Global existence and time decay of small solutions to the landau-ginzburg type equations

We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|^2/n u = 0,x ∃ Rⁿ,t } 0,u(0,x) = u0(x),x ∃ Rⁿ, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|^n-1n^n/2 / ((n + 1)|α|² + α^{2 n/2}. Furthermore, we assume that the initial data u₀ ∃ L^p are such that (1 + |x|)^αu0 ∃ L¹, with sufficiently small norm ∃ = (1 + |x|)^α u₀ 1 + u₀ p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L^∞) ∩ C ([0, ∞); L¹ ∩ L^p) satisfying the time decay estimates for allt0 u(t)||_∞ Cɛt^-n/2(1 + η log 〈t〉)^-n/2, if hg = θ^2/n 2_π ℜδ(α, Β) 0; u(t)||_∞ Cɛt^-n/2(1 + Μ log 〈t〉)^-n/4, if η = 0 and Μ = θ^4/n 4π)² (ℑδ(α, Β))² ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||_∞ Cɛt^-n/2(1 + κ log 〈t〉)^-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31).     

A problem on semigroups satisfying permutation identities

Let σ be a nontrivial permutation of ordern. A semigroupS is said to be σ-permutable ifx ₁ x ₂ ...x _n =x _σ(1) x _σ(2) ...x _σ(n) , for every (x ₁ ,x ₂,...,x _n )∈S ⁿ . A semigroupS is called(r,t)-commutative, wherer,t are in ℕ^*, ifx ₁ ...x _r x _r+1 ...x _r+t =x _r+1 ...x _r+t x ₁ ...x _r , for every (x ₁ ,x ₂,...,x _r+t ∈S ^r+t . According to a result of M. Putcha and A. Yaqub ([11]), if σ is a fixed-point-free permutation andS is a σ-permutable semigroup then there existsk ∈ ℕ^* such thatS is (1,k)-commutative. In [8], S. Lajos raises up the problem to determine the leastk=k(n) ∈ ℕ^* such that, for every fixed-point-free permutation σ of ordern, every σ-permutable semigroup is also (1,k)-commutative. In this paper this problem is solved for anyn less than or equal to eight and also whenn is any odd integer. For doing this we establish that if a semigroup satisfies a permutation identity of ordern then inevitably it also satisfies some permutation identities of ordern+1.     

Superstability of the generalized orthogonality equation on restricted domains

Chmielinski has proved in the paper [4] the superstability of the generalized orthogonality equation |〈f(x), f(y)〉| = |〈x,y〉|. In this paper, we will extend the result of Chmielinski by proving a theorem: LetD _n be a suitable subset of ℝⁿ. If a function f:D _n → ℝⁿ satisfies the inequality ∥〈f(x), f(y)〉| |〈x,y〉∥ ≤ φ(x,y) for an appropriate control function φ(x, y) and for allx, y ∈ D _n, thenf satisfies the generalized orthogonality equation for anyx, y ∈ D _n.     

Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous^*-automorphisms of a Frechet^*-algebraA, it is shown that the enveloping ↡-C ^*-algebraE(S(ℝ, A^∞, α)) of the smooth Schwartz crossed productS(ℝ,A ^∞, α) of the Frechet algebra A^∞ of C^∞-elements ofA is isomorphic to the Σ-C ^*-crossed productC ^*(ℝ,E(A), α) of the enveloping Σ-C ^*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK _*(S(ℝ, A^∞, α)) =K ^*(C ^*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC ^*-algebra defined by densely defined differential seminorms is given.     

A General Theory of Almost Splitting Sets

Let * be a star-operation of finite type on an integral domain D. In this paper, we generalize and study the concept of almost splitting sets. We define a saturated multiplicative subset S of D to be an almost g*-splitting set of D if for each 0 ≠ d ∈ D, there exists an integer n = n(d) ≥1 such that d ⁿ  = st for some s ∈ S and t ∈ D with (t, s′)_* = D for all s′ ∈ S. Among other things, we prove that every saturated multiplicative subset of D is an almost g*-splitting set if and only if D is an almost weakly factorial domain (AWFD) with *-dim (D) = 1. We also give an example of an almost g*-splitting set which is not a g*-splitting set.     

