The Equivalence Theorem of Appoximation for Two-dimensional Baskakov Operators

In this paper, we study the convergence rate of two-dimensional Baakakov operators with Jacobi-weights and the approximation equivalence theorem is obtained, making use of multivariate decompose skills and results of one-dimensional Baskakov operators.

EQUIVALENCE THEOREM ON WEIGHTED SIMULTANEOUS L-APPROXIMATION BY MEANS OF SZASZ-KANTOROVICH OPERATORS AND BASKAKOV-KANTOROVICH OPERATORS

The intention of this paper is to give direct and converse results on weighted simultaneous approximation by means of Sz(?)sz-Kantorovich operators and Baskakov-Kantorovich operators in L_p-norms using the weighted Ditzian-Totik modulus of smoothness.

Helmholtz Theorems, Gauge Transformations, General Covariance and the Empirical Meaning of Gauge Conditions

It is well known that the use of Helmholtz decomposition theorem for static vector fields , when applied to the time dependent vector fields , which represent the electromagnetic field, allows us to obtain instantaneous-like solutions all along . For this reason, some people thought (see e.g. [1] and references therein) that the Helmholtz theorem cannot be applied to time dependent vector fields and some modification is wanted in order to get the retarded solutions. However, the use of the Helmholtz theorem for static vector fields is correct even for time dependent vector fields (see, e.g. [2]), so a relation between the solutions was required, in such a way that a retarded solution can be transformed in an instantaneous one, and conversely. On this paper we want to suggest, following most of the time the mathematical formalism of Woodside in [3], that: 1) there are many Helmholtz decompositions, all equally consistent, 2) each one is naturally related to a space-time structure, 3) when we use the Helmholtz decomposition for the electromagnetic potentials it is equivalent to a gauge transformation, 4) there is a natural methodological criterion for choosing the gauge according to the structure postulated for a global space-time, 5) the Helmholtz decomposition is the manifestation at the level of the fields that a gauge is involved. So, when we relate the retarded solution to the instantaneous one what we do is to change the gauge and the space-time. And, if the Helmholtz decompositions are related to a space-time structure, and are equivalent to gauge transformations, each gauge transformation is natural for a specific space-time. In this way, a Helmholtz decomposition for Euclidean space is equivalent to the Coulomb gauge and a Helmholtz decomposition for the Minkowski space is equivalent to the Lorenz gauge. This leads us to consider that the theories defined by different gauges may be mathematically equivalent, because they can be related by means of a gauge transformation, but they are not empirically equivalent, because they have quite different observational consequences due to the different space-time structure involved.

Approach to a Proof of the Riemann Hypothesis by the Second Mean-Value Theorem of Calculus

By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove that the class of functionswith an integral representation of the form with a real-valued function which is non-increasing and decreases in infinity more rapidly than any exponential functions , possesses zeros only on the imaginary axis. The Riemann zeta function as it is known can be related to an entire functionwith the same non-trivial zeros as . Then after a trivial argument displacement we relate it to a function with a representation of the form where is rapidly decreasing in infinity and satisfies all requirements necessary for the given proof of the position of its zeros on the imaginary axis z=iy by the second mean-value theorem. Besides this theorem we apply the Cauchy-Riemann differential equation in an integrated operator form derived in the Appendix B. All this means that we prove a theorem for zeros of on the imaginary axis z=iy for a whole class of function which includes in this way the proof of the Riemann hypothesis. This whole class includes, in particular, also the modified Bessel functions for which it is known that their zeros lie on the imaginary axis and which affirms our conclusions that we intend to publish at another place. In the same way a class of almost-periodic functions to piece-wise constant non-increasing functions belong also to this case. At the end we give shortly an equivalent way of a more formal description of the obtained results using the Mellin transform of functions with its variable substituted by an operator.

Convolution Theorem of Fractional Fourier Transformation Derived by Representation Transformation in Quantum Mechancis

Based on our previous paper (Commun.Theor.Phys.39 (2003) 417) we derive the convolution theoremof fractional Fourier transformation in the context of quantum mechanics,which seems a convenient and neat way.Generalization of this method to the complex fractional Fourier transformation case is also possible.

STUDY OF THE EQUIVALENT THEOREM OF GENERALIZED VARIATIONAL PRINCIPLES IN ELASTICITY

The relations of all generalized variational principles in elasticity are studied by employing the invariance theorem of field theory. The infinitesimal scale transformation in field theory was employed to investigate the equivalent theorem. Among the results found particularly interesting are those related to that all generalized variational principles in elasticity are equal to each other. Also studied result is that only two variables are independent in the functional and the stress-strain relation is the variational constraint condition for all generalized variational principles in elasticity. This work has proven again the conclusion of Prof. Chien Wei-zang.

