Some new generalized difference sequence spaces defined by a sequence of moduli

The idea of difference sequence sets X( ) = {x = (x k ) : x ∈ X} with X = l ∞ , c and c 0 was introduced by Kizmaz [12]. In this paper, using a sequence of moduli we define some generalized difference sequence spaces and give some inclusion relations.     

Simple C*-Algebras with Unique Tracial States and Quantized Topological Spaces

Let X be a connected finite CW complex and d _X : K ₀(C(X)) →ℤ be the dimension function. We show that, if A is a unital separable simple nuclear C*-algebra of TR(A) = 0 with the unique tracial state and satisfying the UCT such that K ₀(A) = ℚ⊕ kerd _x and K ₁(A) = K ₁(C(X)), then A is isomorphic to an inductive limit of M _{n !}(C(X)). Received April 19, 2001, Accepted April 27, 2001.     

On the average distance property and certain energy integrals

One of our main results is the following: LetX be a compact connected subset of the Euclidean spaceR ⁿ andr(X, d ₂) the rendezvous number ofX, whered ₂ denotes the Euclidean distance inR ⁿ . (The rendezvous numberr(X, d ₂) is the unique positive real number with the property that for each positive integern and for all (not necessarily distinct)x ₁,x ₂,...,x _n inX, there exists somex inX such that .) Then there exists some regular Borel probability measure μ₀ onX such that the value of ∫ _X d ₂(x, y)dμ₀ (y) is independent of the choicex inX, if and only ifr(X, d ₂) = sup_μ ∫ _X ∫ _X d ₂(x, y)dμ(x)dμ(y), where the supremum is taken over all regular Borel probability measures μ onX.     

Generalized bisymmetry on a restricted domain

Let X ⊂ ℝ be an interval of positive length and define the set Δ = {(x, y) ∈ X × X | x ≧ y}. We give the solution of the equation

Oscillation for almost continuity

Let X be a topological space and (Y,d) be a metric space. If f: X → Y is a function then there is a function a _f : X → [0, ∞] such that f is almost continuous at x if and only if a _f (x) = 0. Some properties of this function are investigated. Supported by grant VEGA 2/6087/26 and APVT-51-006904.     

Some path properties of generalized Lévy sheet

Let X(t) be an N parameter generalized Lévy sheet taking values in ℝ^d with a lower index α, ℜ = {(s, t] = ∏ _i=1 ^N (s _i, t _i], s _i < t _i}, E(x, Q) = {t ∈ Q: X(t) = x}, Q ∈ ℜ be the level set of X at x and X(Q) = {x: ∃t ∈ Q such that X(t) = x} be the image of X on Q. In this paper, the problems of the existence and increment size of the local times for X(t) are studied. In addition, the Hausdorff dimension of E(x, Q) and the upper bound of a uniform dimension for X(Q) are also established.     

Subalgebras of the Stanley—Reisner Ring

In [2], Billera proved that the R -algebra of continuous piecewise polynomial functions (C ⁰ splines) on a d -dimensional simplicial complex Δ embedded in R ^d is a quotient of the Stanley—Reisner ring A _Δ of Δ. We derive a criterion to determine which elements of the Stanley—Reisner ring correspond to splines of higher-order smoothness. In [5], Lau and Stiller point out that the dimension of C ^r _k (Δ) is upper semicontinuous in the Zariski topology. Using the criterion, we give an algorithm for obtaining the defining equations of the set of vertex locations where the dimension jumps. Received June 2, 1997, and in revised form December 22, 1997, and March 24, 1998.     

On the hot spots of a catalytic super-Brownian motion

Consider the catalytic super-Brownian motion X ^ϱ (reactant) in ℝ ^d , d≤3, which branching rates vary randomly in time and space and in fact are given by an ordinary super-Brownian motion ϱ (catalyst). Our main object of study is the collision local time L = L _[ϱ,Xϱ] (d(s,x) )of catalyst and reactant. It determines the covariance measure in themartingale problem for X ^ϱ and reflects the occurrence of “hot spots” of reactant which can be seen in simulations of X ^ϱ. In dimension 2, the collision local time is absolutely continuous in time, L(d(s,x) ) = ds K _s (dx). At fixed time s, the collision measures K _s (dx) of ϱ _s and X _s ^ϱ have carrying Hausdorff dimension 2. Spatial marginal densities of L exist, and, via self-similarity, enter in the long-term randomergodic limit of L (diffusiveness of the 2-dimensional model). We alsocompare some of our results with the case of super-Brownian motions withdeterministic time-independent catalysts. Received: 2 December 1998 / Revised version: 2 February 2001 / Published online: 9 October 2001     

Closure Operators and Lattice Extensions

For closure operators Γ and Δ on the same set X, we say that Δ is a weak (resp. strong) extension of Γ if Cl(X, Γ) is a complete meet-subsemilattice (resp. complete sublattice) of Cl(X, Δ). This context is used to describe the extensions of a finite lattice that preserve various properties. This revised version was published online in September 2006 with corrections to the Cover Date.     

