Let {aj}and {adj} be two binary maximal length linear sequences of period 2n?1. The cross-correlation function is defined as for t = 0, 1,…, 2n ? 2. We find the values and the number of occurrences of each value of Cd(t) when (mod 8). 相似文献
Consider the class of retarded functional differential equations , (1) where xt(θ) = x(t + θ), ?1 ? θ ? 0, so xt?C = C([?1, 0], Rn), and . Let 2 ? r ? ∞ and give the appropriate (Whitney) topology. Then the set of such that all fixed points and all periodic solutions of (1) are hyperbolic is residual in . 相似文献
Elementary methods are used to study sums of the form for integers p and t, t > 0, where {x} denotes the fractional part of x. These sums are then used to study sums of the form for integers p and t, t > 0, where Pt(x) = Bt({x}) and Bt(x) are Bernoulli polynomials. some general results on sums of error terms are used to study sums of the form Σn≤xntσa(n) and Σn≤xEt(n) for integers t and a, a ≥ 0, where σa(n) is the sum of the ath powers of the divisors of n and Et(x) is the error term in the sum Σn≤xntσa(n). 相似文献
For a sequence A = {Ak} of finite subsets of N we introduce: , , where A(m) is the number of subsets Ak ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation constitutes a finite semi-group N∪ (semi-group N∩) (group ). For N∪, N∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {Akr}, where Akr | Akr+1 for r = 1, 2, …. In Section 6 we consider for analogues of Rohrbach inequality: , where g(n) = min k over the subsets {a1 < … < ak} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = ai + aj.Pour une série A = {Ak} de sous-ensembles finis de N on introduit les densités: , où A(m) est le nombre d'ensembles Ak ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations , un semi-groupe fini N∪, N∩ ou un groupe N1 respectivement. Pour N∪, N∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {Akr}, où Akr|Akr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N∪, les analogues de l'inégalité de Rohrbach: , où g(n) = min k pour les ensembles {a1 < … < ak} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = ai + aj. 相似文献
Let {Gn} be a sequence of finite transitive graphs with vertex degree d = d(n) and |Gn| = n. Denote by pt(v, v) the return probability after t steps of the non-backtracking random walk on Gn. We show that if pt(v, v) has quasi-random properties, then critical bond-percolation on Gn behaves as it would on a random graph. More precisely, if $\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$then the size of the largest component in p-bond-percolation with ${p =\frac{1+O(n^{-1/3})}{d-1}}Let {Gn} be a sequence of finite transitive graphs with vertex degree d = d(n) and |Gn| = n. Denote by pt(v, v) the return probability after t steps of the non-backtracking random walk on Gn. We show that if pt(v, v) has quasi-random properties, then critical bond-percolation on Gn behaves as it would on a random graph. More precisely, if
It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let An=(aij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ(n)}1?k?n its eigenvalues, and {uk(n)}1?k?n the corresponding (orthonormalized column) eigenvectors. Let , put (bookeeping function for the length of the projections of the new row v1n of An onto the eigenvectors of the preceding matrix An?1), and let finally (empirical distribution function of the eigenvalues of . Suppose (i) , (ii) limnXn(t)=Ct(0<C<∞,0?t?1). Then ,where W is absolutely continuous with (semicircle) density 相似文献