THE SPACES OF CESRO ALMOST CONVERGENT SEQUENCES AND CORE THEOREMS

As known, the method to obtain a sequence space by using convergence field of an infinite matrix is an old method in the theory of sequence spaces. However, the study of convergence field of an infinite matrix in the space of almost convergent sequences is so new （see [15]）. The purpose of this paper is to introduce the new spaces ^ ～f and fo consisting of all sequences whose Ceshro transforms of order one are in the spaces f and ^ ～ f0, respectively. Also, in this paper, we show that ^ ～f and ^ ～f0 are linearly isomorphic to the spaces f and f0, respectively. The β- and γ-duals of the spaces ^ ～f and 2% are computed. Furthermore, the classes （^ ～f： μ） and （μ ： f） of infinite matrices are characterized for any given sequence space μ, and determined the necessary and sufficient conditions on a matrix A to satisfy Bc-core（Ax） K-core（x）, K-core（Ax） Bg-core（x）, Bc-core（Ax） Be-core（x）, Bc-core（Ax） t-core（x） for all x ∈ t∞.

Equivalence Problem of the PainlevéEquations

The manuscript is devoted to the equivalence problem of the Painlevé equations. Conditions which are necessary and sufficient for second-order ordinary differential equations y″=F (x ,y, y′) to be equivalent to the first and second Painlevé equation under a general point transformation are obtained. A procedure to check these conditions is found.

The Conditions of Equivalence of a Class Germs of C~∞ Functions on Banach Spaces

In this paper,we prove the converse of gem is right equivalent is also true in[1] and[2],obtain the necessary and sufficient conditions of equivalence of a class gems of C∞ function on Banach spaces.

ON THE DECOMPOSITION OF COMPLEX VECTOR SPACES AND THE JORDAN CANONICAL FORM OF COMPLEX LINEAR TRANSFORMATIONS

New objects characterizing the structure of complex linear transformations areintroduced. These new objects yield a new result for the decomposition of complexvector spaces relative to complex lrnear transformations and provide all Jordan basesby which the Jordan canonical form is constructed. Accordingly, they can result in thecelebrated Jordan theorem and the third decomposition theorem of space directly. and,moreover, they can give a new deep insight into the exquisite and subtle structure ofthe Jordan form. The latter indicates that the Jordan canonical form of a complexlinear transformation is an invariant structure associated with double arbitrary. choices.

Investigation of Probability Generating Function in an Interdependent <i>M/M/</i>1:(∞;GD) Queueing Model with Controllable Arrival Rates Using Rouche’s Theorem

Present paper deals a M/M/1:(∞;GD) queueing model with interdependent controllable arrival and service rates where- in customers arrive in the system according to poisson distribution with two different arrivals rates-slower and faster as per controllable arrival policy. Keeping in view the general trend of interdependent arrival and service processes, it is presumed that random variables of arrival and service processes follow a bivariate poisson distribution and the server provides his services under general discipline of service rule in an infinitely large waiting space. In this paper, our central attention is to explore the probability generating functions using Rouche’s theorem in both cases of slower and faster arrival rates of the queueing model taken into consideration;which may be helpful for mathematicians and researchers for establishing significant performance measures of the model. Moreover, for the purpose of high-lighting the application aspect of our investigated result, very recently Maurya [1] has derived successfully the expected busy periods of the server in both cases of slower and faster arrival rates, which have also been presented by the end of this paper.

Characterization Theorem of Generalized Operators

In this paper, by using the W-transform of an operator on white noise functionals, we establish a general characterization theorem for operators on white noise functionals in term of growth condition. We also discuss convergence of operator sequences.

基于El-Nabulsi分数阶模型的广义Birkhoff系统Noether对称性研究

为了进一步揭示力学系统的对称性与守恒量之间的内在关系,基于El-Nabulsi分数阶模型提出并研究了广义Birkhoff系统的Noether定理。首先,提出分数阶广义El-Nabulsi-PfaffBirkhoff原理,建立广义El-Nabulsi-Birkhoff方程。其次,基于El-Nabulsi-Pfaff作用量在无限小变换下的不变性,给出广义Birkhoff系统Noether对称性的定义和判据。最后,提出广义Birkhoff系统的Noether定理。该文研究结果可进一步应用于完整和非完整约束系统。

The Noether's Theory of Constrained Birkhoffian System

By using the synmeby and the qusi-symmetry of the infinitesimal transformation of the transformation group G1 and by imposing restrictions of constraints on the transformation, the Noether's theory of constrained Birkhoffian system has been established. The theory includes the generalized Noether's theorem obtaining the first integrals from the known symmetry and quasi-symmetry and its inverse obtaining the corresponding symmetry and quasi-symmetry from the known first integrals for the systerm.

