L2-approximation of real-valued functions with interpolatory constraints

We construct the polynomial p_m,n^* of degree m which interpolates a given real-valued function f L²[a, b] at pre-assigned n distinct nodes and is the best approximant to f in the L₂-sense over all polynomials of degree m with the same interpolatory character. It is shown that the L₂-error p_m,n^* − f → 0 as m → ∞ if f C[a, b].     

Band Toeplitz preconditioners for block Toeplitz systems

We consider the solutions of block Toeplitz systems with Toeplitz blocks by the preconditioned conjugate gradient (PCG) method. Here the block Toeplitz matrices are generated by nonnegative functions f(x,y). We use band Toeplitz matrices as preconditioners. The generating functions g(x,y) of the preconditioners are trigonometric polynomials of fixed degree and are determined by minimizing (f − g)/f∞. We prove that the condition number of the preconditioned system is O(1). An a priori bound on the number of iterations for convergence is obtained.     

Stability of Periodic Orbits and Return Trajectories of Continuous Multi-valued Maps on Intervals

Let I be a compact interval of real axis R, and(I, H) be the metric space of all nonempty closed subintervals of I with the Hausdorff metric H and f : I → I be a continuous multi-valued map. Assume that Pn =(x_0, x_1,..., xn) is a return tra jectory of f and that p ∈ [min Pn, max Pn] with p ∈ f(p). In this paper, we show that if there exist k(≥ 1) centripetal point pairs of f(relative to p)in {(x_i; x_i+1) : 0 ≤ i ≤ n-1} and n = sk + r(0 ≤ r ≤ k-1), then f has an R-periodic orbit, where R = s + 1 if s is even, and R = s if s is odd and r = 0, and R = s + 2 if s is odd and r 0. Besides,we also study stability of periodic orbits of continuous multi-valued maps from I to I.     

Full quadrature sums for pth powers of polynomials with Freud weights

In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r> 0, b (− ∞, 2]. We establish a full quadrature sum estimate

1 p < ∞, for every polynomial P of degree at most n + rn^1/3, where W² is a Freud weight such as exp(−¦x¦), > 1, λ_jn are the Christoffel numbers, x_jn are the zeros of the orthonormal polynomials for the weight W², and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree M = m(n) if m(n) = n + ξ_nn^1/3, where ξ_n → ∞ as n → ∞. Previous estimates could sum only over those x_jn with ¦x_jn¦ σx_1n, some fixed 0 < σ < 1.     

An extremal problem concerning matrices of 0's and 1 's

We determine the smallest number f(n,k) such that every (0,1)-matrix of order n what zero main diagonal which has at least f(n,k) 1's contains an irreducible, principal submatrix of order K. We characterize those matrices with f(n,k)-1 l's having no irreducible, principal submatrix of order k     

On the jordan form of the tensor product over fields of prime characteristic

Let A, B denote the companion matrices of the polynomials x^m,xⁿ over a field F of prime order p and let λ,μ be non-zero elements of an extension field K of F. The Jordan form of the tensor product (λI + A)⊗(μI + B) of invertible Jordan matrices over K is determined via an equivalent study of the nilpotent tranformation S of m × n matrices X over F where(X)S = A ^TX + XB. Using module-theoretic concepts a Jordan basis for S is specified recursively in terms of the representations of m and n in the scale of p, and reduction formulae for the elementary divisors of S are established.     

Degrees in a digraph whose nodes are graphs

By an f-graph we mean an unlabeled graph having no vertex of degree greater than f. Let D(n, f) denote the digraph whose node set is the set of f-graphs of order n and such that there is an arc from the node corresponding to graph H to the node corresponding to the graph K if and only if K is obtainable from H by the addition of a single edge. In earlier work, algorithms were developed which produce exact results about the structure of D(n, f), nevertheless many open problems remain. For example, the computation of the order and size of D(n, f) for a number of values of n and f have been obtained. Formulas for the order and size for f = 2 have also been derived. However, no closed form formulas have been determined for the order and size of D(n, f) for any value of f. Here we focus on questions concerning the degrees of the nodes in D(n,n − 1) and comment on related questions for D(n,f) for 2 f < n − 1.     

