Affine \mathbb{T} -varieties of complexity one and locally nilpotent derivations

Let X = Spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus \mathbbT \mathbb{T} of dimension n. Let also ∂ be a homogeneous locally nilpotent derivation on the normal affine \mathbbZⁿ {\mathbb{Z}^n} -graded domain A, so that ∂ generates a k ₊-action on X that is normalized by the \mathbbT \mathbb{T} -action.     

Harmonic analysis and asymptotic behavior of solutions to the abstract cauchy problem

LetC _ub ( $\mathbb{J}$ , X) denote the Banach space of all uniformly continuous bounded functions defined on $\mathbb{J}$ 2 ε {?⁺, ?} with values in a Banach spaceX. Let ? be a class fromC _ub( $\mathbb{J}$ ,X). We introduce a spectrumsp?(φ) of a functionφ εC _ub (?,X) with respect to ?. This notion of spectrum enables us to investigate all twice differentiable bounded uniformly continuous solutions on ? to the abstract Cauchy problem (*)ω′(t) =Aω(t) +φ(t),φ(0) =x,φ ε ?, whereA is the generator of aC ₀-semigroupT(t) of bounded operators. Ifφ = 0 andσ(A) ∩i? is countable, all bounded uniformly continuous mild solutions on ?⁺ to (*) are studied. We prove the bound-edness and uniform continuity of all mild solutions on ?⁺ in the cases (i)T(t) is a uniformly exponentially stableC ₀-semigroup andφ εC _ub(?,X); (ii)T(t) is a uniformly bounded analyticC ₀-semigroup,φ εC _ub (?,X) andσ(A) ∩i sp(φ) = Ø. Under the condition (i) if the restriction ofφ to ?⁺ belongs to ? = ?(?⁺,X), then the solutions belong to ?. In case (ii) if the restriction ofφ to ?⁺ belongs to ? = ?(?⁺,X), andT(t) is almost periodic, then the solutions belong to ?. The existence of mild solutions on ? to (*) is also discussed.     

On the evolution operator for abstract parabolic equations

We find a new construction of the evolution operatorG(t, s) associated to a family {A(t), 0≦t≦T} of generators of analytic semigroups in a Banach spaceX. We study the dependence ofG (t, s) ont ands, and we give regularity results for the solution of the i.v.p.u′(t)=A(t)u(t)+f(t),u(0)=x.     

Self-similar processes with independent increments

Summary A stochastic process {X _t t 0} onR ^d is called wide-sense self-similar if, for eachc>0, there are a positive numbera and a functionb(t) such that {X _ct } and {aX _t +b(t)} have common finite-dimensional distributions. If {X _t } is widesense self-similar with independent increments, stochastically continuous, andX ₀=const, then, for everyt, the distribution ofX _t is of classL. Conversely, if is a distribution of classL, then, for everyH>0, there is a unique process {X _(H) ^t } selfsimilar with exponentH with independent increments such thatX ₁ has distribution . Consequences of this characterization are discussed. The properties (finitedimensional distributions, behaviors for small time, etc.) of the process {X _(H) ^t } (called the process of classL with exponentH induced by ) are compared with those of the Lévy process {Y _t } such thatY ₁ has distribution . Results are generalized to operator-self-similar processes and distributions of classOL. A process {X _t } onR ^d is called wide-sense operator-self-similar if, for eachc>0, there are a linear operatorA _c and a functionb _c (t) such that {X _ct } and {A _c X _t +b _c (t)} have common finite-dimensional distributions. It is proved that, if {X _t } is wide-sense operator-self-similar and stochastically continuous, then theA _c can be chosen asA _c =c ^Q with a linear operatorQ with some special spectral properties. This is an extension of a theorem of Hudson and Mason [4].     

C-Cosine Functions and the Abstract Cauchy Problem, II

For a bounded linear injectionCon a Banach spaceXand a closed linear operatorA : D(A) X → Xwhich commutes withCwe prove that (1) the abstract Cauchy problem,u″(t) = Au(t),t R,u(0) = Cx,u′(0) = Cy, has a unique strong solution for everyx,y D(A) if and only if (2)A₁ = AD(A²) generates aC₁-cosine function onX₁(D(A) with the graph norm), if (and only if, in caseAhas nonempty resolvent set) (3)Agenerates aC-cosine function onX. HereC₁ = CX₁. Under the assumption thatAis densely defined andC⁻¹AC = A, statement (3) is also equivalent to each of the following statements: (4) the problemv″(t) = Av(t) + C(x + ty) + ∫^t₀ Cg(r) dr,t R,v(0) = v′(0) = 0, has a unique strong solution for everyg L¹_locandx, y X; (5) the problemw″(t) = Aw(t) + Cg(t),t R,w(0) = Cx,w′(0) = Cy, has a unique weak solution for everyg L¹_locandx, y X. Finally, as an application, it is shown that for any bounded operatorBwhich commutes withCand has range contained in the range ofC,A + Bis also a generator.     

