On the Lp–Lq maximal regularity for Stokes equations with Robin boundary condition in a bounded domain

We obtain the L_p–L_q maximal regularity of the Stokes equations with Robin boundary condition in a bounded domain in ?ⁿ (n?2). The Robin condition consists of two conditions: v ? u=0 and αu+β(T(u, p)v – 〈T(u, p)v, v〉v)=h on the boundary of the domain with α, β?0 and α+β=1, where u and p denote a velocity vector and a pressure, T(u, p) the stress tensor for the Stokes flow and v the unit outer normal to the boundary of the domain. It presents the slip condition when β=1 and non‐slip one when α=1, respectively. The slip condition is appropriate for problems that involve free boundaries. Copyright © 2006 John Wiley & Sons, Ltd.     

Global existence for one-dimensional motion of non-isentropic viscous fluids

We study the p-system with viscosity given by v_t ? u_x = 0, u_t + p(v)_x = (k(v)u_x)_x + f(∫ vdx, t), with the initial and the boundary conditions (v(x, 0), u(x,0)) = (v₀, u₀(x)), u(0,t) = u(X,t) = 0. To describe the motion of the fluid more realistically, many equations of state, namely the function p(v) have been proposed. In this paper, we adopt Planck's equation, which is defined only for v > b(> 0) and not a monotonic function of v, and prove the global existence of the smooth solution. The essential point of the proof is to obtain the bound of v of the form b < h(T) ? v(x, t) ? H(T) < ∞ for some constants h(T) and H(T).     

Further Results on the Existence of Splitting BIBDs and Application to Authentication Codes

Let (v,u×c,λ)-splitting BIBD denote a (v,u×c,λ)-splitting balanced incomplete block design of order v with block size u×c and index λ. Necessary conditions for the existence of a (v,u×c,λ)-splitting BIBD are v≥uc, λ(v−1)≡0 (mod c(u−1)) and λ v(v−1)≡0 (mod (c ² u(u−1))). We show in this paper that the necessary conditions for the existence of a (v,3×3,λ)-splitting BIBD are also sufficient with possible exceptions when (1) (v,λ)∈{(55,1),(39,9k):k=1,2,…}, (2) λ≡0 (mod 54) and v≡0 (mod 2). We also show that there exists a (v,3×4,1)-splitting BIBD when v≡1 (mod 96). As its application, we obtain a new infinite class of optimal 4-splitting authentication codes.     

Meyniel's theorem for strongly (p,q) - Hamiltonian digraphs

We give the following theorem: Let D = (V, E) be a strongly (p + q + 1)-connected digraph with n ≥ p + q + 1 vertices, where p and q are nonnegative integers, p ≤ n - 2, n ≥ 2. Suppose that, for each four vertices u, v, w, z (not necessarily distinct) such that {u, v} ∩ {w, z} = Ø, (w, u) ? E, (v, z) ? E, we have id(u) + od(v) + od(w + id(z) ≥ 2 (n + p + q)) + 1. Then D is strongly (p, q)-Hamiltonian.     

Existence of Three Positive Pseudo-symmetric Solutions for a One Dimensional Discrete p-Laplacian

We apply the Five Functionals Fixed Point Theorem to verify the existence of at least three positive pseudo-symmetric solutions for the discrete three point boundary value problem, ?(g(?u(t-1)))+a(t))f(u(t))=0, for t∈{a+1,…,b+1} and u(a)=0 with u(v)=u(b+2) where g(v)=|v| ^p-2 v, p>1, for some fixed v∈{a+1,…,b+1} and σ=(b+2+v)/2 is an integer.     

On Variational Models for Quasi-static Bingham Fluids

We study a quasi-static incompressible flow of Bingham type with constituent law \[ \begin{array}{ll} T = p\left| {\cal E}u\right| ⁁{p-2}{\cal E}u+\beta \frac{{\cal E}u}{\left| {\cal E}u\right| } & \text{if }{\cal E}u\neq 0, \\ \left| T\right| \leq \beta & \text{if }{\cal E}u = 0, \end{array} \] T = p∣ℰu∣^p-2ℰu+β ℰu ∣ℰu∣ if ℰu≠0, ∣T∣⩽β if ℰu = 0, where p≥2 and β>0. Here ℰu denotes the strain velocity and T the corresponding stress. The problem admits a variational formulation in the sense that the velocity field u minimizes the energy I(u) = ∫_Ω∣ℰu∣^p+β∣ℰu∣dx in the space {v∈H^1,p(Ω,ℝⁿ): div v = 0} subject to appropriate boundary conditions. We then show smoothness of u on the set {x∈Ω: ℰu≠0}.     

