Standard tri-point transition function

It is usually difficult to express a family of tri-point transition function (TTF) by a transition matrix as Markov processes with one parameter. In this paper, we define three kinds of connection matrixes on the states of standard tri-point transition function (STTF) and study their essential character, give a constructive method on the constant-value standard tri-point transition function and a general expression of the state-symmetric standard tri-point transition function by a sequence of the transition matrixes of special and simple Markov processes with one parameter.     

Pillai’s generalized function

The distribution of values of Pillai’s function over the ring of Gauss integers $ \mathbb{Z}\left[ i \right] $ is studied. The asymptotic formulas for the summators $ {\varSigma_{{N\left( \alpha \right)\leq x}}}\frac{{g\left( \alpha \right)}}{{{N^{\alpha }}\left( \alpha \right)}} $ , where $ a\in \mathbb{R} $ , are found.     

Integrals of psi—function

To the author's knowledge, among the so—called special functions, the gamma function is a unique one which is defined by a linear difference equation and is a hyper—transcendental function. There exists an another well—known hyper—transcendental function called the psi function, which is merely the logarithmic derivative of the gamma function. In this paper the author consider an extension of the gamma function and then obtain a series of integrals of the psi function.     

Pták function and symmetry

We investigate subadditivity of Pták function as a sufficient and necessary condition for symmetry in the class of involutive spectrally bounded algebras. Among others it is proved an analogue of Raikov's criterion for complete lmc^*-algebras in which the hermitian elements have finite spectral radius. This paper is based on the author's Doctoral Thesis, University of Athens, written under the supervision of Professor M. Fragoulopoulou.     

What is my objective function?

This paper expresses some of the views of the author about OR and his warm thanks to many friends.     

On the differential properties of the support function of the∈-subdifferential of a convex function

We study differential properties of the support function of the-subdifferential of a convex function; applications in algorithmics are also given.     

L_1-norm packings from function fields

In this paper,we study some packings in a cube,namely,how to pack n points in a cube so as to maximize the minimal distance.The distance is induced by the L_1-norm which is analogous to the Hamming distance in coding theory.Two constructions with reasonable parameters are obtained,by using some results from a function field including divisor class group,narrow ray class group,and so on.We also present some asymptotic results of the two packings.     

On characterizations of isometries on function spaces

The paper gives characterization for an isometric isomorphism on little Bloch space, VMOA and holomorphic Besov space over the unit ball B_n in C~n.     

Computing the function GM(υ)

Tables are presented on the function G_M() which describes head waves or waves propagating near a concave surface.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 85–92, 1975.     

The impulse influence function for de-centralized control

Impulse influence matrix function is introduced based on that the de-centralized control analysis is analogous to the sub-structural analysis in structural mechanics. The static sub-structural analysis is analogous to the usual de-centralized control, whereas the dynamic sub-structural analysis corresponds to the de-centralized control theory. The re-     

Strange evaluations of Fox–Wright function

By means of inversion techniques and four known hypergeometric series identities, eight summation formulas for the Fox–Wright function are established. They can give numerous summation formulas for 2-balanced hypergeometric series when the parameters are specified.     

Positivstellens?tze for real function algebras

We look for algebraic certificates of positivity for functions which are not necessarily polynomial functions. Similar questions were examined earlier by Lasserre and Putinar [Positivity and optimization for semi-algebraic functions (to appear), Proposition 1] and by Putinar [A Striktpositivestellensatz for measurable functions (corrected version) (to appear), Theorem 2.1]. We explain how these results can be understood as results on hidden positivity: The required positivity of the functions implies their positivity when considered as polynomials on the real variety of the respective algebra of functions. This variety is however not directly visible in general. We show how algebras and quadratic modules with this hidden positivity property can be constructed. We can then use known results, for example Jacobi’s representation theorem (Jacobi in Math Z 237:259–273, 2001, Theorem 4), or the Krivine-Stengle Positivstellensatz (Marshall in Positive polynomials and sums of squares. Mathematical Surveys and Monographs 146, 2008, page 25), to obtain certificates of positivity relative to a quadratic module of an algebra of real-valued functions. Our results go beyond the results of Lasserre and Putinar, for example when dealing with non-continuous functions. The conditions are also easier to check. We explain the application of our result to various sorts of real finitely generated algebras of semialgebraic functions. The emphasis is on the case where the quadratic module is also finitely generated. Our results also have application to optimization of real-valued functions, using the semidefinite programming relaxation methods pioneered by Lasserre [SIAM J Optim 11(3): 796–817, 2001; Lasserre in Moments, positive polynomials and their applications. Imperial College Press, London, 2009; Lasserre and Putinar in Positivity and optimization for semi-algebraic functions (to appear); Marshall in Positive polynomials and sums of squares. Mathematical Surveys and Monographs 146, 2008, page 25].     

Pólya and Newtonian function fields

Let be a finite function field extension and denote by O _K the integral closure of in K. In this article, we are interested in Pólya fields, that is, fields K, such that the O _K -module Int(O _K ) of integer-valued polynomials over O _K admits a regular basis. We show that the cyclotomic extensions of are Pólya fields, and we characterize some totally imaginary extensions which are Pólya fields. Then, we are interested in Pólya fields K which have a regular basis of the form for some sequences of elements of O _K . For totally imaginary extensions, we show that it is the case if and only if O _K is isomorphic to . This gives a answer to a question raised by Thakur. The author thanks his thesis adviser Jean-Luc Chabert, and Mireille Car for their help, and their valuable advices to do this work. The author thanks also the referee for his valuable remarks.     

Absolutely monotonic functions related to Euler’s gamma function and Barnes’ double and triple gamma function

It is shown that the remainders in asymptotic expansions of the logarithm of Barnes double and triple gamma function and Euler’s gamma function are Laplace transformations of positive multiples of absolutely monotonic functions. Applications concerning positivity of sums involving Bernoulli numbers are given.     

Combinatorial interpretations of Ramanujan’s tau function

We use a $q$-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan’s tau function which involves $t$-cores and a new class of partitions which we call $(m,k)$-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function.     

On some problems involving Hardy’s function

Some problems involving the classical Hardy function