LEIBNIZ′FORMULA OF GENERALIZED DIFFERENCE WITH RESPECT TO A CLASS OF DIFFERENTIAL OPERATORS AND RECURRENCE FORMULA OF THEIR GREEN′S FUNCTION

In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.

ENTIRE FUNCTIONS SHARING ONE SMALL FUNCTION CM WITH THEIR SHIFTS AND DIFFERENCE OPERATORS

In this article, we mainly devote to proving uniqueness results for entire functionssharing one small function CM with their shift and difference operator simultaneously. Letf（z） be a nonconstant entire function of finite order, c be a nonzero finite complex constant, and n be a positive integer. If f（z）, f（z＋c）, and △n cf（z） share 0 CM, then f（z＋c）≡Af（z）, where A（≠0） is a complex constant. Moreover, let a（z）, b（z）（ O） ∈ S（f） be periodic entire functions with period c and if f（z） - a（z）, f（z ＋ c） - a（z）, △cn f（z） - b（z） share 0 CM, then f（z ＋ c） ≡ f（z）.

Graph Laplacian Matrix Learning from Smooth Time-Vertex Signal

In this paper,we focus on inferring graph Laplacian matrix from the spatiotemporal signal which is defined as“time-vertex signal”.To realize this,we first represent the signals on a joint graph which is the Cartesian product graph of the time-and vertex-graphs.By assuming the signals follow a Gaussian prior distribution on the joint graph,a meaningful representation that promotes the smoothness property of the joint graph signal is derived.Furthermore,by decoupling the joint graph,the graph learning framework is formulated as a joint optimization problem which includes signal denoising,timeand vertex-graphs learning together.Specifically,two algorithms are proposed to solve the optimization problem,where the discrete second-order difference operator with reversed sign(DSODO)in the time domain is used as the time-graph Laplacian operator to recover the signal and infer a vertex-graph in the first algorithm,and the time-graph,as well as the vertex-graph,is estimated by the other algorithm.Experiments on both synthetic and real-world datasets demonstrate that the proposed algorithms can effectively infer meaningful time-and vertex-graphs from noisy and incomplete data.

A NOTE ON VALUE DISTRIBUTION OF DIFFERENCE POLYNOMIALS OF MEROMORPHIC FUNCTIONS

In this paper, we investigate the value distribution of the difference counterpart △f（z）- af（z）^n of f′（z）- af（z）^n and obtain an almost direct difference analogue of result of Hayman.

Finite-difference time-domain studies of low-frequency stop band in superconductor-dielectric superlattice

The transmission coefficients of electromagnetic （EM） waves due to a superconductor-dielectric superlattice are numerically calculated. Shift operator finite difference time domain （SO-FDTD） method is used in the analysis. By using the SO-FDTD method, the transmission spectrum is obtained and its characteristics are investigated for different thicknesses of superconductor layers and dielectric layers, from which a stop band starting from zero frequency can be apparently observed. The relation between this low-frequency stop band and relative temperature, and also the London penetration depth at a superconductor temperature of zero degree are discussed, separately. The low-frequency stop band properties of superconductor-dielectric superlattice thus are well disclosed.

Mutual transformations between the P–Q, Q–P, and generalized Weyl ordering of operators

Based on the generalized Weyl quantization scheme, which relies on the generalized Wigner operator Ok （p, q） with a real k parameter and can unify the P-Q, Q-P, and Weyl ordering of operators in k = 1, - 1,0, respectively, we find the mutual transformations between 6 （p - P） （q - Q）, （q - Q） 3 （p - P）, and （p, q）, which are, respectively, the integration kernels of the P-Q, Q-P, and generalized Weyl quantization schemes. The mutual transformations provide us with a new approach to deriving the Wigner function of quantum states. The - and - ordered forms of （p, q） are also derived, which helps us to put the operators into their - and - ordering, respectively.

CHARACTERIZATIONS OF PRINCIPAL SHIFT-INVARIANT SPACES

Shifts-invariant spaces in L 1(R) are investigated. First,based on a study of the system of linearly difference operators,the method of constructing generators with linearly independent shifts is provided. Then the characterizations of the closed shift-invariant subspaces in L 1(R) are given in terms of such generators and the local basis of shift-invariant subspaces.

