Atoms of set systems with a fixed number of pairwise unions

For a finite set system with ground set X, we let

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. An atom of H is a nonempty maximal subset C of X such that for all A H, either C A or C ∩ A = 0. We obtain a best possible upper bound for the number of atoms determined by a set system H with H = k and H H = u for all integers k and u. This answers a problem posed by Sós.     

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].     

An extended result of Kleitman and Saks concerning binary trees

In a recent paper, D.J. Kleitman and M.E. Saks gave a proof of Huang's conjecture on alphabetic binary trees.

Given a set E = {e_i}, I = 0, 1, 2, …, m and assigned positive weights to its elements and supposing the elements are indexed such that w(e₀) ≤ w(e₁) ≤ … ≤w (e_m), where w(e_i) is the weight of e_i, we call the following sequence E^* a ‘saw-tooth’ sequence

E^*=(e₀,e_m,e₁,…,e_j,e_m−j,…).

Huang's conjecture is: E^* is the most expensive sequence for alphabetic binary trees. This paper shows that this property is true for the L-restricted alphabetic binary trees, where L is the maximum length of the leaves and log₂(m + 1) ≤L≤m.     

Asymptotic bounds for some bipartite graph: complete graph Ramsey numbers

The Ramsey number r(H,K_n) is the smallest integer N so that each graph on N vertices that fails to contain H as a subgraph has independence number at least n. It is shown that r(K_2,m,K_n)(m−1+o(1))(n/log n)² and r(C_2m,K_n)c(n/log n)^m/(m−1) for m fixed and n→∞. Also r(K_2,n,K_n)=Θ(n³/log² n) and .     

Limiting values of the variance and the moments of the dimension of a sum or intersection of random vector subspaces

Let W be an n-dimensional vector space over a field F; for each positive integer m, let the m-tuples (U₁, …, U_m) of vector subspaces of W be uniformly distributed; and consider the statistics X_m,1 dim_F(∑_i=1^m U_i) and X_m,2 dim_F (∩_i=1^m U_i). If F is finite of cardinality q, we determine lim E(X_m,1^k), and lim E(X_m,2^k), and hence, lim var(X_m,1) and lim var(X_m,2), for any k > 0, where the limits are taken as q → ∞ (for fixed n). Further, we determine whether these, and other related, limits are attained monotonically. Analogous issues are also addressed for the case of infinite F.     

A characterization on n-critical economical generalized tic-tac-toe games

There is a so called generalized tic-tac-toe game playing on a finite set X with winning sets A₁, A₂,…, A_m. Two players, F and S, take in turn a previous untaken vertex of X, with F going first. The one who takes all the vertices of some winning set first wins the game. Erd s and Selfridge proved that if |A₁|=|A₂|==|A_m|=n and m<2ⁿ⁻¹, then the game is a draw. This result is best possible in the sense that once m=2ⁿ⁻¹, then there is a family A₁, A₂,…, A_m so that F can win. In this paper we characterize all those sets A₁,…, A_2n−1 so that F can win in exactly n moves. We also get similar result in the biased games.     

The Existence of LHMTS(<Emphasis Type="Italic">m</Emphasis><Superscript><Emphasis Type="Italic">v</Emphasis></Superscript>) and LHDTS(<Emphasis Type="Italic">m</Emphasis><Superscript><Emphasis Type="Italic">v</Emphasis></Superscript>)

In this article, we establish the existence of an LHMTS(m^v) for v ≡ 2 (mod 6) and m≡ 3 (mod 6). Thus there exists an LHMTS(m^v) if and only if v(v-1)m² ≡ 0 (mod 3) except possibly for v=6, m≡ 1, 5 (mod 6) and m≠1. In the similar way, the existence of LHDTS(m^v) is completely determined, i.e., there exists an LHDTS(m^v) if and only if v(v-1)m² ≡ 0 (mod 3).     

