Markov chains and a single functional \(f=f_n\). In fact, we show that the condition \(r<\frac{1+{\alpha }}{1-{\alpha }}\) is optimal even in the homogeneous case.

Asymptotic Expansions for Symmetric Statistics with Degenerate Kernels

Asymptotic expansions for U-statistics and V-statistics with degenerate kernels are investigated, respectively, and the remainder term O(n1−p/2), for some p≥4, is shown in both cases. From the results, it is obtained that asymptotic expansions for the Crame´r–von Mises statistics of the uniform distribution U(0,1) hold with the remainder term On1−p/2 for any p≥4. The scheme of the proof is based on three steps. The first one is the almost sure convergence in a Fourier series expansion of the kernel function u(x,y). The key condition for the convergence is the nuclearity of a linear operator Tu defined by the kernel function. The second one is a representation of U-statistics or V-statistics by single sums of Hilbert space valued random variables. The third one is to apply asymptotic expansions for single sums of Hilbert space valued random variables.

Bounds via spectral radius-preserving row sum expansions

We show a simple method for constructing larger dimension nonnegative matrices with somewhat arbitrary entries which can be irreducible or reducible but preserving the spectral radius via row sum expansions. This yields a sufficient criteria for two square nonnegative matrices of arbitrary dimension to have the same spectral radius, a way to compare spectral radii of two arbitrary square nonnegative matrices, and a way to derive new upper and lower bounds on the spectral radius which give the standard row sum bounds as a special case.

Characters and Chromatic Symmetric Functions

Let $P$ be a poset, $\mathrm{inc}(P)$ its incomparability graph, and $X_{\mathrm{inc}(P)}$ the corresponding chromatic symmetric function, as defined by Stanley in Adv. Math., 111 (1995) pp.166–194. Let $\omega$ be the standard involution on symmetric functions. We express coefficients of $X_{\mathrm{inc}(P)}$ and $\omega X_{\mathrm{inc}(P)}$ as character evaluations to obtain simple combinatorial interpretations of the power sum and monomial expansions of $\omega X_{\mathrm{inc}(P)}$ which hold for all posets $P$. Consequences include new combinatorial interpretations of the permanent, induced trivial character immanants, and power sum immanants of totally nonnegative matrices, and of the sum of elementary coefficients in the Shareshian-Wachs chromatic quasisymmetric function $X_{\mathrm{inc}(P),q}$ when $P$ is a unit interval order.

The Top-Degree Part in the Matchings-Jack Conjecture

In 1996 Goulden and Jackson introduced a family of coefficients $( c_{\pi, \sigma}^{\lambda} ) $ indexed by triples of partitions which arise in the power sum expansion of some Cauchy sum for Jack symmetric functions $(J^{(\alpha )}_\pi )$. The coefficients $ c_{\pi, \sigma}^{\lambda} $ can be viewed as an interpolation between the structure constants of the class algebra and the double coset algebra. Goulden and Jackson suggested that the coefficients $ c_{\pi, \sigma}^{\lambda} $ are polynomials in the variable $\beta := \alpha-1$ with non-negative integer coefficients and that there is a combinatorics of matching hidden behind them. This Matchings-Jack Conjecture remains open. Dołȩga and Féray showed the polynomiality of connection coefficients $c^\lambda_{\pi,\sigma}$ and gave an upper bound on the degrees. We show a dual approach to this problem and investigate Jack characters and their connection coefficients. We give a necessary and sufficient condition for the polynomial $ c_{\pi, \sigma}^{\lambda}$ to achieve this bound. We show that the leading coefficient of $ c_{\pi, \sigma}^{\lambda}$ is a positive integer and we present it in the context of Matchings-Jack Conjecture of Goulden and Jackson.

Asymptotic expansion of a finite sum involving harmonic numbers

&lt;abstract&gt;&lt;p&gt;In the paper, we obtain asymptotic expansion of the finite sum of some sequences $ S_{n} = \sum_{k = 1}^{n}\left(n^{2}+k\right) ^{-1} $ by using the Euler's standard one of the harmonic numbers.&lt;/p&gt;&lt;/abstract&gt;

Analytic continuation of harmonic sums near the integer values

We present a simple method for analytic continuation of harmonic sums near negative and positive integer numbers. We provide a precomputed database for the exact expansion of harmonic sums over a small parameter near these integer numbers, along with MATHEMATICA code, which shows the application of the database for actual problems. We also provide the FORM code that was used to obtain the database mentioned above. The applications of the obtained database for the study of evolution equations in the quantum field theory are discussed.

