共查询到20条相似文献,搜索用时 31 毫秒
1.
Michel Talagrand 《Israel Journal of Mathematics》1992,79(2-3):207-224
Consider a setA of symmetricn×n matricesa=(a
i,j)
i,j≤n
. Consider an independent sequence (g
i)
i≤n
of standard normal random variables, and letM=Esupa∈A|Σi,j⪯nai,jgigj|. Denote byN
2(A, α) (resp.N
t(A, α)) the smallest number of balls of radiusα for thel
2 norm ofR
n
2 (resp. the operator norm) needed to coverA. Then for a universal constantK we haveα(logN
2(A, α))1/4≤KM. This inequality is best possible. We also show that forδ≥0, there exists a constantK(δ) such thatα(logN
t≤K(δ)M.
Work partially supported by an N.S.F. grant. 相似文献
2.
The following first order nonlinear differential equation with a deviating argument $ x'(t) + p(t)[x(\tau (t))]^\alpha = 0 $ is considered, where α > 0, α ≠ 1, p ∈ C[t 0; ∞), p(t) > 0 for t ≧ t 0, τ ∈ C[t 0; ∞), lim t→∞ τ(t) = ∞, τ(t) < t for t ≧ t 0. Every eventually positive solution x(t) satisfying lim t→∞ x(t) ≧ 0. The structure of solutions x(t) satisfying lim t→∞ x(t) > 0 is well known. In this paper we study the existence, nonexistence and asymptotic behavior of eventually positive solutions x(t) satisfying lim t→∞ x(t) = 0. 相似文献
3.
Explicit and asymptotic solutions are presented to the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + min1 ? t ? n(αM(t) + βM(n + 1 ? t)) for the cases (1) α + β < 1, is rational, and g(n) = δnI. (2) α + β > 1, min(α, β) > 1, is rational, and (a) g(n) = δn1, (b) g(n) = 1. The general form of this recurrence was studied extensively by Fredman and Knuth [J. Math. Anal. Appl.48 (1974), 534–559], who showed, without actually solving the recurrence, that in the above cases , where γ is defined by α?γ + β?γ = 1, and that does not exist. Using similar techniques, the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + max1 ? t ? n(αM(t) + βM(n + 1 ? t)) is also investigated for the special case α = β < 1 and g(n) = 1 if n is odd = 0 if n is even. 相似文献
4.
For given matrices A(s) and B(s) whose entries are polynomials in s, the validity of the following implication is investigated: ?ylimt → ∞A(D) y(t) = 0 ? limt → ∞B(D) y(t) = 0. Here D denotes the differentiation operator and y stands for a sufficiently smooth vector valued function. Necessary and sufficient conditions on A(s) and B(s) for this implication to be true are given. A similar result is obtained in connection with an implication of the form ?yA(D) y(t) = 0, limt → ∞B(D) y(t) = 0, C(D) y(t) is bounded ? limt → ∞E(D) y(t) = 0. 相似文献
5.
Monika Ludwig 《Mathematische Nachrichten》2001,227(1):99-108
Let Hj(K, ·) be the j – th elementary symmetric function of the principal curvatures of a convex body K in Euclidean d – space. We show that the functionals ∫bd f(Hj(K, x)) dℋ︁d—1(x) depend upper semicontinuously on K, if f : [0, ∞) is concave, limt→0f(t) = 0, and limt→∞f(t)/t = 0. An analogous statement holds for integrals of elementary symmetric functions of the principal radii of curvature. 相似文献
6.
We consider the maximum queue length and the maximum number of idle servers in the classical Erlang delay model and the generalization allowing customer abandonment—the M/M/n+M queue. We use strong approximations to show, under regularity conditions, that properly scaled versions of the maximum queue length and maximum number of idle servers over subintervals [0,t] in the delay models converge jointly to independent random variables with the Gumbel extreme value distribution in the quality-and-efficiency-driven (QED) and ED many-server heavy-traffic limiting regimes as n and t increase to infinity together appropriately; we require that t n →∞ and t n =o(n 1/2?ε ) as n→∞ for some ε>0. 相似文献
7.
Simeon M. Berman 《Stochastic Processes and their Applications》1983,15(3):213-238
Let X(t) be the trigonometric polynomial Σkj=0aj(Utcosjt+Vjsinjt), –∞< t<∞, where the coefficients Ut and Vt are random variables and aj is real. Suppose that these random variables have a joint distribution which is invariant under all orthogonal transformations of R2k–2. Then X(t) is stationary but not necessarily Gaussian. Put Lt(u) = Lebesgue measure {s: 0?s?t, X(s) > u}, and M(t) = max{X(s): 0?s?t}. Limit theorems for Lt(u) and for u→∞ are obtained under the hypothesis that the distribution of the random norm (Σkj=0(U2j+V2j))1 2 belongs to the domain of attraction of the extreme value distribution exp{ e–2}. The results are also extended to the random Fourier series (k=∞). 相似文献
8.
