On K s -free subgraphs in K s+k -free graphs and vertex Folkman numbers

Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers 2 ≤ s < t let f _s,t (n) = min{max{|S|: S ⊆ V (H) and H[S] contains no K _s }}, where the minimum is taken over all K _t -free graphs H of order n. This function attracted a considerable amount of attention but despite that, the gap between the lower and upper bounds is still fairly wide. For example, when t=s+1, the best bounds have been of the form Ω(n ^1/2+o(1)) ≤ f _s,s+1(n) ≤ O(n ^1−ɛ(s)), where ɛ(s) tends to zero as s tends to infinity. In this paper we improve the upper bound by showing that f _s,s+1(n) ≤ O(n ^2/3). Moreover, we show that for every ɛ > 0 and sufficiently large integers 1 ≪ k ≪ s, Ω(n ^1/2−ɛ ) ≤ f _s,s+k (n) ≤ O(n ^1/2+ɛ . In addition, we also discuss some connections between the function f _s,t and vertex Folkman numbers.     

A Higman inequality for regular near polygons

The inequality of Higman for generalized quadrangles of order (s,t) with s>1 states that t≤s ². We generalize this by proving that the intersection number c _i of a regular near 2d-gon of order (s,t) with s>1 satisfies the tight bound c _i ≤(s ²ⁱ −1)/(s ²−1), and we give properties in case of equality. It is known that hemisystems in generalized quadrangles meeting the Higman bound induce strongly regular subgraphs. We also generalize this by proving that a similar subset in regular near 2d-gons meeting the bounds would induce a distance-regular graph with classical parameters (d,b,α,β)=(d,−q,−(q+1)/2,−((−q) ^d +1)/2) with q an odd prime power.     

How many s-subspaces must miss a point set in PG(d, q)

The following result is well-known for finite projective spaces. The smallest cardinality of a set of points of PG(n, q) with the property that every s-subspace has a point in the set is (q ^n+1-s - 1)/(q - 1). We solve in finite projective spaces PG(n, q) the following problem. Given integers s and b with 0 ≤ s ≤ n - 1 and 1 ≤ b ≤ (q ^n+1-s - 1)/(q - 1), what is the smallest number of s-subspaces that must miss a set of b points. If d is the smallest integer such that b ≤ (q ^d+1 - 1)/(q - 1), then we shall see that the smallest number is obtained only when the b points generate a subspace of dimension d. We then also determine the smallest number of s-subspaces that must miss a set of b points of PG(n, q) which do not lie together in a subspace of dimension d. The results are obtained by geometrical and combinatorial arguments that rely on a strong algebraic result for projective planes by T. Szőnyi and Z. Weiner.     

A class of optimal linear codes of length one above the Griesmer bound

In this paper, we determine the smallest lengths of linear codes with some minimum distances. We construct a [g _q (k, d) + 1, k, d] _q code for sq ^k-1 − sq ^k-2 − q ^s  − q ² + 1 ≤ d ≤ sq ^k-1 − sq ^k-2 − q ^s with 3 ≤ s ≤ k − 2 and q ≥ s + 1. Then we get n _q (k, d) = g _q (k, d) + 1 for (k − 2)q ^k-1 − (k − 1)q ^k-2 − q ² + 1 ≤ d ≤ (k − 2)q ^k-1 − (k − 1)q ^k-2, k ≥ 6, q ≥ 2k − 3; and sq ^k-1 − sq ^k-2 − q ^s  − q + 1 ≤ d ≤ sq ^k-1 − sq ^k-2 − q ^s , s ≥ 2, k ≥ 2s + 1 and q ≥ 2s − 1. This work was partially supported by the Com²MaC-SRC/ERC program of MOST/KOSEF (grant ^# R11-1999-054) and was partially supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD)(KRF-2005-214-C00175).     

Kolmogorov and linear widths of classes of <Emphasis Type="Italic">s</Emphasis>-monotone integrable functions

Let s ∈ ℕ and let Δ ₊ ^s be the set of functions x: I ↦ ℝ on a finite interval I such that the divided differences [x; t ₀, ..., t _s ] of order s of these functions are nonnegative for all collections of s + 1 different points t ₀, ..., t _s ∈ I. For the classes Δ ₊ ^s B _p : = Δ ₊ ^s ∩ B _p , where B _p is the unit ball in L _p , we determine the orders of Kolmogorov and linear widths in the spaces _Lq for 1 ≤ q > p ≤ ∞. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1633–1652, December, 2005.     

