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1.
We investigate when the set of finite products of distinct terms of a sequence 〈x n n=1 in a semigroup (S,⋅) is large in any of several standard notions of largeness. These include piecewise syndetic, central, syndetic, central*, and IP*. In the case of a “nice” sequence in (S,⋅)=(ℕ,+) one has that FS(〈x n n=1) has any or all of the first three properties if and only if {x n+1−∑ t=1 n x t :n∈ℕ} is bounded from above. N. Hindman acknowledges support received from the National Science Foundation via Grant DMS-0554803.  相似文献   

2.
A semigroup S is said to be ℛ-commutative if, for all elements a,bS, there is an element xS 1 such that ab=bax. A semigroup S is called a generalized conditionally commutative (briefly, -commutative) semigroup if it satisfies the identity aba 2=a 2 ba. An ℛ-commutative and -commutative semigroup is called an -commutative semigroup. A semigroup S is said to be a right H-semigroup if every right congruence of S is a congruence of S. In this paper we characterize the subdirectly irreducible semigroups in the class of -commutative right H-semigroups. Research supported by the Hungarian NFSR grant No T029525.  相似文献   

3.
For any sequence {ω(n)} n∈ℕ tending to infinity we construct a “quasiquadratic” representation spectrum Λ = {n 2 + o(ω(n))} n∈ℕ: for any almost everywhere (a. e.) finite measurable function f(x) there exists a series in the form $ \mathop \sum \limits_{k \in \Lambda } $ \mathop \sum \limits_{k \in \Lambda } α k ω k (x) that converges a. e. to this function, where {w k (x)} k∈ℕ is the Walsh system. We find representation spectra in the form {n l + o(n l )} n∈ℕ, where l ∈ {2 k } k∈ℕ.  相似文献   

4.
A digital representation of a semigroup (S,⋅) is a family 〈F t tI , 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 tH 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 tI, 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.  相似文献   

5.
It is proved that for any given sequence (σ n ,n ∈ ℕ)=Γ0 ⊂ Γ, where Γ is the set of all directions in ℝ2 (i.e., pairs of orthogonal straight lines) there exists a locally integrable functionf on ℝ2 such that: (1) for almost all directionsσ ∈ Γ\Γ0 the integral ∫f is differentiable with respect to the familyB 2σ of open rectangles with sides parallel to the straight lines fromσ: (2) for every directionσ n ∈ Γ0 the upper derivative of ∫f with respect toB 2σ n equals +∞; (3) for every directionσ ∈ Γ the upper derivative of ∫ |f| with respect toB 2σ equals +∞.  相似文献   

6.
By [4], a semigroupS is called an (n, m)-commutative semigroup (n, m ∈ ?+, the set of all positive integers) if $$x_1 x_2 \cdot \cdot \cdot x_n y_1 y_2 \cdot \cdot \cdot y_m = y_1 y_2 \cdot \cdot \cdot y_m x_1 x_2 \cdot \cdot \cdot x_n $$ holds for allx 1,...,x n ,y 1,...,y m S It is evident that ifS is an (n, m)-commutative semigroup then it is (n′,m′)-commutative for alln′n andm′m. In this paper, for an arbitrary semigroupS, we determine all pairs (n, m) of positive integersn andm for which the semigroupS is (n, m)-commutative. In our investigation a special type of function mapping ?+ into itself plays an important role. These functions which are defined and discussed here will be called permutation functions.  相似文献   

7.
Suppose thatx=|x(n)|n∈ℤ is a sequence of real numbers. For eachp∈ℕ,x p =|x p (n)|n∈ℤis the resulting sequence ofx throughp times median filterings with window 2k+1. It is proved that whenp→∞, bothx (2p) andx(2 p}-1) are convergent. Thus the problem of convergence of the median filters of infinite-length sequences is completely solved. Project supported by the National Natural Science Foundation of China (Grant No. 16971047).  相似文献   

