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1.
2.
Summary Suppose that 1/2 ≦ λ < 1. Balog and Harman proved that for any real θ there exist infinitely many primes p satisfying pλ-θ < p-(1-λ)/2+ ε (with an asymptotic result). In the present paper we establish that for almost all θ in the interval 0 ≦ θ < 1 there exist infinitely many primes p such that {pλ-θ} < p-min{(2-λ)/6,(14-9λ)/32}+ε. Thus we obtain a better result for almost all θ than for a single θ if λ>1/2.  相似文献   

3.
Letf(x)=θ1 x 1 k +...+θ s x s k be an additive form with real coefficients, and ∥α∥ = min {|α-u|:uεℤ} denote the distance fromα to the nearest integer. We show that ifθ 1,…,θ s , are algebraic ands = 4k then there are integersx 1,…,x s , satisfying l ≤x 1,≤ N and ∥f(x)∥ ≤ N E , withE = − 1 + 2/e. Whens = λk, 1 ≤λ ≤ 2k, the exponentE may be replaced byλE/4, and if we drop the condition thatθ 1,…,θ s , be algebraic then the result holds for almost all values of θεℝ s . Whenk ≥ 6 is small a better exponent is obtained using Heath-Brown’s version of Weyl’s estimate.  相似文献   

4.
Let A be the closed unbounded operator inL p(G) that is associated with an elliptic boundary value problem for a bounded domainG. We prove the existence of a spectral projectionE determined by the set Γ = {λ;θ 1≦argλ≦θ 2} and show thatAE is the infinitesimal generator of an analytic semigroup provided that the following conditions hold: 1<p<∞; the boundary ϖΓ of Γ is contained in the resolvent setp(A) ofA;π/2θ<θ 23π/2 ; and there exists a constantc such that (I)││(λ-A)-1││≦c/│λ│ for λ∈ϖΓ. The following consequence is obtained: Suppose that there exist constantsM andc such that λ∈p(A) and estimate (I) holds provided that |λ|≧M and Re λ=0. Then there exist bounded projectionE andE + such thatA is completely reduced by the direct sum decompositionL p(G)=ELp (G) ⊕E+Lp (G) and each of the operatorsAE and—AE + is the infinitestimal generator of an analytic semigroup.  相似文献   

5.
Consider a single server queue with unit service rate fed by an arrival process of the following form: sessions arrive at the times of a Poisson process of rate λ, with each session lasting for an independent integer time τ ⩾ 1, where P(τ = k) = p k with p k ~ αk −(1+α) L(k), where 1 < α < 2 and L(·) is a slowly varying function. Each session brings in work at unit rate while it is active. Thus the work brought in by each arrival is regularly varying, and, because 1 < α < 2, the arrival process of work is long-range dependent. Assume that the stability condition λE[τ] < 1 holds. By simple arguments we show that for any stationary nonpreemptive service policy at the queue, the stationary sojourn time of a typical session must stochastically dominate a regularly varying random variable having infinite mean; this is true even if the duration of a session is known at the time it arrives. On the other hand, we show that there exist causal stationary preemptive policies, which do not need knowledge of the session durations at the time of arrival, for which the stationary sojourn time of a typical session is stochastically dominated by a regularly varying random variable having finite mean. These results indicate that scheduling policies can have a significant influence on the extent to which long-range dependence in the arrivals influences the performance of communication networks. This revised version was published online in June 2006 with corrections to the Cover Date.  相似文献   

6.
Summary For a given fractional 2 m factorial (2 m -FF) designT, the constitution of a block plan to divideT intok (2 r−1<k≦2 r ) blocks withr block factors each at two levels is proposed and investigated. The well-known three norms of the confounding matrix are used as measures for determining a “good” block plan. Some theorems concerning the constitution of a block plan are derived for a 2 m -FF design of odd or even resolution. Two norms which may be preferred over the other norm are slightly modified. For each value ofN assemblies with 11≦N≦26, optimum block plans fork=2 blocks with block sizes [N/2] andN−[N/2] minimizing the two norms are presented forA-optimal balanced 24-FF designs of resolutionV given by Srivastava and Chopra (Technometrics,13, 257–269).  相似文献   

