In this lecture, we will introduce Markov chains and show a potential algorithmic use of Markov chains for sampling from complex distributions. For a finite state space Ω, we say a sequence of random variables (Xt) on Ω is a Markov chain if the sequence is Markovian in the following sense, for all t, all x0, . . . , xt, y ∈ Ω, we require Pr(Xt+1 = y|X0 = x0, X1 = x1, . . . , Xt = xt) = Pr(Xt+1 ...

Using a numerical method based on sub-super solution, we will obtain positive solution to the coupled-system of boundary value problems of the form −∆u(x) = λf(x, u, v) x ∈ Ω −∆v(x) = λg(x, u, v) x ∈ Ω u(x) = 0 = v(x) x ∈ ∂Ω where f , g are C functions with at least one of f(x0, 0, 0) or g(x0, 0, 0) being negative for some x0 ∈ Ω (semipositone).

Generically, a probability distribution corresponds to a function p(x) which defines the likelihood for random variable x. The likelihood that for which the probability of the random variable falling in a small interval dx centered on the value x0 is p(x0)dx. The distribution describes the probability density which, when integrated, yields the probability of x falling between any two values: P ...

We analyze the inclusive b(c) → s(u)μ+μ− and the exclusive B(D+) → K(π+)μ+μ− flavour changing neutral current decays in the light of HyperCP boson X0 of mass 214 MeV recently observed in the hyperon decay Σ+ → p μ+μ−. Using the branching ratio data of the above inclusive and exclusive decays, we obtain constraints on g1 (h1) and g2 (h2), the scalar and pseudo-scalar coupling constants of the b−...

In the nondiagonal case, interpolation spaces for a collection of Besov spaces are described. The results are consequences of the fact that, whenever the convex hull of points (s̄0, η0), . . . , (s̄n, ηn) ∈ Rm+1 includes a ball of Rm+1, we have (l0 q0 ((X0, X1)η0,p0 ), . . . , l s̄n qn ((X0, X1)ηn,pn))θ̄,q = l s̄θ̄ q ((X0,X1)ηθ̄,q), where θ̄ = (θ0, . . . , θn) and (sθ̄ , ηθ̄) = θ0(s̄0, η0) + · · ·+ θn(s̄n,...

This paper concerns the shape theory for triads of spaces which was introduced by the author. More precisely, in the first part, the shape dimension for triads of spaces (X; X0,X1) is introduced, and its upper and lower bounds are given in terms of the shape dimensions of X0, X1, X0 ∩ X1 and X. In the second part, a Whitehead type theorem for triads of spaces and a Mayer-Vietoris type theorem c...

We use the continuous wavelet transform to generalize the multifractal formalism to fractal functions. We report the results of recent applications of the so-called wavelet transform modulus maxima (WTMM) method to fully developed turbulence data and DNA sequences. We conclude by brie y describing some works currently under progress, which are likely to be the guidelines for future research. c ...

One of the main advantages of cloud computing is that it helps the users to save money: instead of buying a lot of computers to cover all their computations, the user can rent the computation time on the cloud to cover the rare peak spikes of computer need. From this viewpoint, it is important to find the optimal division between in-house and in-the-cloud computations. In this paper, we solve t...

In 1979 Stockman introduced the SSS* minimax search algorithm that dominates Alpha-Beta in the number of leaf nodes expanded. Further investigation of the algorithm showed that it had three serious drawbacks, which prevented its use by practitioners: it is difficult to understand, it has large memory requirements, and it is slow. This paper presents an alternate formulation of SSS*, in which it...

Buchmann and Pethő [5] observed that following algebraic integer 10 + 9α + 8α + 7α + 6α + 5α + 4α, with α = 3 is a unit. Since the coefficients form an arithmetic progressions they have found a solution to the Diophantine equation (1) NK/Q(x0 + αx1 + · · ·+ x6α) = ±1, such that (x0, . . . , x6) ∈ Z is an arithmetic progression. Recently Bérczes and Pethő [3] considered the Diophantine equation ...