Automorphisms of higher‐dimensional Hadamard matrices

This article derives from first principles a definition of equivalence for higher‐dimensional Hadamard matrices and thereby a definition of the automorphism group for higher‐dimensional Hadamard matrices. Our procedure is quite general and could be applied to other kinds of designs for which there are no established definitions for equivalence or automorphism. Given a two‐dimensional Hadamard matrix H of order ν, there is a Product Construction which gives an order ν proper n‐dimensional Hadamard matrix P⁽ⁿ⁾(H). We apply our ideas to the matrices P⁽ⁿ⁾(H). We prove that there is a constant c > 1 such that any Hadamard matrix H of order ν > 2 gives rise via the Product Construction to cν inequivalent proper three‐dimensional Hadamard matrices of order ν. This corrects an erroneous assertion made in the literature that ”P⁽ⁿ⁾(H) is equivalent to “P⁽ⁿ⁾(H′) whenever H is equivalent to H′.” We also show how the automorphism group of P⁽ⁿ⁾(H) depends on the structure of the automorphism group of H. As an application of the above ideas, we determine the automorphism group of P⁽ⁿ⁾(Hk) when Hk is a Sylvester Hadamard matrix of order 2^k. For ν = 4, we exhibit three distinct families of inequivalent Product Construction matrices P⁽ⁿ⁾(H) where H is equivalent to H₂. These matrices each have large but non‐isomorphic automorphism groups. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 507–544, 2008     

Global well‐posedness and global attractor of fourth order semilinear parabolic equation

We study the initial boundary value problem of a class of fourth order semilinear parabolic equations. Global existence and nonexistence of solutions with initial data in the potential well are derived. Moreover, by using the iteration technique for regularity estimates, we obtain that for any k ≥ 0, the semilinear parabolic possesses a global attractor in H^k(Ω), which attracts any bounded subsets of H^k(Ω) in the H^k‐norm. Copyright © 2014 John Wiley & Sons, Ltd.     

Superconvergence of the direct discontinuous Galerkin method for a time‐fractional initial‐boundary value problem

A time‐fractional reaction–diffusion initial‐boundary value problem with periodic boundary condition is considered on Q ? Ω × [0, T] , where Ω is the interval [0, l] . Typical solutions of such problem have a weak singularity at the initial time t = 0. The numerical method of the paper uses a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh, with piecewise polynomials of degree k ≥ 2 . In the temporal direction we use the L1 approximation of the Caputo derivative on a suitably graded mesh. We prove that at each time level of the mesh, our L1‐DDG solution is superconvergent of order k + 2 in L²(Ω) to a particular projection of the exact solution. Moreover, the L1‐DDG solution achieves superconvergence of order (k + 2) in a discrete L²(Q) norm computed at the Lobatto points, and order (k + 1) superconvergence in a discrete H¹(Q) seminorm at the Gauss points; numerical results show that these estimates are sharp.     

A two-grid method with expanded mixed element for nonlinear reaction-diffusion equations

Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O(△t + h k+1 + H 2k+2 d/2 ) (k ≥ 1), where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.     

The existence of global attractors for 2D Navier-Stokes equations in <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript> spaces

In this paper, we prove that the 2D Navier-Stokes equations possess a global attractor in Hk（Ω,R2） for any k ≥ 1, which attracts any bounded set of Hk（Ω,R2） in the H^k-norm. The result is established by means of an iteration technique and regularity estimates for the linear semigroup of operator, together with a classical existence theorem of global attractor. This extends Ma, Wang and Zhong＇s conclusion.     

Error estimates of expanded mixed methods for optimal control problems governed by hyperbolic integro‐differential equations

In this article, we investigate the L^∞(L²) ‐error estimates of the semidiscrete expanded mixed finite element methods for quadratic optimal control problems governed by hyperbolic integrodifferential equations. The state and the costate are discretized by the order k Raviart‐Thomas mixed finite element spaces, and the control is approximated by piecewise polynomials of order k(k ≥ 0). We derive error estimates for both the state and the control approximation. Numerical experiments are presented to test the theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Analysis of a combined mixed finite element and discontinuous Galerkin method for incompressible two‐phase flow in porous media

We analyze a combined method consisting of the mixed finite element method for pressure equation and the discontinuous Galerkin method for saturation equation for the coupled system of incompressible two‐phase flow in porous media. The existence and uniqueness of numerical solutions are established under proper conditions by using a constructive approach. Optimal error estimates in L²(H¹) for saturation and in L ^∞ (H(div)) for velocity are derived. Copyright © 2013 John Wiley & Sons, Ltd.     

The existence of global attractor for a fourth-order parabolic equation

This article is concerned with a fourth-order parabolic equation. Based on the regularity estimates for the semigroups and the classical existence theorem of global attractors, we prove that the fourth-order parabolic equation possesses a global attractor in H ^k (0?≤?k?H ^k (Ω) in the H ^k -norm.     

