Finite Sum Of Functions Research Articles

New classes of periodic and non-periodic exact solutions for Newtonian and non-Newtonian fluid flows

Two different families of planar exact solutions for second grade fluid flows are derived in this paper. Exact solutions for Newtonian fluid flows are derived as particular case when normal stress moduli vanish. The novelty of the solutions is that they are obtained by assuming the stream functions as a finite sum of functions with different arguments. An important property of the derived solutions is that solutions in a given family are all superposable flows. The first family of solutions can be considered as a generalization of rectangular array of Taylor vortices as these classical solutions are special cases of the derived exact solutions. A subfamily of spatially periodic exact solutions for both Newtonian and non-Newtonian fluid flows are also derived. The complex structures of the two dimensional vortices are illustrated in several cases. • Two new families of exact solutions for second grade fluid flows are derived. • These solutions are classified as generalized Taylor vortices. • The solutions in a given family are superposable flows. • Subfamily of spatially periodic solutions are constructed. • Exact solutions for Newtonian flows are also derived.

Log-Coulomb Gas with Norm-Density in $$p$$-Fields

The main result of this paper is a formula for the integral $$\int_{K^N}\rho(x)\big(\max_{i<j}|x_i-x_j|\big)^a\big(\min_{i<j}|x_i-x_j|\big)^b\prod_{i<j}|x_i-x_j|^{s_{ij}}|dx|,$$ where $K$ is a $p$-field (i.e., a nonarchimedean local field) with canonical absolute value $|\cdot|$, $N\geq 2$, $a,b\in\mathbb{C}$, the function $\rho:K^N\to\mathbb{C}$ has mild growth and decay conditions and factors through the norm $\|x\|=\max_i|x_i|$, and $|dx|$ is the usual Haar measure on $K^N$. The formula is a finite sum of functions described explicitly by combinatorial data, and the largest open domain of complex tuples $(s_{ij})_{i<j}$ on which the integral converges absolutely is given explicitly in terms of these data and the parameters $a$, $b$, $N$, and $K$. We then specialize the formula to $s_{ij}=\mathfrak{q}_i\mathfrak{q}_j\beta$, where $\mathfrak{q}_1,\mathfrak{q}_2,\dots,\mathfrak{q}_N>0$ represent the charges of an $N$-particle log-Coulomb gas in $K$ with background density $\rho$ and inverse temperature $\beta$. From this specialization we obtain a mixed-charge $p$-field analogue of Mehta's integral formula, as well as formulas and low-temperature limits for the joint moments of $\max_{i<j}|x_i-x_j|$ (the diameter of the gas) and $\min_{i<j}|x_i-x_j|$ (the minimum distance between its particles).

Optimization for the Sum of Finite Functions Over the Solution Set of Split Equality Optimization Problems with Applications

In this paper, we adopt an iterative approach to solve the class of optimization problem for the sum of finite functions over split equality optimization problems for the sum of two functions. This type of problem contains many optimization problems, and bilevel problems, as well as split equality problems, and split feasibility problems as special cases. Here, we are able to establish a strong convergence theorem for an iterative method for solving this problem. As consequences of this convergence theorem, we study the following problems: optimization for the sum of finite functions over the common solution set of optimization problems for the sum of two functions; optimization for the sum of finite functions; optimization for the sum of finite functions with split equality inconsistent feasibility constraints; optimization for the sum of finite functions over the solution set for split equality constrained quadratic signal recovery problem; optimization for the sum of finite functions over the solution set of generalized split equality multiple set feasibility problem, and optimization for the sum of finite functions over the solution set of split equality linear equations problem. We use simultaneous iteration to establish strong convergence theorems for these problems. Our results generalize and improve many existing theorems for these types of problems in the literature and will have applications in nonlinear analysis, optimization problems and signal processing problems.

Approximate local output regulation for nonlinear distributed parameter systems

Output regulation for a class of nonlinear infinite-dimensional systems, called regular nonlinear systems (RNS), is the subject of this work. For the plants in this class, the linearization at the origin is an exponentially stable regular linear system (RLS). The plants are driven by a control input and a disturbance signal. Well-posedness of the plants for small initial states, control inputs and disturbance signals is established and it is shown that if the control input and the disturbance signal for a plant are T-periodic, then so are its state and output (asymptotically). On the basis of this characterization, an approximate local output regulator problem for multi-input multi-output (MIMO) plants in the RNS class is addressed. Given a plant, the regulation objective is to ensure that a finite number of harmonics of a T-periodic reference signal and the plant output are identical whenever the reference signal, the T-periodic disturbance signal for this plant and the initial state are small. An internal model based output feedback control scheme is proposed for an exponentially stable RLS for tracking reference signals, which are a finite sum of functions that are a product of a sinusoid and a polynomial in time. This scheme merely uses the transfer function gains of the RLS at the poles of the Laplace transform of the reference signal and practically requires no other data. Using the proposed control scheme, a linear finite-dimensional controller is designed for a MIMO nonlinear plant in the RNS class using minimal plant information. The resulting closed-loop system is rigorously analyzed to establish that the controller achieves the regulation objective. The efficacy of the control design is illustrated numerically using the model of a cable coupled to a point mass via a nonlinear spring.

