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1.
For any positive integer k ≥ 3, it is easy to prove that the k-polygonal numbers are an(k) = (2n+n(n?1)(k?2))/2. The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet L-functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums S(an(k)ām(k), p) for k-polygonal numbers with 1 ≤ m, np ? 1, and give an interesting computational formula for it.  相似文献   

2.
Given a sequence A = (a 1, …, a n ) of real numbers, a block B of A is either a set B = {a i , a i+1, …, a j } where ij or the empty set. The size b of a block B is the sum of its elements. We show that when each a i ∈ [0, 1] and k is a positive integer, there is a partition of A into k blocks B 1, …, B k with |b i ?b j | ≤ 1 for every i, j. We extend this result in several directions.  相似文献   

3.
In this paper, we investigate the following problem: give a quasi-Boolean function Ψ(x 1, …, x n ) = (aC) ∨ (a 1C 1) ∨ … ∨ (a p C p ), the term (aC) can be deleted from Ψ(x 1, …, x n )? i.e., (aC) ∨ (a 1C 1) ∨ … ∨ (a p C p ) = (a 1C 1) ∨ … ∨ (a p C p )? When a = 1: we divide our discussion into two cases. (1) ?1(Ψ,C) = ø, C can not be deleted; ?1(Ψ,C) ≠ ø, if S i 0 ≠ ø (1 ≤ iq), then C can not be deleted, otherwise C can be deleted. When a = m: we prove the following results: (mC)∨(a 1C 1)∨…∨(a p C p ) = (a 1C 1)∨…∨(a p C p ) ? (mC) ∨ C 1 ∨ … ∨C p = C 1 ∨ … ∨C p . Two possible cases are listed as follows, (1) ?2(Ψ,C) = ø, the term (mC) can not be deleted; (2) ?2(Ψ,C) ≠ ø, if (?i 0) such that \(S'_{i_0 } \) = ø, then (mC) can be deleted, otherwise ((mC)∨C 1∨…∨C q )(v 1, …, v n ) = (C 1 ∨ … ∨ C q )(v 1, …, v n )(?(v 1, …, v n ) ∈ L 3 n ) ? (C 1 ∨ … ∨ C q )(u 1, …, u q ) = 1(?(u 1, …, u q ) ∈ B 2 n ).  相似文献   

4.
Assume that G is a finite non-Dedekind p-group. D. S. Passman introduced the following concept: we say that H1 < H2 < ? < Hk is a chain of nonnormal subgroups of G if each Hi ? G and if |Hi : Hi?1| = p for i = 2, 3,…, k. k is called the length of the chain. chn(G) denotes the maximum of the lengths of the chains of nonnormal subgroups of G. In this paper, finite 2-groups G with chn(G) ? 2 are completely classified up to isomorphism.  相似文献   

5.
A mixed covering array (MCA) of type (v 1, v 2,..., v k ), denoted by MCAλ (N; t, k, (v 1, v 2,..., v k )), is an N × k array with entries in the i-th column from a set V i of v i symbols and has the property that each N × t sub-array covers all the t-tuples at least λ times, where 1 ≤ ik. An MCA λ (N; t, k, (v 1, v 2,..., v k )) is said to be super-simple, if each of its N × (t + 1) sub-arrays contains each (t + 1)-tuple at most once. Recently, it was proved by Tang, Yin and the author that an optimum super-simple MCA of type (a, b, b,..., b) is equivalent to a mixed detecting array (DTA) of type (a, b, b,..., b) with optimum size. Such DTAs can be used to generate test suites to identify and determine the interaction faults between the factors in a component-based system. In this paper, some combinatorial constructions of optimum super-simple MCAs of type (a, b, b,..., b) are provided. By employing these constructions, some optimum super-simple MCAs are then obtained. In particular, the spectrum across which optimum super-simple MCA2(2b 2; 2, 4, (a, b, b, b))′s exist, is completely determined, where 2 ≤ ab.  相似文献   

6.
This note considers the joint replenishment inventory problem for N items under constant demand. The frequently-used cyclic strategy (T; k1, …, k N ) is investigated: a family replenishment is made every T time units and item i is included in each k i th replenishment. Goyal proposed a solution to find the global optimum within the class of cyclic strategies. However, we will show that the algorithm of Goyal does not always lead to the optimal cyclic strategy. A simple correction is suggested.  相似文献   