Degeneration of the Images using System Illustrations and Inserted Zero Tree Calculation to Improve Weight Extent

The outcome is a conclusive property for picking the pressure calculation with best pressure proportion. inside 8 calculations, just two calculations give stable outcomes with the difference in entropy. Jpeg2000 is productive for the most part on pictures with high multifaceted nature and lzma is effective on low unpredictability pictures. Despite the fact that the increase acquired is little, addition will develop with picture goals. Great success rate of the estimation calculation depends on the sum and nature of factual information. with the expansion of sum and assortment of information, calculation will give increasingly exact outcomes for various circumstances. Execution of estimation calculation is likewise founded on utilized realized calculations like jpeg2000 or lzma. With finding diverse unpredictability strategies, wiped out 14 calculations could be additionally utilized in estimation calculation. Utilizing these calculations with various circumstances could deliver progressively effective pressure proportions. a calculation with lossy picture pressure calculations could likewise be created. at the point when a client needs to indicate the record size or measure of misfortune before pressure, the calculation could choose legitimate lossy or lossless calculation with right pressure parameters to acquire wanted outcome.

Exotic tetraquark states with JPC=0+−

We use the Laplace/Borel sum rules (LSR) and the finite energy/local duality sum rules (FESR) to investigate the non-strange $ud\bar u\bar d$ and hidden-strange $us\bar u\bar s$ tetraquark states with exotic quantum numbers $J^{PC}=0^{+-}$ . We systematically construct all eight possible tetraquark currents in this channel without covariant derivative operator. Our analyses show that the $ud\bar u\bar d$ systems have good behaviour of sum rule stability and expansion series convergence in both the LSR and FESR analyses, while the LSR for the $us\bar u\bar s$ states do not associate with convergent OPE series in the stability regions and only the FESR can provide valid results. We give the mass predictions $1.43\pm0.09$ GeV and $1.54\pm0.12$ GeV for the $ud\bar u\bar d$ and $us\bar u\bar s$ tetraquark states, respectively. Our results indicate that the $0^{+-}$ isovector $us\bar u\bar s$ tetraquark may only decay via weak interaction mechanism, e.g. $X_{us\bar{u}\bar{s}}\to K\pi\pi$, since its strong decays are forbidden by kinematics and the symmetry constraints on the exotic quantum numbers. It is predicted to be very narrow, if it does exist. The $0^{+-}$ isoscalar $us\bar u\bar s$ tetraquark is also predicted to be not very wide because its dominate decay mode $X_{us\bar{u}\bar{s}}\to\phi\pi\pi$ is in $P$-wave.

POINT DISTRIBUTIONS IN TWO‐POINT HOMOGENEOUS SPACES

We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known: they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. For all such spaces the best possible bounds for the quadratic discrepancies and sums of pairwise distances are obtained in the paper (Theorems 2.1 and 2.2). Distributions of points of -designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances (Corollary 2.1). Our approach is based on the Fourier analysis on two-point homogeneous spaces and explicit spherical function expansions for discrepancies and sums of distances (Theorems 4.1 and 4.2).

On the Expedient Solution of the Boltzmann Equation by Modified Time Relaxed Monte Carlo (MTRMC) Method

In the present study, a modified time relaxed Monte Carlo (MTRMC) method is developed for numerical solution of the Boltzmann equation in rarefied regimes. Taylor series expansion is employed to obtain a generalized form of the Wild sum expansion and consequently the modified collision functions with fewer inter-molecular interactions are obtained. The proposed algorithm is applied on the lid-driven micro cavity flow with different lid velocities and the results for velocity and shear stress distributions are compared with those from the standard DSMC and TRMC methods. The comparisons show excellent agreement between the results of the MTRMC method with their counterparts from TRMC and DSMC methods. The present study illustrates appreciable improvement in the computational expense of the MTRMC method compared to those from standard TRMC and DSMC methods. The improvement is more pronounced compared to the standard DSMC method. It is observed that up to 56% reduction in CPU time is obtained in the studied cases.

EDGEWORTH CORRECTION FOR THE LARGEST EIGENVALUE IN A SPIKED PCA MODEL.

We study improved approximations to the distribution of the largest eigenvalue of the sample covariance matrix of n zero-mean Gaussian observations in dimension p + 1. We assume that one population principal component has variance ℓ > 1 and the remaining 'noise' components have common variance 1. In the high-dimensional limit p/n → γ > 0, we study Edgeworth corrections to the limiting Gaussian distribution of in the supercritical case . The skewness correction involves a quadratic polynomial, as in classical settings, but the coefficients reflect the high-dimensional structure. The methods involve Edgeworth expansions for sums of independent non-identically distributed variates obtained by conditioning on the sample noise eigenvalues, and the limiting bulk properties and fluctuations of these noise eigenvalues.

On the time relaxed Monte Carlo computations for the flow over a flat nano-plate

In the present study, the first, second and third orders of the TRMC scheme (TRMC1, TRMC2 and TRMC3 schemes) are employed to numerically investigate the Argon flow over a flat nano-plate with different free stream velocities and variety of Knudsen numbers. The simulations cover flow regimes from early slip to early transition regimes (0.00129 ≤ Kn ≤ 0.09). Higher order terms in the Wild sum expansion are considered to obtain higher order collisions. The results are compared with those from the standard DSMC method. Comparisons show that among the studied schemes, the results obtained from the TRMC3 scheme have excellent agreement with the ones from the DSMC method. On the other hand, the results of the TRMC1 and TRMC2 schemes show deviations compared to those from the DSMC method. The deviations are more pronounced for the temperature and pressure distributions. Moreover, the present investigation illustrates that as the Knudsen number increases the accuracy of lower orders of the TRMC scheme improves. It is observed that truncating the Wild sum expansion up to the third order approximation of the TRMC scheme, may be a proper alternative method for the DSMC method in simulating the flow over the nano-plate for Knudsen numbers 0.00129 ≤ Kn ≤ 0.09 with reasonable accuracy and simplicity in mathematics.

