Study on the generator f of the operators I n and I t

Define two operators In and It,the inner product operator In(g)(x) := j∈Zs(g,f(·-j))f(x-j) and the interpolation operator It(g)(x) := j∈Zs g(j)f(x-j),where f belongs to some space and integer s 1.We call f the generator of the operators In and It.It is well known that there are many results on operators In and It.But there remain some important problems to be further explored.For application we first need to find the available generators (that can recover polynomials as It(p) = p or In(p) = p,p ∈Πm-1) for constructing the relative operators.In this paper,we focus on the available generator in the class of spline functions.We shall see that not all spline functions can be used to construct available generators.Fortunately,we do find a spline function in S,of degree m-1,where m is even and S is a class of splines.But for odd m the problem is still open.Results on spline functions in this paper are new.     

A Fisher type inequality

In this paper we study subsets of a finite set that intersect each other in at most one element. Each subset intersects most of the other subsets in exactly one element. The following theorem is one of our main conclusions. Let S₁,… S_m be m subsets of an n-set S with |S₁| ? 2 (l = 1, …,m) and |S_i ∩ S_j| ? 1 (i ≠ j; i, j = 1, …, m). Suppose further that for some fixed positive integer c each S_i has non-empty intersection with at least m ? c of the remaining subsets. Then there is a least positive integer M(c) depending only on c such that either m ? n or m ? M(c).     

Monotonicity of quadrature formulae of Gauss type and comparison theorems for monosplines

Summary This paper deals with quadrature formulae of Gauss type corresponding to subspaces of spline functionsS _{m–1, k} of degreem–1 withk fixed knots. We shall show monotonicity of the quadrature formulae for functions which are contained in the so-called convexity cone ofS _m–1,k Moreover, we apply these results to monosplines and establish comparison theorems for these splines.     

Modular retractions of numerical semigroups

Let S be a numerical semigroup, let m be a nonzero element of S, and let a be a nonnegative integer. We denote ${\rm R}(S,a,m) = \{ s-as \bmod m \mid s \in S \}$ (where asmodm is the remainder of the division of as by m). In this paper we characterize the pairs (a,m) such that ${\rm R}(S,a,m)$ is a numerical semigroup. In this way, if we have a pair (a,m) with such characteristics, then we can reduce the problem of computing the genus of S=〈n ₁,…,n _p 〉 to computing the genus of a “smaller” numerical semigroup 〈n ₁?an ₁modm,…,n _p ?an _p modm〉. This reduction is also useful for estimating other important invariants of S such as the Frobenius number and the type.     

Permutation functions and (n,m)-commutativity of semigroups

By [4], a semigroupS is called an (n, m)-commutative semigroup (n, m ∈ ?⁺, the set of all positive integers) if $$x_1 x_2 \cdot \cdot \cdot x_n y_1 y_2 \cdot \cdot \cdot y_m = y_1 y_2 \cdot \cdot \cdot y_m x_1 x_2 \cdot \cdot \cdot x_n $$ holds for allx ₁,...,x _n ,y ₁,...,y _m ∈S It is evident that ifS is an (n, m)-commutative semigroup then it is (n′,m′)-commutative for alln′≥n andm′≥m. In this paper, for an arbitrary semigroupS, we determine all pairs (n, m) of positive integersn andm for which the semigroupS is (n, m)-commutative. In our investigation a special type of function mapping ?⁺ into itself plays an important role. These functions which are defined and discussed here will be called permutation functions.     

Weighted Reed–Muller codes revisited

We consider weighted Reed–Muller codes over point ensemble S ₁ × · · · × S _m where S _i needs not be of the same size as S _j . For m = 2 we determine optimal weights and analyze in detail what is the impact of the ratio |S ₁|/|S ₂| on the minimum distance. In conclusion the weighted Reed–Muller code construction is much better than its reputation. For a class of affine variety codes that contains the weighted Reed–Muller codes we then present two list decoding algorithms. With a small modification one of these algorithms is able to correct up to 31 errors of the [49,11,28] Joyner code.     