Exponential-cosine operator-valued functional equation in the strong operator topology

LetX be a Banach Space and letB(X) denote the family of bounded linear operators onX. LetR ⁺ = [0, ). A one parameter family of operators {S(t);t R ⁺},S:R ⁺ B(X), is called exponential-cosine operator function ifS(O) =I andS(s +t) – 2S(s)S(t) = (S(2s) – 2S ²(s))S(t –s), for alls, t R ⁺,s t. Let ,fD(A), and ,fD(B). It is shown that for a strongly continuous exponential-cosine operator {S(t)},fD(A ²) implies ₀ ^t (t –u(S(u)fduD(B) and B ₀ ^t (t –u)S(u)fdu =S(t)f –f +tAf – 2A ₀ ^t S(u)fdu + 2A ² ₀ ^t (t –u)S(u)fdu.D(B) is seen to be dense inD(A ²). Some regularity properties ofS(t) have also been obtained.     

The Picard iteration and its application

The convergence behavior of the Picard iteration X_k+1=AX_k+B and the weighted case Y_k=X_k/b_k is investigated. It is shown that the convergence of both these iterations is related to the so-called effective spectrum of A with respect to some matrix. As an application of our convergence results we discuss the convergence behavior of a sequence of scaled triangular matrices {D^NT^N }.     

Characterization of Hermitian and skew-Hermitian maps between matrix algebras

Let D be a division ring with an involution J such that D is finite-dimensional over its center Z and char D≠2. Let T:M_m(D)→M_n(D) be a Z-linear map between matrix rings over D. We show that T satisfies [T(X)]¹=T(X¹) if and only if T(X)=∑±A¹_kXA_k. Similarly, T satisfies [T(X)]¹ = ? T(X¹) if and only if T(X = ∑(A¹_kXB_k ? B¹_kXA_k). The first of these results generalizes and extends a theorem of R.D. Hill [2] on Hermitian-preserving transformations.     

The Algebra of Flows in Graphs

We define a contravariant functorKfrom the category of finite graphs and graph morphisms to the category of finitely generated graded abelian groups and homomorphisms. For a graphX, an abelian groupB, and a nonnegative integerj, an element of Hom(K^j(X), B) is a coherent family ofB-valued flows on the set of all graphs obtained by contracting some (j − 1)-set of edges ofX; in particular, Hom(K¹(X), ) is the familiar (real) “cycle-space” ofX. We show thatK ^· (X) is torsion-free and that its Poincaré polynomial is the specializationt^{n − k}T_X(1/t, 1 + t) of the Tutte polynomial ofX(hereXhasnvertices andkcomponents). Functoriality ofK ^· induces a functorial coalgebra structure onK ^· (X); dualizing, for any ringBwe obtain a functorialB-algebra structure on Hom(K ^· (X), B). WhenBis commutative we present this algebra as a quotient of a divided power algebra, leading to some interesting inequalities on the coefficients of the above Poincaré polynomial. We also provide a formula for the theta function of the lattice of integer-valued flows inX, and conclude with 10 open problems.     

Absolute continuities of exit measures for superdiffusions

Suppose X= X_t, X_T, P_μis a superdiffusion in ℝ^d with general branching mechanism ψ and general branching rate functionA. We discuss conditions onA to guarantee that the exit measure X_TL of the superdiffusionX from bounded smooth domains in ℝ^d have absolutely continuous states.     

On perturbations of differentiable semigroups

LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC ₀-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can be summarized roughly as follows:

On projectively normal embeddings of generalk-gonal curves

Fix integersg, k andt witht>0,k≥3 andtk<g/2−1. LetX be a generalk-gonal curve of genusg andR∈Pic ^k (X) the uniqueg _k ¹ onX. SetL:=K _X⊗(R ^*)^⊗t.L is very ample. Leth _L:X→P(H ⁰(X, L)^*) be the associated embedding. Here we prove thath _L(X) is projectively normal. Ifk≥4 andtk<g/2−2 the curveh _L(X) is scheme-theoretically cut out by quadrics. The author was partially supported by MURST and GNSAGA of CNR (Italy).     

A Functional LIL for m-Fold Integrated Brownian Motion

Abstract Let {X _m (t); t ∈ R ₊} be an m-Fold integrated Brownian motion. In this paper, with the help of small ball probability estimate, a functional law of the iterated logarithm (LIL) for X _m (t) is established. This extends the classic Chung type liminf result for this process. Furthermore, a result about the weighted occupation measure for X _m (t) is also obtained. *Project supported by the National Natural Science Foundation of China (No.10131040) and the Specialized Research Fund for the Doctor Program of Higher Education (No.2002335090).     

