Study of families of monotone continuous functions on Tychonoff spaces

The main object of study is the space of all monotone continuous functions CM(X) on a connected Tychonoff space X endowed with the topology of pointwise (CM _p (X)) or uniform (CM(X)) convergence. Technical questions concerning restriction and extension of monotone functions are considered in Sec. 2. Conditions for CM(X) to separate the points of X and for CM(X) to contain only constant functions are found in Sec. 3. In Sec. 4, the linear structure of CM(X) is studied and all linear subspaces of CM(X) for a certain class of spaces X are described. In Sec. 5, conditions under which CM(X) is closed and nowhere dense in C _p (X) and C(X) are determined. The metrizability of CM _p (X) is considered in Sec. 6; necessary and sufficient metrizability conditions for various classes of spaces X are obtained. In Sec. 7, criteria for σ-compactness and the Hurewicz property in the class of spaces CM _p (X) are given. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 34, General Topology, 2005.     

Three characterizations of non-binary correlation-immune and resilient functions

A functionf(X ₁,X ₂, ...,X _n ) is said to betth-order correlation-immune if the random variableZ=f(X ₁,X ₂,...,X _n ) is independent of every set oft random variables chosen from the independent equiprobable random variablesX ₁,X ₂,...,X _n . Additionally, if all possible outputs are equally likely, thenf is called at-resilient function. In this paper, we provide three different characterizations oft th-order correlation immune functions and resilient functions where the random variable is overGF (q). The first is in terms of the structure of a certain associated matrix. The second characterization involves Fourier transforms. The third characterization establishes the equivalence of resilient functions and large sets of orthogonal arrays.     

K-monotone weighted couples of banach lattices

We give a criterion for the uniform relative K-monotonicity of weighted couples (X,X(w ₁)) and (X,X(w ₂)), where X is some Banach lattice of measurable functions with the Fatou property while w ₁ and w ₂ are weight functions. Using the criterion, we prove some corollaries for sequence spaces and arbitrary Banach lattices.     

Markov Jump Processes Approximating a Non-Symmetric Generalized Diffusion

Consider a non-symmetric generalized diffusion X(⋅) in ℝ ^d determined by the differential operator $A(\mbox{\boldmath{$A(\mbox{\boldmath{. In this paper the diffusion process is approximated by Markov jump processes X _n (⋅), in homogeneous and isotropic grids G _n ⊂ℝ ^d , which converge in distribution in the Skorokhod space D([0,∞),ℝ ^d ) to the diffusion X(⋅). The generators of X _n (⋅) are constructed explicitly. Due to the homogeneity and isotropy of grids, the proposed method for d≥3 can be applied to processes for which the diffusion tensor $\{a_{ij}(\mbox{\boldmath{$\{a_{ij}(\mbox{\boldmath{ fulfills an additional condition. The proposed construction offers a simple method for simulation of sample paths of non-symmetric generalized diffusion. Simulations are carried out in terms of jump processes X _n (⋅). For piece-wise constant functions a _ij on ℝ ^d and piece-wise continuous functions a _ij on ℝ² the construction and principal algorithm are described enabling an easy implementation into a computer code.     

Numerical peak holomorphic functions on Banach spaces

We introduce the notion of numerical (strong) peak function and investigate the denseness of the norm and numerical peak functions on complex Banach spaces. Let Ab(BX:X) be the Banach space of all bounded continuous functions f on the unit ball BX of a Banach space X and their restrictions to the open unit ball are holomorphic. In finite dimensional spaces, we show that the intersection of the set of all norm peak functions and the set of all numerical peak functions is a dense Gδ-subset of Ab(BX:X). We also prove that if X is a smooth Banach space with the Radon-Nikodým property, then the set of all numerical strong peak functions is dense in Ab(BX:X). In particular, when X=Lp(μ)(1<p<∞) or X=?₁, it is shown that the intersection of the set of all norm strong peak functions and the set of all numerical strong peak functions is a dense Gδ-subset of Ab(BX:X). As an application, the existence and properties of numerical boundary of Ab(BX:X) are studied. Finally, the numerical peak function in Ab(BX:X) is characterized when X=C(K) and some negative results on the denseness of numerical (strong) peak holomorphic functions are given.     

A Note on the Convergence of Integral Functionals of Diffusion Processes. An Application to Strong Convergence

For a diffusion type process dX_t = dW_i + a(t, X)dt and a sequence (f_n) of nonnegative functions necessary and sufficient conditions to the f_n are established which guarantee the a.s. convergence of f_n(X_t)dt to zero. This result is applied to derive simple necessary and sufficient conditions for the strong convergence of distributions of diffusion processes formulated in terms of the corresponding drift functions.     

