The distribution of Brownian motion in Rn at a natural stopping time

For a measure μ on Rⁿ let ((B_t, P^μ) be Brownian motion in Rⁿ with initial distribution μ. Let D be an open subset of Rⁿ with exit time ζ ≡ inf {t > 0: B_t ? D}. In the case where D is a Green region with Green function G and μ is a measure in D such that G_μ is not identically infinite on any component of D, we have given necessary and sufficient conditions for a measure ν in D to be of the form ν(dx) = P^μ(B_T ? dx, T <ζ), where T is some natural stopping time for (B_t), and we have applied this characterization to show that a measure ν in D satisfies G_ν ? G_μ iff ν is of the form ν(dx) = P^α(B_T ? dx, T <ζ) + β(dx), where T is some natural stopping time for (B_t) and α and β are measures in D such that α + β = μ and β lives on a polar set. We have proved analogous results in the case where D = R² and μ is a finite measure on R² such that ∫ log⁺ ∥x∥ du(x) < ∞, and applied this to give a characterization of the stopping times T for Brownian motion in R² such that (log⁺ ∥B_T∧t∥)_0<t<∞ is P^μ-uniformly integrable.     

Polaroid operators satisfying Weyl’s theorem

A Banach space operator T is polaroid and satisfies Weyl’s theorem if and only if T is Kato type at points λ ∈ iso σ(T) and has SVEP at points λ not in the Weyl spectrum of T. For such operators T, f(T) satisfies Weyl’s theorem for every non-constant function f analytic on a neighborhood of σ(T) if and only if f(T^∗) satisfies Weyl’s theorem.     

On the global solvability in Gevrey classes on the n-dimensional torus

Let P be a linear partial differential operator with coefficients in the Gevrey class G^s(Tⁿ), where Tⁿ is the n-dimensional torus and s?1. We prove a necessary condition for the s-global solvability of P on Tⁿ. We also apply this result to give a complete characterization for the s-global solvability for a class of formally self-adjoint operators with nonconstant coefficients.     

On homotopy varieties

Given an algebraic theory T, a homotopy T-algebra is a simplicial set where all equations from T hold up to homotopy. All homotopy T-algebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties.     

Lie algebras generated by bounded linear operators on Hilbert spaces

It is proved that the operator Lie algebra ε(T,T^∗) generated by a bounded linear operator T on Hilbert space H is finite-dimensional if and only if T=N+Q, N is a normal operator, [N,Q]=0, and dimA(Q,Q^∗)<+∞, where ε(T,T^∗) denotes the smallest Lie algebra containing T,T^∗, and A(Q,Q^∗) denotes the associative subalgebra of B(H) generated by Q,Q^∗. Moreover, we also give a sufficient and necessary condition for operators to generate finite-dimensional semi-simple Lie algebras. Finally, we prove that if ε(T,T^∗) is an ad-compact E-solvable Lie algebra, then T is a normal operator.     

Large forbidden trade volumes and edge packings of random graphs

Let G be a graph. A G-trade of volume m is a pair (T,T^′), where each of T and T^′ consists of m graphs, pairwise edge-disjoint, isomorphic to G, such that T∩T^′=∅ and the union of the edge sets of the graphs in T is identical to the union of the edge sets of the graphs in T^′. Let X(G) be the set of non-negative integers m such that no G-trade of volume m exists. In this paper we prove that, for G∈G holds asymptotically almost surely, where c=log(4/3)/88.     

On T-Spaces in a Relatively Free Two-Generated Lie Nilpotent Associative Algebra of Index 4

In this work, we continue to study T-spaces in Lie nilpotent algebras of index l > 3. The focus is on the two-generated algebra of index 4, where simple finite systems of generators are specified for a broad class of T-spaces in T ⁽³⁾ /T ⁽⁴⁾.     