New upper bounds on the minimum size of covering designs

Let D = {B₁, B₂,…, B_b} be a finite family of k-subsets (called blocks ) of a v-set X(v) = {1, 2,…, v} (with elements called points ). Then D is a (v, k, t) covering design or covering if every t-subset of X(v) is contained in at least one block of D. The number of blocks, b, is the size of the covering, and the minimum size of the covering is called the covering number , denoted C(v, k, t). This article is concerned with new constructions of coverings. The constructions improve many upper bounds on the covering number C(v, k, t) © 1998 John Wiley & Sons, Inc. J Combin Designs 6:21–41, 1998     

Difference sets and inverting the difference operator

For a setA of non-negative numbers, letD(A) (the difference set ofA) be the set of nonnegative differences of elements ofA, and letD ^k be thek-fold iteration ofD. We show that for everyk, almost every set of non-negative integers containing 0 arises asD ^k (A) for someA. We also give sufficient conditions for a setA to be the unique setX such that 0X andD ^k (X)=D ^k (A). We show that for eachm there is a setA such thatD(X)=D(A) has exactly 2 ^m solutionsX with 0X.This work was supported by grants DMS 92-02833 and DMS 91-23478 from the National Science Foundation. The first author acknowledges the support of the Hungarian National Science Foundation under grants, OTKA 4269, and OTKA 016389, and the National Security Agency (grant No. MDA904-95-H-1045).Lee A. Rubel died March 25, 1995. He is very much missed by his coauthors.     

On the symbolic calculus in homogeneous Banach algebras

We construct a strongly homogeneous Banach algebraB so that for an appropriate positive integern, (1)A(T) ⊊B ⊊C(T); (2)n-times continuously differentiable functions operate onB. This work was supported in part by NSF Grant MCS75-08552-A02.     

Eigenvalues and power growth

LetX be a complex Banach space and letT be a bounded linear operator onX. Denote by σ _p (T) the point spectrum ofT and by the unit circle. We investigate how the growth of the sequence ‖T ⁿ ‖ is influenced by the size of the set (T) and by the geometry of the spaceX. We also prove analogous results forC ₀-semigroups(T _t )_t≥0. Research partially supported by grants from NSERC, FQRNT and the Canada research chairs program.     

Maximal subspaces for solutions of the second order abstract cauchy problem

For a continuous, increasing function ω: R → R \{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]-1u(t,x) is uniformly continues on R , and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A|z(A,ω) generates an O(ω(t))strongly continuous cosine operator function family.     

Asymptotic Behaviour of C0–Semigroups with Bounded Local Resolvents

Let {T(t)}_t≥0 be a C₀–semigroup on a Banach space X with generator A, and let H^∞_T be the space of all x ∈ X such that the local resolvent λ ↦ R(λ, A)x has a bounded holomorphic extension to the right half–plane. For the class of integrable functions ϕ on [0, ∞) whose Fourier transforms are integrable, we construct a functional calculus ϕ ↦ T_ϕ, as operators on H^∞_T. Weshow that each orbit T(·)T_ϕx is bounded and uniformly continuous, and T(t)T_ϕx → 0 weakly as t → ∞, and we give a new proof that ∥T(t)R(μ, A)x∥ = O(t). We also show that ∥T(t)T_ϕx∥ → 0 when T is sun –reflexive, and that ∥T(t)R(μ, A)x∥ = O(ln t) when T is a positive semigroup on a normal ordered space X and x is a positive vector in H^∞_T.     

Multiple recurrence and nilsequences

Aiming at a simultaneous extension of Khintchine(X,X,m,T)(X,\mathcal{X},\mu,T) and a set A ? XA\in\mathcal{X} of positive measure, the set of integers n such that A T^2nA T^knA)(A)^k+1-\mu(A{\cap} T^{n}A{\cap} T^{2n}A{\cap} \ldots{\cap} T^{kn}A)>\mu(A)^{k+1}-\epsilon is syndetic. The size of this set, surprisingly enough, depends on the length (k+1) of the arithmetic progression under consideration. In an ergodic system, for k=2 and k=3, this set is syndetic, while for kòf(x)f(Tⁿx)f(T²ⁿx)? f(T^knx)  dm(x)\int{f(x)f(T^{n}x)f(T^{2n}x){\ldots} f(T^{kn}x) \,d\mu(x)} , where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d^*(E)>0 and for all {n ? \mathbbZ\colon d^*(E?(E+n)?(E+2n)?(E+3n)) > d^*(E)⁴-e}\big\{n\in\mathbb{Z}{\colon} d^*\big(E\cap(E+n)\cap(E+2n)\cap(E+3n)\big) > d^*(E)^4-\epsilon\big\}