On based-free actions of compact lie groups on the hilbert cube

It is proved that a based-free action α of a given compact Lie groupG on the Hilbert cubeQ is equivalent to the standard based-free action σ if and only if the orbit spaceQ ₀/α of the free partQ ₀=Q* is aQ-manifold having the proper homotopy type of the orbit spaceQ ₀/σ. The existence of an equivariant retraction (Q ₀, σ)→(Q ₀, α) is established. It is proved that for any TikhonovG-spaceX the family of all equivariant mapsX→ conG separates the points and the closed sets inX. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 163–174, February, 1999.     

Homogenous projective factors for actions of semi-simple Lie groups

We analyze the structure of a continuous (or Borel) action of a connected semi-simple Lie group G with finite center and real rank at least 2 on a compact metric (or Borel) space X, using the existence of a stationary measure as the basic tool. The main result has the following corollary: Let P be a minimal parabolic subgroup of G, and K a maximal compact subgroup. Let λ be a P-invariant probability measure on X, and assume the P-action on (X,λ) is mixing. Then either λ is invariant under G, or there exists a proper parabolic subgroup Q⊂G, and a measurable G-equivariant factor map ϕ:(X,ν)→(G/Q,m), where ν=∫ _K kλdk and m is the K-invariant measure on G/Q. Furthermore, The extension has relatively G-invariant measure, namely (X,ν) is induced from a (mixing) probability measure preserving action of Q. Oblatum 14-X-1997 & 18-XI-1998 / Published online: 20 August 1999     

The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion

Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established.     

Multidimensional version of M. A. Krasnoseľskii’s generalized contraction principle

Let M be a complete K-metric space with n-dimensional metric ρ(x, y): M × M → R ⁿ , where K is the cone of nonnegative vectors in R ⁿ . A mapping F: M → M is called a Q-contraction if ρ (Fx,Fy) ⩽ Qρ (x,y), where Q: K → K is a semi-additive absolutely stable mapping. A Q-contraction always has a unique fixed point x* in M, and ρ(x*,a) ⩽ (I - Q)^-1 ρ(Fa, a) for every point a in M. The point x* can be obtained by the successive approximation method x _k = Fx _k-1, k = 1, 2,..., starting from an arbitrary point x ₀ in M, and the following error estimates hold: ρ (x*, x _k ) ⩽ Q ^k (I - Q)^-1ρ(x ₁, _x ⁰) ⩽ (I - Q)^-1 Q ^k ρ(x ₁, x ₀), k = 1, 2,.... Generally the mappings (I - Q)^-1 and Q ^k do not commute. For n = 1, the result is close to M. A. Krasnosel’skii’s generalized contraction principle.     

Rationally convex sets on the unit sphere in ℂ2

Let X be a rationally convex compact subset of the unit sphere S in ?², of three-dimensional measure zero. Denote by R(X) the uniform closure on X of the space of functions P/Q, where P and Q are polynomials and Q≠0 on X. When does R(X)=C(X)? Our work makes use of the kernel function for the $\bar{\delta}_{b}Let X be a rationally convex compact subset of the unit sphere S in ℂ², of three-dimensional measure zero. Denote by R(X) the uniform closure on X of the space of functions P/Q, where P and Q are polynomials and Q≠0 on X. When does R(X)=C(X)? Our work makes use of the kernel function for the operator on S, introduced by Henkin in [5] and builds on results obtained in Anderson–Izzo–Wermer [3]. We define a real-valued function ε _X on the open unit ball intB, with ε _X (z,w) tending to 0 as (z,w) tends to X. We give a growth condition on ε _X (z,w) as (z,w) approaches X, and show that this condition is sufficient for R(X)=C(X) (Theorem 1.1). In Section 4, we consider a class of sets X which are limits of a family of Levi-flat hypersurfaces in intB. For each compact set Y in ℂ², we denote the rationally convex hull of Y by . A general reference is Rudin [8] or Aleksandrov [1].     