Quasi‐embeddings of Steiner triple systems,or Steiner triple systems of different orders with maximum intersection

In this paper, we present a conjecture that is a common generalization of the Doyen–Wilson Theorem and Lindner and Rosa's intersection theorem for Steiner triple systems. Given u, v ≡ 1,3 (mod 6), u < v < 2u + 1, we ask for the minimum r such that there exists a Steiner triple system such that some partial system can be completed to an STS , where |?| = r. In other words, in order to “quasi‐embed” an STS(u) into an STS(v), we must remove r blocks from the small system, and this r is the least such with this property. One can also view the quantity (u(u ? 1)/6) ? r as the maximum intersection of an STS(u) and an STS(v) with u < v. We conjecture that the necessary minimum r = (v ? u) (2u + 1 ? v)/6 can be achieved, except when u = 6t + 1 and v = 6t + 3, in which case it is r = 3t for t ≠ 2, or r = 7 when t = 2. Using small examples and recursion, we solve the cases v ? u = 2 and 4, asymptotically solve the cases v ? u = 6, 8, and 10, and further show for given v ? u > 2 that an asymptotic solution exists if solutions exist for a run of consecutive values of u (whose required length is no more than v ? u). Some results are obtained for v close to 2u + 1 as well. The cases where ≈ 3u/2 seem to be the hardest. © 2004 Wiley Periodicals, Inc.     

On local connectivity of graphs with given clique number

For a vertex v of a graph G, we denote by d(v) the degree of v. The local connectivity κ(u, v) of two vertices u and v in a graph G is the maximum number of internally disjoint u –v paths in G, and the connectivity of G is defined as κ(G)=min{κ(u, v)|u, v∈V(G)}. Clearly, κ(u, v)?min{d(u), d(v)} for all pairs u and v of vertices in G. Let δ(G) be the minimum degree of G. We call a graph G maximally connected when κ(G)=δ(G) and maximally local connected when for all pairs u and v of distinct vertices in G. In 2006, Hellwig and Volkmann (J Graph Theory 52 (2006), 7–14) proved that a connected graph G with given clique number ω(G)?p of order n(G) is maximally connected when As an extension of this result, we will show in this work that these conditions even guarantee that G is maximally local connected. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 192–197, 2010     

Standard Bigraded Hilbert Functions

We show Macaulay-type bounds and persistence results for bigraded Hilbert function along rays. Global behavior of the bigraded Hilbert function and very supportive sets for bigraded Hilbert polynomials are also discussed.

We indicate by several results that direct generalizations of the standard graded theory are not true. In particular, we give a counterexample to an immediate generalization of Gotzmann's Regularity Theorem and Persistence Theorem. It is also made clear that maximal growth of a bigraded Hilbert function from bidegree (u, v) to bidegree (u + 1, v) usually is obtained at the expense of the growth from bidegree (u, v) to bidegree (u, v + 1).     

Brooks-type theorems for choosability with separation

We consider the following type of problems. Given a graph G = (V, E) and lists L(v) of allowed colors for its vertices v ∈ V such that |L(v)| = p for all v ∈ V and |L(u) ∩ L(v)| ≤ c for all uv ∈ E, is it possible to find a “list coloring,” i.e., a color f(v) ∈ L(v) for each v ∈ V, so that f(u) ≠ f(v) for all uv ∈ E? We prove that every of maximum degree Δ admits a list coloring for every such list assignment, provided p ≥ . Apart from a multiplicative constant, the result is tight, as lists of length may be necessary. Moreover, for G = K_n (the complete graph on n vertices) and c = 1 (i.e., almost disjoint lists), the smallest value of p is shown to have asymptotics (1 + o(1)) . For planar graphs and c = 1, lists of length 4 suffice. ˜© 1998 John Wiley & Sons, Inc. J Graph Theory 27: 43–49, 1998     