On Some Class of n-Normed Generalized Difference Sequences Related to l_p-Space

In this paper, we introduce the class of n-normed generalized difference sequences related to lp-space. Some properties of this sequence space like solidness, symmetricity, convergence-free etc. are studied. We obtain some inclusion relations involving this sequence space.

截断误差最小化的优化空间有限差分算子(英文)

Highly accurate spatial discretization is essentially required to perform numerical climate and weather prediction. The difference between the differential and the finite-difference operator is however a primitive error source in the numerics. This paper presents an optimization of centered finite-difference operator based on the principle of constrained cost function, which can reduce the truncation error to minimum. In the optimization point of view, such optimal operator is in fact an attempt to minimize spatial truncation er-rors in atmospheric modeling, in a simple way and indeed a quite innovative way to implement Variational Continuous Assimilation (VCA) technique. Furthermore, the optimizing difference operator is consciously designed to be meshing-independent, so that it can be used for most Arakawa-mesh configurations, such as un-staggered (Arakawa-A) or com-monly staggered (Arakawa-B, Arakawa-C, Arakawa-D) mesh. But for the calibration purpose, the pro-posed operator is implemented on an un-staggered mesh in which the truncation oscillation is mostly ex-cited, and it thus makes a severe and indeed a benchmark test for the proposed optimal scheme. Both theo-retical investigation and practical modeling indicate that the aforementioned numerical noise can be significantly eliminated.

NOTES ON THE SPECTRAL PROPERTIES OF THE WEIGHTED MEAN DIFFERENCE OPERATOR G(u,v;?) OVER THE SEQUENCE SPACE ?_1

In the study by Baliarsingh and Dutta [Internat. J.Anal., Vol.2014（2014）, Article ID 786437], the authors computed the spectrum and the fine spectrum of the product operator G （u, v; A） over the sequence space e1. The product operator G （u, v; △） over l1 is defined by （G（u,v;△）x）k=^k∑i=0ukvi（xi- xi-1） with xk = 0 for all k 〈 0, where x = （xk）∈e1,and u and v axe either constant or strictly decreasing sequences of positive real numbers satisfying certain conditions. In this article we give some improvements of the computation of the spectrum of the operator G （u, v; △） on the sequence space gl.

On a Difference Matrix and Its Properties

In the present paper, a new difference matrix via difference operator D is introduced. Let x = （xk） be a sequence of real numbers, then the difference operatorD is defined by D（x）n =∑kn=0（-1）k（n-kn）xk,where n = 0,1,2,3,.... Several interestingproperties of the new operator D are discussed.

Separation of Waves by Interpolation Method

The separation of waves by an interpolation method is presented in detail. The composite wave sequences measured with two wave gauges in the wave flume are separated very quickly into two series of incident and reflected waves in time domain via the simple interpolation and difference operations. Then, the reflection coefficient can be estimated easily and accurately without calculation of wave heights and phases. The intial phase of reflection can also be detected easily for improvement of the accuracy of calculation. The present method is applicable to both regular and irregular trains of waves based on the linear wave theory which are proved to be accurate through numerical sample tests. Physical experiments are conducted and compared with Goda′s method and analytical method with satisfactory results. Furthermore, the present method can be used for the absorbing wave maker to extract reflected waves in real time.

ELEMENT FUNCTIONS OF DISCRETE OPERATOR DIFFERENCE METHOD

The discrete scheme called discrete operator difference for differential equations was given. Several difference elements for plate bending problems and plane problems were given. By investigating these elements, the ability of the discrete forms expressing to the element functions was talked about. In discrete operator difference method, the displacements of the elements can be reproduced exactly in the discrete forms whether the displacements are conforming or not. According to this point, discrete operator difference method is a method with good performance.

Determination of Threshold Voltage and Mobility ofMOSFET by Proportional Difference Operator

Proportional Difference Operator (PDO) method is proposed for the first time to determine the key parameters of a MOSFET, including the threshold voltage and ca rrier mobility.This method is applied to the transfer characteristic of a MOSFET first, and then the effect of gate voltage on carrier mobility is considered. The dependence of carrier mobility on the gate voltage is obtained.

