Let w, w,2 and w2 be the fitnesses of genotypes A,A, AlA2 and A2A2 in an infinite diploid population, and let p. be the A, gene frequency in the nth generation. If fitness varies independently from generation to generation, then pn is a Markov process with a continuum of states. If E[ln(w/w2,)] <0 for i = 1,2, then there is a unique stationary probability, and the distribution of p. converges t...

We show that if T is a bounded linear operator on a complex Hilbert space, then 1 2 ‖T‖ ≤ √ w2(T ) 2 + w(T ) 2 √ w2(T )− c2(T ) ≤ w(T ), where w(·) and c(·) are the numerical radius and the Crawford number, respectively. We then apply it to prove that for each t ∈ [0, 12 ) and natural number k, (1 + 2t) 1 2k 2 1 k m(T ) ≤ w(T ), where m(T ) denotes the minimum modulus of T . Some other related ...

For an ordered set W = {w1, w2, . . . , wk} of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W ) := (d(v, w1), d(v, w2), . . . , d(v, wk)) is called the metric representation of v with respect to W , where d(x, y) is the distance between vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W ...

1. I n t r o d u c t i o n 1. Let M be a real-analytic manifold, Vect M the Lie algebra of analytic vector fields on M. We consider a vector field f E Vect M as a differential operator of first order on the algebra of smooth functions C ~ (M) which satisfies the differentiation rule f(W1W2) -" (fw1)W2 + wI(fW2) VWI, W2 E Coo(M). A point z E M will be identified with the corresponding homomorphi...

The present paper is a continuation of [1]. In the previous paper we dealt with the purely asymptotic estimations, whereas now our goal is to provide some estimates valid for quite a substantial part of the tail. We shall deal with the following simple case. An investor operating on an emerging market, has in his portfolio two currencies which are highly dependent, for example euros (EUR) and S...

In this paper we study the asymptotic distribution of the moments of (nonnormalized) traces Tr(w1),Tr(w2), . . . ,Tr(wr), where w1, w2, . . . , wr are reduced words in unitaries in the group U(N). We prove that as N → ∞ these variables are distributed as normal gaussian variables √ j1Z1, . . . , √ Zr , where j1, . . . , jr are the number of cyclic rotations of the words w1, . . . , ws leaving t...

In the context of paracontact geometry, η-Ricci solitons are considered on manifolds satisfying certain curvature conditions: R(ξ,X) · S = 0, S · R(ξ,X) = 0, W2(ξ,X) · S = 0 and S · W2(ξ,X) = 0. We prove that on a para-Kenmotsu manifold (M,φ, ξ, η, g), the existence of an η-Ricci soliton implies that (M, g) is quasi-Einstein and if the Ricci curvature satisfies R(ξ,X) · S = 0, then (M, g) is Ei...

휂-Ricci solitons on Lorentzian 훽-Kenmotsu manifold are considered an manifolds satisfying certain curvature Conditions, R(ξ,X).S=0, S(ξ,X).R=0, W2(ξ,X).S=0,S(ξ,X).W2=0 We proved that in β-Kenmotsu (M,φ,ξ,η,푔). Then the existence of implies M is Einstein and if Ricci tensor satisfies, then steady. If condition 휇=0, 휆=0, which shows 휆is steady

We calculate the nondegenerate bound electronic nonlinear refractive index n2 ( w1; w2) (i.e., an index change at frequency w1 due to the presence of a beam at frequency w2) in semiconductors. We calculate this nonlinearity and its disper~ sion using a Kramers-Kronig transformation on the calculated nondegenerate nonlinear absorption spectrum due to two-photon absorption, electronic Raman and o...

We study the sets of values taken by the exponents of quadratic approximation w2 and w ∗ 2 evaluated at real numbers whose sequence of partial quotients is generated by a finite automaton. Among other results, we show that these sets contain every sufficiently large rational number and also some transcendental numbers. Résumé. Nous étudions les ensembles des valeurs prises par les exposants d’a...