7.
It is shown that for doubling weights, the zeros of the associated orthogonal polynomials are uniformly spaced in the sense that if cos θ m,k , θ m,k ∈[0,π] are the zeros of the m-th orthogonal polynomial associated with w, then θ m,k θ m,k+1∼1/m. It is also shown that for doubling weights, neighboring Cotes numbers are of the same order. Finally, it is proved that these two properties are actually equivalent to the doubling property of the weight function.  相似文献   

8.
It is shown that if the subsetsX 1,...,X v of a setX form a (v, k, λ)-design, then there does not exist another subsetX v+1 ofX havingany cardinalityk 1 and intersecting each of theX j, 1≦jv, inany number λ1 of elements, where 0<k 1<v and 0<λ1<k (in order to avoid uninteresting cases).  相似文献   

9.
Ifμ is a positive measure, andA 2, ...,A n are measurable sets, the sequencesS 0, ...,S n andP [0], ...,P [n] are related by the inclusion-exclusion equalities. Inequalities among theS i are based on the obviousP [k]≧0. Letting =the average average measure of the intersection ofk of the setsA i , it is shown that (−1) k Δ k M i ≧0 fori+kn. The casek=1 yields Fréchet’s inequalities, andk=2 yields Gumbel’s and K. L. Chung’s inequalities. Generalizations are given involvingk-th order divided differences. Using convexity arguments, it is shown that forS 0=1, whenS 1N−1, and for 1≦k<Nn andv=0, 1, .... Asymptotic results asn → ∞ are obtained. In particular it is shown that for fixedN, for all sequencesM 0, ...,M n of sufficiently large length if and only if for 0<t<1.  相似文献   

10.
Let Ω[ξ] denote the polynomial algebra (with 1) in commutative indeterminates {ie65-1}, 1 ≦i, jn, 1 ≦k < ∞, over a commutative ring Ω. Thealgebra of generic matrices Ω [Y] is defined to be the Ω-subalgebra ofM n (Ω[ξ]) generated by the matricesY k=({ie65-2}), 1 ≦i, jn, 1 ≦k < ∞. This algebra has been studied extensively by Amitsur and by Procesi in particular Amitsur has used it to construct a finite dimensional, central division algebra Ω (Y) which is not a crossed product. In this paper we shall prove, for Ω a domain, that Ω(Y) has exponentn in the Brauer group (Amitsur may already know this fact); consequently, for Ω an infinite field andn a multiple of 4, iff(X 1, …,X m) is a polynomial linear in all theX i but one (similar to Formanek’s central polynomials for matrix rings) andf 2 is central forM n (Ω), thenf is central forM n (Ω). (The existence of a polynomial not central forM n (Ω), but whose square is central forM n(Ω) is equivalent to every central division algebra of degreen containing a quadratic extension of its center; well-known theory immediately shows this is the case of 4‖n and 8χn.) Also, information is obtained about Ω(Y) for arbitary Ω, most notably that the Jacobson radical is the set of nilpotent elements. Partial support for this work was provided by National Science Foundation grant NSF-GP 33591.  相似文献   

11.
The main result is that for 2≦qp<∞ the only subspaces of the Lorentz function spaceL pq [0, 1] which are isomorphic to r.i. function spaces on [0, 1] are, up to equivalent renormings,L pq [0, 1] andL 2[0, 1].  相似文献   

12.
Summary A sequential sampling rule is given for selecting the most probable event from a multinomial distribution withk cells. A random number of observations is taken from the given multinomial distribution at each stage of sampling, where the number is distributed according to a Poisson distribution with mean λ. The sampling is stopped when the count in any cell is greater than or equal to a given positive integerN. The cell with the highest count is selected for the most probable event. The mathematical analysis of the problem is simplified as a result of the statistical independence of the cell frequencies due to the randomization of the sample number. The expected value of the stage when the sampling terminates is decreasing in λ. The sequential sampling scheme in which one observation is taken at a time until the highest cell count is equal toN, corresponds to λ→0. A table is given showing some properties of the given selection procedure.  相似文献   

