Abstract:
We show that a concavity property of the exponential function is a direct consequence of the convexity of the continued Erlang loss function.

Abstract:
The definition of the spline functions as solutions of a variational problem ispresented in the preliminaries of this paper and are shown some theorems regarding to theexistence and uniqueness. The main result of this article consists of a property verified by thespline functions in connection with the spaces of functions used.

Abstract:
We study totally positive (TP) functions of finite type and exponential B-splines as window functions for Gabor frames. We establish the connection of the Zak transform of these two classes of functions and prove that the Zak transforms have only one zero in their fundamental domain of quasi-periodicity. Our proof is based on the variation-diminishing property of shifts of exponential B-splines. For the exponential B-spline B_m of order m, we determine a large set of lattice parameters a,b>0 such that the Gabor family of time-frequency shifts is a frame for L^2(R). By the connection of its Zak transform to the Zak transform of TP functions of finite type, our result provides an alternative proof that TP functions of finite type provide Gabor frames for all lattice parameters with ab<1. For even two-sided exponentials and the related exponential B-spline of order 2, we find lower frame-bounds A, which show the asymptotically linear decay A (1-ab) as the density ab of the time-frequency lattice tends to the critical density ab=1.

Abstract:
We present an exponential B-spline collocation method for solving convection-diffusion equation with Dirichlet’s type boundary conditions. The method is based on the Crank-Nicolson formulation for time integration and exponential B-spline functions for space integration. Using the Von Neumann method, the proposed method is shown to be unconditionally stable. Numerical experiments have been conducted to demonstrate the accuracy of the current algorithm with relatively minimal computational effort. The results showed that use of the present approach in the simulation is very applicable for the solution of convection-diffusion equation. The current results are also seen to be more accurate than some results given in the literature. The proposed algorithm is seen to be very good alternatives to existing approaches for such physical applications.

Abstract:
Shifted surface spline is a frequently used radial function for scattered data interpolation. The most frequently used error bounds for this radial function are the one raised by Wu and Schaback in [17] and the one raised byMadych and Nelson in [14]. Both are O(dl) as d → 0, where l is a positive integer and d is the well-known fill-distance which roughly speaking measures the spacing of the data points. Then RBF people found that there should be an error bound of the form O(ω 1 d ) because shifted surface spline is smooth and every smooth function shares this property. This only problem was that the value of the cucial constant ω was unknown. Recently Luh raised an exponential-type error bound with convergence rate O(ω 1 d ) as d → 0 where 0 < ω < 1 is a fixed constant which can be accurately computed [11]. Although the exponential-type error bound converges much faster than the algebraic-type error bound, the constant ω is intensely influenced by the dimension n in the sense ω → 1 rapidly as n → ∞. Here the variable x of both the interpolated and interpolating functions lies in Rn. In this paper we present an error bound which is O(√dω′1d ) where 0 < ω′ < 1 is a fixed constant for any fixed n, and is only mildly influenced by n. In other words, ω′ → 1 very slowly as n → ∞, and ω′

Abstract:
In this paper, the maximum entropy property of the discrete-time first-order stable spline kernel is studied. The advantages of studying this property in discrete-time domain instead of continuous-time domain are outlined. One of such advantages is that the differential entropy rate is well-defined for discrete-time stochastic processes. By formulating the maximum entropy problem for discrete-time stochastic processes we provide a simple and self-contained proof to show what maximum entropy property the discrete-time first-order stable spline kernel has.

Abstract:
We extend the concept of exponential B-spline to complex orders. This extension contains as special cases the class of exponential splines and also the class of polynomial B-splines of complex order. We derive a time domain representation of a complex exponential B-spline depending on a single parameter and establish a connection to fractional differential operators defined on Lizorkin spaces. Moreover, we prove that complex exponential splines give rise to multiresolution analyses of $L^2(\mathbb{R})$ and define wavelet bases for $L^2(\mathbb{R})$.

Abstract:
For the generalized Dedekind sums s_{ij}(p,q) defined in association with the x^{i}y^{j}-coefficient of the Todd power series of the lattice cone in R^2 generated by (1,0) and (q,p), we associate an exponential sum. We obtain this exponential sum using the cocycle property of the Todd series of 2d cones and the nonsingular cone decomposition along with the continued fraction of q/p. Its Weil bound is given for the modulus q, applying the purity theorem of the cohomology of the related l-adic sheaf due to Denef and Loeser. The Weil type bound of Denef and Loeser fulfills the Weyl's equidistribution criterion for R(i,j)q^{i+j-2} s_{ij}(p,q). As a special case, we recover the equidistribution result of the classical Dedekind sums multiplied by 12 not using the modular weight of the Dedekind's \eta(\tau).

Abstract:
Confiscation has existed in the domestic legal system for more than half a century. Considering limitations in practical implementation of this principle, espe-cially in the context of fighting against organized crime, the domestic legislator has recently offered new solutions for “criminal property” confiscation in the form of criminal property confiscation procedure regulated by a special law. In this sense, this paper ana-lyzes the specific characteristics of organized crime phenomena, which require a different approach compared to the standard solutions in this area, criticizing such solutions, as well as the state of the local law and practice that preceded the adoption of the above mentioned regulation. The author concludes that criminal property confiscation may be considered as a desirable instrument in the fight against organized crime, whereby we should be careful in creating the related normative and legal framework, thus avoiding numerous negative effects that may challenge its creators. He also points out the international element that has significant influence in the design and practical implementation of the national models. [Projekat Ministarstva nauke Republike Srbije, br. 179045: Razvoj institucionalnih kapaciteta, standarda i procedura za suprotstavljanje organizovanom kriminalu i terorizmu u uslovima me unarodnih integracija]

Abstract:
The complex exponential function is a local homeomorphism and gives therefore rise to an 'etale groupoid and a C*-algebra. We determine this algebra, as well as the alge bra of the complex conjugate of the exponential function.