Good points for diophantine approximation

Given a sequence (x _n ) _n=1^∞ of real numbers in the interval [0, 1) and a sequence (δ _n ) _n=1^∞ of positive numbers tending to zero, we consider the size of the set of numbers in [0, 1] which can be ‘well approximated’ by terms of the first sequence, namely, those y ∈ [0, 1] for which the inequality |y − x _n | < δ _n holds for infinitely many positive integers n. We show that the set of ‘well approximable’ points by a sequence (x _n ) _n=1^∞, which is dense in [0, 1], is ‘quite large’ no matter how fast the sequence (δ _n ) _n=1^∞ converges to zero. On the other hand, for any sequence of positive numbers (δ _n ) _n=1^∞ tending to zero, there is a well distributed sequence (x _n ) _n=1^∞ in the interval [0, 1] such that the set of ‘well approximable’ points y is ‘quite small’.     

Convergence in law of partial sum processes in p-variation norm

Let X ₁, X ₂, … be a sequence of independent identically distributed real-valued random variables, S _n be the nth partial sum process S _n (t) ≔ X ₁ + ⋯ X _⌊tn⌋, t ∈ [0, 1], W be the standard Wiener process on [0, 1], and 2 < p < ∞. It is proved that n ^−1/2 S _n converges in law to σW as n → ∞ in p-variation norm if and only if EX ₁ = 0 and σ ² = EX ₁² < ∞. The result is applied to test the stability of a regression model. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-21/07     

On Hamilton Cycle Decomposition of 6-regular Circulant Graphs

The circulant graph 〈S〉_n, where S⊆Z_n∖{0}, has vertex set Z_n and edge set {{x,x+s}|x ∈ Z_n,s ∈ S}. It is shown that there is a Hamilton cycle decomposition of every 6-regular circulant graph 〈S〉_n in which S has an element of order n.     

Applications of operator semigroups to Fourier analysis

Of concern are semigroups of linear norm one operators on Hilbert space of the form (discrete case)T={T ⁿ /n=0,1,2,...} or (continuous case)T={T(t)/t=≥0}. Using ergodic theory and Hilbert-Schmidt operators, the Cesàro limits (asn→∞) of |〈T ⁿ f,f〉|², |〈T (n)f,f〉|² are computed (withn∈ℤ⁺ orn∈ℤ⁺). Specializing the Hilbert space to beL ²(T,μ) (discrete case) orL ²(ℝ,μ) (continuous case) where μ is a Borel probability measure on the circle group or the line, the Cesàro limit of (asn→±∞, with,n∈ℤ orn∈ℝ) is obtained and interpreted. Extensions toT ^M , and ℝ ^M are given. Finally, we discuss recent operator theoretic extensions from a Hilbert to a Banach space context. Partially supported by an NSF grant     

On Large Deviations of Multivariate Heavy-Tailed Random Walks

Let {S _n ;n=1,2,…} be a random walk in R ^d and E(S ₁)=(μ ₁,…,μ _d ). Let a _j >μ _j for j=1,…,d and A=(a ₁,∞)×⋅⋅⋅×(a _d ,∞). We are interested in the probability P(S _n /n∈A) for large n in the case where the components of S ₁ are heavy tailed. An objective is to associate an exact power with the aforementioned probability. We also derive sharper asymptotic bounds for the probability and show that in essence, the occurrence of the event {S _n /n∈A} is caused by large single increments of the components in a specific way.      

