We generalize some of the central results in automata theory to the abstraction level of coalgebras and thus lay out the foundations of the theory of coalgebra automata. In particular, we prove the following results for any functor F that preserves weak pullbacks. We show that the class of recognizable languages of F-coalgebras is closed under taking unions, intersections, and projections. We a...

We give necessary and sufficient conditions for the Hardy operator to be bounded on a rearrangement invariant quasi-Banach space in terms of its Boyd indices. MAIN RESULTS A rearrangement invariant space X on R is a set of measurable functions (modulo functions equal almost everywhere) with a complete quasi-norm ‖ · ‖X such that the following holds: i) if g ≤ f ∗ and f ∈ X, then g ∈ X with ‖g‖X...

We show that for certain bounded cylinder functions of the form F(x) = μ̂((h1,x)∼, . . . ,(hn,x)∼), x ∈ B, where μ̂ :Rn → C is the Fourier-transform of the complexvalued Borel measure μ on (Rn), the Borel σ -algebra of Rn with ‖μ‖ < ∞, the analytic Feynman integral of F exists, although the analytic Feynman integral, limz→−iq Iaw(F ;z)= limz→−iq(z/2π) ∫ Rn f( →u)exp{−(z/2)| →u|2}d →u, do not alwa...

Let f be an analytic function in the open unit disk normalized such that f(0) = f ′(0) − 1 = 0. In this paper the modulus and the real part of the linear combination of f ′(z) and f(z)/z is studied and conditions when f is with bounded turning are obtained.

Let F be any family of graphs of fixed treewidth and bounded degree. We construct a quadratic-time algorithm for calculating the genus distribution of the graphs in F . Within a post-order traversal of the decomposition tree, the algorithm involves a full-powered upgrading of production rules and root-popping. This algorithm for calculating genus distributions in quadratic time complements an a...

In a Euclidean space , the Lebesgue-Stieltjes integral of set-valued stochastic processes d R       , 0, t F F t T    with respect to real valued finite variation process       , 0, t A t T   t is defined directly by employing all integrably bounded selections instead of taking the decomposable closure appearing in some existed references. We shall show that this kind of integr...

Let a, b ∈ Q be such that exactly one of a and b is an algebraic integer, and let ft(z) := z 2 + t be a family of polynomials parametrized by t ∈ Q. We prove that the set of all t ∈ Q for which there exist m,n ≥ 0 such that f t (a) = f t (b) has bounded height. This is a special case of a more general result supporting a new bounded height conjecture in arithmetic dynamics.

In stochastic convex optimization the goal is to minimize a convex function F (x) . = Ef∼D[f(x)] over a convex set K ⊂ R where D is some unknown distribution and each f(·) in the support of D is convex over K. The optimization is commonly based on i.i.d. samples f, f, . . . , f from D. A standard approach to such problems is empirical risk minimization (ERM) that optimizes FS(x) . = 1 n ∑ i≤n f...

Let X be a metric space and define the category B(X) of manifolds bounded over X in the usual way [FW]. The objects of B(X) are manifolds M equipped with proper2 maps p : M → X. These maps p need not be continuous. A morphism (M1, p1) → (M2, p2) between objects of B(X) is a continuous map f : M1 →M2 which is bounded over X in the sense that there exists k > 0 for which d(p1(m), p2(f(m))) < k fo...

We prove that entire bounded monotone solutions to fully nonlinear equations in R of the form F (Du) = f(u) are one-dimensional, under appropriate compatibility conditions for F and f. In the particular case when F = ∆ and f(u) = u − u, our result gives a new (non-variational) proof of the well known De Giorgi’s conjecture.