The functional equation for the Hurwitz Zeta function zeta(s,a) is used to obtain formulas for derivatives of zeta(s,a) at negative odd s and rational a. For several of these rational arguments, closed-form expression...
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The functional equation for the Hurwitz Zeta function zeta(s,a) is used to obtain formulas for derivatives of zeta(s,a) at negative odd s and rational a. For several of these rational arguments, closed-form expressions are given in terms of simpler transcendental functions, like the logarithm, the polygamma function, and the Riemann Zeta function. (C) 1998 Elsevier Science B.V. All rights reserved.
We consider a class, denoted by Q, of the nonlinear control systems which can be densely represented as a subsystem of a certain kind of quadratic system, namely a quadratic target. We say that a system in Q undergoes...
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We consider a class, denoted by Q, of the nonlinear control systems which can be densely represented as a subsystem of a certain kind of quadratic system, namely a quadratic target. We say that a system in Q undergoes a globally exact quadratization. Here "globally" adds up to a slight extension of the notion of C-infinity immersion (of systems), namely a dense immersion, which amounts to saying that it is defined on the whole manifold of the system states, except possibly a zero-measure set. It is proven that the class Q includes all systems characterized by vector fields whose components are analytic integral closed-form functions (ICFFs). The result is first proven for algebraic system functions, by means of a constructive proof, and next extended up to analytic ICFFs. For nonanalytic ICFFs a weaker result is proven as well. Also the case of a partially observed system is considered, as well as the internal structure of every quadratic representation, which is proven to be always a feedback interconnection of bilinear systems. Finally, examples are presented for which the constructive proof given earlier is turned into a quadratization algorithm, which can be carried out by hand, and the resulting differential equations of the quadratic representation are presented.
The performances of orthogonal space-time block codes (OSTBCs) over Rician-Nakagami channels are investigated. In particular, we derive closed-form symbol error probability (SEP) expressions for OSTBC systems in which...
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The performances of orthogonal space-time block codes (OSTBCs) over Rician-Nakagami channels are investigated. In particular, we derive closed-form symbol error probability (SEP) expressions for OSTBC systems in which M-ary phase-shift-keying modulation and M-ary quadrature-amplitude modulation are used. These SEP results are expressed in terms of Lauricella's multivariate hypergeometric functions, which can be easily evaluated numerically. When the Rician-Nakagarm channel degenerates to the Rician-Rayleigh channel, or equivalently the Rayleigh fading channel, the closed-form SEP expressions are rewritten in terms of higher transcendental functions, that is, Gauss hypergeometric function and Appell hypergeometric function.
This paper considers the stability of the zero solution of impulsive delay differential equations with impulses at variable times. By means of Lyapunov functions and the Razumikhin technique, some sufficient condition...
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This paper considers the stability of the zero solution of impulsive delay differential equations with impulses at variable times. By means of Lyapunov functions and the Razumikhin technique, some sufficient conditions of uniform stability and uniform asymptotic stability for the delay differential equation with impulses at variable times are obtained.
This article presents the results of some new research on a new class of hyperbolic functions that unite the characteristics of the classical hyperbolic functions and the recurring Fibonacci and Lucas series. The hype...
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This article presents the results of some new research on a new class of hyperbolic functions that unite the characteristics of the classical hyperbolic functions and the recurring Fibonacci and Lucas series. The hyperbolic Fibonacci and Lucas functions, which are the being extension of Binet's formulas for the Fibonacci and Lucas numbers in continuous domain, transform the Fibonacci numbers theory into "continuous" theory because every identity for the hyperbolic Fibonacci and Lucas functions has its discrete analogy in the framework of the Fibonacci and Lucas numbers. Taking into consideration a great role played by the hyperbolic functions in geometry and physics, ("Lobatchevski's hyperbolic geometry", "Four-dimensional Minkowski's world", etc.), it is possible to expect that the new theory of the hyperbolic functions will bring to new results and interpretations on mathematics, biology, physics, and cosmology. In particular, the result is vital for understanding the relation between transfinitness i.e. fractal geometry and the hyperbolic symmetrical character of the disintegration of the neural vacuum, as pointed out by El Naschie [Chaos Solitons & Fractals 17 (2003) 631]. (C) 2004 Elsevier Ltd. All rights reserved.
We seek the optimal boundary control of vibrations of a spherical layer in the spherically symmetric case. This paper continues the series of papers by V.A. Il'in and his students and, unlike the previous papers, ...
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We seek the optimal boundary control of vibrations of a spherical layer in the spherically symmetric case. This paper continues the series of papers by V.A. Il'in and his students and, unlike the previous papers, uses a more general control optimality criterion. We obtain closed formulas for the controls;these formulas are consistent with Il'in's earlier results.
For a smooth and projective variety over a number field with torsion-free geometric Picard group and finite transcendental Brauer group we show that only the archimedean places, the primes of bad reduction and the pri...
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For a smooth and projective variety over a number field with torsion-free geometric Picard group and finite transcendental Brauer group we show that only the archimedean places, the primes of bad reduction and the primes dividing the order of the transcendental Brauer group can turn up in the description of the Brauer-Manin set.
We give a general formulation of the algorithm of Fokas and Ablowitz, which then allows us to obtain transformations for nth order ordinary differential equations, to equations of the same order but perhaps of higher ...
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We give a general formulation of the algorithm of Fokas and Ablowitz, which then allows us to obtain transformations for nth order ordinary differential equations, to equations of the same order but perhaps of higher degree. Previously this algorithm has been used to obtain transformations for the six second order equations defining new transcendental functions discovered by Painleve and co-workers, either to other equations in the Painleve classification or to equations of second order and second degree. As an example of our approach we consider a new fourth order ordinary differential equation due to Cosgrove which is believed to define a new transcendent. We obtain transformations relating this equation to other fourth order ordinary differential equations, of degrees greater than or equal to2. All of these transformations, as well as the corresponding higher degree differential equations, all of which have the Painleve property, are new. (C) 2001 American Institute of Physics.
The Euler-Poisson-Darboux equation can be represented in one of the followingforms: where a is a real parameter. Equation (1.1) is an elliptic Euler-Poisson-Darboux *** also referred to as a generalized axisymmetric L...
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The Euler-Poisson-Darboux equation can be represented in one of the followingforms: where a is a real parameter. Equation (1.1) is an elliptic Euler-Poisson-Darboux *** also referred to as a generalized axisymmetric Laplace equation [1, 2]. For a - 1, it is tlie tisymmetric Laplace equation, which was studied in [3]. Equations (1.2) and (1.3) are a hypertocEuler-Poisson-Darboux equation. [Equation (1.3) is represented via the characteristic variableEquations (1.1)-(1.3) specify classes of Euler-Poisson-Darboux equations determined by the i rametera.
In the present paper, we solve the following problem. Consider an arbitraryC∞ system F of ordinary differential equations in Rn, n ≥ 2, that has a periodic trajectory Lo ofthe type of a simple saddle-node. Further, ...
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In the present paper, we solve the following problem. Consider an arbitraryC∞ system F of ordinary differential equations in Rn, n ≥ 2, that has a periodic trajectory Lo ofthe type of a simple saddle-node. Further, we introduce the two-dimensional system Fn that is therestriction of the original system F to the center manifold Wc (Lo) of the cycle Lo. We areinterested in the simplest form to which the vector field Fo can be reduced in some sufficientlysmall neighborhood u is contained in Wc (L0) of the cycle L0.
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