On the Blum-Hanson theorem for quantum quadratic processes

In this paper an analog of the Blum-Hanson theorem for quantum quadratic processes on the von Neumann algebra is proved, i.e., it is established that the following conditions are equivalent:

Topological *-algebras withC*-enveloping algebras II

UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC _c (G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC ^∞-elementsC ^∞(A), the analytic elementsC ^ω(A) as well as the entire analytic elementsC ^є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI _α is constructed satisfyingA =C*-ind limI _α; and the locally convex inductive limit ind limI _α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK _a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.     

The isometric extension of the into mapping from the unit sphere S 1( E ) to S 1( l ∞(Γ))

This paper considers the isometric extension problem concerning the mapping from the unit sphere S ₁(E) of the normed space E into the unit sphere S ₁(l ^∞(Γ)). We find a condition under which an isometry from S ₁(E) into S ₁(l ^∞(Γ)) can be linearly and isometrically extended to the whole space. Since l ^∞(Γ) is universal with respect to isometry for normed spaces, isometric extension problems on a class of normed spaces are solved. More precisely, if E and F are two normed spaces, and if V ₀: S ₁(E) → S ₁(F) is a surjective isometry, where c ₀₀(Γ) ⊆ F ⊆ l ^∞(Γ), then V ₀ can be extended to be an isometric operator defined on the whole space. This work is supported by Natural Science Foundation of Guangdong Province, China (Grant No. 7300614)     

Limit theorems for large deviations probabilities of certain quadratic forms

Let (ξ _k ,F _k ) be a martingale difference sequence. The paper concerns the tail behavior of the quadratic formS _n = ∑ _k=1 ⁿ ∑ _j=1 ^k−1 β _n ^k−j χ _k χ _j , where β _n asn→∞. The main conclusions aboutP}n ⁻¹ S _n >x _n }, wherex _n →∞, asn→∞, are obtained using the tail behavior of a martingale with values in a certain Hilbert space. Vilnius University, Naugarduko 24; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 4, pp. 532–549, October–December, 1997.     

Digital representation of semigroups and groups

A digital representation of a semigroup (S,⋅) is a family 〈F _t 〉 _t∈I , where I is a linearly ordered set, each F _t is a finite non-empty subset of S and every element of S is uniquely representable in the form Π _t∈H   x _t where H is a finite subset of I, each x _t ∈F _t and products are taken in increasing order of indices. (If S has an identity 1, then Π _t∈∅   x _t =1.) A strong digital representation of a group G is a digital representation of G with the additional property that for each t∈I, for some x _t ∈G and some m _t >1 in ℕ where m _t =2 if the order of x _t is infinite, while, if the order of x _t is finite, then m _t is a prime and the order of x _t is a power of m _t . We show that any free semigroup has a digital representation with each | F _t |=1 and that each Abelian group has a strong digital representation. We investigate the problem of whether all groups, or even all finite groups have strong digital representations, obtaining several partial results. Finally, we give applications to the algebra of the Stone-Čech compactification of a discrete group and the weakly almost periodic compactification of a discrete semigroup. Dedicated to Karl Heinrich Hofmann on the occasion of his 75th birthday. Stefano Ferri was partially supported by a research grant of the Faculty of Sciences of Universidad de los Andes. The support is gratefully acknowledged. Neil Hindman acknowledges support received from the National Science Foundation via Grant DMS-0554803.     

A large-deviation result for the range of random walk and for the Wiener sausage

Let {S _n } be a random walk on ℤ ^d and let R _n be the number of different points among 0, S ₁,…, S _{n −1}. We prove here that if d≥ 2, then ψ(x) := lim _{n →∞}(−:1/n) logP{R _n ≥nx} exists for x≥ 0 and establish some convexity and monotonicity properties of ψ(x). The one-dimensional case will be treated in a separate paper. We also prove a similar result for the Wiener sausage (with drift). Let B(t) be a d-dimensional Brownian motion with constant drift, and for a bounded set A⊂ℝ ^d let Λ _t = Λ _t (A) be the d-dimensional Lebesgue measure of the `sausage' ∪_{0≤ s ≤ t} (B(s) + A). Then φ(x) := lim _t→∞: (−1/t) log P{Λ _t ≥tx exists for x≥ 0 and has similar properties as ψ. Received: 20 April 2000 / Revised version: 1 September 2000 / Published online: 26 April 2001     