高阶单步A-稳定指数拟合法

In this paper,the necessary and sutlicient conditions for general one-step methoos to be exponentially fitted at q0∈C are given, A class of multtderivative hybrid one-step methods of order at least s 1 is constructed with s 1 parameters,where s is the order of derivative. The necessary and sufficient conditions for these methods to be A-stable and exponentially fitted is proved, Furthermore,a class of A-stable 2 parameters hybrid one-step methods of order at least 8 are constructed,which use 4th order derivative,These methods are exponentially fitted at q0 if and only its fitted function f(q) satisfies f(q0)= 0, Finally,an A-stable exponentlally fitted method of order 8 is obtained.     

On a k-Tree Containing Specified Leaves in a Graph

A k-tree of a graph is a spanning tree with maximum degree at most k. We give sufficient conditions for a graph G to have a k-tree with specified leaves: Let k,s, and n be integers such that k≥2, 0≤s≤k, and n≥s+1. Suppose that (1) G is (s+1)-connected and the degree sum of any k independent vertices of G is at least |G|+(k−1)s−1, or (2) G is n-connected and the independence number of G is at most (n−s)(k−1)+1. Then for any s specified vertices of G, G has a k-tree containing them as leaves. We also discuss the sharpness of the results. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists, 15740077, 2005 This research was partially supported by the Japan Society for the Promotion of Science for Young Scientists.     

Delay-dependent stability of high order Runge–Kutta methods

This paper is concerned with the study of the delay-dependent stability of Runge–Kutta methods for delay differential equations. First, a new sufficient and necessary condition is given for the asymptotic stability of analytical solution. Then, based on this condition, we establish a relationship between τ(0)-stability and the boundary locus of the stability region of numerical methods for ordinary differential equations. Consequently, a class of high order Runge–Kutta methods are proved to be τ(0)-stable. In particular, the τ(0)-stability of the Radau IIA methods is proved.     

Many partition relations below density

We force 2 ^λ to be large, and for many pairs in the interval (λ, 2 ^λ ) a strong version of the polarized partition relations holds. We apply this to problems in general topology. For example, consistently, every 2 ^λ is the successor of a singular and for every Hausdorff regular space X, hd(X) ≤ s(X)⁺³, hL(X) ≤ s(X)⁺³ and better when s(X) is regular, via a halfgraph partition relations. For the case s(X) = ℵ ₀ we get hd(X), hL(X) ≤ N ₂.     

Approximation of Poisson integrals by de la Valleé-Poussin sums in uniform and integral metrics

On the classes of Poisson integrals of functions belonging to the unit balls of the spaces L _s , 1 ≤ s ≤ ∞, we establish asymptotic equalities for upper bounds of approximations by de la Vallée-Poussin sums in the uniform metric. Asymptotic equalities are also obtained for the case of approximation by de la Vallée-Poussin sums in the metrics of the spaces L _s , 1 ≤ s ≤ ∞, on the classes of Poisson integrals of functions belonging to the unit ball of the space L ₁.     

Stability of ZakianI MN recursions for linear delay differential equations

Long sequences of linear delay differential equations (DDEs) frequently occur in the design of control systems with delays using iterative-numerical methods, such as the method of inequalities. ZakianI _MN recursions for DDEs are suitable for solving this class of problems, since they are reliable and provide results to the desired accuracy, economically even if the systems are stiff. This paper investigates the numerical stability property of theI _MN recursions with respect to Barwell's concept ofP-stability. The result shows that the recursions using full gradeI _MN approximants areP-stable if, and only if,N−2≤M≤N−1.     

On generalized energy equality of the Navier-Stokes equations

We show that for every initial dataa εL ²(Ω) there exists a weak solutionu of the Navier-Stokes equations satisfying the generalized energy inequality introduced by Caffarelli-Kohn-Nirenberg forn=3. We also show that if a weak solutionu εL ^s (0,T;L ^q (Ω)) with 2/q + 2/s ≤ 1 and 3/q + 1/s ≤ 1 forn=3, or 2/q + 2/s ≤ 1 andq ≥ 4 forn ≥ 4, thenu satisfies both the generalized and the usual energy equalities. Moreover we show that the generalized energy equality holds only under the local hypothesis thatu εL ^s (ε, T; L ^q (K)) for all compact setsK ⊂ ⊂ Ω and all 0 <ε <T with the same (q, s) as above when 3 ≤n ≤ 10.     