The W-convexity and normal structure in banach spaces

Let X be a Banach space, S(X) - x ε X : #x02016; = 1 be the unit sphere of X.The parameter, modulus of W^*-convexity, W^*(ε) = inf <(x − y)/2, f_x> : x, y S(X), x − y ≥ ε, f_x Δ_x , where 0 ≤ ε ≤ 2 and Δ_x S(X^*) be the set of norm 1 supporting functionals of S(X) at x, is investigated_ The relationship among uniform nonsquareness, uniform normal structure and the parameter W^*(ε) are studied, and a known result is improved. The main result is that for a Banach space X, if there is ε, where 0 < ε < 1/2, such that W^*(1 + ε) > ε/2 where W^*(1 + ε) = lim_→ε W^* (1 + ), then X has normal structure.     

Strong continuity of the relative rearrangement maps and application to a Galerkin approach for nonlocal problems

We study the strong continuity of the map u   (b_*u, b_*u(| > u(·)|)). Here,

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for σ]0 means Ω[, u_* (respectively, (b|_{u=u*(σ)})*) denotes the decreasing rearrangement of u (respectively b restricted to the set {u = u_*(σ)}) and |E| denotes the Lebesgue measure of a set E included in a domain Ω. The results are useful for solving plasmas physics equations or any nonlocal problems involving the monotone rearrangement, its inverse or its derivatives.     

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.     

New constructions of divisible designs

A construction is given for a (p^2a(p+1),p²,p^2a+1(p+1),p^2a+1,p^2a(p+1)) (p a prime) divisible difference set in the group H×Z²_pa+1 where H is any abelian group of order p+1. This can be used to generate a symmetric semi-regular divisible design; this is a new set of parameters for λ₁≠0, and those are fairly rare. We also give a construction for a (p^a−1+p^a−2+…+p+2,p^a+2, p^a(p^a+p^a−1+…+p+1), p^a(p^a−1+…+p+1), p^a−1(p^a+…+p²+2)) divisible difference set in the group H×Z_p2×Z^a_p. This is another new set of parameters, and it corresponds to a symmetric regular divisible design. For p=2, these parameters have λ₁=λ₂, and this corresponds to the parameters for the ordinary Menon difference sets.     

Some explicit formulas for the polynomial decomposition of the matrix exponential and applications

This paper is devoted to the study of some formulas for polynomial decomposition of the exponential of a square matrix A. More precisely, we suppose that the minimal polynomial M_A(X) of A is known and has degree m. Therefore, e^tA is given in terms of P₀(A),…,P_m−1(A), where the P_j(A) are polynomials in A of degree less than m, and some explicit analytic functions. Examples and applications are given. In particular, the two cases m=5 and m=6 are considered.     

Beurling Type Theorem on the Hilbert Space Generated by a Positive Sequence

Let H²(γ) be the Hilbert space over the bidisk D² generated by a positive sequence γ={γ_nm}_n,m ≥ 0. In this paper, we prove that the Beurling type theorem holds for the shift operator on H²(γ) with γ={γ_nm}_n,m ≥ 0 satisfying certain series of inequalities. As a corollary, we give several applications to a class of classical analytic reproducing kernel Hilbert spaces over the bidisk D².     

Asymptotic and Partial Asymptotic Hankel Operators on $$H^2(\mathbb{D}^n)$$ H 2 ( D n ) )

In this paper, we generalize the concept of asymptotic Hankel operators on H²(D) to the Hardy space H²(Dⁿ) (over polydisk) in terms of asymptotic Hankel and partial asymptotic Hankel operators and investigate some properties in case of its weak and strong convergence. Meanwhile, we introduce i^th-partial Hankel operators on H²(Dⁿ) and obtain a characterization of its compactness for n > 1. Our main results include the containment of Toeplitz algebra in the collection of all strong partial asymptotic Hankel operators on H²(Dⁿ). It is also shown that a Toeplitz operator with symbol φ is asymptotic Hankel if and only if φ is holomorphic function in L^∞(Tⁿ).     