Some characterizations and properties of the “distance to ill-posedness” and the condition measure of a conic linear system

A conic linear system is a system of the form?P(d): find x that solves b - Ax∈C _Y , x∈C _X ,? where C _X and C _Y are closed convex cones, and the data for the system is d=(A,b). This system is“well-posed” to the extent that (small) changes in the data (A,b) do not alter the status of the system (the system remains solvable or not). Renegar defined the “distance to ill-posedness”, ρ(d), to be the smallest change in the data Δd=(ΔA,Δb) for which the system P(d+Δd) is “ill-posed”, i.e., d+Δd is in the intersection of the closure of feasible and infeasible instances d’=(A’,b’) of P(·). Renegar also defined the “condition measure” of the data instance d as C(d):=∥d∥/ρ(d), and showed that this measure is a natural extension of the familiar condition measure associated with systems of linear equations. This study presents two categories of results related to ρ(d), the distance to ill-posedness, and C(d), the condition measure of d. The first category of results involves the approximation of ρ(d) as the optimal value of certain mathematical programs. We present ten different mathematical programs each of whose optimal values provides an approximation of ρ(d) to within certain constants, depending on whether P(d) is feasible or not, and where the constants depend on properties of the cones and the norms used. The second category of results involves the existence of certain inscribed and intersecting balls involving the feasible region of P(d) or the feasible region of its alternative system, in the spirit of the ellipsoid algorithm. These results roughly state that the feasible region of P(d) (or its alternative system when P(d) is not feasible) will contain a ball of radius r that is itself no more than a distance R from the origin, where the ratio R/r satisfies R/r≤c ₁ C(d), and such that r≥ and R≤c ₃ C(d), where c ₁,c ₂,c ₃ are constants that depend only on properties of the cones and the norms used. Therefore the condition measure C(d) is a relevant tool in proving the existence of an inscribed ball in the feasible region of P(d) that is not too far from the origin and whose radius is not too small. Received November 2, 1995 / Revised version received June 26, 1998?Published online May 12, 1999     

The spectral projections and the resolvent for scattering metrics

In this paper, we consider a compact manifold with boundaryX equipped with a scattering metricg as defined by Melrose [9]. That is,g is a Riemannian metric in the interior ofX that can be brought to the formg=x ⁻⁴ dx²+x^{−2 h’} near the boundary, wherex is a boundary defining function andh’ is a smooth symmetric 2-cotensor which restricts to a metrich on ϖX. LetH=Δ+V, whereV∈x ²C^∞ (X) is real, soV is a ‘short-range’ perturbation of Δ. Melrose and Zworski started a detailed analysis of various operators associated toH in [11] and showed that the scattering matrix ofH is a Fourier integral operator associated to the geodesic flow ofh on ϖX at distance π and that the kernel of the Poisson operator is a Legendre distribution onX×ϖX associated to an intersecting pair with conic points. In this paper, we describe the kernel of the spectral projections and the resolvent,R(σ±i0), on the positive real axis. We define a class of Legendre distributions on certain types of manifolds with corners and show that the kernel of the spectral projection is a Legendre distribution associated to a conic pair on the b-stretched productX _b ² (the blowup ofX ² about the corner, (ϖX)²). The structure of the resolvent is only slightly more complicated. As applications of our results, we show that there are ‘distorted Fourier transforms’ forH, i.e., unitary operators which intertwineH with a multiplication operator and determine the scattering matrix; we also give a scattering wavefront set estimate for the resolventR(σ±i0) applied to a distributionf.     

Quantitative form of the Luzin <Emphasis Type="Italic">C</Emphasis>-property

We prove the following statement, which is a quantitative form of the Luzin theorem on C-property: Let (X, d, μ) be a bounded metric space with metric d and regular Borel measure μ that are related to one another by the doubling condition. Then, for any function f measurable on X, there exist a positive increasing function η ∈ Ω (η(+0) = 0 and η(t)t ^−a decreases for a certain a > 0), a nonnegative function g measurable on X, and a set E ⊂ X, μE = 0 , for which

Distinguishing labeling of the actions of almost simple groups

Suppose Γ is a group acting on a set X, written as (Γ,X). An r-labeling f: X→{1,2, ..., r} of X is called distinguishing for (Γ,X) if for all σ∈Γ,σ≠1, there exists an element x∈X such that f(x)≠f(x ^σ ). The distinguishing number d(Γ,X) of (Γ,X) is the minimum r for which there is a distinguishing r-labeling for (Γ,X). If Γ is the automorphism group of a graph G, then d(Γ,V (G)) is denoted by d(G), and is called the distinguishing number of the graph G. The distinguishing set of Γ-actions is defined to be D*(Γ)={d(Γ,X): Γ acts on X}, and the distinguishing set of Γ-graphs is defined to be D(Γ)={d(G): Aut(G)≅Γ}. This paper determines the distinguishing set of Γ-actions and the distinguishing set of Γ-graphs for almost simple groups Γ.     

