On domination and independent domination numbers of a graph

For a graph G, the definitions of domination number, denoted γ(G), and independent domination number, denoted i(G), are given, and the following results are obtained:Theorem.If G does not have an induced subgraph isomorphic to K_1,3, thenγ(G) = i(G).Corollary 1.For any graph G, γ(L(G))=i(L(G)), where L(G) is the line graph of G. (This extends the result γ(L(T))=i(L(T)), where T is a tree. Hedetniemi and Mitchell, S. E. Conf. Baton Rouge, 1977.)Corollary 2.For any Graph G, γ(M(G))=i(M(G)), where M is the middle graph of G.     

Second order BVPs with state dependent impulses via lower and upper functions

The paper deals with the following second order Dirichlet boundary value problem with p ∈ ? state-dependent impulses: z″(t) = f (t,z(t)) for a.e. t ∈ [0, T], z(0) = z(T) = 0, z′(τ _i +) ? z′(τ _i ?) = I _i (τ _i , z(τ _i )), τ _i = γ _i (z(τ _i )), i = 1,..., p. Solvability of this problem is proved under the assumption that there exists a well-ordered couple of lower and upper functions to the corresponding Dirichlet problem without impulses.     

Равносходимость разложений в кратный ряд и интеграл фурье, «прямоугольные частичные суммы» которых рассматриваются по некоторой подпоследовательности

In this paper we investigate the problem of the equiconvergence on T ^N = [-π, π) ^N of the expansions in multiple trigonometric series and Fourier integral of functions f ∈ L _p (T ^N ) and g ∈ L _p (? ^N ), where p > 1, N ≥ 3, g(x) = f(x) on T ^N , in the case when the “rectangular partial sums” of the indicated expansions, i.e.,– _n (x; f) and J _α(x; g), respectively, have indices n ∈ ? ^N and α ∈ ? ^N (n _j = [α _j ], j = 1,...,N, [t] is the integer part of t ∈ ?¹), in those certain components are the elements of “lacunary sequences”.     

Stability of Friedrichs's scheme in the maximum norm for hyperbolic systems in one space dimension

We are concerned with Friedrichs's scheme for an initial value problem u_t(t, x) = A(t, x)u_x(t, x), u(0, x) = u₀(x), where u₀(x) belongs to L_∞, not to L₂. We show that Friedrichs's scheme is stable in the maximum norm ·_L∞, provided that the system is regularly hyperbolic and that the eigenvalues d_i(t, x) (i = 1,2,..., N) of the N XN matrix A(t, x) satisfy the conditions 1±λd_i(t, x)?0 (i = 1,2,..., N), where λ is a mesh ratio.     

Greedy bases in L p spaces

We consider a weighted L ^p space L ^p (w) with a weight function w. It is known that the Haar system H _p normalized in L ^p is a greedy basis of L ^p , 1 < p < ∞. We study a question of when the Haar system H _p ^w normalized in L ^p (w) is a greedy basis of L ^p (w), 1 < p < ∞. We prove that if w is such that H _p ^w is a Schauder basis of L ^p (w), then H _p ^w is also a greedy basis of L ^p (w), 1 < p < ∞. Moreover, we prove that a subsystem of the Haar system obtained by discarding finitely many elements from it is a Schauder basis in a weighted norm space L ^p (w); then it is a greedy basis.     

Equiconvergence of expansions in multiple Fourier series and in fourier integrals with “lacunary sequences of partial sums”

We investigate the equiconvergence on T^N = [?π, π)^N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L_p(T^N) and g ∈ L_p(R^N), p > 1, N ≥ 3, g(x) = f(x) on T^N, in the case where the “partial sums” of these expansions, i.e., S_n(x; f) and J_α(x; g), respectively, have “numbers” n ∈ Z^N and α ∈ R^N (n_j = [α_j], j = 1,..., N, [t] is the integral part of t ∈ R¹) containing N ? 1 components which are elements of “lacunary sequences.”     

Relativistic Lamé functions: Completeness vs. polynomial asymptotics

In earlier work we introduced and studied two commuting generalized Lamé operators, obtaining in particular joint eigenfunctions for a dense set in the natural parameter space. Here we consider these difference operators and their eigenfunctions in relation to the Hilbert space L²((0, π/r), w(x)dx), with r > 0 and the weight function w(x) a ratio of elliptic gamma functions. In particular, we show that the previously known pairwise orthogonal joint eigenfunctions need only be supplemented by finitely many new ones to obtain an orthogonal base. This completeness property is derived by exploiting recent results on the large-degree Hilbert space asymptotics of a class of orthonormal polynomials. The polynomials p_n(cos(rx)), n ε , that are relevant in the Lamé setting are orthonormal in L²((0, π/r), w_P(x)dx), with w_p(x) closely related to w(x).     