Smoothability and order bound for AS semigroups

In this paper we consider numerical semigroups S generated by arithmetic sequences m ₀,??,m _n (AS-semigroups). First we state some results on the module $T^{1}_{k[S]}$ ; further in the cases m ₀??1 and m ₀??n (modulo n), we prove these semigroups are Weierstrass by showing that the associated monomial curves $X=\operatorname {Spec}{k[S]}$ are smoothable. Finally for each semigroup S generated by an arithmetic sequence we evaluate the so-called ??order bounds??: when S is Weierstrass, these invariants are good approximations for the minimum distance of the related one-point codes.     

Functions with derivatives given by polynomials in the function itself or a related function

We construct the set of holomorphic functions S ₁ = {f: Ω_f ? ? → ?} whose members have n-th order derivatives which are given by a polynomial of degree n+1 in the function itself. We also construct the set of holomorphic functions S ₂ = {g: Ω_g ? ? → ?} whose members have n-th order derivatives which are given by the product of the function itself with a polynomial of degree n in an element of S ₁. The union S ₁ ∪ S ₂ contains all the hyperbolic and trigonometric functions. We study the properties of the polynomials involved and derive explicit expressions for them. As particular results, we obtain explicit and elegant formulas for the n-th order derivatives of the hyperbolic functions tanh, sech, coth and csch and the trigonometric functions tan, sec, cot and csc.     

Asymptotic expansions for distributions of latent roots in multivariate analysis

Asymptotic expansions are given for the distributions of latent roots of matrices in three multivariate situations. The distribution of the roots of the matrix S₁(S₁ + S₂)^?1, where S₁ is $\text{W}{}_{\mathrm{<mtext>m</mtext>}}\text{(n}{}_1\text{, Σ, Ω)}$ and S₂ is W_m(n₂, Σ), is studied in detail and asymptotic series for the distribution are obtained which are valid for some or all of the roots of the noncentrality matrix Ω large. These expansions are obtained using partial-differential equations satisfied by the distribution. Asymptotic series are also obtained for the distributions of the roots of n^?1S, where S in W_m(n, Σ), for large n, and S₁S₂^?1, where S₁ is W_m(n₁, Σ) and S₂ is W_m(n₂, Σ), for large n₁ + n₂.     

A deficient spline function approximation to systems of first order differential equations

The stability of the vector-valued spline function approximations S(x) of degree m deficiency 3, i.e., S∈C^m?3, to systems of first order differential equations are investigated. The method will be shown to be A-stable for m=4, unstable and hence divergent for m?6. The method is stable form=5.     

Nonzero decomposable symmetrized tensors

Suppose k₁ ? ? ? k_t ? 1, m₁ ? ?? m_r ? 1, k₁+ ? +k_t = m₁+ ? +m_r = m. Let λ=(k₁,…,k_t) be a character of the symmetric group S_m. The restriction of λ to S_m1X…XS_mr contains the principal character as a component if and only if λ majorizes (m₁,…,m_r). This result is used to characterize the index set of the nonzero decomposable symmetrized tensors, corresponding to S_m and λ, which are induced from a basis of the underlying vector space.     

Error estimates for Hermite interpolation on spheres

In this paper, we prove convergence rates for spherical spline Hermite interpolation on the sphere S^d−1 via an error estimate given in a technical report by Luo and Levesley. The functionals in the Hermite interpolation are either point evaluations of pseudodifferential operators or rotational differential operators, the desirable feature of these operators being that they map polynomials to polynomials. Convergence rates for certain derivatives are given in terms of maximum point separation.     

Symmetric multi-box splines for higher degree

We consider the space S _n =S _n (v ₀,…,v _n+r ) of compactly supported C ^n?1 piecewise polynomials on a mesh M of lines through ?² in directions v ₀,…,v _n+r . A sequence ψ=(ψ ₁,…,ψ _r ) of elements of S _n is called a multi-box spline if every element of S _n is a finite linear combination of shifts of (the components of) ψ. For the case n=2, 3 we give some examples for multi-box splines and show that they are not always stable. It is further shown that any C ^n?1 piecewise polynomial of degree n≥2 on M, is possibly a symmetric multi-box spline.     