Some results on distance two labelling of outerplanar graphs

Let G be an outerplanar graph with maximum degree △. Let χ（G^2） and A（G） denote the chromatic number of the square and the L（2, 1）-labelling number of G, respectively. In this paper we prove the following results： （1） χ（G^2） = 7 if △= 6; （2） λ（G） ≤ △ ＋5 if △ ≥ 4, and ）,（G）≤ 7 if △ = 3; and （3） there is an outerplanar graph G with △ = 4 such that ）λ（G） = 7. These improve some known results on the distance two labelling of outerplanar graphs.     

Almost continuous orbit equivalence for non-singular homeomorphisms

Let X and Y be Polish spaces with non-atomic Borel measures μ and ν of full support. Suppose that T and S are ergodic non-singular homeomorphisms of (X, μ) and (Y, ν) with continuous Radon-Nikodym derivatives. Suppose that either they are both of type III ₁ or that they are both of type III _λ, 0 < λ < 1 and, in the III _λ case, suppose in addition that both ‘topological asymptotic ranges’ (defined in the article) are log λ · ℤ. Then there exist invariant dense G _δ-subsets X′ ⊂ X and Y′ ⊂ Y of full measure and a non-singular homeomorphism ϕ: X′ → Y′ which is an orbit equivalence between T| _X′ and S| _Y′, that is ϕ{T ⁱ x} = {S ⁱ ϕx} for all x ∈ X′. Moreover, the Radon-Nikodym derivative dν ∘ ϕ/dμ is continuous on X′ and, letting S′ = ϕ ⁻¹ Sϕ, we have T _x = S′ ^n(x) x and S′x = T ^m(x) x where n and m are continuous on X′.     

On the fundamental group of real toric varieties

LetX (Δ) be the real toric variety associated to a smooth fan Δ. The main purpose of this article is: (i) to determine the fundamental group and the universal cover ofX (Δ), (ii) to give necessary and sufficient conditions on Δ under which π₁(X(Δ)) is abelian, (iii) to give necessary and sufficient conditions on Δ under whichX(Δ) is aspherical, and when Δ is complete, (iv) to give necessary and sufficient conditions forC _Δ to be aK (π, 1) space whereC _Δ is the complement of a real subspace arrangement associated to Δ.     

Resolvability and monotone normality

A space X is said to be κ-resolvable (resp., almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp., almost disjoint over the ideal of nowhere dense subsets). X is maximally resolvable if and only if it is Δ(X)-resolvable, where Δ(X) = min{|G| : G ≠ open}. We show that every crowded monotonically normal (in short: MN) space is ω-resolvable and almost μ-resolvable, where μ = min{2 ^ω , ω ₂}. On the other hand, if κ is a measurable cardinal then there is a MN space X with Δ(X) = κ such that no subspace of X is ω ₁-resolvable. Any MN space of cardinality < ℵ _ω is maximally resolvable. But from a supercompact cardinal we obtain the consistency of the existence of a MN space X with |X| = Δ(X) = ℵ _ω such that no subspace of X is ω ₂-resolvable. The preparation of this paper was supported by OTKA grant no. 61600     

On some inequalities of probabilistic number theory

Let Q ₊ denote the set of positive rational numbers. We define discrete probability measures ν _x on the σ-algebra of subsets of Q ₊.We introduce additive functions ƒ: Q ₊ → G and obtain a bound for ν_x(ƒ (r) ∉ X+X−X) using a probability related to some independent random variables. This inequality is an analogue to that proved by I. Ruzsa for additive arithmetical functions. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 256–266, April–June, 2006.     

Approximation of the effective conductivity of ergodic media by periodization

 This paper is concerned with the approximation of the effective conductivity σ(A, μ) associated to an elliptic operator ∇ _xA (x,η)∇ _x where for xℝ ^d , d≥1, A(x,η) is a bounded elliptic random symmetric d×d matrix and η takes value in an ergodic probability space (X, μ). Writing A ^N (x, η) the periodization of A(x, η) on the torus T ^d _N of dimension d and side N we prove that for μ-almost all η