Remarks on the overlapping-sphere method for molecular orbitals

In preference to the overlapping-sphere multiple-scattering method for solving the self-consistent-field problem for a molecule or atomic cluster, the authors recommend a cellular method. Each nucleus is surrounded by a cell, and the cells taken together fill all space and do not overlap. The cells in such a case as a diatomic molecule extend out to infinity. For an approximation one can assume a spherically symmetrical potential in each cell. The general solution of Schroedinger's equation within a cell is then set up as a linear combination of solutions of the spherical problem for a single energy, but for different l and m values. In order to have the functions regular at the nucleus and at infinity, one must join two independent solutions for each l value, one regular at the origin and the other at infinity, over the surface of a sphere of radius r/sub 1/, on which the functions are made continuous, but the normal derivatives are discontinuous. The coefficients of the functions are so chosen that the sum is continuous with continuous derivative over the spheres of radius r/sub 1/, and over the planes separating neighboring atoms. In practice, one uses a finite sum of functions, perhapsmore » for i = 0, 1, ..., 10, and applies the conditions of continuity at enough points to determine the coefficients. By expressing the potential and the radial functions as power series in r, or in r - r/sub 0/, one can get a convenient analytic solution of Schroedinger's equation. The various steps involved in this method have been studied, but they have not yet been combined into a computer program, so that this paper is regarded as a progress report.« less

Shannon capacity of STBC in Rayleigh fading channels

Space–time block coding (STBC) is a powerful transmit diversity scheme for multiple antenna systems. A closed-form expression for the Shannon capacity of MIMO systems is derived using STBC under independent and quasi-static flat Rayleigh fading channels. The Shannon capacity of STBC over such channels is expressed as a finite sum of functions that are easy to evaluate, thereby avoiding the need for numerical integration or Monte Carlo simulation.

A Decomposition Theorem for the Solutions to the Interface Problems of Quasi–Linear Elliptic Equations

Interface problems for second order quasi–linear elliptic partial differential equations in a two–dimensional space are studied. We prove that each weak solution can be decomposed into two parts near singular points, one of which is a finite sum of functions of the form cr α log m rϕ (θ), where the coefficients c depend on the H 1–norm of the solution, the C 0,δ–norm of the solution, and the equation only; and the other one of which is a regular one, the norm of which is also estimated.

A method for solving the Schrodinger equation

The author discusses a method for solving the Schrodinger equation with spherically symmetric potentials. The method is based on the use of a certain integral transform for the radial wavefunction. In the representation defined by this transform the Schrodinger equation becomes an integral equation of Volterra type. In a number of cases this equation can be solved exactly without any use of perturbation theory. The resulting solution is represented by a locally finite sum of functions, which can be easily calculated since they are defined by integrals involving simple algebraic functions. For this reason, the method is well suited for practical calculations. The author derives exact solutions for the Yukawa and exponential potentials. As an application the author calculates the bound state spectrum and the scattering cross-sections.

On the values at integers of the Dedekind zeta function of a real quadratic field