Fréchet Spaces of Holomorphic Functions without Copies of l1

Let X be a Banach space. Let H_w*(X*) the Fréchet space whose elements are the holomorphic functions defined on X* whose restrictions to each multiple mB(X*), m = 1,2, …, of the closed unit ball B(X*) of X* are continuous for the weak-star topology. A fundamental system of norms for this space is the supremum of the absolute value of each element of H_w*(X*) in mB(X*), m = 1,2,…. In this paper we construct the bidual of l₁ when this space contains no copy of l¹. We also show that if X is an Asplund space, then H_w*(X*) can be represented as the projective limit of a sequence of Banach spaces that are Asplund.     

A lattice-theoretic approach to arbitrary real functions on frames

In this paper we model discontinuous extended real functions in pointfree topology following a lattice-theoretic approach, in such a way that, if L is a subfit frame, arbitrary extended real functions on L are the elements of the Dedekind-MacNeille completion of the poset of all extended semicontinuous functions on L. This approach mimicks the situation one has with a T₁-space X, where the lattice F?(X) of arbitrary extended real functions on X is the smallest complete lattice containing both extended upper and lower semicontinuous functions on X. Then, we identify real-valued functions by lattice-theoretic means. By construction, we obtain definitions of discontinuous functions that are conservative for T₁-spaces. We also analyze semicontinuity and introduce definitions which are conservative for T₀-spaces.     

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) _t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C_b(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C_b(Ω). We also prove that the class of bi-continuous semigroups on C_b(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.     

Uniformity over primes of tamely ramified splittings

We fix a prime p and let f(X) vary over all monic integer polynomials of fixed degree n. Given any possible shape of a tamely ramified splitting of p in an extension of degree n, we prove that there exists a rational function φ(X)∈ℚ(X) such that the density of the monic integer polynomials f(X) for which the splitting of p has the given shape in ℚ[X]/f(X) is φ(p) (here reducible polynomials can be neglected). As a corollary, we prove that, for p≥n, the density of irreducible monic polynomials of degree n in ℤ _p [X] is the value at p of a rational function φ _n (X)∈ℚ(X). All rational functions involved are effectively computable. Received: 15 September 1998 / Revised version: 21 October 1999     

Unbounded Disjointness Preserving Linear Functionals

Let X be a locally compact Hausdorff space and C ₀(X) the Banach space of continuous functions on X vanishing at infinity. In this paper, we shall study unbounded disjointness preserving linear functionals on C ₀(X). They arise from prime ideals of C ₀(X), and we translate it into the cozero set ideal setting. In particular, every unbounded disjointness preserving linear functional of c ₀ can be constructed explicitly through an ultrafilter on complementary to a cozero set ideal. This ultrafilter method can be extended to produce many, but in general not all, such functionals on C ₀(X) for arbitrary X. We also make some remarks where C ₀(X) is replaced by a non-commutative C*-algebra.     

A generalized Dantzig—Wolfe decomposition principle for a class of nonconvex programming problems

Since Dantzig—Wolfe's pioneering contribution, the decomposition approach using a pricing mechanism has been developed for a wide class of mathematical programs. For convex programs a linear space of Lagrangean multipliers is enough to define price functions. For general mathematical programs the price functions could be defined by using a subclass of nondecreasing functions. However the space of nondecreasing functions is no longer finite dimensional. In this paper we consider a specific nonconvex optimization problem min {f(x):h _j (x)g(x),j=1, ,m, xX}, wheref(·),h _j (·) andg(·) are finite convex functions andX is a closed convex set. We generalize optimal price functions for this problem in such a way that the parameters of generalized price functions are defined in a finite dimensional space. Combining convex duality and a nonconvex duality we can develop a decomposition method to find a globally optimal solution.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.     

Remarks on Linear Homeomorphisms of Function Spaces

It is proved that whenever X and Y are completely regular -spaces of pointwise countable type and the spaces C _p(X) and C _p(Y) of real-valued continuous functions on X and Y, respectively, endowed with the topology of pointwise convergence, are linearly homeomorphic, the X is locally compact iff Y is locally compact. This extends the McCoy and Ntantu result.     

Hyperspaces of Normed Linear Spaces with the Attouch–Wets Topology

Let Cld _AW (X) be the hyperspace of nonempty closed subsets of a normed linear space X with the Attouch–Wets topology. It is shown that the space Cld _AW (X) and its various subspaces are AR's. Moreover, if X is an infinite-dimensional Banach space with weight w(X) then Cld _AW (X) is homeomorphic to a Hilbert space with weight 2 ^w(X).     