涉及整函数差分的唯一性研究

In this paper,we study the uniqueness of entire functions and prove the following theorem.Let f be a transcendental entire function of finite order.Then there exists at most one positive integer k,such that f(z)△^(k)_(c)f(z)-R(z)has finitely many zeros,where R(z)is a non-vanishing rational function and c is a nonzero complex number.Our result is an improvement of the theorem given by Andasmas and Latreuch[1].

A Numerical Algorithm for the Caputo Tempered Fractional Advection-Diffusion Equation

By transforming the Caputo tempered fractional advection-diffusion equation into the Riemann–Liouville tempered fractional advection-diffusion equation,and then using the fractional-compact Grünwald–Letnikov tempered difference operator to approximate the Riemann–Liouville tempered fractional partial derivative,the fractional central difference operator to discritize the space Riesz fractional partial derivative,and the classical central difference formula to discretize the advection term,a numerical algorithm is constructed for solving the Caputo tempered fractional advection-diffusion equation.The stability and the convergence analysis of the numerical method are given.Numerical experiments show that the numerical method is effective.

Uniqueness on Difference Operators of Meromorphic Functions of Infinite Order

We investigate the uniqueness problems of meromorphic functions and their difference operators by using a new method.It is proved that if a non-constant meromorphic function f shares a non-zero constant and ∞ counting multiplicities with its difference operators Δcf(z) and Δ_(c)^(2)f(z), thenΔcf(z)≡Δ_(c)^(2)f(z).In particular,we give a difference analogue of a result of Jank-Mues-Volkmann.Our method has two distinct features:(ⅰ) It converts the relations between functions into the corresponding vectors.This makes it possible to deal with the uniqueness problem by linear algebra and combinatorics.(ⅱ) It circumvents the obstacle of the difference logarithmic derivative lemma for meromorphic functions of infinite order,since this method does not depend on the growth of the functions.Furthermore,the idea in this paper can also be applied to the case for several variables.

Meromorphic Functions Sharing Two Values

Let f and g be two nonconstant meromorphic functions,and n be a positive integer.If f^(n)(z)and g^(n)(z)share 1 CM(counting multiplicities),f(z)and g(z)share∞IM(ignoring multiplicities),and N(r,1f)+N(r,1g)=S(r,f)+S(r,g),then either f(z)≡tg(z)or f(z)g(z)≡t,where t^(n)=1.As an application,shared values problems of a meromorphic function related to its shifts and difference operators are also investigated.

The Fine Spectra of the Difference Operator △ Over the Sequence Space bv_p,(1≤p<∞)

The sequence space bvp consisting of all sequences （xk） such that （xk -xk-1） belongs to the space gp has recently been introduced by Basar and Altay [Ukrainian Math. J., 55（1）, 136-147（2003）]; where 1 ≤ p ≤ ∞. In the present paper, some results concerning with the continuous dual and f-dual, and the AD-property of the sequence space bvp have been given and the norm of the difference operator A acting on the sequence space bvp has been found. The fine spectrum with respect to the Goldberg＇s classification of the difference operator △ over the sequence space bvp has been determined, where 1≤p〈∞.

A Generalized Cubic Functional Equation

In this paper, we determine the general solution of the functional equation f1 （2x ＋ y） ＋ f2（2x - y） = f3（x ＋ y） ＋ f4（x - y） ＋ f5（x） without assuming any regularity condition on the unknown functions f1,f2,f3, f4, f5 ： R→R. The general solution of this equation is obtained by finding the general solution of the functional equations f（2x ＋ y） ＋ f（2x - y） = g（x ＋ y） ＋ g（x - y） ＋ h（x） and f（2x ＋ y） - f（2x - y） = g（x ＋ y） - g（x - y）. The method used for solving these functional equations is elementary but exploits an important result due to Hosszfi. The solution of this functional equation can also be determined in certain type of groups using two important results due to Szekelyhidi.