The hyperspace of the regions below continuous maps with the fell topology

For a Tychonoff space X,we use ↓USC F(X) and ↓C F(X) to denote the families of the hypographs of all semi-continuous maps and of all continuous maps from X to I = [0,1] with the subspace topologies of the hyperspace Cld F(X × I) consisting of all non-empty closed sets in X × I endowed with the Fell topology.In this paper,we shall show that there exists a homeomorphism h:↓USC F(X) → Q = [1,1] ω such that h(↓CF(X))=c0 = {(xn)∈Q| lim n→∞ x n = 0} if and only if X is a locally compact separable metrizable space and the set of isolated points is not dense in X.     

Idealization of a decomposition theorem

In 1986, Tong [13] proved that a function f : (X,τ)→(Y,φ) is continuous if and only if it is α-continuous and A-continuous. We extend this decomposition of continuity in terms of ideals. First, we introduce the notions of regular-I-closed sets, A _I-sets and A _I -continuous functions in ideal topological spaces and investigate their properties. Then, we show that a function f : (X,τ,I)→(Y, φ) is continuous if and only if it is α-I-continuous and A _I-continuous. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the finiteness of a field-theoretic invariant for commutative rings

If T is a (commutative unital) ring extension of a ring R, then Λ(T /R) is defined to be the supremum of the lengths of chains of intermediate fields between R _P /P R _P and T _Q /QT _Q , where Q varies over Spec(T) and P:= Q ∩ R. The invariant σ(R):= sup Λ(T/R), where T varies over all the overrings of R. It is proved that if Λ(S/R)< ∞ for all rings S between R and T, then (R, T) is an INC-pair; and that if (R, T) is an INC-pair such that T is a finite-type R-algebra, then Λ(T/R)< ∞. Consequently, if R is a domain with σ(R) < ∞, then the integral closure of R is a Prüfer domain; and if R is a Noetherian G-domain, then σ(R) < ∞, with examples showing that σ(R) can be any given non-negative integer. Other examples include that of a onedimensional Noetherian locally pseudo-valuation domain R with σ(R)=∞.     

A topological position of the set of continuous maps in the set of upper semicontinuous maps

Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0, 1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points, then (↓USCC(X), ↓CC(X)) ≈ (Q, c0 ∪ (Q \ Σ)), i.e., there is a homeomorphism h :↓USCC(X) → Q such that h(↓CC(X)) = c0 ∪ (Q \ Σ...     

Test for properness using residue calculus

We will consider global problems in the ringK[X ₁, …,X _n] on the polynomials with coefficients in a subfieldK ofC. LetP=(P ₁, …,P _n):K ⁿ →K ⁿ be a polynomial map such that (P ₁,…,P _n) is a quasi-regular sequence generating a proper ideal, the main thing we do is to use the algebraic residues theory (as described in [5]) as a computational tool to give some result to test when a map (P ₁, …,P _n) is a proper map by computing a finite number of residue symbols.     

Sufficient conditions for the existence of a left quotient ring of a ring decomposed into a direct sum of left ideals

Let R=P₁⊕P₂⊕…⊕P_n be a decomposition of a ring into a direct sum of indecomposable left ideals. Assume that these ideals possess the following properties: (1) any nonzero homomorphisms ϕ: P_i→P_j is a monomorphism; (2) if subideals Q₁, Q₂ of the ideal P_j are isomorphic to the ideal P_i, then there exists a subideal Q₃⊆Q₁⊎Q₂, which is also isomorphic to P_i. It is proved that, under these asumptions, a left quotient ring of the ring R exists. This left quotient ring inherits properties(1), (2) and satisfies condition (3): any nonzero homomorphism ϕ: P_i→P_i is an automorphism of the ideal P_i. Bibliography: 2 titles. Translated fromZapiski Nauchnyk Seminarov POMI, Vol. 227, 1995, pp. 9–14.     

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.     

A non-Bernoulli skew product which is loosely Bernoulli

LetT: Y→Y be the Bernoulli two shift with independent generatorQ={Q ₀,Q ₁} and letS: X→X be a measure preserving bijection. If (S, X) is ergodic then the skew product onX×Y defined by {fx339-1} is aK-automorphism. IfŜ is also Bernoulli we sayS is pre-Bernoulli. J. Feldman showed that ifS is pre-Bernoulli thenS must be loosely Bernoulli. We construct an example to show the converse is false, i.e. anS that is loosely Bernoulli but not pre-Bernoulli.     