Biased partitions and judicious <Emphasis Type="Italic">k</Emphasis>-partitions of graphs

Let G = (V,E) be a graph with m edges. For reals p ∈ [0, 1] and q = 1- p, let m_p(G) be the minimum of qe(V₁) +pe(V₂) over partitions V = V₁ ∪ V₂, where e(V_i) denotes the number of edges spanned by V_i. We show that if mp(G) = pqm-δ, then there exists a bipartition V₁, V₂ of G such that e(V₁) ≤ p²m - δ + p√m/2 + o(√m) and e(V₂) ≤ q²m - δ + q √m/2 + o(√m) for δ = o(m^2/3). This is sharp for complete graphs up to the error term o(√m). For an integer k ≥ 2, let fk(G) denote the maximum number of edges in a k-partite subgraph of G. We prove that if fk(G) = (1 - 1/k)m + α, then G admits a k-partition such that each vertex class spans at most m/k² - Ω(m/k^7.5) edges for α = Ω(m/k⁶). Both of the above improve the results of Bollobás and Scott.     

A class of solutions to the gossip problem, part III

We complete the study of NOHO-graphs, begun in Parts I and II of this paper. NOHO- graphs correspond to solutions to the gossip problem where No One Hears his Own information. These are graphs with a linear ordering on their edges such that an increasing path exists from each vertex to every other, but from no vertex to itself. We discard the two such graphs with no 2-valent vertices. In Part I, we translated these graphs into quadruples of integer sequences. In Part II, we characterized and enumerated the realizable quadruples and various subclasses of them. In Part III, we eliminate the overcounting of isomorphic graphs and obtain recurrence relations and generating functions to enumerate the non-isomorphic NOHO-graphs. If u_m=(1,1,2,…) satisfies u_m=3u_m-1–u_m-3, then the number of non-isomorphic NOHO- graphs on 2m+2 vertices is (u_m + u_[m/2]+1 + u_[m/2]+1 - u_[m/2]). We also examine some re lated questions.     

On application of the Lanczos method to solution of some partial differential equations

Let A be a square symmetric n × n matrix, φ be a vector from ⁿ, and f be a function defined on the spectral interval of A. The problem of computation of the vector u = f(A)φ arises very often in mathematical physics.

We propose the following method to compute u. First, perform m steps of the Lanczos method with A and φ. Define the spectral Lanczos decomposition method (SLDM) solution as u_m = φ Qf(H)e₁, where Q is the n × m matrix of the m Lanczos vectors and H is the m × m tridiagonal symmetric matrix of the Lanczos method. We obtain estimates for u − u_m that are stable in the presence of computer round-off errors when using the simple Lanczos method.

We concentrate on computation of exp(− tA)φ, when A is nonnegative definite. Error estimates for this special case show superconvergence of the SLDM solution. Sample computational results are given for the two-dimensional equation of heat conduction. These results show that computational costs are reduced by a factor between 3 and 90 compared to the most efficient explicit time-stepping schemes. Finally, we consider application of SLDM to hyperbolic and elliptic equations.     

Wandering subspaces and quasi-wandering subspaces in the Hardy–Sobolev spaces

In this paper, we prove that for-1/2 ≤β≤0.suppose M is an invariant subspaces of the Hardy Sobolev spaces H_β~2(D) for T_z~β, then M() zM is a generating wandering subspace of M, that is,M=[MzM]_T_z~β Moreover, any non-trivial invariant subspace M of H_β~2(D) is also generated by the quasi-wandering subspace P_MT_z~βM~⊥ that is,M=[P_MT_z~βM~⊥]_(T_z~β).     

Nonexistence of a code for

In this paper, we shall prove that there is no [3q⁴-q³-q²-3q-1,5,3q⁴-4q³-2q+1]_q code over the finite field for q11. Thus, we conclude the nonexistence of a [g_q(5,d),5,d]_q code for 3q⁴-4q³-2q+1d3q⁴-4q³-q.