13.
We consider a single server loss system in which arrivals occur according to a doubly stochastic Poisson process with a stationary ergodic intensity functionλ t . The service times are independent, exponentially distributed r.v.'s with meanμ −1, and are independent of arrivals. We obtain monotonicity results for loss probabilities under time scaling as well as under amplitude scaling ofλ t . Moreover, using these results we obtain both lower and upper bounds for the loss probability.  相似文献   

14.
LetK be an algebraically closed field of characteristic zero. ForAK[x, y] let σ(A) = {λ ∈K:A − λ is reducible}. For λ ∈ σ(A) letA − λ = ∏ i=1 n(λ) A iλ k μ whereA iλ are distinct primes. Let ϱλ(A) =n(λ) − 1 and let ρ(A) = Σλɛσ(A)ϱλ(A). The main result is the following: Theorem.If A ∈ K[x, y] is not a composite polynomial, then ρ(A) < degA.  相似文献   

15.
Summary Let1≦k≦n−1, 2≦n. This paper examines the vectors δp(Lk), where Lk is a k-dimensional subspace of an n-dimensional space, and the co-ordinates of δp(Lk) are given below by (1.1). For fixed k, the set of such vectors as Lk varies is determined for p=2. For general p, information is given on upper and lower bounds for the sum of the co-ordinates of δp(Lk). Dedicated to the sixtieth birthday of Prof. Edgar R. Lorch This research was supported in part by the Office of Naval Research under contract number Nonr 3775(09), NR 047040.  相似文献   

16.
In this article we introduce the generalized lacunary difference sequence spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and [N_θ,M,Δ~m]_∞using m~(th)- difference.We study their properties like completeness,solidness,symmetricity.Also we obtain some inclusion relations involving the spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and[N_θ,M,Δ~m]_∞and the Cesàro summable and strongly Cesàro summable sequences.  相似文献   

17.
Letn cells be arranged in a ring, or alternatively, in a row. Initially, all cells are unmarked. Sequentially, one of the unmarked cells is chosen at random and marked until, aftern steps, each cell is marked. After thekth cell has been marked the configuration of marked cells defines some number of islands: maximal sets of adjacent marked cells. Let ξ k denote the random number of islands afterk cells have been marked. We give explicit expressions for moments of products of ξ k ’s and for moments of products of 1/ξ k ’s. These are used in a companion paper to prove that if a random graph on the natural number is made by drawing an edge betweeni≧1 andj>i with probabilityλ/j, then the graph is almost surely connected ifλ>1/4 and almost surely disconnected ifλ≦1/4.  相似文献   

18.
Let Λ = (λ k ) be a sequence of non-zero complex numbers. In this paper we introduce the strongly almost convergent generalized difference sequence spaces associated with multiplier sequences i.e. w 0[A m ,Λ,p], w 1[A m ,Λ,p], w [A m ,Λ,p] and study their different properties. We also introduce Δ Λ m -statistically convergent sequences and give some inclusion relations between w 1 m ,λ,p] convergence and Δ Λ m -statistical convergence. Communicated by Pavel Kostyrko  相似文献   

19.
A non-simpliciald-polytope is shown to have strictly fewerk-faces ([(d−1)/2]≦kd−1) then some simpliciald-polytope with the same number of vertices; actual numerical bounds are given. This provides a strong affirmative answer to a problem of Klee.  相似文献   

20.
Two inverse problems for the Sturm-Liouville operator Ly = s-y″ + q(x)y on the interval [0, fy] are studied. For θ ⩾ 0, there is a mapping F:W 2θl B θ, F(σ) = {s k }1, related to the first of these problems, where W 2 = W 2[0, π] is the Sobolev space, σ = ∫ q is a primitive of the potential q, and l B θ is a specially constructed finite-dimensional extension of the weighted space l 2θ, where we place the regularized spectral data s = {s k }1 in the problem of reconstruction from two spectra. The main result is uniform lower and upper bounds for ∥σ - σ1θ via the l B θ-norm ∥s − s1θ of the difference of regularized spectral data. A similar result is obtained for the second inverse problem, that is, the problem of reconstructing the potential from the spectral function of the operator L generated by the Dirichlet boundary conditions. The result is new even for the classical case qL 2, which corresponds to θ = 1.  相似文献   

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