8.
For the equation L 0 x(t) + L 1 x (1)(t) + ... + L n x (n)(t) = 0, where L k, k = 0, 1, ... , n, are operators acting in a Banach space, we formulate conditions under which a solution x(t) that satisfies some nonlocal homogeneous boundary conditions is equal to zero.  相似文献   

9.
We consider an infinite tandem queueing network consisting of ·/GI/1/∞ stations with i.i.d. service times. We investigate the asymptotic behavior of t(n, k), the inter-arrival times between customers n and n + 1 at station k, and that of w(n, k), the waiting time of customer n at station k. We establish a duality property by which w(n, k) and the “idle times”y(n, k) play symmetrical roles. This duality structure, interesting by itself, is also instrumental in proving some of the ergodic results. We consider two versions of the model: the quadrant and the half-plane. In the quadrant version, the sequences of boundary conditions {w(0,k), k∈ℕ} and {t(n, 0), n∈ℕ}, are given. In the half-plane version, the sequence {t(n, 0), n∈ℕ} is given. Under appropriate assumptions on the boundary conditions and on the services, we obtain ergodic results for both versions of the model. For the quadrant version, we prove the existence of temporally ergodic evolutions and of spatially ergodic ones. Furthermore, the process {t(n, k), n∈ℕ} converges weakly with k to a limiting distribution, which is invariant for the queueing operator. In the more difficult half plane problem, the aim is to obtain evolutions which are both temporally and spatially ergodic. We prove that 1/n k=1 n w(0, k) converges almost surely and in L 1 to a finite constant. This constitutes a first step in trying to prove that {t(n,k), n∈ℤ} converges weakly with k to an invariant limiting distribution. Received: 23 March 1999 / Revised version: 5 January 2000 / Published online: 12 October 2000  相似文献   

10.
Let A be an Artin group with standard generating set {σ s :sS}. Tits conjectured that the only relations in A amongst the squares of the generators are consequences of the obvious ones, namely that σ s 2 and σ t 2 commute whenever σ s and σ t commute, for s,tS. In this paper we prove Tits’ conjecture for all Artin groups. In fact, given a number m s ≥2 for each sS, we show that the elements {T s s ms :sS} generate a subgroup that has a finite presentation in which the only defining relations are that T s and T t commute if σ s and σ t commute. Oblatum 21-III-2000 & 1-XII-2000?Published online: 5 March 2001  相似文献   

11.
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  相似文献   

12.
We consider one-dimensional Gibbs measures on spin configurations σ ∈ {–1,+1}. For N ∈ ℕ let l N denote the length of the longest interval of consecutive spins of the same kind in the interval [0,N]. We show that the distribution of a suitable continuous modification l c (N) of l N converges to the Gumbel distribution, i.e., for some α, β ∈ (0, ∞) and γ ∈ ℝ, lim N →∞ ℙ(l c (N) ≤ α log N + βx + γ) = e –e –x . Received: 2 September 2002  相似文献   

13.
By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation
, on a time scale . The results improve some oscillation results for neutral delay dynamic equations and in the special case when = ℝ our results cover and improve the oscillation results for second-order neutral delay differential equations established by Li and Liu [Canad. J. Math., 48 (1996), 871–886]. When = ℕ, our results cover and improve the oscillation results for second order neutral delay difference equations established by Li and Yeh [Comp. Math. Appl., 36 (1998), 123–132]. When =hℕ, = {t: t = q k , k ∈ ℕ, q > 1}, = ℕ2 = {t 2: t ∈ ℕ}, = = {t n = Σ k=1 n , n ∈ ℕ0}, ={t 2: t ∈ ℕ}, = {√n: n ∈ ℕ0} and ={: n ∈ ℕ0} our results are essentially new. Some examples illustrating our main results are given.   相似文献   

