Application of Gaussian Beam Summation Migration in Reflected In-seam Wave Imaging

The geological conditions for coal mining in China are complex,with various structural issues such as faults and collapsed columns seriously compromising the safety of coal mine production.In-seam wave exploration is an effective technique for acquiring detailed information on geological structures in coal seam working faces.However,the existing reflected in-seam wave imaging technique can no longer meet the exploration precision requirements,making it imperative to develop a new reflected in-seam wave imaging technique.This study applies the Gaussian beam summation(GBS)migration method to imaging coal seams'reflected in-seam wave data.Firstly,with regard to the characteristics of the reflected in-seam wave data,methods such as wavefield removal and enveloped superposition are employed for the corresponding wavefield separation,wave train compression and other processing of reflected in-seam waves.Thereafter,imaging is performed using the GBS migration technique.The feasibility and effectiveness of the proposed method for reflected in-seam wave imaging are validated by conducting GBS migration tests on 3D coal-seam fault models with different dip angles and throws.By applying the method to reflected in-seam wave data for an actual coal seam working face,accurate imaging of a fault structure is obtained,thereby validating its practicality.

Estimates of three classical summations on the spaces F_p^(α,q)(R^n)(0 < p ≤ 1)

We obtain the boundedness on Fp^α,q（R^n）for the Poisson summation and Gauss summation. Their maximal operators are proved to be bounded from Fp^α,q（R^n）to L∞（R^n）For the maximal operator of the Bochner-Riesz summation, we prove that it is bounded from Fp^α,q（R^n）to L n-pα/pn,（0〈p≤ 1）∞（R^n）.

A self-cited pixel summation based image encryption algorithm

In this paper, a novel image encryption algorithm is presented based on self-cited pixel summation. With the classical mechanism of permutation plus diffusion, a pixel summation of the plain image is employed to make a gravity influence on the pixel positions in the permutation stage. Then, for each pixel in every step of the diffusion stage, the pixel summation calculated from the permuted image is updated. The values from a chaotic sequence generated by an intertwining logistic map are selected by this summation. Consequently, the keystreams generated in both stages are dependent on both the plain image and the permuted image. Because of the sensitivity of the chaotic map to its initial conditions and the plain-imagedependent keystreams, any tiny change in the secret key or the plain image would lead to a significantly different cipher image. As a result, the proposed encryption algorithm is immune to the known plaintext attack（KPA） and the chosen plaintext attack（CPA）. Moreover, experimental simulations and security analyses show that the proposed permutationdiffusion encryption scheme can achieve a satisfactory level of security.

Loudness Summation and Weightings for Loudness and Environmental Noise Assessment

A stationary loudness model has been built up on the basis of the former ISO 226: 1987 concerning equal-loudness-level contours. The loudness and loudness level expressions derived in the study include the same parameters as used when determining the equal-loudness-level contours of the former ISO standard. However, as an additional main idea, a loudness summation rule has been proposed in the study. Moreover, the loudness expressions have been normalised to give the same values for people who have a similar sense of hearing. It has also been found that the loudness expressions include basically two different weightings. The first weighting is a conservative frequency weighting in the domain of sound pressure level, and the second weighting consists of coefficients applied to the weighted sound pressure levels. The latter have the greatest effect on the very low-frequency range. Finally, the paper includes a new way to use the A-weighting which takes into account the compressed character of the equal-loudness-level contours at the low frequency range. This method remarkably transforms the character of the A-weighting as a measure for low-frequency environmental noise.

ULAM-HYERS-RASSIAS STABILITY AND EXISTENCE OF SOLUTIONS TO NONLINEAR FRACTIONAL DIFFERENCE EQUATIONS WITH MULTIPOINT SUMMATION BOUNDARY CONDITION

The purpose of this study is to acquire some conditions that reveal existence and stability for solutions to a class of difference equations with non-integer orderμ∈(1,2].The required conditions are obtained by applying the technique of contraction principle for uniqueness and Schauder’s fixed point theorem for existence.Also,we establish some conditions under which the solution of the considered class of difference equations is generalized Ulam-Hyers-Rassias stable.Example for the illustration of results is given.