On the approximate behavior of the posterior distribution for an extreme multivariate observation

The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let X = (X₁, X₂, …, X_n) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ₁, Θ₂, …, Θ_n). Let θ_x^* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θ_x^* given X = x to converge to a multivariate normal with mean vector 0 as |x| tends to infinity. These same conditions imply that E(Θ | X = x) − θ_x^* converges to the zero vector as |x| tends to infinity.The posterior for an observation X = (X₁, X₂, …, X_n is considered for a location vector Θ = (Θ₁, Θ₂, …, Θ_n) as x gets large along a path, γ, in Rⁿ. Sufficient conditions are given for the distribution of γ(t) − Θ given X = γ(t) to converge in law as t → ∞. Slightly stronger conditions ensure that γ(t) − E(Θ | X = γ(t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ − E(Θ | X = γ(t)) given X = γ(t) converges in law to a symmetric distribution as t → ∞, it is shown that δ(γ(t)) − E(Θ | X = γ(t)) → 0 as t → ∞.     

Aging in two-dimensional Bouchaud's model

Let E _x be a collection of i.i.d. exponential random variables. Symmetric Bouchaud's model on ℤ² is a Markov chain X(t) whose transition rates are given by w _xy = ν exp (−βE _x ) if x, y are neighbours in ℤ². We study the behaviour of two correlation functions: ℙ[X(t _w +t) = X(t _w )] and ℙ[X(t') = X(t _w ) ∀ t'∈ [t _w , t _w + t]]. We prove the (sub)aging behaviour of these functions when β > 1.     

Algebraically constructible functions and signs of polynomials

LetW be a real algebraic set. We show that the following families of integer-valued functions onW coincide: (i) the functions of the formω →λ(X _ω ), where X _ω are the fibres of a regular morphismf :X →W of real algebraic sets, (ii) the functions of the formω →χ(X _ω ), where X _ω are the fibres of a proper regular morphismf :X →W of real algebraic sets, (iii) the finite sums of signs of polynomials onW. Such functions are called algebraically constructible onW. Using their characterization in terms of signs of polynomials we present new proofs of their basic functorial properties with respect to the link operator and specialization. Research partially supported by an Australian Research Council Small Grant. Second author also partially supported by KBN 610/P3/94.     

Superposition of Spatially Interacting Aggregated Continuous Time Markov Chains

A system s{ X(t)} = {X ₁(t),X ₂(t),..., X _N(t)} of N interacting time reversible continuous time Markov chains is considered. The state space of each of the processes {X _i(t)} (i = 1, 2,...,N) is partitioned into two aggregates. Interaction between the processes {X _i(t)},{X ₂(t)},...,{X _N(t)} is introduced by allowing the transition rates of an individual process at time t to depend on the configuration of aggregates occupied by the other N - 1 processes at that time. The motivation for this work comes from ion channel modeling, where {(X}(t)} describes the gating mechanisms of N channels and the partitioning of the state space of {X _i(t)} correspond to whether the channel is conducting or not. Let S(t) denote the number of conducting channels at time t. For a time-reversible class of such processes, expressions are derived for the mean and probability density function of the sojourns of {S(t)} at its different levels when {X(t)} is in equilibrium. Particular attention is paid to the situation when the N channels are located on a circle with nearest neighbor interaction. Necessary and sufficient conditions for a general co-operative multiple channel system to be time reversible are derived.     

On spaces of p-adic vector measures

LetX be a Hausdorff zero-dimensional topological space,K(X) the algebra of all clopen subsets of X, E a Hausdorff locally convex space over a non-Archimedean valued field and C _b (X) the space of all bounded continuous -valued functions on X. The space M(K(X),E), of all bounded finitely-additive measures m: K(X) → E, is investigated. If we equip C _b (X) with the topologies β _o , β, β _u , τ _b or β _ob , it is shown that, for E (compete, the corresponding spaces of continuous linear operators from C _b (X) to E (are algebraically isomorphic to certain subspaces of M(K(X),E). The text was submitted by the author in English.     

Asymptotic behavior of self-normalized trimmed sums: Nonnormal limits II

Let {X _j} be independent, identically distributed random variables which are symmetric about the origin and have a continuous nondegenerate distributionF. Let {X _n(1),...,X _n(n)} denote the arrangement of {X ₁,...,X _n} in decreasing order of magnitude, so that with probability one, |X _n(1)|>|X _n(2)|>...> |X _n(n)|. For initegersr _n such thatr _n/n0, define the self-normalized trimmed sumT _n= _i=rn ⁿ X _n(i)/{ _i=rn ⁿ X _n ² (i)}^1/2. Hahn and Weiner⁽⁶⁾ showed that under a probabilistically meaningful analytic condition generalizing the asymptotic normality criterion forT _n, various nonnormal limit laws forT _n arise which are represented by means of infinite random series. The analytic condition is now extended and the previous approach is refined to obtain limits which are mixtures of a normal, a Rademacher, and a law represented by a more general random series. Each such limit law actually arises as can be seen from the construction of a single distribution whose correspondingL(T _n ) generates all of the law along different subsequences, at least if {r _n} grows sufficiency fast. Another example clarifies the limitations of the basic approach.