7.
Motivated by a question of Sárközy, we study the gaps in the product sequence B = A · A = {b 1 < b 2 < …} of all products a i a j with a i , a j A when A has upper Banach density α > 0. We prove that there are infinitely many gaps b n+1 ? b n ? α ?3 and that for t ≥ 2 there are infinitely many t-gaps b n+t ? b n ? t 2 α ?4. Furthermore, we prove that these estimates are best possible.We also discuss a related question about the cardinality of the quotient set A/A = {a i /a j , a i , a j A} when A ? {1, …, N} and |A| = αN.  相似文献   

8.
Let G be an abelian group of order n. The sum of subsets A1,...,Ak of G is defined as the collection of all sums of k elements from A1,...,Ak; i.e., A1 + A2 + · · · + Ak = {a1 + · · · + ak | a1A1,..., akAk}. A subset representable as the sum of k subsets of G is a k-sumset. We consider the problem of the number of k-sumsets in an abelian group G. It is obvious that each subset A in G is a k-sumset since A is representable as A = A1 + · · · + Ak, where A1 = A and A2 = · · · = Ak = {0}. Thus, the number of k-sumsets is equal to the number of all subsets of G. But, if we introduce a constraint on the size of the summands A1,...,Ak then the number of k-sumsets becomes substantially smaller. A lower and upper asymptotic bounds of the number of k-sumsets in abelian groups are obtained provided that there exists a summand Ai such that |Ai| = n logqn and |A1 +· · ·+ Ai-1 + Ai+1 + · · ·+Ak| = n logqn, where q = -1/8 and i ∈ {1,..., k}.  相似文献   

9.
Let p be an odd prime and c a fixed integer with (c, p) = 1. For each integer a with 1 ≤ ap ? 1, it is clear that there exists one and only one b with 0 ? b ? p ? 1 such that abc (mod p). Let N(c, p) denote the number of all solutions of the congruence equation abc (mod p) for 1 ? a, b ? p?1 in which a and \(\overline b \) are of opposite parity, where \(\overline b \) is defined by the congruence equation b\(\overline b \) ≡ 1 (mod p). The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet L-functions to study the hybrid mean value problem involving N(c, p)?½φ(p) and the Dedekind sums S(c, p), and to establish a sharp asymptotic formula for it.  相似文献   

10.
We present a tight bound on the exact maximum complexity of Minkowski sums of polytopes in ?3. In particular, we prove that the maximum number of facets of the Minkowski sum of k polytopes with m 1,m 2,…,m k facets, respectively, is bounded from above by \(\sum_{1\leq i. Given k positive integers m 1,m 2,…,m k , we describe how to construct k polytopes with corresponding number of facets, such that the number of facets of their Minkowski sum is exactly \(\sum_{1\leq i. When k=2, for example, the expression above reduces to 4m 1 m 2?9m 1?9m 2+26.  相似文献   

11.
Let N0 be the set of natural numbers whose binary expansions have an even number of 1’s, and let N1 = N\N0. In this paper, we obtain asymptotic formulas for the number of primes p not exceeding X and such that p ∈ Ni, p + 1 ∈ Nj, where i and j take values 0 and 1 independently of each other.  相似文献   

12.
Let x 0, x 1,? , x n , be a set of n + 1 distinct real numbers (i.e., x i x j , for ij) and y i, k , for i = 0,1,? , n, and k = 0 ,1 ,? , n i , with n i ≥ 1, be given of real numbers, we know that there exists a unique polynomial p N ? 1(x) of degree N ? 1 where \(N={\sum }_{i=0}^{n}(n_{i}+1)\), such that \(p_{N-1}^{(k)}(x_{i})=y_{i,k}\), for i = 0,1,? , n and k = 0,1,? , n i . P N?1(x) is the Hermite interpolation polynomial for the set {(x i , y i, k ), i = 0,1,? , n, k = 0,1,? , n i }. The polynomial p N?1(x) can be computed by using the Lagrange polynomials. This paper presents a new method for computing Hermite interpolation polynomials, for a particular case n i = 1. We will reformulate the Hermite interpolation polynomial problem and give a new algorithm for giving the solution of this problem, the Matrix Recursive Polynomial Interpolation Algorithm (MRPIA). Some properties of this algorithm will be studied and some examples will also be given.  相似文献   