On the best convergence order of a new class of triangular summation operators

In this paper, a new class of triangular summation operators based on the equidistant nodes was constructed. It is proved that this class of operators converges uniformly to arbitrary continuous fimctions with the period 2π on the whole axis, Fttrthermore, the best approximation order and the highest convergence order are obtained. In contrast to certain operators constructed by Bernstein and Kis in the previous works, the convergence properties of the new operator constructed in this paper are superior.

SUMMATION OF TRIGONOMETRIC SERIES BY FOURIER TRANSFORMS

This paper gives the theorems concerning the summation of trigonometric series with the help of Fourier transforms. By means of the known results of Fourier transforms, many difficult and complex problems of summation of trigonometric series can be solved. This method is a comparatively unusual way to find the summation of trigonometric series, and has been used to establish the comprehensive table of summation of trigonometric series. In this table 10 thousand scries arc given, and most of them are new.

Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators

In this article,discrete variants of several results from vector calculus are studied for clas-sical finite difference summation by parts operators in two and three space dimensions.It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields cannot hold discretely because of grid oscillations,which are characterised explicitly.This results in a non-vanishing remainder associated with grid oscillations in the discrete Helmholtz Hodge decomposition.Nevertheless,iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposition via orthogonal projections are pro-posed and applied successfully.In numerical experiments,the discrete remainder vanishes and the potentials converge with the same order of accuracy as usual in other first-order partial differential equations.Motivated by the successful application of the Helmholtz Hodge decomposition in theoretical plasma physics,applications to the discrete analysis of magnetohydrodynamic(MHD) wave modes are presented and discussed.

On a New Family of Trigonometric Summation Polynomials of Bernstein Type

A new family of trigonometric summation polynomials, Gn,r（f; θ）, of Bernstein type is constructed. In contrast to other trigonometric summation polynomials, the convergence properties of the new polynomials are superior to others. It is proved that Gn,r（f; θ） converges to arbitrary continuous functions with period 2π uniformly on （-∞ ＋∞） as n→ ∞. In particular, Gn,r（f; θ） has the best convergence order, and its saturation order is 1/n^2r＋4.

OPTIMAL SUMMATION INTERVAL AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A DISCRETE SYSTEM

In this paper, we are concerned with properties of positive solutions of the following Euler-Lagrange system associated with the weighted Hardy-Littlewood-Sobolev inequality in discrete form{uj =∑ k ∈Zn vk^q/（1 ＋ ｜j｜）^α（1 ＋ ｜k- j｜）^λ（1 ＋ ｜k｜）^β,（0.1）vj =∑ k ∈Zn uk^p/（1 ＋ ｜j｜）^β（1 ＋ ｜k- j｜）^λ（1 ＋ ｜k｜）^α,where u, v 〉 0, 1 〈 p, q 〈 ∞, 0 〈 λ 〈 n, 0 ≤α ＋ β≤ n- λ,1/p＋1〈λ＋α/n and 1/p＋1＋1/q＋1≤λ＋α＋β/n：=λ^-/n. We first show that positive solutions of（0.1） have the optimal summation interval under assumptions that u ∈ l^p＋1（Z^n） and v ∈ l^q＋1（Z^n）. Then we show that problem（0.1） has no positive solution if 0 〈λˉ pq ≤ 1 or pq 〉 1 and max{（n-λ^-）（q＋1）/pq-1,（n-λ^-）（p＋1）/pq-1} ≥λ^-.

The Proof and Application of a Summation Formula

In this paper, some conclusions related to the prime number theorem, such as the Mertens formula are improved by the improved Abelian summation formula, and some problems such as “Dirichlet” function and “W(n)” function are studied.

On the Implementation of Exponential B-Splines by Poisson Summation Formula

Polynomial splines have played an important role in image processing, medical imaging and wavelet theory. Exponential splines which are of more general concept have been recently investigated.We focus on cardinal exponential splines and develop a method to implement the exponential B-splines which form a Riesz basis of the space of cardinal exponential splines with finite energy.

