Equicontinuity of maps on a dendrite with finite branch points

Let(T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote byω(x,f) and P(f) the ω-limit set of x under f and the set of periodic points of,respectively. Write Ω(x,f) = {y| there exist a sequence of points x_k E T and a sequence of positive integers n_1 n_2 … such that lim_(k→∞)x_k=x and lim_(k→∞)f~(n_k)(x_k) =y}. In this paper, we show that the following statements are equivalent:(1) f is equicontinuous.(2) ω(x, f) = Ω(x,f) for any x∈T.(3) ∩_(n=1)~∞f~n(T) = P(f),and ω(x,f)is a periodic orbit for every x ∈ T and map h : x→ω(x,f)(x ET)is continuous.(4) Ω(x,f) is a periodic orbit for any x∈T.     

Sobolev spaces and capacities theory on path spaces over a compact Riemannian manifold

We introduce Sobolev spaces and capacities on the path space P _{m 0} (M) over a compact Riemannian manifold M. We prove the smoothness of the Itô map and the stochastic anti-development map in the sense of stochastic calculus of variation. We establish a Sobolev norm comparison theorem and a capacity comparison theorem between the Wiener space and the path space P _{m 0} (M). Moreover, we prove the tightness of (r, p)-capacities on P _{m 0} (M), \(\), which generalises a result due to Airault-Malliavin and Sugita on the Wiener space. Finally, we extend our results to the fractional Hölder continuous path space \(\).     

Chowla’s cosine problem

A continuous linear map T from a Banach algebra A into another B approximately preserves the zero products if ‖T(a)T(b)‖ ≤ α‖a‖‖b‖ (a,b ∈ A, ab = 0) for some small positive α. This paper is mainly concerned with the question of whether any continuous linear surjective map T: A → B that approximately preserves the zero products is close to a continuous homomorphism from A onto B with respect to the operator norm. We show that this is indeed the case for amenable group algebras.     

Semiclassical resonances associated with a periodic orbit

We consider resonances for a h-pseudo-differential operator H(x, hD _x; h) induced by a periodic orbit of hyperbolic type. We generalize the framework of Gérard and Sjöstrand, in the sense that we allow hyperbolic and elliptic eigenvalues of the Poincarémap, and look for so-called semi-excited resonances with imaginary part of magnitude ?h log h, or h ^δ, with 0 < δ < 1.     

Embedding asymptotically expansive systems

A topological dynamical system is said asymptotically expansive when entropy and periodic points grow subexponentially at arbitrarily small scales. We prove a Krieger like embedding theorem for asymptotically expansive systems with the small boundary property. We show that such a system (X, T) embeds in the K-full shift if \( h_{top}(T)<\log K\) and \(\sharp Per_n(X,T)\le K^n\) for any integer n. The embedding is in general not continuous (unless the system is expansive and X is zero-dimensional) but the induced map on the set of invariant measures is a topological embedding. It is shown that this property implies asymptotical expansiveness. We prove also that the inverse of the embedding map may be continuously extended to a faithful principal symbolic extension.     

<Emphasis Type="Italic">T</Emphasis>-optimal designs for discrimination between rational and polynomial models

This paper considers the problem of the analytical construction of experimental designs optimal with respect to the popular T-optimality criterion proposed by A.C. Atkinson and V.V. Fedorov in 1975 for discrimination between the simplest rational and polynomial regression models. It is shown how the classical results from approximation theory can be used to derive explicit formulas describing the behavior of support points and weights of T-optimal designs for different fixed parameter values. An applied discrimination problem for rational and polynomial regression models is considered as an example. For this models the numerical construction of experimental designs optimal with respect to robust analogues of T-criterion is also briefly discussed.     

Global Well-posedness of the Stochastic Generalized Kuramoto-Sivashinsky Equation with Multiplicative Noise

Global well-posedness of the initial-boundary value problem for the stochastic generalized Kuramoto- Sivashinsky equation in a bounded domain D with a multiplicative noise is studied. It is shown that under suitable sufficient conditions, for any initial data u₀ ∈ L₂(D × Ω), this problem has a unique global solution u in the space L₂(Ω, C([0, T], L₂(D))) for any T >0, and the solution map u₀ ? u is Lipschitz continuous.     

An Optimal Algorithm for the <Emphasis Type="Italic">k</Emphasis>-Fixed-Endpoint Path Cover on Proper Interval Graphs

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.     

