Volume 8 (2012) Issue 1

Abstract: We recall first some basic facts on weighted homogeneous functions and filtrations in the ring A of formal power series. We introduce next their analogues for weighted homogeneous diffeomorphisms and vector fields. We show that the Milnor algebra is a complete invariant for the classification of weighted homogeneous polynomials with respect to right-equivalence, i.e. change of coordinates in the source and target by diffeomorphism.

Abstract: Let \(G\) be a connected graph. Let p be the number of pebbles distributed on the vertices of \(G\). A pebbling move is defined by removing two pebbles from one vertex and put a pebble on an adjacent vertex. The covering cover pebbling number, \(σ(G)\), is the least p such that after a sequence of pebbling moves, the set of vertices should form a covering for \(G\) from every configuration of p pebbles on the vertices of \(G\). In this paper, we determine the covering cover pebbling number for square of a cycle.

Abstract: If \(G\) is a connected graph, the distance d(u, v) between two vertices \(u, v ∈ V (G)\) is the length of a shortest path between them. Let \(W = {w_1, w_2, …., w_k}\) be an ordered set of vertices of \(G\) and let \(v\) be a vertex of \(G\). The representation r(v|W) of \(v\) with respect to \(W\) is the k-tuple \((d(v, w_1), d(v, w_2), ….., d(v, w_k))\). If distinct vertices of \(G\) have distinct representations with respect to W, then W is called a resolving set or locating set for \(G\). A resolving set of minimum cardinality is called a basis for \(G\) and this cardinality is the metric dimension of \(G\), denoted by \(dim(G)\). A family G of connected graphs is a family with constant metric dimension if \(dim(G)\) does not depend upon the choice of \(G\) in \(G\). In this paper, we show that the graphs (D^{∗}_{p}\) and \(D^{n}_{p}\), obtained from prism graph have constant metric dimension.

Abstract: In this paper we consider the Garman-Kohlhagen model for the American foreign exchange put option in one-dimensional diffusion model where the volatility and the domestic and foreign currency risk-free interest rates are constants. First we make preliminary estimate regarding the optimal exercise boundary of the American foreign exchange put option and then the continuity estimate with respect to volatility for the value functions of the corresponding options. Finally we establish the continuity estimate for the optimal exercise boundary of the American foreign exchange put option with respect to the volatility parameter.

Abstract: In this paper we introduce the concept of accretive operators, discuss some properties of resolvents of an accretive operator in 2- probabilistic normed spaces and focusing on the results of convex sets in 2-probabilistic normed spaces.

Abstract: Lanczos methods for solving \(Ax = b\) consist in constructing a sequence of vectors \((x_k)\), \(k = 1, …\) such that \(r_k = b − Ax_k = P_k(A)r_0\), where \(P_k\) is the orthogonal polynomial of degree at most k with respect to the linear functional c defined as c(ξ^i) = (y, A^ir_0)\). Let \(P^(1)_k\) be the regular monic polynomial of degree k belonging to the family of formal orthogonal polynomials (FOP) with respect to \(c^(1)\) defined as c^(1)(ξ ^{i}) = c^{(ξi+1)}\). All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for \(P_k\) and one for \(P^{(1)}_k\). We shall study some new recurrence relations involving these two polynomials and their possible combinations to obtain new Lanczos-type algorithms. We will show that some recurrence relations exist, but cannot be used to derive Lanczos-type algorithms, while others do not exist at all.

Abstract: In this paper we use radial basis functions in one of the projection methods to solve integral equations of the second kind with two or more variables. This method implemented without needing any introductory algorithms. Relatively good error bound and the numerical experiments show the accuracy of the method.

Abstract: The hyperbolic distances from the origin are changed under the radial selfmapping \(x → |x|^{1/K−1}x\), \(K > 1\) of the unit ball. Here author gives the power mean inequality of the hyperbolic metric under the radial mapping.

Abstract: We calculate the Boyd indices for the sum and intersection of two quasi-normed rearrangement invariant spaces. An application to Lorentz type spaces is also given.

Abstract: We show that if \(F_n = 22^n+ 1\) is the nth Fermat number, then the binary digit sum of \(π(F_n)\) tends to infinity with \(n\), where \(π(x)\) is the counting function of the primes \(p ≤ x\). We also show that if \(F_n\) is not prime, then the binary expansion of \(φ(F_n)\) starts with a long string of 1’s, where \(φ\) is the Euler function. We also consider the binary expansion of the counting function of irreducible monic polynomials of degree a given power of 2 over the field \(\mathbb{F}_{2}\). Finally, we relate the problem of the irrationality of Euler constant with the binary expansion of the sum of the divisor function.