Single-frequency upconverted laser generation by phase summation

The phase summation effect in sum-frequency mixing process is utilized to avoid a nonlinearity obstacle in the power scaling of single-frequency visible or ultraviolet lasers.Two single-frequency fundamental lasers are spectrally broadened by phase modulation to suppress stimulated Brillouin scattering in fiber amplifier and achieve higher power.After sum-frequency mixing in a nonlinear optical crystal,the upconverted laser returns to single frequency due to phase summation,when the phase modulations on two fundamental lasers have a similar amplitude but opposite sign.The method was experimentally proved in a Raman fiber amplifier-based laser system,which generated a power-scalable sideband-free single-frequency 590 nm laser.The proposal manifests the importance of phase operation in wave-mixing processes for precision laser technology.

Abel’s lemma on summation by parts and partial q-series transformations

The partial sums of basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several transformation and summation formulae for well-poised, quadratic, cubic and quartic q-series are established.

POINTWISE ESTIMATE OF BOUNDED SOLUTIONS TO SOME DISCRETE INEQUALITIES INVOLVING INFINITE SUMMATION

Explicit bounds on bounded solutions to a new class of Volterra-type linear and nonlinear discrete inequalities involving infinite sums are established. These inequalities can be viewed as discrete analogues of some Volterra-type inequalities having improper integral functionals,which are new to the literature.

A refined Poisson summation formula for certain Braverman-Kazhdan spaces

Braverman and Kazhdan(2000)introduced influential conjectures aimed at generalizing the Fourier transform and the Poisson summation formula.Their conjectures should imply that quite general Langlands L-functions have meromorphic continuations and functional equations as predicted by Langlands'functoriality conjecture.As an evidence for their conjectures,Braverman and Kazhdan(2002)considered a setting related to the so-called doubling method in a later paper and proved the corresponding Poisson summation formula under restrictive assumptions on the functions involved.The connection between the two papers is made explicit in the work of Li(2018).In this paper,we consider a special case of the setting in Braverman and Kazhdan's later paper and prove a refined Poisson summation formula that eliminates the restrictive assumptions of that paper.Along the way we provide analytic control on the Schwartz space we construct;this analytic control was conjectured to hold(in a slightly different setting)in the work of Braverman and Kazhdan(2002).

On a Linear Summation Problem of the Double Fourier Series

It this paper we construct a double Fourier series with a new linear summation factor, for the arbitrary continuous periodic function f(x,y)with period 2л, it converges to the function(fx,y) uniformly on total oplane,and its convergence order is the best one.

On a Kind of Summation Formula Using Howard's Generalized Stirling Numbers

It is shown that Howard＇s degenerate weighted Stirling numbers can be used to construct a fruitful summation formula for a class of formal power series involving generalized factorials.This is achieved with the aid of a series-transformation formula due to He,Hsu and Shiue,and several identities involving generalized Stirling numbers and Bell numbers are given to illustrate the application of the formula obtained.

A MULTIPLE q-EXPONENTIAL DIFFERENTIAL OPERATIONAL IDENTITY

Using Hartogs’fundamental theorem for analytic functions in several complex variables and q-partial differential equations,we establish a multiple q-exponential differential formula for analytic functions in several variables.With this identity,we give new proofs of a variety of important classical formulas including Bailey’s 6ψ6 series summation formula and the Atakishiyev integral.A new transformation formula for a double q-series with several interesting special cases is given.A new transformation formula for a 3ψ3 series is proved.

Reachable set estimation for discrete-time Markovian jump neural networks with unified uncertain transition probability

This paper focuses on the reachable set estimation for Markovian jump neural networks with time delay.By allowing uncertainty in the transition probabilities,a framework unifies and enhances the generality and realism of these systems.To fully exploit the unified uncertain transition probabilities,an equivalent transformation technique is introduced as an alternative to traditional estimation methods,effectively utilizing the information of transition probabilities.Furthermore,a vector Wirtinger-based summation inequality is proposed,which captures more system information compared to existing ones.Building upon these components,a novel condition that guarantees a reachable set estimation is presented for Markovian jump neural networks with unified uncertain transition probabilities.A numerical example is illustrated to demonstrate the superiority of the approaches.