Fuqing Gao 《Journal of Mathematical Analysis and Applications》2003,285(1):1-7
Let X0,X1,… be i.i.d. random variables with E(X0)=0, E(X20)=1 and E(exp{tX0})<∞ for any |t|<t0. We prove that the weighted sums V(n)=∑j=0∞aj(n)Xj, n?1 obey a moderately large deviation principle if the weights satisfy certain regularity conditions. Then we prove a new version of the Erdös-Rényi-Shepp laws for the weighted sums. 相似文献
9.
Let {ø n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior ofø n (dμ) when logμ′∈L 1. In what follows we will discuss the asymptotic behavior of the ratioø n (dμ 2)/ø n (dμ 1) on the unit circle whendμ 1 anddμ 2 are close in a sense (e.g.,dμ 2=gdμ 1, where g≥0 is such thatQ(e it )g(t) andQ(e it )/g(t) are bounded for a suitable polynomialQ) and μ 1 ′ >0 almost everywhere or (a somewhat weaker requirement) lim n→∞Φ n (dμ 1,0)=0 for the monic polynomial Φ n . The asymptotic behavior of the same fraction outside the unit circle was discussed in an earlier paper. 相似文献
10.
Hans Riesel 《BIT Numerical Mathematics》1988,28(4):839-851
Letg be a primitive λ-root modn. Then the powersg t mod,n, fort=1, 2, ..., λ(n) represent a (cyclic) subgroupC λ(n) (of order λ(n)) ofM n , the group of order ?(n), representingall primitive residue classes modn. To computet backwards fromg t modn is calledthe discrete logarithm problem inC λ(n), also expressible by $$a \equiv g^t \bmod n \Leftrightarrow t \equiv \log _g a\bmod \lambda \left( n \right).$$ The purpose of this paper is to point out some cases, in which this problem can be solved by astraight-forward formula only, with no element of guessing or collecting factorizations or the like involved at all, taking timeO(log3 n) or less to compute (orO(log2+ε n), if fast multiplication of multiple precision integers is used).—One very simple case is given by the modulusn=10 s , fors≥4. To give just one instance of this case: Forn=108, if the primitive λ-rootg=317 ≡ 29140163 modn is chosen, anda=g t modn, then $$a \equiv 4962873 \cdot \frac{{a^{250000} - 1}}{{5000000}}\bmod 5000000.$$ 相似文献
11.
A.M. Frieze 《Discrete Applied Mathematics》1985,10(1):47-56
Suppose we are given a complete graph on n vertices in which the lenghts of the edges are independent identically distributed non-negative random variables. Suppose that their common distribution function F is differentiable at zero and D = F′ (0) > 0 and each edge length has a finite mean and variance. Let Ln be the random variable whose value is the length of the minimum spanning tree in such a graph. Then we will prove the following: limn → ∞E(Ln) = ζ(3)/D where ζ(3) = Σk = 1∞ 1/k3 = 1.202… and for any ε > 0 limn → ∞ Pr(|Ln?ζ(3)/D|) > ε) = 0. 相似文献
12.
The paper deals with the strong summability of Marcinkiewicz means with a variable power. Let $$H_n \left( {f,x,y,A_n } \right): = \tfrac{1} {n}\sum\nolimits_{l = 1}^n {\left( {e^{\left. {A_n } \right|\left. {S_{ll} \left( {f,x,y} \right) - f\left( {x,y} \right)} \right|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } - 1} \right)} .$$ It is shown that if A n ↑ ∞ arbitrary slowly, there exists f ∈ C(I 2) such that lim n→∞ H n (f, 0, 0, A n ) = +∞. At the same time, for every f ∈ C (I 2) there exists A n (f) ↑ ∞ such that lim n→∞ H n (f, x, y, A n ) = 0 uniformly on I 2. 相似文献
13.
Donald E Knuth 《Journal of Combinatorial Theory, Series A》1974,16(3):398-400
A simple proof is given that limn?t8(log2 log2gn)/n = 1, where gn denotes the number of distinct combinatorial geometries on n point 相似文献
14.
N.H. Guersenzvaig 《Linear algebra and its applications》2010,432(10):2691-2700
Let f,g∈Z[X] be monic polynomials of degree n and let C,D∈Mn(Z) be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra Z〈C,D〉 to be a sublattice of finite index in the full integral lattice Mn(Z), in which case we compute the exact value of this index in terms of the resultant of f and g. If R is a commutative ring with identity we determine when R〈C,D〉=Mn(R), in which case a presentation for Mn(R) in terms of C and D is given. 相似文献
15.
