Companions of the inequalities of Fejér--Jackson and Young

Summary Applications of some well-known theorems of Jackson and Young lead to the sharp inequalities -1<ⁿ_k-1Σ(cos(kx)+sin(kx))/k (n ≥1; 1<x<π) and -1/2Si(π)<ⁿ_k-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≤ⁿ_k-1Σ(cos(kx)-sin(kx))/k where the sign of equality holds if and only if n =2 and x = π /2.     

On minimal non-orientable matroids with elements and rank

We describe an infinite family M_n,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, M_n,k is isomorphic to the Bland–Las Vergnas matroid M_n. For every 2≤k≤n−3 a new minimal non-orientable matroid is obtained. All proper minors of the matroids M_n,k are representable over .     

Every T-space is equivalent to a T-space of continuous functions

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.     

Strong and weak convergence theorems for asymptotically nonexpansive mappings

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 {k_n}_n1[1,∞), limk_n=1, F(T):={xK: Tx=x}≠. Suppose {x_n}_n1 is generated iteratively by

where {α_n}_n1(0,1) is such that ε<1−α_n<1−ε for some ε>0. It is proved that (I−T) is demiclosed at 0. Moreover, if ∑_n1(k_n²−1)<∞ and T is completely continuous, strong convergence of {x_n} 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 {x_n} to some x^*F(T) is obtained.     

On the L convergence of Lagrange interpolating entire functions of exponential type

Let f: be a continuous, 2π-periodic function and for each n ε let t_n(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 lim_{n → ∞} ∝₀^2π¦f(θ) − t_n(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.     

Subgraphs with Restricted Degrees of their Vertices in Large Polyhedral Maps on Compact Two-manifolds

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.     

Reflection sieve obstructions in finite groups of type dn

In [4] we constructed certain homology representations of a finite group G of type A_n, B_n or C_n, 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 D_n, 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).     

Positive Bernstein-Sheffer Operators

Let h(t) = Σ_{n ≥ 1}h_nt_n, h₁ > 0, and exp(xh(t)) = Σ_{n ≥ 0}P_n(x) tⁿ/n!. For f C[0,1], the associated Bernstein-Sheffer operator of degree n is defined by B^h_nf(x) = P_n^{− 1} Σⁿ_{k = 0}f(k/n)(ⁿ_k) P_k(x) P_{n − k}(1 − x) where p_n = p_n(1). We characterize functions h for which B^h_n is a positive operator for all n ≥ 0. Then we give a necessary and sufficient condition insuring the uniform convergence of B^h_nf to f. When h is a polynomial, we give an upper bound for the error f − B^h_nf _∞. We also discuss the behavior of B^h_nf when h is a series with a finite or infinite radius of convergence.     

Representations for the first associated q-classical orthogonal polynomials

Let {p_k(x; q)} be any system of the q-classical orthogonal polynomials, and let be the corresponding weight function, satisfying the q-difference equation D_q(σ)=τ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let {p_k⁽¹⁾(x;q)} be associated polynomials of the polynomials {p_k(x; q)}. Explicit forms of the coefficients b_n,k and c_n,k in the expansions

are given in terms of basic hypergeometric functions. Here _k(x) equals x^k 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 p_n−1⁽¹⁾(x;q) in a special form, recurrence relations (in k) for b_n,k and c_n,k are obtained in terms of σ and τ.     

A Coding Theory Bound and Zero-Sum Square Matrices

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⁾⁾.     

Trace cocharacters and the Kronecker products of Schur functions

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 M_r(F) of r×r matrices over a field F of characteristic zero are given by TC_r,n=∑_λΛr(n)χ^λχ^λ where χ^λχ^λ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λ_r(n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ=(λ₁λ_k) with kr. We study the behavior of the sequence of trace cocharacters TC_r,n. In particular, we study the behavior of the coefficient of χ^(ν,n−m) in TC_r,n as a function of n where ν=(ν₁ν_k) is some fixed partition of m and n−mν_k. Our main result shows that such coefficients always grow as a polynomial in n of degree r−1.     

Discrete Approximation of Unbounded Operators and Approximation of their Spectra

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 ((A_n, (A_n)))_n of operators A_n where each A_n is densely defined in the Banach space F_n. Let σ_a(A) be the approximate point spectrum of A and let σ(A_n) denote the -pseudospectrum of A_n. Generalizing our own result, we show that σ_a(A)lim inf σ(A_n)=_n ∩_kn σ(A_k) holds for every >0. We deduce that then for every compact set K lim_n dist(σ_a(A)∩K, σ_a(A_n))=0 provided there exists M>0 such that (λ−A_n)⁻¹M dist(λ, σ(A_n))⁻¹ holds for every n and every λ in the resolvent set ρ(A_n) of A_n. We finally treat the problem under which conditions σ_a(A) can be approximated from below. More precisely we investigate the problem: Under which assumptions does ∩_>0 ∩_{n kn} σ_, a(A_k)σ_a(A) hold where σ_, a(A) denotes the -approximate pseudospectrum?     

Strong limit theorems for quasi-orthogonal random fields

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 ζ_mn = 1/(λ₁(m) λ₂(n)) Σ_i=1^m Σ_k=1ⁿ X_ik as either max{m, n} → ∞ or min{m, n} → ∞. Here {X_ik: 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.     

On the asymptotic behaviour of a semigroup of linear operators

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 ω_Lp(T) of weak p-integrability of T (1 ≤ p < ∞), the abscissa ω^p_R(A) of p-boundedness of the resolvent of the generator A of T (1 ≤ p ≤ ∞), and the growth bounds ω_β(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 ω_{1 − ε}(T) < 0 for some ε > 0.

2. (ii) Let T be a C₀-semigroup on L^p such that the resolvent of the generator is uniformly bounded in the right half plane. Then ω_β(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 L^p-stable C₀-semigroup on a Banach space X. Then for all β>1/p we have ω_β(T) ≤ 0.

Further, we give sufficient conditions in terms of ω^q_R(A) for the existence of L^p-solutions and W^1,p-solutions (1 ≤ p ≤ ∞) of the abstract Cauchy problem for a general class of operators A on X.     

Idempotence preserving maps between matrix spaces

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 Φ_𝔽(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?∈?Φ_𝔽(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.     

On improved estimators of the generalized variance

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 W_p(n, Σ) and an independent normal random matrix X: p × k N(ξ, Σ I_k) 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.     

Examples of associative algebras for which the <Emphasis Type="Italic">T</Emphasis>-space of central polynomials is not finitely based

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.     

Path numbers of tournaments

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.     

Extremal Regular Graphs with Prescribed Odd Girth

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 x_e 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.     

Approximations of Two-Place Functions by Finite Sums of Terms with Separable Variables

It is known that if a smooth function h in two real variables x and y belongs to the class Σ_n of all sums of the form Σⁿ_k=1ƒ_k(x) g_k(y), then its (n + 1)th order "Wronskian" det[h_xⁱy^j]ⁿ_i,j=0 is identically equal to zero. The present paper deals with the approximation problem h(x, y) Σⁿ_k=1ƒ_k(x) g_k(y) with a prescribed n, for general smooth functions h not lying in Σ_n. Two natural approximation sums T=T(h) Σ_n, S=S(h) Σ_n 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.