Corank 1 Singularities of Stable Smooth Maps and Special Tangent Hyperplanes to a Space Curve

Let γ be a smooth generic curve in ?P ₃. Denote by C the number of its flattening points, and by T the number of planes tangent to γ at three distinct points. Consider the osculating planes to γ at the flattening points. Let N denote the total number of points where γ intersects these osculating plane transversally. Then T ≡ [N + θ(γ)C]/2 (mod 2), where θ(γ) is the number of noncontractible components of γ. This congruence generalizes the well-known Freedman theorem, which states that if a smooth connected closed generic curve in ?³ has no flattening points, then the number of its triple tangent planes is even. We also give multidimensional analogs of this formula and show that these results follow from certain general facts about the topology of codimension 1 singularities of stable maps between manifolds having the same dimension.     

LELONG-DEMAILLY NUMBERS IN TERMS OF CAPACITY AND WEAK CONVERGENCE FOR CLOSED POSITIVE CURRENTS

In this paper we give a new definition of the Lelong-Demailly number in terms of the CT-capacity, where T is a closed positive current and CT is the capacity associated to T. This derived from some esimate on the growth of the CT-capacity of the sublevel sets of a weighted plurisubharmonic （psh for short） function. These estimates enable us to give another proof of the Demailly＇s comparaison theorem as well as a generalization of some results due to Xing concerning the characterization of bounded psh functions. Another problem that we consider here is related to the existence of a psh function v that satisfies the equality CT（K） ： fK T ∧ （dd^cu）^p, where K is a compact subset. Finally, we give some conditions on the capacity CT that guarantees the weak convergence ukTk → uT, for positive closed currents T, Tk and psh functions uk, u.     

The Boundedness of Calderón-Zygmund Operators by Wavelet Characterization

This article deals with the boundedness properties of Calderón-Zygmund operators on Hardy spaces Hp(Rn). We use wavelet characterization of Hp(Rn) to show that a Calderón-Zygmund operator T with T*1 = 0 is bounded on Hp(Rn), n/n+ε p ≤ 1, where ε is the regular exponent of kernel of T . This approach can be applied to the boundedness of operators on certain Hardy spaces without atomic decomposition or molecular characterization.     

Sets of all periodic points of a toral automorphism

In this article, subsets of T² that can arise as sets of all periodic points of a continuous 2-dimensional toral automorphism are characterized. Here the torus T² is viewed as [0,1)×[0,1) as a group under coordinatewise addition modulo 1.     

Nonlinear conditions for weighted composition operators between Lipschitz algebras

Let T:Lip₀(X)→Lip₀(Y) be a surjective map between pointed Lipschitz ^∗-algebras, where X and Y are compact metric spaces. On the one hand, we prove that if T satisfies the non-symmetric norm ^∗-multiplicativity condition:

     

On Buzyakova's example; quotients of the Cantor trees

Let T be the Cantor tree and let A be a subset of the ωth level of T (= Cantor set C). Buzyakova considered the quotient space TAT^′ obtained from T×2 by identifying two points 〈a,0〉 and 〈a,1〉 for each a∈A to construct an example of a non-submetrizable space of countable extent with a Gδ-diagonal. We prove that the space TAT^′ is submetrizable if and only if C?A is an Fσ-set in C with the Euclidean topology. This improves Buzyakova's Lemma.     

Contractive multifunctions, fixed point inclusions and iterated multifunction systems

We study the properties of multifunction operators that are contractive in the Covitz-Nadler sense. In this situation, such operators T possess fixed points satisfying the relation x∈Tx. We introduce an iterative method involving projections that guarantees convergence from any starting point x₀∈X to a point x∈XT, the set of all fixed points of a multifunction operator T. We also prove a continuity result for fixed point sets XT as well as a “generalized collage theorem” for contractive multifunctions. These results can then be used to solve inverse problems involving contractive multifunctions. Two applications of contractive multifunctions are introduced: (i) integral inclusions and (ii) iterated multifunction systems.     