Pairs of alternating forms and products of two skew-symmetric matrices

Let k be a field, and let S,T,S₁,T₁ be skew-symmetric matrices over k with S,S₁ both nonsingular (if k has characteristic 2, a skew-symmetric matrix is a symmetric one with zero diagonal). It is shown that there exists a nonsingular matrix P over k with P'SP = S₁, P'TP = T₁ (where P' denotes the transpose of P) if and only if S^-1T and S^-1₁T₁ are similar. It is also shown that a 2m×2m matrix over k can be factored as ST, with S,T skew-symmetric and S nonsingular, if and only if A is similar to a matrix direct sum B⊕B where B is an m×m matrix over k. This is equivalent to saying that all elementary divisors of A occur with even multiplicity. An extension of this result giving necessary and sufficient conditions for a square matrix to be so expressible, without assuming that either S or T is nonsingular, is included.     

Prime interchange graphs of classes of matrices of zeros and ones

Let R = (r₁,…, r_m) and S = (s₁,…, s_n) be nonnegative integral vectors, and let $\text{U}$(R, S) denote the class of all m × n matrices of 0's and 1's having row sum vector R and column sum vector S. An invariant position of $\text{U}$(R, S) is a position whose entry is the same for all matrices in $\text{U}$(R, S). The interchange graph G(R, S) is the graph where the vertices are the matrices in $\text{U}$(R, S) and where two matrices are joined by an edge provided they differ by an interchange. We prove that when 1 ≤ r_i ≤ n ? 1 (i = 1,…, m) and 1 ≤ s_j ≤ m ? 1 (j = 1,…, n), G(R, S) is prime if and only if $\text{U}$(R, S) has no invariant positions.     

A probabilistically attained set of polynomials that generate stirling numbers of the second kind

Using probabilistic arguments, we derive a sequence of polynomials in one variable which generate the Stirling numbers of the second kind. Specifically, S^m_c=(c!/m!)P_c-m(c), where S^m_c is the desired Stirling number and P_c-m(·) is the polynomial of degree c-m.     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ^?1. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ^?1 and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

A minimization problem concerning subsets of a finite set

Let F be the set of subsets of a finite set S, and for H ? F, let H′ denote the elements of F which are contained in some element of H. Given integers m_l and m_l+1 does there exist a subset H of F consisting of exactly m_ll-element subsets of S and m_l+1 (l+1)-element subsets of S such that no two elements of H are related by set-wise inclusion, and if such sets H do exist what the smallest |(l?1)(H′)| can be, where |(l?1)(H′)| is the number of (l?1)-element subsets of S in H′? A generalization of this problem, which was posed by G. Katona, is solved in this paper with the help of the generalized Macaulay theorem [2].     

Inverted orders for monotone scoring rules

When each of n judges ranks a set A of m objects from best to worst, and s=(s₁,…,s_m) is a decreasing sequence of real numbers, the collective ranking determined by s orders the objects in A according to their total scores. The total score of x equals s_p times the number of judges who rank x in pth place, summed over p.For normalization purposes, let S_m denote the set of all decreasing s=(s₁,…,s_m) for which s_{m ? 1}=1 and s_m=0. Given any m ? 3, we show firstthat if s and s′ in S_m are not identical, then some profile of judges' rankings yields a linear collective order for s′ that is the reverse or dual of the linear collective order for s.We then consider reversals in collective rankings when one object is removed from A. Suppose s is in S_m and t is in S_{m ? 1}, with m ≥ 3. A simple constructive proof shows that there is a profile of judges' rankings on A which yields a collective linear order for s such that, when any pre-specified object in A is removed, t yields the reverse ranking on the remaining m ? 1 objects. More detailed results are derived for m=3, and shown to depend on the nature of s=(s₁, 1, 0). In particular, the sum-of-ranks procedure with s₁=2 permits fewer reversals than any other s₁>1.