共查询到20条相似文献,搜索用时 31 毫秒
1.
Horst Alzer 《Analysis Mathematica》2005,31(2):75-84
Summary Applications of some well-known theorems of Jackson and Young lead to the sharp inequalities -1<nk-1Σ(cos(kx)+sin(kx))/k (n ≥1; 1<x<π) and -1/2Si(π)<nk-1Σ(cos(kx)·sin(kx))/k (n ≥1; xЄR) We prove that the following counterpart is valid for all integers n ≥1 and real numbers xЄ (0, π): -3/2≤nk-1Σ(cos(kx)-sin(kx))/k where the sign of equality holds if and only if n =2 and x = π /2. 相似文献
2.
We describe an infinite family Mn,k, with n≥4 and 1≤k≤n−2, of minimal non-orientable matroids of rank n on a set with 2n elements. For k=1,n−2, Mn,k is isomorphic to the Bland–Las Vergnas matroid Mn. For every 2≤k≤n−3 a new minimal non-orientable matroid is obtained. All proper minors of the matroids Mn,k are representable over . 相似文献
3.
A T-space U of degree k is a (k + 1)-dimensional vector space over
(the real line) of real-valued functions defined on a linearly ordered set, satisfying the condition: for every nonzero u ε U, Z(u), the number of distinct zeros of u and -(u), the number of alternations in sign of u(t) with increasing t, each do not exceed k. It is demonstrated that given a T-space U of degree k > 0 on an arbitrary linearly ordered set T, there is a subset T′ of the real line and a nonsingular linear map L:U→ C(T′), the set of continuous functions on T′, such that the following hold: L(U) is a T-space of degree k; for u ε U, Z(u) = Z(L(u)), S−(u) = S−(L(u); and for some order-preserving bijection Θ:T → T′, u(t) = O if and only if L(u)(Θ(t) = 0. It is also shown that a T-space on a subset T
can be extended to a T-space on the closure of T in ]inf T, sup T], provided that there are no “interval gaps” in T. Examples show that, in general, a T-space cannot be extended across an “interval gap” in its domain, and cannot be extended to both the infimum and supremum of its domain. Conditions for a T-space to be Markov, and to admit an adjoined function are derived. 相似文献
4.
C. E. Chidume E. U. Ofoedu H. Zegeye 《Journal of Mathematical Analysis and Applications》2003,280(2):364-374
Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T :K→E be an asymptotically nonexpansive nonself-map with sequence {kn}n1[1,∞), limkn=1, F(T):={xK: Tx=x}≠. Suppose {xn}n1 is generated iteratively by where {αn}n1(0,1) is such that ε<1−αn<1−ε for some ε>0. It is proved that (I−T) is demiclosed at 0. Moreover, if ∑n1(kn2−1)<∞ and T is completely continuous, strong convergence of {xn} to some x*F(T) is proved. If T is not assumed to be completely continuous but E also has a Fréchet differentiable norm, then weak convergence of {xn} to some x*F(T) is obtained. 相似文献
5.
Let f:
be a continuous, 2π-periodic function and for each n ε
let tn(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that limn → ∞ ∝02π¦f(θ) − tn(f θ)¦p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz. 相似文献
6.
Let k ≥ 2, be an integer and M be a closed two-manifold with Euler characteristic χ(M) ≤ 0. We prove that each polyhedral map G onM , which has at least (8 k2 + 6 k − 6)|χ (M)| vertices, contains a connected subgraph H of order k such that every vertex of this subgraph has, in G, the degree at most 4 k + 4. Moreover, we show that the bound 4k + 4 is best possible. Fabrici and Jendrol’ proved that for the sphere this bound is 10 ifk = 2 and 4 k + 3 if k ≥ 3. We also show that the same holds for the projective plane. 相似文献
7.