Possible limit laws for entrance times of an ergodic aperiodic dynamical system

LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ ₀ ^∞ G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and assume (X, T, μ) to be aperiodic. LetU⊂X be such that μ(U)>0. Let τ _U (x)=inf{k⩾1:T ^k xεU}, and defineG _U (t)=1/u(U)u({xεU:u(U)τU(x)>t),tεℝ We prove that for μ-a.e.x∈X, there exists a sequence (U _n ) _n≥1 of neighbourhoods ofx such that {x}=∩ _n U _n , and for anyG ∈G, there exists a subsequence (n _k ) _k≥1 withG _{U n k} ↑U weakly. We also construct a uniquely ergodic Toeplitz flowO(x ^∞,S, μ), the orbit closure of a Toeplitz sequencex ^∞, such that the above conclusion still holds, with moreover the requirement that eachU _n be a cylinder set. In memory of Anzelm Iwanik     

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:

Generalized Tractability for Multivariate Problems Part II: Linear Tensor Product Problems, Linear Information, and Unrestricted Tractability

We continue the study of generalized tractability initiated in our previous paper “Generalized tractability for multivariate problems, Part I: Linear tensor product problems and linear information”, J. Complex. 23:262–295, 2007. We study linear tensor product problems for which we can compute linear information which is given by arbitrary continuous linear functionals. We want to approximate an operator S _d given as the d-fold tensor product of a compact linear operator S ₁ for d=1,2,…, with ‖S ₁‖=1 and S ₁ having at least two positive singular values. Let n(ε,S _d ) be the minimal number of information evaluations needed to approximate S _d to within ε∈[0,1]. We study generalized tractability by verifying when n(ε,S _d ) can be bounded by a multiple of a power of T(ε ⁻¹,d) for all (ε ⁻¹,d)∈Ω⊆[1,∞)×ℕ. Here, T is a tractability function which is non-decreasing in both variables and grows slower than exponentially to infinity. We study the exponent of tractability which is the smallest power of T(ε ⁻¹,d) whose multiple bounds n(ε,S _d ). We also study weak tractability, i.e., when . In our previous paper, we studied generalized tractability for proper subsets Ω of [1,∞)×ℕ, whereas in this paper we take the unrestricted domain Ω ^unr=[1,∞)×ℕ. We consider the three cases for which we have only finitely many positive singular values of S ₁, or they decay exponentially or polynomially fast. Weak tractability holds for these three cases, and for all linear tensor product problems for which the singular values of S ₁ decay slightly faster than logarithmically. We provide necessary and sufficient conditions on the function T such that generalized tractability holds. These conditions are obtained in terms of the singular values of S ₁ and mostly asymptotic properties of T. The tractability conditions tell us how fast T must go to infinity. It is known that T must go to infinity faster than polynomially. We show that generalized tractability is obtained for T(x,y)=x ^1+ln y . We also study tractability functions T of product form, T(x,y)=f ₁(x)f ₂(x). Assume that a _i =lim inf  _x→∞(ln ln f _i (x))/(ln ln x) is finite for i=1,2. Then generalized tractability takes place iff

Moments and Distribution of the Local Times of a Transient Random Walk on ℤ d

Consider an arbitrary transient random walk on ℤ ^d with d∈ℕ. Pick α∈[0,∞), and let L _n (α) be the spatial sum of the αth power of the n-step local times of the walk. Hence, L _n (0) is the range, L _n (1)=n+1, and for integers α, L _n (α) is the number of the α-fold self-intersections of the walk. We prove a strong law of large numbers for L _n (α) as n→∞. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimensions recently derived by Černy (Stoch. Proc. Appl. 117:262–270, 2007). Although these assertions are certainly known to experts, we could find no proof in the literature in this generality.