Giant components in Kronecker graphs

Let \begin{align*}n\in\mathbb{N}\end{align*}, 0 <α,β,γ< 1. Define the random Kronecker graph K(n,α,γ,β) to be the graph with vertex set \begin{align*}\mathbb{Z}_2^n\end{align*}, where the probability that u is adjacent to v is given by p_u,v =α ^{u ? v} γ^{( 1‐u )?( 1‐v )}β^{n‐ u ? v ‐( 1‐u )?( 1‐v )}. This model has been shown to obey several useful properties of real‐world networks. We establish the asymptotic size of the giant component in the random Kronecker graph.© 2011 Wiley Periodicals, Inc. Random Struct. Alg.,2011     

Weighted inequalities and a.e. convergence for Poisson integrals in light-cones

We show that the Poisson maximal operator for the tube over the light-cone, P ^*, is bounded in the weighted space L ^p (w) if and only if the weight w(x) belongs to the Muckenhoupt class A _p . We also characterize with a geometric condition related to the intrinsic geometry of the cone the weights v(x) for which P ^* is bounded from L ^p (v) into L ^p (u), for some other weight u(x) > 0. Some applications to a.e. restricted convergence of Poisson integrals are given.     

An Explicit Computation of a Family of Trivializing Étale Covers

We explicitly compute étale covers of the smooth Fermat curves Y _p+1 = Proj k[u, v, w]/(u ^p+1 + v ^p+1 ? w ^p+1) which trivialize the vector bundles Syz(u ², v ², w ²)(3), where k is a field of characteristic p ≥ 3.     

Octahedral designs

An oriented octahedral design of order v, or OCT(v), is a decomposition of all oriented triples on v points into oriented octahedra. Hanani [H. Hanani, Decomposition of hypergraphs into octahedra, Second International Conference on Combinatorial Mathematics (New York, 1978), Annals of the New York Academy of Sciences, 319, New York Academy of Science, New York, 1979, pp. 260–264.] settled the existence of these designs in the unoriented case. We show that an OCT(v) exists if and only if v≡1, 2, 6 (mod 8) (the admissible numbers), and moreover the constructed OCT(v) are unsplit, i.e. their octahedra cannot be paired into mirror images. We show that an OCT(v) with a subdesign OCT(U) exists if and only if v and u are admissible and v≥u+4. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:319–327, 2010     

A probabilistic Poisson representation for positive solutions of Δu = u2 in a planar domain

We use the stochastic process called the Brownian snake to investigate solutions of the partial differential equation Δu = u² in a domain D of class C² of the plane. We prove that nonnegative solutions are in one-to-one correspondence with pairs (K, v) where K is a closed subset of ∂D and v is a Radon measure on ∂D\K. Both Kand v are determined from the boundary behavior of the solution u. On the other hand, u can be expressed in terms of the pair (K, v) by an explicit probabilistic representation formula involving the Brownian snake. © 1997 John Wiley & Sons, Inc.     

Pan-connectedness of graphs with large neighborhood unions

Let G be a simple graph with n vertices. For any v ? V(G){v \in V(G)} , let N(v)={u ? V(G): uv ? E(G)}{N(v)=\{u \in V(G): uv \in E(G)\}} , NC(G) = min{|N(u) èN(v)|: u, v ? V(G){NC(G)= \min \{|N(u) \cup N(v)|: u, v \in V(G)} and uv \not ? E(G)}{uv \not \in E(G)\}} , and NC₂(G) = min{|N(u) èN(v)|: u, v ? V(G){NC_2(G)= \min\{|N(u) \cup N(v)|: u, v \in V(G)} and u and v has distance 2 in E(G)}. Let l ≥ 1 be an integer. A graph G on n ≥ l vertices is [l, n]-pan-connected if for any u, v ? V(G){u, v \in V(G)} , and any integer m with l ≤ m ≤ n, G has a (u, v)-path of length m. In 1998, Wei and Zhu (Graphs Combinatorics 14:263–274, 1998) proved that for a three-connected graph on n ≥ 7 vertices, if NC(G) ≥ n − δ(G) + 1, then G is [6, n]-pan-connected. They conjectured that such graphs should be [5, n]-pan-connected. In this paper, we prove that for a three-connected graph on n ≥ 7 vertices, if NC ₂(G) ≥ n − δ(G) + 1, then G is [5, n]-pan-connected. Consequently, the conjecture of Wei and Zhu is proved as NC ₂(G) ≥ NC(G). Furthermore, we show that the lower bound is best possible and characterize all 2-connected graphs with NC ₂(G) ≥ n − δ(G) + 1 which are not [4, n]-pan-connected.     