Lower bound inequalities for norms of symmetrized tensor powers

Let V be a complex inner product space of positive dimension m with inner product 〈·,·〉, and let T_n(V) denote the set of all n-linear complex-valued functions defined on V×V×?×V (n-copies). By S_n(V) we mean the set of all symmetric members of T_n(V). We extend the inner product, 〈·,·〉, on V to T_n(V) in the usual way, and we define multiple tensor products A₁⊗A₂⊗?⊗A_n and symmetric products A₁·A₂?A_n, where q₁,q₂,…,q_n are positive integers and A_i∈T_qi(V) for each i, as expected. If A∈S_n(V), then A^k denotes the symmetric product A·A?A where there are k copies of A. We are concerned with producing the best lower bounds for ‖A^k‖², particularly when n=2. In this case we are able to show that ‖A^k‖² is a symmetric polynomial in the eigenvalues of a positive semi-definite Hermitian matrix, M_A, that is closely related to A. From this we are able to obtain many lower bounds for ‖A^k‖². In particular, we are able to show that if ω denotes 1/r where r is the rank of M_A, and , then

     

Positive solutions of singular problems with sign changing Carathéodory nonlinearities depending on x′

We consider the singular boundary value problem for the differential equation x″+f(t,x,x′)=0 with the boundary conditions x(0)=0, w(x(T),x′(T))+?(x)=0. Here f is a Carathéodory function on which may by singular at the value x=0 of the phase variable x and f may change sign, w is a continuous function, and ? is a continuous nondecreasing functional on C⁰([0,T]). The existence of positive solutions on (0,T] in the classes AC¹([0,T]) and C⁰([0,T])∩AC¹_loc((0,T]) is considered. Existence results are proved by combining the method of lower and upper functions with Leray-Schauder degree theory.     

Strongly simplicial vertices of powers of trees

For a tree T and an integer k?1, it is well known that the kth power T^k of T is strongly chordal and hence has a strong elimination ordering of its vertices. In this note we obtain a complete characterization of strongly simplicial vertices of T^k, thereby characterizing all strong elimination orderings of the vertices of T^k.     

Enumeration of perfect matchings of a type of Cartesian products of graphs

Let G be a graph and let Pm(G) denote the number of perfect matchings of G.We denote the path with m vertices by P_m and the Cartesian product of graphs G and H by G×H. In this paper, as the continuance of our paper [W. Yan, F. Zhang, Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians, Adv. Appl. Math. 32 (2004) 175-188], we enumerate perfect matchings in a type of Cartesian products of graphs by the Pfaffian method, which was discovered by Kasteleyn. Here are some of our results:1. Let T be a tree and let C_n denote the cycle with n vertices. Then Pm(C₄×T)=∏(2+α²), where the product ranges over all eigenvalues α of T. Moreover, we prove that Pm(C₄×T) is always a square or double a square.2. Let T be a tree. Then Pm(P₄×T)=∏(1+3α²+α⁴), where the product ranges over all non-negative eigenvalues α of T.3. Let T be a tree with a perfect matching. Then Pm(P₃×T)=∏(2+α²), where the product ranges over all positive eigenvalues α of T. Moreover, we prove that Pm(C₄×T)=[Pm(P₃×T)]².     

Points of Weak-Norm Continuity in the Unit Ball of Banach Spaces

Using the M-structure theory, we show that several classical function spaces and spaces of operators on them fail to have points of weak-norm continuity for the identity map on the unit ball. This gives a unified approach to several results in the literature that establish the failure of strong geometric structure in the unit ball of classical function spaces. Spaces covered by our result include the Bloch spaces, dual of the Bergman space L¹_a and spaces of operators on them, as well as the space C(T)/A, where A is the disc algebra on the unit circle T. For any unit vector f in an infinite-dimensional function algebra A we explicitly construct a sequence {f_n} in the unit ball of A that converges weakly to f but not in the norm.     

Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation

In this work we study the period function T of solutions to the conservative equation x^″(t)+f(x(t))=0. We present conditions on f that imply the monotonicity and convexity of T. As a consequence we obtain the criterium established by C. Chicone and find conditions easier to apply. We also get a condition obtained by Cima, Gasull and Mañosas about monotonicity and, following some of their calculations, present results on the period function of Hamiltonian systems where H(x,y)=F(x)+n^-1|y|ⁿ. Using the monotonicity of T, we count the homogeneous solutions to the semilinear elliptic equation Δu=γu^γ-1 in two dimensions.