14.
15.
On weak positive supercyclicity   总被引:1,自引:0,他引:1  
A bounded linear operator T on a separable complex Banach space X is called weakly supercyclic if there exists a vector xX such that the projective orbit {λT n x: n ∈ ℕ λ ∈ ℂ} is weakly dense in X. Among other results, it is proved that an operator T such that σ p (T *) = 0, is weakly supercyclic if and only if T is positive weakly supercyclic, that is, for every supercyclic vector xX, only considering the positive projective orbit: {rT n x: n ∈ ℂ, r ∈ ℝ+} we obtain a weakly dense subset in X. As a consequence it is established the existence of non-weakly supercyclic vectors (non-trivial) for positive operators defined on an infinite dimensional separable complex Banach space. The paper is closed with concluding remarks and further directions. Partially supported by MEC MTM2006-09060 and MTM2006-15546, Junta de Andalucía FQM-257 and P06-FQM-02225. Partially supported by Junta de Andalucía FQM-257, and P06-FQM-02225  相似文献   

16.
An abelian *-semigroup S is perfect (resp. Stieltjes perfect) if every positive definite (resp. completely so) function on S admits a unique disintegration as an integral of hermitian multiplicative functions (resp. nonnegative such). We prove that every Stieltjes perfect semigroup is perfect. The converse has been known for semigroups with neutral element, but is here shown to be not true in general. We prove that an abelian *-semigroup S is perfect if for each sS there exist tS and m, n ∈ ℕ0 such that m + n ≥ 2 and s + s* = s* + mt + nt*. This was known only with s = mt + nt* instead. The equality cannot be replaced by s + s* + s = s + s* + mt + nt* in general, but for semigroups with neutral element it can be replaced by s + p(s + s*) = p(s + s*) + mt + nt* for arbitrary p ∈ ℕ (allowed to depend on s).  相似文献   

17.
Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<( n n+x ). Here we prove that the order x Veronese embedding ofP n is not weakly (k−1)-defective, i.e. for a general SP n such that #(S) = k+1 the projective space | I 2S (x)| of all degree t hypersurfaces ofP n singular at each point of S has dimension ( n /n+x )−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I 2S (x)| has an ordinary double point at each PS and Sing (F)=S. The author was partially supported by MIUR and GNSAGA of INdAM (Italy).  相似文献   

18.
A set S={x 1,...,x n } of n distinct positive integers is said to be gcd-closed if (x i , x j ) ∈ S for all 1 ⩽ i, jn. Shaofang Hong conjectured in 2002 that for a given positive integer t there is a positive integer k(t) depending only on t, such that if nk(t), then the power LCM matrix ([x i , x j ] t ) defined on any gcd-closed set S={x 1,...,x n } is nonsingular, but for nk(t) + 1, there exists a gcd-closed set S={x 1,...,x n } such that the power LCM matrix ([x i , x j ] t ) on S is singular. In 1996, Hong proved k(1) = 7 and noted k(t) ⩾ 7 for all t ⩾ 2. This paper develops Hong’s method and provides a new idea to calculate the determinant of the LCM matrix on a gcd-closed set and proves that k(t) ⩾ 8 for all t ⩾ 2. We further prove that k(t) ⩾ 9 iff a special Diophantine equation, which we call the LCM equation, has no t-th power solution and conjecture that k(t) = 8 for all t ⩾ 2, namely, the LCM equation has t-th power solution for all t ⩾ 2.  相似文献   

19.
Let S be a commutative inverse semigroup and let E be its subsemigroup of idempotents. In this paper we define the n-th module cohomology group of Banach algebras and we show that H2l1(E)(l1(S),l1(S)(n))\mathcal {H}^{2}_{\ell^{1}(E)}(\ell^{1}(S),\ell^{1}(S)^{(n)}) is a Banach space for every odd n∈ℕ.  相似文献   

20.
This paper is concerned with the approximate solution of functional differential equations having the form: x′(t) = αx(t) + βx(t - 1) + γx(t + 1). We search for a solution x, defined for t ∈ [−1, k], k ∈ ℕ, which takes given values on intervals [−1, 0] and (k-1, k]. We introduce and analyse some new computational methods for the solution of this problem. Numerical results are presented and compared with the results obtained by other methods.   相似文献   

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