On circuits and pancyclic line graphs

Clark proved that L(G) is hamiltonian if G is a connected graph of order n ≥ 6 such that deg u + deg v ≥ n – 1 – p(n) for every edge uv of G, where p(n) = 0 if n is even and p(n) = 1 if n is odd. Here it is shown that the bound n – 1 – p(n) can be decreased to (2n + 1)/3 if every bridge of G is incident with a vertex of degree 1, which is a necessary condition for hamiltonicity of L(G). Moreover, the conclusion that L(G) is hamiltonian can be strengthened to the conclusion that L(G) is pancyclic. Lesniak-Foster and Williamson proved that G contains a spanning closed trail if |V(G)| = n ≥ 6, δ(G) ≥ 2 and deg u + deg v ≥ n – 1 for every pair of nonadjacent vertices u and v. The bound n – 1 can be decreased to (2n + 3)/3 if G is connected and bridgeless, which is necessary for G to have a spanning closed trail.     

On the critical graph conjecture

Gol'dberg has recently constructed an infinite family of 3-critical graphs of even order. We now prove that if there exists a p(≥4)-critical graph K of odd order such that K has a vertex u of valency 2 and another vertex v ≠ u of valency ≤(p + 2)/2, then there exists a p-critical graph of even order.     

Some paranormed sequence spaces of non-absolute type derived by weighted mean

The sequence spaces ?_∞(p), c(p) and c₀(p) were introduced and studied by Maddox [I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64 (1968) 335-340]. In the present paper, the sequence spaces λ(u,v;p) of non-absolute type which are derived by the generalized weighted mean are defined and proved that the spaces λ(u,v;p) and λ(p) are linearly isomorphic, where λ denotes the one of the sequence spaces ?_∞, c or c₀. Besides this, the β- and γ-duals of the spaces λ(u,v;p) are computed and the basis of the spaces c₀(u,v;p) and c(u,v;p) is constructed. Additionally, it is established that the sequence space c₀(u,v) has AD property and given the f-dual of the space c₀(u,v;p). Finally, the matrix mappings from the sequence spaces λ(u,v;p) to the sequence space μ and from the sequence space μ to the sequence spaces λ(u,v;p) are characterized.     

Asymptotic limit of initial boundary value problems for conservation laws with relaxational extensions

We study the boundary layer effect in the small relaxation limit to the equilibrium scalar conservation laws in one space dimension for the relaxation system proposed in [6]. First, it is shown that for initial and boundary data satisfying a strict version of the subcharacteristic condition, there exists a unique global (in time) solution, (u^ε, v^ε), to the relaxation system (1.4) for each ε > 0. The spatial total variation of (u^ε, v^ε) is shown to be bounded independently of ε, and consequently, a subsequence of (u^ε, v^ε) converges to a limit (u, v) as ε → 0⁺. Furthermore, u(x, t) is a weak solution to the scalar conservation law (1.5) and v = f(u). Next, we prove that for data that are suitably small perturbations of a nontransonic state, the relaxation limit function satisfies the boundary-entropy condition (2.11). Finally, the weak solutions to (1.5) with the boundary-entropy condition (2.11) is shown to be unique. Consequently, the relaxation limit of solutions to (1.4) is unique, and the whole sequence converges to the unique limit. One consequence of our analysis shows that the boundary layer occurs only in the u-component in the sense that v^ε(0, ·) converges strongly to γ ○ v = f(γ ○ u), the trace of f(u) on the t-axis. © 1998 John Wiley & Sons, Inc.