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1.
《Discrete Mathematics》1985,54(3):301-311
For each sequence q = {qi} = ± 1, i = 1, …, n−1 let Nq = the number of permutations σ of 1, 2, …, n with up-down sequence sgn(σi+1 − σi) = qi, i = 1,…, n−1. Clearly Σq (Nq/n!) =1 but what is the probability pn = Σq (Nq/n!)2 that two random permutations have the same up-down sequence? We show that pn = (Kn−11,1) where 1 = 1(x, y) ≡ 1 and Kn−1 is the iterated integral operator with Kφ(x, y) = ∫01∫01 K(x, y; x′, y′)φ(x′, y′) dx′ dy′ on L2[0, 1] × [0, 1] where K(x, y; x′, y′) is 1 if (x− x′)(y − y′) > 0 otherwise, and (f, g) = ∫01∫01fg. The eigenexpression of K yeilds pn ∼ cαn as n → ∞, where c ≈ 1.6, α ≈ 0.55.We also give a recursion formula for a polynomial whose coefficients are the frequencies of all the possible forms. 相似文献
2.
Hock Ong 《Linear algebra and its applications》1976,15(2):119-151
Let F be a field, F1 be its multiplicative group, and = {H:H is a subgroup of F1 and there do not exist a, b?F1 such that Ha+b?H}. Let Dn be the dihedral group of degree n, H be a nontrivial group in , and τn(H) = {α = (α1, α2,…, αn):αi?H}. For σ?Dn and α?τn(H), let P(σ, α) be the matrix whose (i,j) entry is αiδiσ(j) (i.e., a generalized permutation matrix), and . Let Mn(F) be the vector space of all n×n matrices over F and P(Dn, H) = {T:T is a linear transformation on Mn (F) to itself and T(P(Dn, H)) = P(Dn, H)}. In this paper we classify all T in P(Dn, H) and determine the structure of the group P(Dn, H) (Theorems 1 to 4). An expository version of the main results is given in Sec. 1, and an example is given at the end of the paper. 相似文献
3.
Taras Banakh Yaroslav Kholyavka Oles Potyatynyk Michał Machura Katarzyna Kuhlmann 《Central European Journal of Mathematics》2014,12(8):1239-1248
We prove that for every n ∈ ? the space M(K(x 1, …, x n ) of ?-places of the field K(x 1, …, x n ) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x 1, …, x n )) ≤ n. For n = 2 the space M(K(x 1, x 2)) has covering and integral dimensions dimM(K(x 1, x 2)) = dim? M(K(x 1, x 2)) = 2 and the cohomological dimension dim G M(K(x 1, x 2)) = 1 for any Abelian 2-divisible coefficient group G. 相似文献
4.
The following Theorem is proved:Let K be a finitely generated field over its prime field. Then, for almost all e-tuples (σ)=(σ 1, …,σ e)of elements of the abstract Galois group G(K)of K we have:
- If e=1,then E tor(K(σ))is infinite. Morover, there exist infinitely many primes l such that E(K(σ))contains points of order l.
- If e≧2,then E tor(K(σ))is finite.
- If e≧1,then for every prime l, the group E(K(σ))contains only finitely many points of an l-power order.
5.
The set Vkn of all n-tuples (x1, x2,…, xn) with xi?, k is considered. The problem treated in this paper is determining σ(n, k), the minimum size of a set W ? Vkn such that for each x in Vkn, there is an element in W that differs from x in at most one coordinate. By using a new constructive method, it is shown that σ(n, p) ? (p ? t + 1)pn?r, where p is a prime and for some integers t and r. The same method also gives σ(7, 3) ? 216. Another construction gives the inequality σ(n, kt) ? σ(n, k)tn?1 which implies that σ(q + 1, qt) = qq?1tq when q is a prime power. By proving another inequality σ(np + 1, p) ? σ(n, p)pn(p?1), σ(10, 3) ? 5 · 36 and σ(16, 5) ? 13 · 512 are obtained. 相似文献
6.
Stephen M Tanny 《Journal of Combinatorial Theory, Series A》1976,21(2):196-202
Let π = (π(1), π(2),…, π(n)) be a permutation on {1, 2, …, n}. A succession (respectively, 1-succession) in π is any pair π(i), π(i + 1), where π(i + 1) = π(i) + 1 (respectively, π(i + 1) ≡ π(i) + 1 (mod n)), i = 1, 2, …, n ? 1. Let R(n, k) (respectively, ) be the number of permutations with k successions (respectively, 1-successions). In this note we determine R(n, k) and . In addition, these notions are generalized to the case of circular permutations, where analogous results are developed. 相似文献
7.