The explicit expression of the smallest constantC satisfying $$\mathop {lim}\limits_{\lambda \to \infty } \left| {t_{n(\lambda )}^{(1)} - t_{m(\lambda )}^{(2)} } \right| \leqq C. \mathop {lim sup}\limits_{n \to \infty } \left| {d_n } \right|$$ for all sequences {sn} satisfying lim supn→∞ |dn| <∞, where {tn(1)}, {tn(2)} are two generalised Hausdorff transforms of {sn}, {dn} is the generalised (C, α)-transform (0≦α≦1) of {λnan} andn(λ,m(λ) are suitably related, is obtained. These results are obtained by using new properties of positive approximation operators and generalised Bernstein approximation operators. 相似文献
A set {b1,b2,…,bi} ? {1,2,…,N} is said to be a difference intersector set if {a1,a2,…,as} ? {1,2,…,N}, j > ?N imply the solvability of the equation ax ? ay = b′; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b1,b2,…,bi} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the equations (, , , (where () denotes the Legendre symbol) and to show that “almost all” sets form both difference and sum intersector sets. 相似文献
Let {Xn}n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let Sn = Σj=1nXj and . In this paper, we prove that (1) lim?→0+?α(r?1)E{N∞(r, t, ?)} = K(r, t) if E(X1) = 0, Var(X1) = 1, and E(| X1 |t) < ∞, where 2 ≤ t < 2r ≤ 2t, , and ; (2) if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(|X1|t) < ∞, where G(t, ?) = E{N∞(t, t, ?)} = Σn=1∞nt?2P{| Sn | > ?n} → ∞ as ? → 0+ and , i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(| X1 |t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution. 相似文献
A weighted translation semigroup {St} on L2(+) is defined by for x ? t and 0 otherwise, where φ is a continuous nonzero scalar-valued function on +. It is shown that {St} is subnormal if and only if φ2 is the product of an exponential function and the Laplace-Stieltjes transform of an increasing function of total variation one. A necessary and sufficient condition for similarity of weighted translation semigroups is developed. 相似文献
A t-spread set [1] is a set of (t + 1) × (t + 1) matrices over GF(q) such that ∥C∥ = qt+1, 0 ? C, I?C, and det(X ? Y) ≠ 0 if X and Y are distinct elements of . The amount of computation involved in constructing t-spread sets is considerable, and the following construction technique reduces somewhat this computation. Construction: Let be a subgroup of GL(t + 1, q), (the non-singular (t + 1) × (t + 1) matrices over GF(q)), such that ∥G∥|at+1, and det (G ? H) ≠ 0 if G and H are distinct elements of . Let A1, A2, …, An?GL(t + 1, q) such that det(Ai ? G) ≠ 0 for i = 1, …, n and all G?G, and det(Ai ? AjG) ≠ 0 for i > j and all G?G. Let , and ∥C∥ = qt+1. Then is a t-spread set. A t-spread set can be used to define a left V ? W system over V(t + 1, q) as follows: x + y is the vector sum; let e?V(t + 1, q), then xoy = yM(x) where M(x) is the unique element of with x = eM(x). Theorem: Letbe a t-spread set and F the associatedV ? Wsystem; the left nucleus = {y | CM(y) = C}, and the middle nucleus = }y | M(y)C = C}. Theorem: Forconstructed as aboveG ? {M(x) | x?Nλ}. This construction technique has been applied to construct a V ? W system of order 25 with ∥Nλ∥ = 6, and ∥Nμ∥ = 4. This system coordinatizes a new projective plane. 相似文献
Let {Xn, n ≥ 1} be a real-valued stationary Gaussian sequence with mean zero and variance one. Let Mn = max{Xt, i ≤ n} and Hn(t) = (M[nt] ? bn)an?1 be the maximum resp. the properly normalised maximum process, where , and . We characterize the almost sure limit functions of (Hn)n≥3 in the set of non-negative, non-decreasing, right-continuous, real-valued functions on (0, ∞), if r(n) (log n)3?Δ = O(1) for all Δ > 0 or if r(n) (log n)2?Δ = O(1) for all Δ > 0 and r(n) convex and fulfills another regularity condition, where r(n) is the correlation function of the Gaussian sequence. 相似文献