Series of sums of products of higher-order Bernoulli functions

It is shown in a previous work that Faber-Pandharipande-Zagier’s and Miki’s identities can be derived from a polynomial identity, which in turn follows from the Fourier series expansion of sums of products of Bernoulli functions. Motivated by and generalizing this, we consider three types of functions given by sums of products of higher-order Bernoulli functions and derive their Fourier series expansions. Moreover, we express each of them in terms of Bernoulli functions.

Sums of finite products of Genocchi functions

In a previous work, it was shown that Faber-Pandharipande-Zagier and Miki’s identities can be derived from a polynomial identity which in turn follows from a Fourier series expansion of sums of products of Bernoulli functions. Motivated by this work, we consider three types of sums of finite products of Genocchi functions and derive Fourier series expansions for them. Moreover, we will be able to express each of them in terms of Bernoulli functions.

Sharp large deviations for sums of bounded from above random variables

We show large deviation expansions for sums of independent and bounded from above random variables. Our moderate deviation expansions are similar to those of Cramer (1938), Bahadur and Ranga Rao (1960), and Sakhanenko (1991). In particular, our results extend Talagrand’s inequality from bounded random variables to random variables having finite (2 + δ)-th moments, where δ ∈ (0; 1]. As a consequence, we obtain an improvement of Hoeffding’s inequality. Applications to linear regression, self-normalized large deviations and t-statistic are also discussed.

Moments of the Hermitian Matrix Jacobi Process

In this paper, we compute the expectation of traces of powers of the Hermitian matrix Jacobi process for a large enough but fixed size. To proceed, we first derive the semi-group density of its eigenvalues process as a bilinear series of symmetric Jacobi polynomials. Next, we use the expansion of power sums in the Schur polynomial basis and the integral Cauchy–Binet formula in order to determine the partitions having nonzero contributions after integration. It turns out that these are hooks of bounded weight and the sought expectation results from the integral of a product of two Schur functions with respect to a generalized beta distribution. For special values of the parameters on which the matrix Jacobi process depends, the last integral reduces to the Cauchy determinant and we close the paper with the investigation of the asymptotic behavior of the resulting formula as the matrix size tends to infinity.

On the time relaxed Monte Carlo computations for the lid-driven micro cavity flow

In the present study, the first, second and third orders of the TRMC scheme (TRMC1, TRMC2 and TRMC3 schemes) are employed to numerically investigate the lid-driven micro cavity flow with different lid velocities and Knudsen numbers. The simulations cover flow regimes from early slip to early transition regimes (0.005≤Kn≤0.1). Higher order terms in the Wild sum expansion are considered to obtain higher order collisions. The results are compared with those from the standard DSMC method. Comparisons show that among the studied schemes, the results obtained from the TRMC3 scheme have excellent agreement with the ones from the DSMC method. On the other hand, the results of TRMC1 and TRMC2 schemes show deviations compared to those from the DSMC method. The deviations are more pronounced for the velocity distributions. Moreover, the present investigation illustrates that as the Knudsen number increases the accuracy of lower orders of the TRMC scheme improves. It is observed that truncating the Wild sum expansion up to the third order approximation of the TRMC scheme, may be a proper alternative method for the DSMC method in simulating the lid-driven micro cavity problem for Knudsen numbers 0.005≤Kn≤0.1 with reasonable accuracy and simplicity in mathematics.

On the Matchings-Jack Conjecture for Jack Connection Coefficients Indexed by Two Single Part Partitions

This article is devoted to the study of Jack connection coefficients, a generalization of the connection coefficients of the classical commutative subalgebras of the group algebra of the symmetric group closely related to the theory of Jack symmetric functions. First introduced by Goulden and Jackson (1996) these numbers indexed by three partitions of a given integer $n$ and the Jack parameter $\alpha$ are defined as the coefficients in the power sum expansion of some Cauchy sum for Jack symmetric functions. Goulden and Jackson conjectured that they are polynomials in $\beta = \alpha-1$ with non negative integer coefficients of combinatorial significance, the Matchings-Jack conjecture.In this paper we look at the case when two of the integer partitions are equal to the single part $(n)$. We use an algebraic framework of Lasalle (2008) for Jack symmetric functions and a bijective construction in order to show that the coefficients satisfy a simple recurrence formula and prove the Matchings-Jack conjecture in this case. Furthermore we exhibit the polynomial properties of more general coefficients where the two single part partitions are replaced by an arbitrary number of integer partitions either equal to $(n)$ or $[1^{n-2}2]$.