We present here a proof that a certain rational function Cn(q,t) which has come to be known as the “q,t-Catalan” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. The precise form of the conjecture is given in Garsia and Haiman (J. Algebraic Combin. 5(3) (1996) 191), where it is further conjectured that Cn(q,t) is the Hilbert series of the diagonal harmonic alternants in the variables (x1,x2,…,xn;y1,y2,…,yn). Since Cn(q,t) evaluates to the Catalan number at t=q=1, it has also been an open problem to find a pair of statistics a(π),b(π) on Dyck paths π in the n×n square yielding Cn(q,t)=∑πta(π)qb(π). Our proof is based on a recursion for Cn(q,t) suggested by a pair of statistics a(π),b(π) recently proposed by Haglund. Thus, one of the byproducts of our developments is a proof of the validity of Haglund's conjecture. It should also be noted that our arguments rely and expand on the plethystic machinery developed in Bergeron et al. (Methods and Applications of Analysis, Vol. VII(3), 1999, p. 363). 相似文献
8.
Chu Yung-ching 《Discrete Mathematics》1980,29(3):251-255
Given n weights, w1, w2,…, wn, such that 0?w1?w2???w1, we examine a property of permutation π1, where π1=(w1, wn, w2, wn?1,…), concerning alphabetical binary trees.For each permutation π of these n weights, there is an optimal alphabetical binary tree corresponding to π, we denote it's cost by V(π). There is also an optimal almost uniform alphabetical binary tree, corresponding to π, we denote it's cost by Vu(π).This paper asserts that Vu(π1)?Vu(π)?V(π) for all π. This is a preliminary result concerning the conjecture of T.C. Hu. Hu's conjecture is V(π1)?V(π) for all π. 相似文献
9.
We determine the structure of the ideal of identities of degreen +1 in theT-ideals To(sn(x1,…,xn)) and To(un(x1,…,xn)), wheres n is the standard identity and un the unitary identity of degreen. 相似文献
10.
Wolfgang Haas 《Journal of Combinatorial Theory, Series A》2009,116(2):478-484
Let Kq(n,R) denote the minimal cardinality of a q-ary code of length n and covering radius R. Let σq(n,s;r) denote the minimal cardinality of a q-ary code of length n, which is s-surjective with radius r. In order to lower-bound Kq(n,n−2) and σq(n,s;s−2) we introduce partition matrices and their transversals. Our approach leads to a short new proof of a classical bound of Rodemich on Kq(n,n−2) and to the new bound Kq(n,n−2)?3q−2n+2, improving the first iff 5?n<q?2n−4. We determine Kq(q,q−2)=q−2+σ2(q,2;0) if q?10. Moreover, we obtain the new powerful recursive bound Kq+1(n+1,R+1)?min{2(q+1),Kq(n,R)+1}. 相似文献
11.
Laurie B. Hopkins William T. Trotter Douglas B. West 《Discrete Applied Mathematics》1984,8(2):163-187
The interval number of a graph G, denoted i(G), is the least positive integer t for which G is the intersection graph of a family of sets each of which is the union of at most t closed intervals of the real line . Trotter and Harary showed that the interval number of the complete bipartite graph K(m,n) is . Matthews showed that the interval number of the complete multipartite graph K(n1,n2,…,np) was the same as the interval number of K(n1,n2) when n1 = n2 = ? = np. Trotter and Hopkins showed that i(K(n1,n2,…,np)) ≤ 1 + i(K(n1,n2)) whenever p ≥ 2 and n1≥n2≥ ? ≥np. West showed that for each n ≥ 3, there exists a constant cn so that if p ≥ cn,n1 = n2?n ?1, and n2 = n3 = ? np = n, then i(K(n1,n2,…,np) = 1 + i(K(n1, n2)). In view of these results, it is natural to consider the problem of determining those pairs (n1,n2) with n1 ≥ n2 so that i(K(n2,…,np)) = i(K(n1,n2)) whenever p ≥ 2 and n2 ≥ n3 ≥ ? ≥ np. In this paper, we present constructions utilizing Eulerian circuits in directed graphs to show that the only exceptional pairs are (n2 ? n ? 1, n) for n ≥ 3 and (7,5). 相似文献
12.
The least absolute deviation estimates L(N), from N data points, of the autoregressive constants a = (a1, …, aq)′ for a stationary autoregressive model, are shown to have the property that Nσ(L(N) ? a) converge to zero in probability, for , where the disturbances are i.i.d., attracted to a stable law of index α, 1 ≤ α < 2, and satisfy some other conditions. 相似文献
13.