David B. Surowski 《Journal of Algebra》1983,80(2):552-558
In [4] we constructed certain homology representations of a finite group G of type An, Bn or Cn, and showed that these representations can be used to sift out the reflection compound characters of G. In the present note, we show that for a group G of type Dn, each reflection compound character π(k), 2 k n − 2, determines a unique “obstruction” character θ(k), which occurs with positive multiplicity in every homology representation containing π(k). 相似文献
8.
Let h(t) = Σn ≥ 1hntn, h1 > 0, and exp(xh(t)) = Σn ≥ 0Pn(x) tn/n!. For f C[0,1], the associated Bernstein-Sheffer operator of degree n is defined by Bhnf(x) = Pn− 1 Σnk = 0f(k/n)(nk) Pk(x) Pn − k(1 − x) where pn = pn(1). We characterize functions h for which Bhn is a positive operator for all n ≥ 0. Then we give a necessary and sufficient condition insuring the uniform convergence of Bhnf to f. When h is a polynomial, we give an upper bound for the error f − Bhnf ∞. We also discuss the behavior of Bhnf when h is a series with a finite or infinite radius of convergence. 相似文献
9.
Stanis
aw Lewanowicz 《Journal of Computational and Applied Mathematics》2003,150(2):193-327
Let {pk(x; q)} be any system of the q-classical orthogonal polynomials, and let be the corresponding weight function, satisfying the q-difference equation Dq(σ)=τ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let {pk(1)(x;q)} be associated polynomials of the polynomials {pk(x; q)}. Explicit forms of the coefficients bn,k and cn,k in the expansions are given in terms of basic hypergeometric functions. Here k(x) equals xk if σ+(0)=0, or (x;q)k if σ+(1)=0, where σ+(x)σ(x)+(q−1)xτ(x). The most important representatives of those two classes are the families of little q-Jacobi and big q-Jacobi polynomials, respectively.Writing the second-order nonhomogeneous q-difference equation satisfied by pn−1(1)(x;q) in a special form, recurrence relations (in k) for bn,k and cn,k are obtained in terms of σ and τ. 相似文献
10.
For a code C=C(n,M) the level
k code of C, denoted C
k
, is the set of all vectors resulting from a linear combination of precisely k distinct codewords of C. We prove that if k is any positive integer divisible by 8, and n=k, M=k2k then there is a codeword in C
k
whose weight is either 0 or at most
. In particular, if <(4–2)2/48 then there is a codeword in C
k
whose weight is n/2–(n). The method used to prove this result enables us to prove the following: Let k be an integer divisible by p, and let f(k,p) denote the minimum integer guaranteeing that in any square matrix over Z
p
, of order f(k,p), there is a square submatrix of order k such that the sum of all the elements in each row and column is 0. We prove that lim inf f(k,2)/k<3.836. For general p we obtain, using a different approach, that f(k,p)p(
k
/ ln
k
)(1+
o
k
(1)). 相似文献
11.
It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra Mr(F) of r×r matrices over a field F of characteristic zero are given by TCr,n=∑λΛr(n)χλχλ where χλχλ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λr(n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ=(λ1λk) with kr. We study the behavior of the sequence of trace cocharacters TCr,n. In particular, we study the behavior of the coefficient of χ(ν,n−m) in TCr,n as a function of n where ν=(ν1νk) is some fixed partition of m and n−mνk. Our main result shows that such coefficients always grow as a polynomial in n of degree r−1. 相似文献
12.
M. P. H. Wolff 《Journal of Approximation Theory》2001,113(2):229
Let E be a Banach space over
and let the densely defined closed linear operator A:
(A)E→E be discretely approximated by the sequence ((An,
(An)))n
of operators An where each An is densely defined in the Banach space Fn. Let σa(A) be the approximate point spectrum of A and let σ(An) denote the -pseudospectrum of An. Generalizing our own result, we show that σa(A)lim inf σ(An)=n
∩kn σ(Ak) holds for every >0. We deduce that then for every compact set K
limn dist(σa(A)∩K, σa(An))=0 provided there exists M>0 such that (λ−An)−1M dist(λ, σ(An))−1 holds for every n and every λ in the resolvent set ρ(An) of An. We finally treat the problem under which conditions σa(A) can be approximated from below. More precisely we investigate the problem: Under which assumptions does ∩>0 ∩n
kn σ, a(Ak)σa(A) hold where σ, a(A) denotes the -approximate pseudospectrum? 相似文献
13.