On the exponential map of borei subgroups in a semi-simple lie group

Let G be a connected, complex, semi-simple Lie group Let g be an element in G. Let B be a Borel subgroup of G and g in B. Let m and n be the least positive integers such that the element g^m lies on a one-parameter subgroup in G and the element gⁿ lies on a one-parameter subgroup in B. We denote these integers by ind_G(g) and ind_B(g). In this note we prove the conjecture ind_G(g) = ind_B(g), if g is regular.     

The location of zeros of polynomials

Let A be a complex n×n matrix. p an equilibrated vectonal norm and x(A) the spectrial abscissa of A. Then, it is known [5] x(A)≤x(γ_p(A)) where γp is the matricial logarithmic derivative induced by p. We will make use of the above inequality to obtain regions in the plane which contain the zeros of complex polynomials.     

Induced subgraphs of given sizes

We say (n, e) → (m, f), an (m, f) subgraph is forced, if every n-vertex graph of size e has an m-vertex spanned subgraph with f edges. For example, as Turán proved, (n,e)→(k,(^k₂)) for e> t_{k − 1}(n) and (n,e) (^k₂)), otherwise. We give a number of constructions showing that forced pairs are rare. Using tools of extremal graph theory we also show infinitely many positive cases. Several problems remain open.     

On some properties of the series ∑k=0 kx and the Stirling numbers of the second kind

We partially characterize the rational numbers x and integers n 0 for which the sum ∑_k=0^∞ kⁿx^k assumes integers. We prove that if ∑_k=0^∞ kⁿx^k is an integer for x = 1 − a/b with a, b> 0 integers and gcd(a,b) = 1, then a = 1 or 2. Partial results and conjectures are given which indicate for which b and n it is an integer if a = 2. The proof is based on lower bounds on the multiplicities of factors of the Stirling number of the second kind, S(n,k). More specifically, we obtain for all integers k, 2 k n, and a 3, provided a is odd or divisible by 4, where v_a(m) denotes the exponent of the highest power of a which divides m, for m and a> 1 integers.

New identities are also derived for the Stirling numbers, e.g., we show that ∑_k=0²ⁿk! S(2n, k) , for all integers n 1.     

Subsets of an interval whose product is a power

We form squares from the product of integers in a short interval [n, n + t_n], where we include n in the product. If p is prime, p|n, and (₂^p) > n, we prove that p is the minimum t_n. If no such prime exists, we prove t_n √5n when n> 32. If n = p(2p − 1) and both p and 2p − 1 are primes, then t_n = 3p> 3 √n/2. For n(n + u) a square > n², we conjecture that a and b exist where n < a < b < n + u and nab is a square (except n = 8 and N = 392). Let g₂(n) be minimal such that a square can be formed as the product of distinct integers from [n, g₂(n)] so that no pair of consecutive integers is omitted. We prove that g₂(n) 3n − 3, and list or conjecture the values of g₂(n) for all n. We describe the generalization to kth powers and conjecture the values for large n.     

Matrices with zero trace as commutators of nilpotents

Let C ∈ F^n×n have minimum polynomial m(x). Suppose C is of zero trace and m(x) splits over F. Then, except when n = 2 and m(x) = (x - c)² or when n = 3 and m(x) = x - c)² with c ≠ 0, there exist nilpotents A, B ∈ F^n×n such that C = AB - BA.     

On the number of edge-disjoint one factors and the existence of k-factors in complete multipartite graphs

In this paper we use Tutte's f-factor theorem and the method of amalgamations to find necessary and sufficient conditions for the existence of a k-factor in the complete multipartite graph K(p(1), …, p(n)), conditions that are reminiscent of the Erdös-Gallai conditions for the existence of simple graphs with a given degree sequence. We then use this result to investigate the maximum number of edge-disjoint 1-factors in K(p(1), …, p(n)), settling the problem in the case where this number is greater than δ - p(2), where p(1) p(2) … p(n).     

Prime power divisors of binomial coefficients

It is known that for sufficiently large n and m and any r the binomial coefficient (ⁿ_m) which is close to the middle coefficient is divisible by p^r where p is a ‘large’ prime. We prove the exact divisibility of (ⁿ_m) by p^r for p> c(n). The lower bound is essentially the best possible. We also prove some other results on divisibility of binomial coefficients.     