L. Carlitz 《Discrete Mathematics》1977,17(2):133-138
Let a=(α1, α2, α3, …) be a sequence of positive integers. The sequence (c1, c2, …, c3) is a-alternating if 1 ? c1 < c2 < … < ck ? n and in additionthe first α1 elements have the same parity, the next α2 elements have opposite parity, the next α3 elements have the parity of the first group, and so on. The final group of elements of like parity is permitted to have fewer elements than the required number. Let denote the number of a-alternating sequences of length k. An explicit formula for is obtained. 相似文献
14.
In Part I we obtained results about the embedding of (0, α)-geometries in PG(3, q). Here we determine all (0, α)-geometries with q+1 points on a line, which are embedded in PG(n, q), n>3 and q>2. As a particular case all semi partial geometries with parameters s=q,t,α(>1),μ, which are embeddable in PG(n, q), q≠2, are obtained. We also prove some theorems about the embedding of (0, 2)-geometries in PG(n, 2): we show that without loss of generality we may restrict ourselves to reduced (0, 2)-geometries, we determine all (0, 2)-geometries in PG(4, 2), and we describe an unusual embedding of U2,3(9) in PG(5, 2). 相似文献
15.
Michio Ozeki 《Journal of Number Theory》1977,9(1):112-120
Let F1(x, y),…, F2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of Fi(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field ((?q)1/2). 相似文献
16.
Meirong Zhang 《Journal of Differential Equations》2002,185(1):74-96
In this paper, we study the Fu?ik spectrum of the problem: (*) ?+(λ++q+(t))x++(λ−+q−(t))x−=0 with the 2π-periodic boundary condition, where q±(t) are 2π-periodic. After introducing a rotation number function ρ(λ+, λ−) for (*), we prove using the Hamiltonian structure and the positive homogeneity of (*) that for any positive integer n, the two boundary curves of the domain ρ−1(n/2) in the (λ+, λ−)-plane are Fu?ik curves of (*). The result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional Fu?ik spectrum with the Dirichlet condition. In particular, it remains open if the Fu?ik spectrum of (*) is composed of only these curves. 相似文献
17.
《Journal of Number Theory》1986,23(2):183-194
Several effective results are proved about oscillatory properties of π(x, q, l1) − π(x, q, l2) and related functions assuming the General Riemann Hypothesis and the absence of real zeros of L-functions. 相似文献
18.
Yasuhiro Takeuchi Norihiko Adachi 《Journal of Mathematical Analysis and Applications》1981,79(1):141-162
This paper presents sufficient conditions for the existence of a nonnegative and stable equilibrium point of a dynamical system of Volterra type, (1) , for every q = (q1,…, qn)T?Rn. Results of a nonlinear complementarity problem are applied to obtain the conditions. System (1) has a nonnegative and stable equilibrium point if (i) f(x) = (f1(x),…,fn(x))T is a continuous and differentiable M-function and it satisfies a certain surjectivity property, or (ii), f(x) is continuous and strongly monotone on R+0n. 相似文献
19.
V. V. Kochergin 《Journal of Applied and Industrial Mathematics》2007,1(3):328-342
The computational complexity of integer linear forms is studied. By l 2(A) we denote the minimal number of the additions and subtractions required for computing the system of p linear forms in q variables x 1, x 2, …, x q that are defined by an integer matrix A of size p × q (repeated use of the results of intermediate computation is permitted). We show that l 2(A) ? log D(A), where D(A) is the maximum of the absolute values of the minors of A over all minors from order 1 to order min (p, q) (Theorem 1). Moreover, for each sequence of matrices A(n) of size p(n) × q(n) satisfying the condition p + q = o ((log log D(A))1/2) as n → ∞ the bound l 2(A) ? log D(A) + o(log D(A)) is valid (Theorem 2). Hence, for all fixed (and even weakly increasing) sizes of matrices that determine a system of integer linear forms, the upper bound on the computational complexity of this system is asymptotically equal to the lower bound. 相似文献
20.
Kristopher Lee 《Journal of Mathematical Analysis and Applications》2011,375(1):108-117
Let A and B be uniform algebras on first-countable, compact Hausdorff spaces X and Y, respectively. For f∈A, the peripheral spectrum of f, denoted by σπ(f)={λ∈σ(f):|λ|=‖f‖}, is the set of spectral values of maximum modulus. A map T:A→B is weakly peripherally multiplicative if σπ(T(f)T(g))∩σπ(fg)≠∅ for all f,g∈A. We show that if T is a surjective, weakly peripherally multiplicative map, then T is a weighted composition operator, extending earlier results. Furthermore, if T1,T2:A→B are surjective mappings that satisfy σπ(T1(f)T2(g))∩σπ(fg)≠∅ for all f,g∈A, then T1(f)T2(1)=T1(1)T2(f) for all f∈A, and the map f?T1(f)T2(1) is an isometric algebra isomorphism. 相似文献