The Chow ring of a singular surface

We construct the Chow ringCH*(X) =CH ⁰ (X)⊕CH ¹ (X)⊕CH ² (X) of a reduced, quasi-projective surfaceX, together with Chern class mapsc _i :K ₀ (X) → CH ⁱ (X), with the usual properties. As a consequence, we show that the cycle mapCH ² (X)→ F ₀ K ₀ (X) is an isomorphism. Our treatment is greatly influenced by an unpublished 1983 preprint of Levine’s, but is much simpler, since we deal only with surfaces.     

On A(X) → TC(X) for some X

Let X be a connected based space and p be a two-regular prime number. If the fundamental group of X has order p, we compute the two-primary homotopy groups of the homotopy fiber of the trace map A(X) → TC(X) relating algebraic K-theory of spaces to topological cyclic homology. The proof uses a theorem of Dundas and an explicit calculation of the cyclotomic trace map K(ℤ[C_p])→ TC(ℤ[C_p]).     

Chow group of 1-cycles on the moduli space of vector bundles of rank 2 over a curve

Let X be a non-singular complex projective curve of genus ≥3. Choose a point x ∈ X. Let M_x be the moduli space of stable bundles of rank 2 with determinant We prove that the Chow group CH^Q₁(M_x) of 1-cycles on M_x with rational coefficients is isomorphic to CH^Q₀(X). By studying the rational curves on M_x, it is not difficult to see that there exits a natural homomorphism CH₀(J)→CH₁(M_x) where J denotes the Jacobian of X. The crucial point is to show that this homomorphism induces a homomorphism CH₀(X)→CH₁(M_x), namely, to go from the infinite dimensional object CH₀(J) to the finite dimensional object CH₀(X). This is proved by relating the degeneration of Hecke curves on M_x to the second term I^*2 of Bloch's filtration on CH₀(J). Insong Choe was supported by KOSEF (R01-2003-000-11634-0).     

Additive summable processes and their stochastic integral

We define and study a class of summable processes, called additive summable processes, that is larger than the class used by Dinculeanu and Brooks [D-B]. We relax the definition of a summable processesX:Ω×ℝ₊→E⊂L(F, G) by asking for the associated measureI _X to have just an additive extension to the predictableσ-algebra ℘, such that each of the measures (I _X) _z , forz∈(L _G ^p )*, beingσ-additive, rather than having aσ-additive extension. We define a stochastic integral with respect to such a process and we prove several properties of the integral. After that we show that this class of summable processes contains all processesX:Ω×ℝ₊→E⊂L(F, G) with integrable semivariation ifc ₀ ∋G.     

Homotopy invariants of mappings to the circle

Homotopy classes of mappings of a space X to the circle T form an Abelian group B(X) (the Bruschlinsky group). If a: X → T is a continuous mapping, then [a] denotes the homotopy class of a, and I _r (a): (X × T) _r → \mathbbZ \mathbb{Z} is the indicator function of the rth Cartesian power of the graph of a. Let C be an Abelian group and let f: B(X) → C be a mapping. By definition, f has order not greater than r if the correspondence I _r (a) → f([a]) extends to a (partly defined) homomorphism from the Abelian group of Z-valued functions on (X × T) ^r to C. It is proved that the order of f equals the algebraic degree of f. (A mapping between Abelian groups has degree at most r if all of its finite differences of order r +1 vanish.) Bibliography: 2 titles.     

Universal spaces for the subdifferentials of sublinear operators with values in the spaces of continuous functions

Given a continuous sublinear operator P: V → C(X) from a Hausdorff separable locally convex space V to the Banach space C(X) of continuous functions on a compact set X we prove that the subdifferential ∂P at zero is operator-affinely homeomorphic to the compact subdifferential ∂ ^c Q, i.e., the subdifferential consisting only of compact linear operators, of some compact sublinear operator Q: ł² → C(X) from a separable Hilbert space ł², where the spaces of operators are endowed with the pointwise convergence topology. From the topological viewpoint, this means that the space L ^c (ł², C(X)) of compact linear operators with the pointwise convergence topology is universal with respect to the embedding of the subdifferentials of sublinear operators of the class under consideration.