For 1 ≦ l ≦ j, let l = ?h=1q(l){alh + Mv: v = 0, 1, 2,…}, where j, M, q(l) and the alh are positive integers such that j > 1, al1 < … < alq(2) ≦ M, and let l′ = l ∪ {0}. Let p(n : ) be the number of partitions of n = (n1,…,nj) where, for 1 ≦ l ≦ j, the lth component of each part belongs to l and let p1(n : ) be the number of partitions of n into different parts where again the lth component of each part belongs to l. Asymptotic formulas are obtained for p(n : ), p1(n : ) where all but one nl tend to infinity much more rapidly than that nl, and asymptotic formulas are also obtained for p(n : ′), p1(n ; ′), where one nl tends to infinity much more rapidly than every other nl. These formulas contrast with those of a recent paper (Robertson and Spencer, Trans. Amer. Math. Soc., to appear) in which all the nl tend to infinity at approximately the same rate. 相似文献
Let Xt be the Brownian motion in d. The random set Γ = {(t1,…, tn, z): Xtl = ··· = Xtn = z} in d + n is empty a.s. except in the following cases: (a) n = 1, d = 1, 2,…; (b) d = 2, n = 2, 3,…; (c) d = 3, n = 2. In each of these cases, a family of random measures Mλ concentrated on Γ is constructed (λ takes values in a certain class of measures on d). Measures Mλ characterize the time-space location of self-intersections for Brownian paths. If n = d = 1, then Mλ(dt, dz) = λ(dz) Nz(dt) where N2 is the local time at z. In the case n = 2, the set Γ can be identified with the set of Brownian loops. The measure Mλ “explodes” on the diagonal {t1 = t2} and, to study small loops, a random distribution which regularizes Mλ is constructed. 相似文献
Let Pij and qij be positive numbers for i ≠ j, i, j = 1, …, n, and consider the set of matrix differential equations x′(t) = A(t) x(t) over all A(t), where aij(t) is piecewise continuous, aij(t) = ?∑i ≠ jaij(t), and pij ? aij(t) ? qij all t. A solution x is also to satisfy ∑i = 1nxi(0) = 1. Let Ct denote the set of all solutions, evaluated at t to equations described above. It is shown that , the topological closure of Ct, is a compact convex set for each t. Further, the set valued function , of t is continuous and . 相似文献
Let {pn(t)}n=0t8 be a system of algebraic polynomials orthonormal on the segment [?1, 1] with a weight p(t); let {xn,ν(p)}ν=1n be zeros of a polynomial pn(t) (xx,ν(p) = cosθn,ν(p); 0 < θn,1(p) < θn,2(p) < ... < θn,n(p) < π). It is known that, for a wide class of weights p(t) containing the Jacobi weight, the quantities θn,1(p) and 1 ? xn,1(p) coincide in order with n?1 and n?2, respectively. In the present paper, we prove that, if the weight p(t) has the form p(t) = 4(1 ? t2)?1{ln2[(1 + t)/(1 ? t)] + π2}?1, then the following asymptotic formulas are valid as n → ∞:
For fixed p (0 ≤ p ≤ 1), let {L0, R0} = {0, 1} and X1 be a uniform random variable over {L0, R0}. With probability p let {L1, R1} = {L0, X1} or = {X1, R0} according as ; with probability 1 ? p let {L1, R1} = {X1, R0} or = {L0, X1} according as , and let X2 be a uniform random variable over {L1, R1}. For n ≥ 2, with probability p let {Ln, Rn} = {Ln ? 1, Xn} or = {Xn, Rn ? 1} according as , with probability 1 ? p let {Ln, Rn} = {Xn, Rn ? 1} or = {Ln ? 1, Xn} according as , and let Xn + 1 be a uniform random variable over {Ln, Rn}. By this iterated procedure, a random sequence {Xn}n ≥ 1 is constructed, and it is easy to see that Xn converges to a random variable Yp (say) almost surely as n → ∞. Then what is the distribution of Yp? It is shown that the Beta, (2, 2) distribution is the distribution of Y1; that is, the probability density function of Y1 is g(y) = 6y(1 ? y) I0,1(y). It is also shown that the distribution of Y0 is not a known distribution but has some interesting properties (convexity and differentiability). 相似文献