13.
For 1 ? c ? p ? 1, let E 1,E 2, …,E m be fixed numbers of the set {0, 1}, and let a 1, a 2, …, a m (1 ? a i ? p, i = 1, 2, …,m) be of opposite parity with E 1,E 2, …,E m respectively such that a 1 a 2a m c (mod p). Let $$N(c,m,p) = {1 \over {{2^{m - 1}}}}\mathop {\sum\limits_{{a_1} = 1}^{p - 1} {\sum\limits_{{a_2} = 1}^{p - 1} \ldots } }\limits_{{a_1}{a_2} \ldots \equiv c{\rm{ (}}\bmod {\rm{ }}p)} \sum\limits_{{a_m} = 1}^{p - 1} {(1 - {{( - 1)}^{{a_1} + {E_1}}})(1 - {{( - 1)}^{{a_2} + {E_2}}}) \ldots } (1 - {( - 1)^{{a_m} + {E_m}}}).$$ We are interested in the mean value of the sums $$\sum\limits_{c = 1}^{p - 1} {{E^2}} (c,m,p),$$ where E(c, m, p) = N(c,m, p)?((p ? 1) m?1)/(2 m?1) for the odd prime p and any integers m ? 2. When m = 2, c = 1, it is the Lehmer problem. In this paper, we generalize the Lehmer problem and use analytic method to give an interesting asymptotic formula of the generalized Lehmer problem.  相似文献   

14.
We discuss aggregation of random-coefficient AR(1) processes X i,t = a i X i,t?1 + ε t , i = 1,…,N, with i.i.d. coefficients a i ∈ (?1, 1) and common i.i.d. innovations {ε t } belonging to the domain of attraction of an α-stable law (0 < α ≤ 2). Particular attention is given to the case of slope coefficient having probability density growing regularly to infinity at points a = 1 and a = ?1. We obtain conditions under which the limit aggregate \( {\bar X_t} = {\lim_{N \to \infty }}{N^{ - 1}}\sum\nolimits_{i = 1}^N {{X_{i,t}}} \) exists and exhibits long memory in a certain sense. In particularly, we show that suitably normalized partial sums of the \( {\bar X_t} \)’s tend to a fractional α-stable motion and that \( \left\{ {{{\bar X}_t}} \right\} \) satisfies the long-range dependence (sample Allen variance) property of Heyde and Yang. We also extend some results of Zaffaroni from the finite variance case to the infinite variance case.  相似文献   

15.
In this article, we study the equation
$\frac{\partial }{\partial t}u(x,t)=c^{2}\Diamond _{B}^{k}u(x,t)$
with the initial condition u(x,0)=f(x) for x∈? n + . The operator ? B k is named to be Bessel diamond operator iterated k-times and is defined by
$\Diamond _{B}^{k}=\bigl[(B_{x_{1}}+B_{x_{2}}+\cdots +B_{x_{p}})^{2}-(B_{x_{p+1}}+\cdots +B_{x_{p+q}})^{2}\bigr]^{k},$
where k is a positive integer, p+q=n, \(B_{x_{i}}=\frac{\partial ^{2}}{\partial x_{i}^{2}}+\frac{2v_{i}}{x_{i}}\frac{\partial }{\partial x_{i}},\) 2v i =2α i +1,\(\;\alpha _{i}>-\frac{1}{2}\), x i >0, i=1,2,…,n, and n is the dimension of the ? n + , u(x,t) is an unknown function of the form (x,t)=(x 1,…,x n ,t)∈? n + ×(0,∞), f(x) is a given generalized function and c is a positive constant (see Levitan, Usp. Mat. 6(2(42)):102–143, 1951; Y?ld?r?m, Ph.D. Thesis, 1995; Y?ld?r?m and Sar?kaya, J. Inst. Math. Comput. Sci. 14(3):217–224, 2001; Y?ld?r?m, et al., Proc. Indian Acad. Sci. (Math. Sci.) 114(4):375–387, 2004; Sar?kaya, Ph.D. Thesis, 2007; Sar?kaya and Y?ld?r?m, Nonlinear Anal. 68:430–442, 2008, and Appl. Math. Comput. 189:910–917, 2007). We obtain the solution of such equation, which is related to the spectrum and the kernel, which is so called Bessel diamond heat kernel. Moreover, such Bessel diamond heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.
  相似文献   