The Fermi–Dirac distribution as a model of a thermodynamically ideal liquid. Phase transition of the first kind for neutral gases (corresponding to nonpolar molecules)

In this paper, we introduce the notions of enlarged number theory and of thermodynamically ideal liquid and calculate the temperature below which it appears. This temperature is T = 0.84T _c , where T _c is the critical temperature of a gas whose molecules are nonpolar. For such a gas, in a sufficiently wide neighborhood of the binodal, the isotherms of a gas and of a thermodynamically ideal liquid coincide with those of a van der Waals gas for the critical value of the compressibility factor Z _c = 3/8. In this sense, for T ≤ 0.84T _c and the particular case Z _c = 3/8, the developed theory is a generalization of the van der Waals model. A new phase transition of the second kind at the point of zero activity is described.     

Algebraic Theory of Causal Double Products

Corresponding to each “rectangular” double product in the form of a formal power series R[h] with coefficients in the tensor product ?(?)⊙ ? (?) with itself of the Itô Hopf algebra, we construct “triangular” elements T[h] of ?(?) satisfying ΔT[h] = T[h]⁽¹⁾ R[h]T{h]⁽²⁾. In Fock space representations of ?(?) by iterated quantum stochastic integrals when ? is the algebra of Itô differentials of the calculus, these correspond to “causal” double product integrals in a single Fock space.     

Functional estimation of activity criticality indices and sensitivity analysis of expected project completion time

For a PERT network, a new method is developed for estimating the criticality index of activity i (ACI _i ) as a function of the expected duration of activity i (μ _i ) and for the sensitivity analysis of the expected project completion time (μ _T ) with respect to μ _i . The proposed method evaluates the frequency of activity i being on the critical path, and thereby its ACI _i using Monte Carlo simulation or a Taguchi orthogonal array experiment at several values of μ _i , fits a logistic regression model for estimating ACI _i as a function of μ _i , and then, using the estimated ACI _i function, evaluates the amount of change in μ _T when μ _i is changed by a given amount. Unlike the previous works, the proposed method models ACI _i as a nonlinear (ie, logistic) function of μ _i , which can be used to estimate the amount of change in μ _T for a variety of changes in μ _i . Computational results indicate that the performance of the proposed method is comparable to that of direct Monte Carlo simulation.     

Detecting fixed points of nonexpansive maps by illuminating the unit ball

We give necessary and sufficient conditions for a nonexpansive map on a finite-dimensional normed space to have a nonempty, bounded set of fixed points. Among other results we show that if f: V → V is a nonexpansive map on a finite-dimensional normed space V, then the fixed point set of f is nonempty and bounded if and only if there exist w₁,..., w _m in V such that {f(w _i ) ? w _i : i = 1,..., m} illuminates the unit ball. This yields a numerical procedure for detecting fixed points of nonexpansive maps on finite-dimensional spaces. We also discuss applications of this procedure to certain nonlinear eigenvalue problems arising in game theory and mathematical biology.     

Cohomogeneity one Kähler and Kähler–Einstein manifolds with one singular orbit I

Let M be a cohomogeneity one manifold of a compact semisimple Lie group G with one singular orbit \(S_0 = G/H\). Then M is G-diffeomorphic to the total space \(G \times _H V\) of the homogeneous vector bundle over \(S_0\) defined by a sphere transitive representation of G in a vector space V. We describe all such manifolds M which admit an invariant Kähler structure of standard type. This means that the restriction \(\mu : S = Gx = G/L \rightarrow F = G/K \) of the moment map of M to a regular orbit \(S=G/L\) is a holomorphic map of S with the induced CR structure onto a flag manifold \(F = G/K\), where \(K = N_G(L)\), endowed with an invariant complex structure \(J^F\). We describe all such standard Kähler cohomogeneity one manifolds in terms of the painted Dynkin diagram associated with \((F = G/K,J^F)\) and a parameterized interval in some T-Weyl chamber. We determine which of these manifolds admit invariant Kähler–Einstein metrics.     