F. Mricz 《Journal of multivariate analysis》1989,30(2)
This is a systematic and unified treatment of a variety of seemingly different strong limit problems. The main emphasis is laid on the study of the a.s. behavior of the rectangular means ζmn = 1/(λ1(m) λ2(n)) Σi=1m Σk=1n Xik as either max{m, n} → ∞ or min{m, n} → ∞. Here {Xik: i, k ≥ 1} is an orthogonal or merely quasi-orthogonal random field, whereas {λ1(m): m ≥ 1} and {λ2(n): n ≥ 1} are nondecreasing sequences of positive numbers subject to certain growth conditions. The method applied provides the rate of convergence, as well. The sufficient conditions obtained are shown to be the best possible in general. Results on double subsequences and 1-parameter limit theorems are also included. 相似文献
14.
Let T = {T(t)}t ≥ 0 be a C0-semigroup on a Banach space X. In this paper, we study the relations between the abscissa ωLp(T) of weak p-integrability of T (1 ≤ p < ∞), the abscissa ωpR(A) of p-boundedness of the resolvent of the generator A of T (1 ≤ p ≤ ∞), and the growth bounds ωβ(T), β ≥ 0, of T. Our main results are as follows.
- 1. (i) Let T be a C0-semigroup on a B-convex Banach space such that the resolvent of its generator is uniformly bounded in the right half plane. Then ω1 − ε(T) < 0 for some ε > 0.
- 2. (ii) Let T be a C0-semigroup on Lp such that the resolvent of the generator is uniformly bounded in the right half plane. Then ωβ(T) < 0 for all β>¦1/p − 1/p′¦, 1/p + 1/p′ = 1.
- 3. (iii) Let 1 ≤ p ≤ 2 and let T be a weakly Lp-stable C0-semigroup on a Banach space X. Then for all β>1/p we have ωβ(T) ≤ 0.
15.
Suppose 𝔽 is an arbitrary field of characteristic not 2 and 𝔽?≠?𝔽3. Let M n (𝔽) be the space of all n?×?n full matrices over 𝔽 and P n (𝔽) the subset of M n (𝔽) consisting of all n?×?n idempotent matrices and GL n (𝔽) the subset of M n (𝔽) consisting of all n?×?n invertible matrices. Let Φ𝔽(n,?m) denote the set of all maps from M n (𝔽) to M m (𝔽) satisfying A???λB?∈?P n (𝔽)???φ(A)???λφ(B)?∈?P m (𝔽) for every A,?B?∈?M n (𝔽) and λ?∈?𝔽, where m and n are integers with 3?≤?n?≤?m. It is shown that if φ?∈?Φ𝔽(n,?m), then there exists T?∈?GL m (𝔽) such that φ(A)?=?T?[A???I p ?⊕?A t ???I q ?⊕?0]T??1 for every A?∈?M n (𝔽), where I 0?=?0. This improves the results of some related references. 相似文献
16.