16.
Let M be a commutative, cancellative, atomic monoid and x a nonunit in M. We define ω(x)=n if n is the smallest positive integer with the property that whenever xa 1???a t , where each a i is an atom, there is a T?{1,2,…,t} with |T|≤n such that x∣∏kT a k . The ω-function measures how far x is from being prime in M. In this paper, we give an algorithm for computing ω(x) in any numerical monoid. Simple formulas for ω(x) are given for numerical monoids of the form 〈n,n+1,…,2n?1〉, where n≥3, and 〈n,n+1,…,2n?2〉, where n≥4. The paper then focuses on the special case of 2-generator numerical monoids. We give a formula for computing ω(x) in this case and also necessary and sufficient conditions for determining when x is an atom. Finally, we analyze the asymptotic behavior of ω(x) by computing \(\lim_{x\rightarrow \infty}\frac{\omega(x)}{x}\).  相似文献   

17.
The paper considers a simple Errors-in-Variables (EiV) model Yi = a + bXi + εξi; Zi= Xi + σζi, where ξi, ζi are i.i.d. standard Gaussian random variables, Xi ∈ ? are unknown non-random regressors, and ε, σ are known noise levels. The goal is to estimates unknown parameters a, b ∈ ? based on the observations {Yi, Zi, i = 1, …, n}. It is well known [3] that the maximum likelihood estimates of these parameters have unbounded moments. In order to construct estimates with good statistical properties, we study EiV model in the large noise regime assuming that n → ∞, but \({\epsilon ^2} = \sqrt n \epsilon _ \circ ^2,{\sigma ^2} = \sqrt n \sigma _ \circ ^2\) with some \(\epsilon_\circ^2, \sigma_\circ^2>0\). Under these assumptions, a minimax approach to estimating a, b is developed. It is shown that minimax estimates are solutions to a convex optimization problem and a fast algorithm for solving it is proposed.  相似文献   

18.
Let Ω = {t0, t1, …, tN} and ΩN = {x0, x1, …, xN–1}, where xj = (tj + tj + 1)/2, j = 0, 1, …, N–1 be arbitrary systems of distinct points of the segment [–1, 1]. For each function f(x) continuous on the segment [–1, 1], we construct discrete Fourier sums Sn, N( f, x) with respect to the system of polynomials {p?k,N(x)} k=0 N–1 , forming an orthonormal system on nonuniform point systems ΩN consisting of finite number N of points from the segment [–1, 1] with weight Δtj = tj + 1tj. We find the growth order for the Lebesgue function Ln,N (x) of the considered partial discrete Fourier sums Sn,N ( f, x) as n = O(δ N ?2/7 ), δN = max0≤ jN?1 Δtj More exactly, we have a two-sided pointwise estimate for the Lebesgue function Ln, N(x), depending on n and the position of the point x from [–1, 1].  相似文献   

19.
We investigate the equiconvergence on TN = [?π, π)N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions fLp(TN) and gLp(RN), p > 1, N ≥ 3, g(x) = f(x) on TN, in the case where the “partial sums” of these expansions, i.e., Sn(x; f) and Jα(x; g), respectively, have “numbers” n ∈ ZN and α ∈ RN (nj = [αj], j = 1,..., N, [t] is the integral part of t ∈ R1) containing N ? 1 components which are elements of “lacunary sequences.”  相似文献   

20.
For any x ∈ [0, 1), let x = [? 1, ? 2, …,] be its dyadic expansion. Call r n (x):= max{j ? 1: ? i+1 = … = ? i+j = 1, 0 ? i ? n ? j} the n-th maximal run-length function of x. P.Erdös and A.Rényi showed that \(\mathop {\lim }\limits_{n \to \infty } \) r n (x)/log2 n = 1 almost surely. This paper is concentrated on the points violating the above law. The size of sets of points, whose runlength function assumes on other possible asymptotic behaviors than log2 n, is quantified by their Hausdorff dimension.  相似文献   

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