A generalization of the definition of combinatorial species is given by considering functors whose domains are categories of finite sets, with various classes of relations as moronisms. Two cases in particular correspond to species for which one has notions of restriction and quotient of structures. Coalgebras and/or Hopf algebras can be associated to such species, the duals of which provide an algebraic framework for studying invariants of structures.

In this paper, we are concerned with the basic problem defined in [9]. Formulas for δV(0)and δ∞V(0),respectively the generalized and asymptotic gradient of the value function at zero, corresponding to an L2 -additive perturbation of dynamics are given. Under the normality condition, δV(0)turns out to be a compact subset of L2, formed entirely of arcs, and V is locally finite and Lipschitz at 0. Moreover, estimations of the generalized directional derivative and Dini's derivative of V at 0 are derived. Supplementary conditions imply that Dini's derivative of V at 0 exists, and V is actually strictly differentiate at this point.

This paper uses the notion of the quantum dimension to obtain new results on the cohomology and representation theory of quantum groups at a root of unity. In particular, we consider the elementary theory of support varieties for quantum groups.

We study the family of elliptic curves Em X3 + Y3 = m where m is a cubefree integer.

The elliptic curves Em with even analytic rank and those with odd analytic rank are proved to be equally distributed. It is proved that the number of cubefree integers m ≤ X such that the analytic rank of Em is even and ≥ 2 is at least CX2/3-ε, where ε is arbitrarily small and C is a positive constant, for X large enough. Therefore, if we assume the Birch and Swinnerton-Dyer conjecture, the number of all cubefree integers m ≤ X such that the equation X3 + Y3 = m have at least two independent rational solutions is at least CX2/3-ε.

For a measure preserving system (X, ℬ, μ, T) with a factor (Y, C, v, T) and an infinite sequence {tn}, one can define conditional sequence entropy. We present two theorems which characterize rigid and mildly mixing extensions by conditional sequence entropy. Properties of IP-systems are used to prove our main theorems.

Let be the class of all polynomials/? of degree at most n such that In view of the example zn it follows from Bernstein's inequality for polynomials that at each point z0 of the unitcirle. It was shown by A. Giroux and Q. I. Rahman [2] that if denotes the subclass of polynomials in which vanish at 1, then

where c1 and c2 are constants not depending on n. Here we find the exact value of which has some special significance and also at certain other points of the unit circle.

Let Σ be a countably generated left amenable semigroup and ﹛Tσ|σ ∈ Σ﹜ be a representation of Σ as a semigroup of positive linear operators on a weakly sequentially complete Banach lattice E with a weak unit e. It is assumed Tσ are uniformly bounded. It is shown that a necessary and sufficient condition for the existence of a weak unit invariant under ﹛Tσ | σ ∈ Σ﹜ is that inf σ∈Σ H(Tσe) > 0 for all nonzero H in the positive dual cone of E.

In this note, we will consider the heat propagation on locally finite graph networks which satisfy a skew condition on vertices (See Definition of Section 2). For several periodic models, we will construct the heat kernels Pt with the skew condition explicitly, and derive the decay order of Pt as time goes to infinity.

When an optimization problem is represented by its essential objective function, which incorporates constraints through infinite penalties, first- and secondorder conditions for optimality can be stated in terms of the first- and second-order epi-derivatives of that function. Such derivatives also are the key to the formulation of subproblems determining the response of a problem's solution when the data values on which the problem depends are perturbed. It is vital for such reasons to have available a calculus of epi-derivatives. This paper builds on a central case already understood, where the essential objective function is the composite of a convex function and a smooth mapping with certain qualifications, in order to develop differentiation rules covering operations such as addition of functions and a more general form of composition. Classes of "amenable" functions are introduced to mark out territory in which this sharper form of nonsmooth analysis can be carried out.

A geometric characterization is given for a collection of points in ℝd to produce a singular l1 -distance matrix. Some quantitative results are established in terms of "characteristic matrices". The results in this paper generalize those of Dyn, Light and Cheney and have application to ridge function interpolation.

Let f define a baker's transformation (Tf, Pf). We find necessary and sufficient conditions on f for (Tf, Pf) to be an N(ω)-step random Markov chain. Using this result, we give a simplified proof of Bose's results on Holder continuous baker's transformations where f is bounded away from zero and one. We extend Bose's results to show that, for the class of baker's transformations which are random Markov chains where TV has finite expectation, a sufficient condition for weak Bernoullicity is that the Lebesgue measure λ{x f(x) = 0 or f(x) = 1} = 0. We also examine random Markov chains satisfying a strictly weaker condition, those for which the differences between the entropy of the process and the conditional entropy given the past to time n form a summable sequence; and we show that a similar result holds. A condition is given on/ which is weaker than Holder continuity, but which implies that the entropy difference sequence is summable. Finally, a particular baker's transformation is exhibited as an easy example of a weakly Bernoulli transformation on which the supremum of the measure of atoms indexed by n-strings decays only as the reciprocal of n.

We use recent advances in the theory of non-symmetric Dirichlet forms to study a class of Banach space valued Ornstein-Uhlenbeck processes. As an example, we look at Walsh's stochastic model of neural response and show that it is a continuous process in any Sobolev space Hα(α < ½), and that it takes values only among functions with unbounded variation.

In this paper we show that any (respectively α є Ωm) can be represented by a closed smooth (respectively closed, oriented smooth) manifold Mm admitting a smooth (Z/2)m (respectively S1)-action with a finite stationary set. We also completely determine the Grassman manifolds which are oriented boundaries as well as those which represent non-torsion elements in Ω*.

We prove that any holomorphic, homogenous vector bundle admits a homogenous minimal metric—a metric for which the Hermitian-Einstein tensor is diagonal in a suitable sense. The concept of minimality depends on the choice of the Jordan-Holder filtration of the corresponding parabolic module. We show that the set of all admissible Hermitian-Einstein tensors of certain class of minimal metrics is a convex subset of the euclidean space. As an application, we obtain an algebraic criterion for semistability of homogenous holomorphic vector bundles.