Bimal Kumar Sinha 《Journal of multivariate analysis》1976,6(4):617-625
Treated in this paper is the problem of estimating with squared error loss the generalized variance | Σ | from a Wishart random matrix S: p × p Wp(n, Σ) and an independent normal random matrix X: p × k N(ξ, Σ Ik) with ξ(p × k) unknown. Denote the columns of X by X(1) ,…, X(k) and set ψ(0)(S, X) = {(n − p + 2)!/(n + 2)!} | S |, ψ(i)(X, X) = min[ψ(i−1)(S, X), {(n − p + i + 2)!/(n + i + 2)!} | S + X(1) X′(1) + + X(i) X′(i) |] and Ψ(i)(S, X) = min[ψ(0)(S, X), {(n − p + i + 2)!/(n + i + 2)!}| S + X(1) X′(1) + + X(i) X′(i) |], i = 1,…,k. Our result is that the minimax, best affine equivariant estimator ψ(0)(S, X) is dominated by each of Ψ(i)(S, X), i = 1,…,k and for every i, ψ(i)(S, X) is better than ψ(i−1)(S, X). In particular, ψ(k)(S, X) = min[{(n − p + 2)!/(n + 2)!} | S |, {(n − p + 2)!/(n + 2)!} | S + X(1)X′(1)|,…,| {(n − p + k + 2)!/(n + k + 2)!} | S + X(1)X′(1) + + X(k)X′(k)|] dominates all other ψ's. It is obtained by considering a multivariate extension of Stein's result (Ann. Inst. Statist. Math. 16, 155–160 (1964)) on the estimation of the normal variance. 相似文献
17.
In 1988 (see [7]), S. V. Okhitin proved that for any field k of characteristic zero, the T-space CP(M
2(k)) is finitely based, and he raised the question as to whether CP(A) is finitely based for every (unitary) associative algebra A for which 0 ≠ T(A) ⊊ CP(A). V. V. Shchigolev (see [9], 2001) showed that for any field of characteristic zero, every T-space of k
0〈X〉 is finitely based, and it follows from this that every T-space of k
1〈X〉 is also finitely based. This more than answers Okhitin’s question (in the affirmative) for fields of characteristic zero. 相似文献
18.
Brian Alspach David W. Mason Norman J. Pullman 《Journal of Combinatorial Theory, Series B》1976,20(3):222-228
A family
of simple (that is, cycle-free) paths is a path decomposition of a tournament T if and only if
partitions the acrs of T. The path number of T, denoted pn(T), is the minimum value of |
| over all path decompositions
of T. In this paper it is shown that if n is even, then there is a tournament on n vertices with path number k if and only if n/2 k n2/4, k an integer. It is also shown that if n is odd and T is a tournament on n vertices, then (n + 1)/2 pn(T) (n2 − 1)/4. Moreover, if k is an integer satisfying (i) (n + 1)/2 k n − 1 or (ii) n < k (n2 − 1)/4 and k is even, then a tournament on n vertices having path number k is constructed. It is conjectured that there are no tournaments of odd order n with odd path number k for n k < (n2 − 1)/4. 相似文献
19.
The odd girth of a graph G gives the length of a shortest odd cycle in G. Let ƒ(k, g) denote the smallest n such that there exists a k-regular graph of order n and odd girth g. It is known that ƒ(k, g) ≥ kg/2 and that ƒ(k, g) = kg/2 if k is even. The exact values of ƒ(k, g) are also known if k = 3 or g = 5. Let xe denote the smallest even integer no less than x, δ(g) = (−1)g − 1/2, and s(k) = min {p + q | k = pq, where p and q are both positive integers}. It is proved that if k ≥ 5 and g ≥ 7 are both odd, then [formula] with the exception that ƒ(5, 7) = 20. 相似文献
20.
Simsa J. 《Journal of Approximation Theory》1994,76(3)
It is known that if a smooth function h in two real variables x and y belongs to the class Σn of all sums of the form Σnk=1ƒk(x) gk(y), then its (n + 1)th order "Wronskian" det[hxiyj]ni,j=0 is identically equal to zero. The present paper deals with the approximation problem h(x, y) Σnk=1ƒk(x) gk(y) with a prescribed n, for general smooth functions h not lying in Σn. Two natural approximation sums T=T(h) Σn, S=S(h) Σn are introduced and the errors |h-T|, |h-S| are estimated by means of the above mentioned Wronskian of the function h. The proofs utilize the technique of ordinary linear differential equations. 相似文献