Holomorphic maps preserving parts of the local spectrum

Let x ₀ be a nonzero vector in \({\mathbb{C}^{n}}\) , and let \({U\subseteq \mathcal{M}_{n}}\) be a domain containing the zero matrix. We prove that if φ is a holomorphic map from U into \({\mathcal{M}_{n}}\) such that the local spectrum of T ∈ U at x ₀ and the local spectrum of φ(T) at x ₀ have always a common value, then T and φ(T) have always the same spectrum, and they have the same local spectrum at x ₀ a.e. with respect to the Lebesgue measure on U. If \({\varphi \colon U\rightarrow \mathcal{M}_{n}}\) is holomorphic with φ(0) = 0 such that the local spectral radius of T at x ₀ equals the local spectral radius of φ(T) at x ₀ for all T ∈ U, there exists \({\xi \in \mathbb{C}}\) of modulus one such that ξT and φ(T) have the same spectrum for all T in U. We also prove that if for all T ∈ U the local spectral radius of φ(T) coincides with the local spectral radius of T at each vector x, there exists \({\xi \in \mathbb{C}}\) of modulus one such that φ(T) = ξT on U.     

Tangential polynomials and matrix KdV elliptic solitons

Let (X, q) be an elliptic curve marked at the origin. Starting from any cover π: Γ → X of an elliptic curve X marked at d points {π _i } of the fiber π ^?1(q) and satisfying a particular criterion, Krichever constructed a family of d × d matrix KP solitons, that is, matrix solutions, doubly periodic in x, of the KP equation. Moreover, if Γ has a meromorphic function f: Γ → P¹ with a double pole at each p _i , then these solutions are doubly periodic solutions of the matrix KdV equation U _t = 1/4(3UU _x + 3U _x U + U _xxx ). In this article, we restrict ourselves to the case in which there exists a meromorphic function with a unique double pole at each of the d points {p _i }; i.e. Γ is hyperelliptic and each pi is a Weierstrass point of Γ. More precisely, our purpose is threefold: (1) present simple polynomial equations defining spectral curves of matrix KP elliptic solitons; (2) construct the corresponding polynomials via the vector Baker–Akhiezer function of X; (3) find arbitrarily high genus spectral curves of matrix KdV elliptic solitons.     

Unbounded disjointness preserving linear functionals and operators

Let E and F be Banach lattices. We show first that the disjointness preserving linear functionals separate the points of any infinite dimensional Banach lattice E, which shows that in this case the unbounded disjointness preserving operators from \({E\to F}\) separate the points of E. Then we show that every disjointness preserving operator \({T:E\to F}\) is norm bounded on an order dense ideal. In case E has order continuous norm, this implies that every unbounded disjointness preserving map \({T:E\to F}\) has a unique decomposition T = R + S, where R is a bounded disjointness preserving operator and S is an unbounded disjointness preserving operator, which is zero on a norm dense ideal. For the case that E = C(X), with X a compact Hausdorff space, we show that every disjointness preserving operator \({T:C(X)\to F}\) is norm bounded on a norm dense sublattice algebra of C(X), which leads then to a decomposition of T into a bounded disjointness preserving operator and a finite sum of unbounded disjointness preserving operators, which are zero on order dense ideals.     

Space of invariant bilinear forms

Let \({\mathbb {F}}\) be a field, V a vector space of dimension n over \({\mathbb {F}}\). Then the set of bilinear forms on V forms a vector space of dimension \(n^2\) over \({\mathbb {F}}\). For char \({\mathbb {F}}\ne 2\), if T is an invertible linear map from V onto V then the set of T-invariant bilinear forms, forms a subspace of this space of forms. In this paper, we compute the dimension of T-invariant bilinear forms over \({\mathbb {F}}\). Also we investigate similar type of questions for the infinitesimally T-invariant bilinear forms (T-skew symmetric forms). Moreover, we discuss the existence of nondegenerate invariant (resp. infinitesimally invariant) bilinear forms.     

The Gibbs phenomenon for general Franklin systems

The paper is devoted to the Gibbs phenomenon for series by general Franklin systems. The general Franklin system corresponding to a given dense sequence of points T = (t _n , n ≥ 0) in [0, 1] is a sequence of orthonormal piecewise linear functions with knots from T, that is, the nth function from the system has knots t ₀,..., t _n . The main result of this paper is that the Gibbs phenomenon for Fourier series by general Franklin systems occurs for almost all points of [0, 1].     

Existence of an invariant form under a linear map

Let F be a field of characteristic different from 2 and V be a vector space over F. Let J: α → α ^J be a fixed involutory automorphism on F. In this paper we answer the following question: given an invertible linear map T: V → V, when does the vector space V admit a T-invariant nondegenerate J-hermitian, resp. J-skew-hermitian, form?