Stephen James Wolfe 《Stochastic Processes and their Applications》1982,12(3):301-312
Let B1, B2, ... be a sequence of independent, identically distributed random variables, letX0 be a random variable that is independent ofBn forn?1, let ρ be a constant such that 0<ρ<1 and letX1,X2, ... be another sequence of random variables that are defined recursively by the relationshipsXn=ρXn-1+Bn. It can be shown that the sequence of random variablesX1,X2, ... converges in law to a random variableX if and only ifE[log+¦B1¦]<∞. In this paper we let {B(t):0≦t<∞} be a stochastic process with independent, homogeneous increments and define another stochastic process {X(t):0?t<∞} that stands in the same relationship to the stochastic process {B(t):0?t<∞} as the sequence of random variablesX1,X2,...stands toB1,B2,.... It is shown thatX(t) converges in law to a random variableX ast →+∞ if and only ifE[log+¦B(1)¦]<∞ in which caseX has a distribution function of class L. Several other related results are obtained. The main analytical tool used to obtain these results is a theorem of Lukacs concerning characteristic functions of certain stochastic integrals. 相似文献
16.
R.S. Singh 《Journal of multivariate analysis》1976,6(2):338-342
Let Xj = (X1j ,…, Xpj), j = 1,…, n be n independent random vectors. For x = (x1 ,…, xp) in Rp and for α in [0, 1], let Fj(x) = αI(X1j < x1 ,…, Xpj < xp) + (1 ? α) I(X1j ≤ x1 ,…, Xpj ≤ xp), where I(A) is the indicator random variable of the event A. Let Fj(x) = E(Fj(x)) and Dn = supx, α max1 ≤ N ≤ n |Σ0n(Fj(x) ? Fj(x))|. It is shown that P[Dn ≥ L] < 4pL exp{?2(L2n?1 ? 1)} for each positive integer n and for all L2 ≥ n; and, as n → ∞, with probability one. 相似文献
17.
Ian Deters 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2014,49(3):117-125
In [2] a cyclic diagonal operator on the space of functions analytic on the unit disk with eigenvalues (λ n ) is shown to admit spectral synthesis if and only if for each j there is a sequence of polynomials (p n ) such that lim n→∞ p n (λ k ) = δ j,k and lim sup n→∞ sup k>j |p n (λ k )|1/k ≤ 1. The author also shows, through contradiction, that certain classes of cyclic diagonal operators are synthetic. It is the intent of this paper to use the aforementioned equivalence to constructively produce examples of synthetic diagonal operators. In particular, this paper gives two different constructions for sequences of polynomials that satisfy the required properties for certain sequences to be the eigenvalues of a synthetic operator. Along the way we compare this to other results in the literature connecting polynomial behavior ([4] and [9]) and analytic continuation of Dirichlet series ([1]) to the spectral synthesis of diagonal operators. 相似文献
18.
Awi Federgruen 《Stochastic Processes and their Applications》1981,11(2):187-192
Recent papers have shown that Π∞k = 1P(k) = limm→∞ (P(m) ? P(1)) exists whenever the sequence of stochastic matrices {P(k)}∞k = 1 exhibits convergence to an aperiodic matrix P with a single subchain (closed, irreducible set of states). We show how the limit matrix depends upon P(1).In addition, we prove that limm→∞limn→∞ (P(n + m) ? P(m + 1)) exists and equals the invariant probability matrix associated with P. The convergence rate is determined by the rate of convergence of {P(k)}∞k = 1 towards P.Examples are given which show how these results break down in case the limiting matrix P has multiple subchains, with {P(k)}∞k = 1 approaching the latter at a less than geometric rate. 相似文献
19.
Joan Bagaria 《Archive for Mathematical Logic》2012,51(3-4):213-240
For each natural number n, let C (n) be the closed and unbounded proper class of ordinals α such that V α is a Σ n elementary substructure of V. We say that κ is a C (n) -cardinal if it is the critical point of an elementary embedding j : V → M, M transitive, with j(κ) in C (n). By analyzing the notion of C (n)-cardinal at various levels of the usual hierarchy of large cardinal principles we show that, starting at the level of superstrong cardinals and up to the level of rank-into-rank embeddings, C (n)-cardinals form a much finer hierarchy. The naturalness of the notion of C (n)-cardinal is exemplified by showing that the existence of C (n)-extendible cardinals is equivalent to simple reflection principles for classes of structures, which generalize the notions of supercompact and extendible cardinals. Moreover, building on results of Bagaria et?al. (2010), we give new characterizations of Vopeňka’s Principle in terms of C (n)-extendible cardinals. 相似文献
20.
Gelasio Salazar 《Journal of Graph Theory》1998,28(3):163-170
We show that the M-crossing number crM(Cm × Cn) of Cm × Cn behaves asymptotically according to limn→∞ {crM(Cm × Cn)/((m − 2)n)} = 1, for each m ≥ 3. This result reinforces the conjecture cr(Cm × Cn) = (m − 2)n if 3 ≤ m ≤ n, which has been proved only for m ≤ 6. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 163–170, 1998 相似文献