A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations

In Burrage and Burrage [1] it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In this present paper, general order results are proven about the maximum attainable strong order of these stochastic Runge-Kutta methods (SRKs) in terms of the order of the Stratonovich integrals appearing in the Runge-Kutta formulation. In particular, it will be shown that if ans-stage SRK contains Stratonovich integrals up to orderp then the strong order of the SRK cannot exceed min{(p+1)/2, (s−1)/2},p≥2,s≥3 or 1 ifp=1.     

The Linear 2-Arboricity of Planar Graphs

 Let G be a planar graph with maximum degree Δ and girth g. The linear 2-arboricity la ₂(G) of G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose component trees are paths of length at most 2. We prove that (1) la ₂(G)≤⌈(Δ+1)/2⌉+12; (2) la ₂(G)≤⌈(Δ+1)/2⌉+6 if g≥4; (3) la ₂(G)≤⌈(Δ+1)/2⌉+2 if g≥5; (4) la ₂(G)≤⌈(Δ+1)/2⌉+1 if g≥7. Received: June 28, 2001 Final version received: May 17, 2002 Acknowledgments. This work was done while the second and third authors were visiting the Institute of Mathematics, Academia Sinica, Taipei. The financial support provided by the Institute is greatly appreciated.     

Discrepancy of Certain Kronecker Sequences Concerning Transcendental Numbers

Let ω1,..., ωs be a set of real transcendental numbers satisfying a certain Diophantine inequality. The upper bound for the discrepancy of the Kronecker sequence （{nω1},..., {nωs}）（1 ≤ n ≤ N） is given. In particular, some low-discrepancy sequences are constructed.     

Topological minors in bipartite graphs

For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s and t such that 2 ≤ s ≤ t, 0 ≤ m − s ≤ n − t, and m + n ≤ 2s + t − 1, we prove that if G has at least mn − (2(m − s) + n − t) edges then it contains a subdivision of the complete bipartite K _(s,t) with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem.     

Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W_p^k(ℝ^s) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space W_p^k(ℝ^s) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the L_p (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector φ∈L_p(ℝ^s) (φ∈C(ℝ^s) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz space Lip(ν,L_p(ℝ^s)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167) in the univariate setting to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C20, 41A25, 39B12. Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant G121210654.     

An <Emphasis Type="Italic">s</Emphasis>-Hamiltonian Line Graph Problem

For an integer k > 0, a graph G is k-triangular if every edge of G lies in at least k distinct 3-cycles of G. In (J Graph Theory 11:399–407 (1987)), Broersma and Veldman proposed an open problem: for a given positive integer k, determine the value s for which the statement “Let G be a k-triangular graph. Then L(G), the line graph of G, is s-hamiltonian if and only L(G) is (s + 2)-connected” is valid. Broersma and Veldman proved in 1987 that the statement above holds for 0 ≤ s ≤ k and asked, specifically, if the statement holds when s = 2k. In this paper, we prove that the statement above holds for 0 ≤ s ≤ max{2k, 6k − 16}.     

Small fractional parts of additive forms

Letf(x)=θ₁ x ₁ ^k +...+θ _s x _s ^k be an additive form with real coefficients, and ∥α∥ = min {|α-u|:uεℤ} denote the distance fromα to the nearest integer. We show that ifθ ₁,…,θ _s , are algebraic ands = 4k then there are integersx ₁,…,x _s , satisfying l ≤x ₁,≤ N and ∥f(x)∥ ≤ N ^E , withE = − 1 + 2/e. Whens = λk, 1 ≤λ ≤ 2k, the exponentE may be replaced byλE/4, and if we drop the condition thatθ ₁,…,θ _s , be algebraic then the result holds for almost all values of θεℝ ^s . Whenk ≥ 6 is small a better exponent is obtained using Heath-Brown’s version of Weyl’s estimate.