Some results on integral sum graphs

Let Z denote the set of all integers. The integral sum graph of a finite subset S of Z is the graph (S,E) with vertex set S and edge set E such that for u,vS, uvE if and only if u+vS. A graph G is called an integral sum graph if it is isomorphic to the integral sum graph of some finite subset S of Z. The integral sum number of a given graph G, denoted by ζ(G), is the smallest number of isolated vertices which when added to G result in an integral sum graph. Let x denote the least integer not less than the real x. In this paper, we (i) determine the value of ζ(K_n−E(K_r)) for r2n/3−1, (ii) obtain a lower bound for ζ(K_n−E(K_r)) when 2r<2n/3−1 and n5, showing by construction that the bound is sharp when r=2, and (iii) determine the value of ζ(K_r,r) for r2. These results provide partial solutions to two problems posed by Harary (Discrete Math. 124 (1994) 101–108). Finally, we furnish a counterexample to a result on the sum number of K_r,s given by Hartsfiedl and Smyth (Graphs and Matrices, R. Rees (Ed.), Marcel, Dekker, New York, 1992, pp. 205–211).     

Oscillations in some nonlinear economic relationships

In this paper we consider a simple family of nonlinear dynamical systems generated by smooth functions. Some theorems for the existence and the uniqueness of the limit cycles of the systems are presented. If f and g are generating functions with unique limit cycles and xf(x) < xg(x), for all x ≠ 0, then according to the ‘bounding theorem’ proved in the paper, the limit cycle of the system generated by f is bounded by the limit cycle of the system generated by g. This gives us a method to estimate the amplitude of the oscillations also for systems for which we do not know the generating function exactly. As an application we extend the nonlinear business cycle model proposed by Tönu Puu (1989).     

Profile minimization on triangulated triangles

Given graph G=(V,E) on n vertices, the profile minimization problem is to find a one-to-one function f:V→{1,2,…,n} such that ∑_vV(G){f(v)−min_xN[v] f(x)} is as small as possible, where N[v]={v}{x: x is adjacent to v} is the closed neighborhood of v in G. The trangulated triangle T_l is the graph whose vertices are the triples of non-negative integers summing to l, with an edge connecting two triples if they agree in one coordinate and differ by 1 in the other two coordinates. This paper provides a polynomial time algorithm to solve the profile minimization problem for trangulated triangles T_l with side-length l.     

A special k-coloring for a connected k-chromatic graph

For each positive integer k we consider the smallest positive integer f(k) (dependent only on k) such that the following holds: Each connected graph G with chromatic number χ(G) = k can be properly vertex colored by k colors so that for each pair of vertices x_o and x_p in any color class there exist vertices x₁, x₂, …, x_p-1 of the same class with dist(x_i, x_i+1) f(k) for each i, 0 i p − 1. Thus, the graph is k-colorable with the vertices of each color class placed throughout the graph so that no subset of the class is at a distance > f(k) from the remainder of the class.

We prove that f(k) < 12k when the order of the graph is k(k − 2) + 1.     

Pontryagin–Thom construction for approximation of mappings by embeddings

Let n3 and be positive integers, f :Sⁿ→Sⁿ be a C⁰-mapping, and denote the standard embedding. As an application of the Pontryagin–Thom construction in the special case of the two-point configuration space, we construct complete algebraic obstructions O(f) and to discrete and isotopic realizability (realizability as an embedding) of the mapping Jf. The obstructions are described in terms of stable (equivariant) homotopy groups of neighborhoods of the singular set Σ(f)={(x,y)Sⁿ×Sⁿf(x)=f(y), x≠y}.

A standard method of solving problems in differential topology is to translate them into homotopy theory by means of bordism theory and Pontryagin–Thom construction. By this method we give a generalization of the van-Kampen–Skopenkov obstruction to discrete realizability of f and the van-Kampen–Melikhov obstruction to isotopic realizability of f. The latter are complete only in the case d=0 and are the images of our obstructions under a Hurewicz homomorphism.

We consider several examples of computation of the obstructions.