Weighted inequalities for fractional type operators with some homogeneous kernels

In this paper, we study integral operators of the form Tαf(x)=∫Rn|x-A1y|-α1 ··· |x-Amy|-αmf(y)dy,where Ai are certain invertible matrices, αi 0, 1 ≤ i ≤ m, α1 + ··· + αm = n-α, 0 ≤α n. For 1/q = 1/p-α/n , we obtain the Lp (Rn, wp)-Lq(Rn, wq) boundedness for weights w in A(p, q) satisfying that there exists c 0 such that w(Aix) ≤ cw(x), a.e. x ∈ Rn , 1 ≤ i ≤ m.Moreover, we obtain theappropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators.     

A subdeterminant inequality for normal matrices

Let A be an n × n normal matrix over $\text{C}$, and Q_m, _n be the set of strictly increasing integer sequences of length m chosen from 1,…,n. For α, β ? Q_m, _n denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k ? {0, 1,…, m} we write |α∩β| = k if there exists a rearrangement of 1,…, m, say i₁,…, i_k, i_k+1,…, i_m, such that α(i_j) = β(i_j), i = 1,…, k, and {α(i_k+1),…, α(i_m) } ∩ {β(i_k+1),…, β(i_m) } = ?. A new bound for |detA[α|β ]| is obtained in terms of the eigenvalues of A when 2m = n and |α∩β| = 0.Let $\text{U}$_n be the group of n × n unitary matrices. Define the nonnegative number

where | α ∩ β| = k. It is proved that

Let A be semidefinite hermitian. We conjecture that ρ₀(A) ? ρ₁(A) ? ··· ? ρ_m(A). These inequalities have been tested by machine calculations.     

On moment-discretization and least-squares solutions of linear integral equations of the first kind

Let K(s, t) be a continuous function on [0, 1] × [0, 1], and let $\text{K}$ be the linear integral operator induced by the kernel K(s, t) on the space $\text{L}$₂[0, 1]. This note is concerned with moment-discretization of the problem of minimizing 6K_x?y6 in the $\text{L}$₂-norm, where y is a given continuous function. This is contrasted with the problem of least-squares solutions of the moment-discretized equation: ∝₀¹K(s_i, t) x(t) dt = y(s_i), i = 1, 2,h., n. A simple commutativity result between the operations of “moment-discretization” and “least-squares” is established. This suggests a procedure for approximating $\text{K2y}$ (where $\text{K}$² is the generalized inverse of $\text{K}$), without recourse to the normal equation $\text{K1Kx = K1y}$, that may be used in conjunction with simple numerical quadrature formulas plus collocation, or related numerical and regularization methods for least-squares solutions of linear integral equations of the first kind.     

Thompson’s conjecture for the alternating group of degree 2<Emphasis Type="Italic">p</Emphasis> and 2<Emphasis Type="Italic">p</Emphasis>+1

For a finite group G denote by N(G) the set of conjugacy class sizes of G. In 1980s, J.G.Thompson posed the following conjecture: If L is a finite nonabelian simple group, G is a finite group with trivial center and N(G) = N(L), then G ? L. We prove this conjecture for an infinite class of simple groups. Let p be an odd prime. We show that every finite group G with the property Z(G) = 1 and N(G) = N(A _i ) is necessarily isomorphic to A _i , where i ∈ {2p, 2p + 1}.     

On the equivalence of the centered Gaussian measure in L 2 with the correlation operator (−d 2/dx 2)−1 and the conditional Wiener measure

Let w and µ be respectively the conditional Wiener measure in C ₀([0, 1]) and the centered Gaussian measure in L ₂[0, 1] with the correlation operator (?d ²/dx ²)^?1. We prove the equivalence of these two measures in the following sense: for any Borel set A ? L ₂[0, 1] the set A ∩ C ₀([0, 1]) is a Borel subset of C ₀([0, 1]) and µ(A) = w(A∩C ₀([0, 1])).     