The Numerical Scheme Development of a Simplified Frozen Soil Model

In almost all frozen soil models used currently, three variables of temperature, ice content and moisture content are used as prognostic variables and the rate term, accounting for the contribution of the phase change between water and ice, is shown explicitly in both the energy and mass balance equations. The models must be solved by a numerical method with an iterative process, and the rate term of the phase change needs to be pre-estimated at the beginning in each iteration step. Since the rate term of the phase change in the energy equation is closely related to the release or absorption of the great amount of fusion heat, a small error in the rate term estimation will introduce greater error in the energy balance, which will amplify the error in the temperature calculation and in turn, cause problems for the numerical solution convergence. In this work, in order to first reduce the trouble, the methodology of the variable transformation is applied to a simplified frozen soil model used currently, which leads to new frozen soil scheme used in this work. In the new scheme, the enthalpy and the total water equivalent are used as predictive variables in the governing equations to replace temperature, volumetric soil moisture and ice content used in many current models. By doing so, the rate terms of the phase change are not shown explicitly in both the mass and energy equations and its pre-estimation is avoided. Secondly, in order to solve this new scheme more functionally, the development of the numerical scheme to the new scheme is described and a numerical algorithm appropriate to the numerical scheme is developed. In order to evaluate the new scheme of the frozen soil model and its relevant algorithm, a series of model evaluations are conducted by comparing numerical results from the new model scheme with three observational data sets. The comparisons show that the results from the model are in good agreement with these data sets in both the change trend of variables and their magnitude values, and the new scheme, together with the algorithm, is more efficient and saves more computer time.

Application of the method of equivalent edge currents to composite scattering from the cone-cylinder above a dielectric rough sea surface

Compared with scattering from a rough surface only, composite scattering from a target above a rough surface has become so practical that it is a subject of great interest. At present, this problem has been solved by some numerical methods which will produce an enormous calculation amount. In order to overcome this shortcoming, the reciprocity theorem （RT） and the method of equivalent edge currents （MEC） are used in this paper. Due to the advantage of RT, the difficulty in computing the secondary scattered fields is reduced. Simultaneously, MEC, a high-frequency method with edge diffraction considered, is used to calculate the scattered field from the cone-cylinder target with a high accuracy and efficiency. The backscattered field and the polarization currents of the rough sea surface are evaluated by the Kirchhoff approximation （KA） method and physical optics （PO） method, respectively. The effects of the backscattering radar cross section （RCS） and the Doppler spectrum on the size of the target and the windspeed of the sea surface for different incident angles are analysed in detail.

Theory and application of equivalent transformation relationships between plane wave and spherical wave

Based on the governing equations and the equivalent models, we propose an equivalent transformation relationships between a plane wave in a one-dimensional medium and a spherical wave in globular geometry with radially inhomogeneous properties. These equivalent relationships can help us to obtain the analytical solutions of the elastodynamic issues in an inhomogeneous medium. The physical essence of the presented equivalent transformations is the equivalent relationships between the geometry and the material properties. It indicates that the spherical wave problem in globular geometry can be transformed into the plane wave problem in the bar with variable property fields, and its inverse transformation is valid as well. Four different examples of wave motion problems in the inhomogeneous media are solved based on the presented equivalent relationships. We obtain two basic analytical solution forms in Examples I and II, investigate the reflection behavior of inhomogeneous half-space in Example III, and exhibit a special inhomogeneity in Example IV, which can keep the traveling spherical wave in constant amplitude. This study implies that our idea makes solving the associated problem easier.

Design and Analysis of Transmitter Charge Switch Assembly Load on Weather Radar Test Platform

This paper provides a new method for designing the load of transmitters switch components on provincial radar test platform. Based on the calculation of the specific parameters related to the actual charge switch components, the load-related data model is obtained, the simulation is carried out, and the results are consistent with the actual test results. Through the theoretical numerical analysis of the load of the charging switch components of the transmitter device in the new generation of weather radar maintenance test platform, radar maintenance personnel at all levels can deepen the thorough understanding of the CINRAD/SA transmitter and have some enlightenment and improvement on the maintenance guarantee ability of the new generation of weather radar.

Computation of the Smith Form for Multivariate Polynomial Matrices Using Maple

In this paper we show how the transformations associated with the reduction to the Smith form of some classes of mul-tivariate polynomial matrices are computed. Using a Maple implementation of a constructive version of the Quillen-Suslin Theorem, we present two algorithms for the reduction to a particular Smith form often associated with the simplification of linear systems of multidimensional equations.

The Functional Gradient Description Method of Space Coordinate Transformation

The 3-dimension coordinate datum transformation is very important in geomatics.Generally,there are small angle coordinate transformation model (i.e.,Bursa model,et al) and large angle coordinate transformation model (i.e.,Rodrigo matrix model,et al).However,these papers only focus on the fixed and static situation regardless of the small or large angle.This paper proposes the conception of coordinate transformation gradient field,which can realize the space coordinate transformation from small angle to arbitrary angle,and from static to dynamical.Based on the equivalent characteristics of the unit quaternion rotation matrix and the Rodrigo matrix,through studying the mathematical relationship between the spatial coordinate transformation and the functional gradient,we derive a functional gradient expression of the arbitrary transformation formula in space.The study shows that the essence of spatial coordinate transformation is a kind of potential field mathematically.This potential field conception can unify all the space coordinate transformations.It is the theoretical foundation for the further study of time continuous space coordinate transformation,and this study gives a new solution for the attitude determination of motion carriers.