In 1976 Shintani gave a decomposition of the Dedekind zeta function, ζ κ ( s ) \zeta \kappa (s) , of a totally real number field into a finite sum of functions, each given by a Dirichlet series whose meromorphic continuation assumes rational values at negative integers. He obtained a formula for these values, thereby giving an expression for ζ κ ( − n ) , \zeta \kappa ( - n), , n = 0 , 1 , 2 , … n = 0,\,1,\,2, \ldots . Earlier, Zagier had studied the special case of ζ ( A , s ) \zeta (A,\,s) , the narrow ideal class zeta function for a real quadratic field. He decomposes ζ ( A , s ) \zeta (A,\,s) into Σ A Z Q ( s ) {\Sigma _A}{Z_Q}(s) , where Z Q ( s ) {Z_Q}(s) is given as a Dirichlet series associated to a binary quadratic form Q ( x , y ) = a x 2 + b x y + c y 2 Q(x,\,y) = a{x^2} + bxy + c{y^2} , and the summation is over a canonically given finite cycle of “reduced” quadratic forms associated to a narrow ideal class A A . He then obtains a formula for Z Q ( − n ) {Z_Q}( - n) as a rational function in the coefficients of the form Q Q . Since the denominator of ζ ( A , − n ) \zeta (A,\, - n) is known not to depend on the class A A , whereas the coefficients of reduced forms attain arbitrarily large values, it is natural to ask whether the rational function in Zagier’s formula might be replaced by a polynomial. In this paper such a result is obtained. For example, Zagier gives \[ 15120 ζ ( A , − 2 ) = ∑ A b 5 − 10 a b 3 c + 30 a 2 b c 2 a 3 + b 5 − 10 a b 3 c + 30 a 2 b c 2 c 3 − 21 b ( a + c ) 15120\zeta (A, - 2) = \sum \limits _A {\frac {{{b^5} - 10a{b^3}c + 30{a^2}b{c^2}}}{{{a^3}}}} + \frac {{{b^5} - 10a{b^3}c + 30{a^2}b{c^2}}}{{{c^3}}} - 21b(a + c){\text { }} \] while our result is \[ 15120 ζ ( A , − 2 ) = 1 2 ( ∑ A − ∑ θ A ) ( 60 a 2 − 117 a b + 76 a c + 38 b 2 − 117 b c + 60 c 2 ) 15120\zeta (A,\, - 2) = \frac {1} {2}\left ( {\sum \limits _A - \sum \limits _{\theta A} } \right )\,(60{a^2} - 117ab + 76ac + 38{b^2} - 117bc + 60{c^2}) \] , where θ \theta is the narrow ideal class consisting of principal ideals generated by elements of negative norm. Starting with a representation of Z Q ( 1 + n ) {Z_Q}(1 + n) due to Shanks and Zagier for n = 1 , 2 , 3 , … n = 1,\,2,\,3, \ldots as a certain transcendental function of the coefficients of Q Q , we also obtain the result that ζ ( A , 1 + n ) \zeta (A,\,1 + n) is given as the same sum of reduced quadratic forms as in the formula for ζ ( A , − n ) \zeta (A,\, - n) , times the appropriate “gamma factor.” This gives a new proof of the functional equation of ζ ( A , s ) \zeta (A,\,s) at integer values of s s , and suggests the possibility that one might be able to prove the functional equation for all s s by finding some relation between Z Q ( s ) {Z_Q}(s) and Z Q ( 1 − s ) {Z_Q}(1 - s) . So far we have not found such a relation.

On the Values at Integers of the Dedekind Zeta Function of a Real Quadratic Field

In 1976 Shintani gave a decomposition of the Dedekind zeta function, $\zeta \kappa (s)$, of a totally real number field into a finite sum of functions, each given by a Dirichlet series whose meromorphic continuation assumes rational values at negative integers. He obtained a formula for these values, thereby giving an expression for $\zeta \kappa ( - n),$, $n = 0, 1, 2, \ldots$. Earlier, Zagier had studied the special case of $\zeta (A, s)$, the narrow ideal class zeta function for a real quadratic field. He decomposes $\zeta (A, s)$ into ${\Sigma _A}{Z_Q}(s)$, where ${Z_Q}(s)$ is given as a Dirichlet series associated to a binary quadratic form $Q(x, y) = a{x^2} + bxy + c{y^2}$, and the summation is over a canonically given finite cycle of “reduced” quadratic forms associated to a narrow ideal class $A$. He then obtains a formula for ${Z_Q}( - n)$ as a rational function in the coefficients of the form $Q$. Since the denominator of $\zeta (A, - n)$ is known not to depend on the class $A$, whereas the coefficients of reduced forms attain arbitrarily large values, it is natural to ask whether the rational function in Zagier’s formula might be replaced by a polynomial. In this paper such a result is obtained. For example, Zagier gives \[ 15120\zeta (A, - 2) = \sum \limits _A {\frac {{{b^5} - 10a{b^3}c + 30{a^2}b{c^2}}}{{{a^3}}}} + \frac {{{b^5} - 10a{b^3}c + 30{a^2}b{c^2}}}{{{c^3}}} - 21b(a + c){\text { }}\] while our result is \[ 15120\zeta (A, - 2) = \frac {1} {2}\left ( {\sum \limits _A - \sum \limits _{\theta A} } \right ) (60{a^2} - 117ab + 76ac + 38{b^2} - 117bc + 60{c^2})\], where $\theta$ is the narrow ideal class consisting of principal ideals generated by elements of negative norm. Starting with a representation of ${Z_Q}(1 + n)$ due to Shanks and Zagier for $n = 1, 2, 3, \ldots$ as a certain transcendental function of the coefficients of $Q$, we also obtain the result that $\zeta (A, 1 + n)$ is given as the same sum of reduced quadratic forms as in the formula for $\zeta (A, - n)$, times the appropriate “gamma factor.” This gives a new proof of the functional equation of $\zeta (A, s)$ at integer values of $s$, and suggests the possibility that one might be able to prove the functional equation for all $s$ by finding some relation between ${Z_Q}(s)$ and ${Z_Q}(1 - s)$. So far we have not found such a relation.

Complete separability of the Stark problem in hydrogen

The authors prove that every fixed m resonance eigenfunction of hydrogen in a constant electric field is a finite sum of functions which are products in squared parabolic coordinates. This shows that the standard ansatz yields all resonances.