An embedding theorem for partial algebras and the free completion of a free partial algebra within a primitive class

Let U be any nontrivial primitive class of partial algebras, i.e. there existsA ∈ U with |A|≥2, and U is closed with respect to homomorphic images (in the weak sense), subalgebras (on closed subsets) and cartesian products of U-algebras, and let U _f denote the—also nontrivial and primitive—class of all full U-algebras. Then every U-algebra with at least two elements is a relative algebra of some U _f -algebra. For any U-algebraAsetU _A =U _i εI({i}×(A ^K _i—domf _i ^A )), where (K _i) _i εI is the type under consideration. Furthermore let F(N, U) denote any U-algebra U-freely generated by some setN (and let F (M, U _f ) be similarly defined). Then for every nonempty setM there exists a setN satisfyingM ?N such that there exists a bijective mapping σ:U _{F(N, U)} →N ?M satisfying σ((i, α)) ? α(K _i ) for all (i, α) ∈U _{F (N, U)}, and, for the structureg=(g _i)_iεI defined by ,g _i : =f _i ^{F(N, U)} ∪ {(α, σ((i, α))) | (i, α ∈U _{F(N, U)}} id _M induces an isomorphism betweenF(M, U _f ), and (F(N, U)g).     

The Exponent of Discrepancy Is at Least 1.0669

LetP⊂[0, 1]^dbe ann-point set and letw: P→[0, ∞) be a weight function withw(P)=∑_z∈P w(z)=1. TheL₂-discrepancy of the weighted set (P, w) is defined as theL₂-average ofD(x)=vol(B_x)−w(P∩B_x) overx∈[0, 1]^d, where vol(B_x) is the volume of thed-dimensional intervalB_x=∏^d_k=1 [0, x_k). The exponent of discrepancyp* is defined as the infimum of numberspsuch that for all dimensionsd⩾1 and allε>0 there exists a weighted set of at mostKε^−ppoints in [0, 1]^dwithL₂-discrepancy at mostε, whereK=K(p) is a suitable number independent ofεandd. Wasilkowski and Woźniakowski proved thatp*⩽1.4779, by combining known bounds for the error of numerical integration and using their relation toL₂-discrepancy. In this note we observe that a careful treatment of a classical lower- bound proof of Roth yieldsp*⩾1.04882, and by a slight modification of the proof we getp*⩾1.0669. Determiningp* exactly seems to be quite a difficult problem.     

Sharpness of the Calderón-Lozanovskii interpolation construction

We present a short proof of the sharpness of the Calderón-Lozanovskii interpolation construction in couples of weighted L _p spaces in the “lower triangle,” i.e., for operators from a couple { L _p0 (V ₀), L _p1 (V ₁)} to a couple {L _q0 (U ₀), L _q1 (U ₁)} with p ₀ ? q ₀ and p ₁ ? q ₁. This generalizes the well-known result due to Dmitriev and Semenov on the sharpness of the Riesz-Thorin interpolation theorem in the “lower triangle” for L _p spaces on intervals.     

On the relation between two Minkowski functions

The definition of Minkowski's “Fragefunktion”? (x) is recapitulated. This function is compared to a function L(x) introduced by the author in 1926. It is shown that the inverse function P(w) = x to the function L(x) is related to the Fragefunktion through ?(w) = P(w)?1.     

Jackson and smoothness theorems for freud weights

We prove Jackson, realization, and converse theorems for Freud weights inL _p, 0<p ≤ ∞. Even forp ≥ 1, our conditions on the weight in our Jackson theorems are far less restrictive than those previously imposed. Moreover, the method—of first approximating by a spline, and then by a polynomial—is new in this context, and of intrinsic interest, since it avoids the use of orthogonal polynomials for Freud weights. We establish some properties of the modulus of smoothness, valid inL _p for 0<p ≤ ∞. Since theK-functional is identically zero inL _p,p<1, the analysis of the modulus of continuity involves a different tool, namely, realization, which works inL _p for all 0<p ≤ ∞. We deduce Marchaud-type inequalities.     

Multiplication operators on vector measure Orlicz spaces

Let m be a countably additive vector measure with values in a real Banach space X, and let L¹(m) and L_w(m) be the spaces of functions which are, correspondingly, integrable and weakly integrable with respect to m. Given a Young's function Φ, we consider the vector measure Orlicz spaces L^Φ(m) and L^Φ_w(m) and establish that the Banach space of multiplication operators going from W = L^Φ(m) into Y = L¹ (m) is M = L^Ψ_w (m) with an equivalent norm; here Ψ is the conjugated Young's function for Φ. We also prove that when W = L^Φ_w(m), Y = L¹(m) we have M = L^Ψ_w (m), and when W = L^Φ_w(m), Y = L¹(m) we have M = L^Ψ (m).