Nearly perfect matchings in regular simple hypergraphs

For every fixedk≥3 there exists a constantc _k with the following property. LetH be ak-uniform,D-regular hypergraph onN vertices, in which no two edges contain more than one common vertex. Ifk>3 thenH contains a matching covering all vertices but at mostc _k ND ^−1/(k−1). Ifk=3, thenH contains a matching covering all vertices but at mostc ₃ ND ^−1/2ln^3/2 D. This improves previous estimates and implies, for example, that any Steiner Triple System onN vertices contains a matching covering all vertices but at mostO(N ^1/2ln^3/2 N), improving results by various authors. Research supported in part by a USA-Israel BSF grant. Research supported in part by a USA-Israel BSF Grant.     

The minimum degree of Ramsey‐minimal graphs

We write H → G if every 2‐coloring of the edges of graph H contains a monochromatic copy of graph G. A graph H is G‐minimal if H → G, but for every proper subgraph H′ of H, H′ ? G. We define s(G) to be the minimum s such that there exists a G‐minimal graph with a vertex of degree s. We prove that s(K_k) = (k ? 1)² and s(K_a,b) = 2 min(a,b) ? 1. We also pose several related open problems. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 167–177, 2007     

Two‐coloring random hypergraphs

A 2‐coloring of a hypergraph is a mapping from its vertex set to a set of two colors such that no edge is monochromatic. Let H=H(k, n, p) be a random k‐uniform hypergraph on a vertex set V of cardinality n, where each k‐subset of V is an edge of H with probability p, independently of all other k‐subsets. Let $ m = p{{n}\choose{k}}$ denote the expected number of edges in H. Let us say that a sequence of events ?_n holds with high probability (w.h.p.) if lim_n→∞Pr[?_n]=1. It is easy to show that if m=c2^kn then w.h.p H is not 2‐colorable for c>ln 2/2. We prove that there exists a constant c>0 such that if m=(c2^k/k)n, then w.h.p H is 2‐colorable. © 2002 Wiley Periodicals, Inc. Random Struct. Alg. 20: 249–259, 2002     

Tight products and graph expansion

In this article, we study a new product of graphs called tight product. A graph H is said to be a tight product of two (undirected multi) graphs G₁ and G₂, if V(H) = V(G₁) × V(G₂) and both projection maps V(H)→V(G₁) and V(H)→V(G₂) are covering maps. It is not a priori clear when two given graphs have a tight product (in fact, it is NP‐hard to decide). We investigate the conditions under which this is possible. This perspective yields a new characterization of class‐1 (2k+ 1)‐regular graphs. We also obtain a new model of random d‐regular graphs whose second eigenvalue is almost surely at most O(d^3/4). This construction resembles random graph lifts, but requires fewer random bits. © 2011 Wiley Periodicals, Inc. J Graph Theory     

List multicoloring problems involving the k‐fold Hall numbers

We show that the four‐cycle has a k‐fold list coloring if the lists of colors available at the vertices satisfy the necessary Hall's condition, and if each list has length at least ?5k/3?; furthermore, the same is not true with shorter list lengths. In terms of h^(k)(G), the k ‐fold Hall number of a graph G, this result is stated as h^(k)(C₄)=2k??k/3?. For longer cycles it is known that h^(k)(C_n)=2k, for n odd, and 2k??k/(n?1)?≤h^(k)(C_n)≤2k, for n even. Here we show the lower bound for n even, and conjecture that this is the right value (just as for C₄). We prove that if G is the diamond (a four‐cycle with a diagonal), then h^(k)(G)=2k. Combining these results with those published earlier we obtain a characterization of graphs G with h^(k)(G)=k. As a tool in the proofs we obtain and apply an elementary generalization of the classical Hall–Rado–Halmos–Vaughan theorem on pairwise disjoint subset representatives with prescribed cardinalities. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 16–34, 2010.     

Semiclassical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Estimates of Quasimodes on Curved Hypersurfaces

Let M be a compact manifold of dimension n, P=P(h) a semiclassical pseudodifferential operator on M, and u=u(h) an L ² normalized family of functions such that P(h)u(h) is O(h) in L ²(M) as h↓0. Let H⊂M be a compact submanifold of M. In a previous article, the second-named author proved estimates on the L ^p norms, p≥2, of u restricted to H, under the assumption that the u are semiclassically localized and under some natural structural assumptions about the principal symbol of P. These estimates are of the form Ch ^−δ(n,k,p) where k=dim H (except for a logarithmic divergence in the case k=n−2, p=2). When H is a hypersurface, i.e., k=n−1, we have δ(n,n−1, 2)=1/4, which is sharp when M is the round n-sphere and H is an equator.     

On the general additive divisor problem

We obtain a new upper bound for the sum Σ _h≤H Δ _k (N, h) when 1 ≤ H ≤ N, k ∈ ℕ, k ≥ 3, where Δ _k (N, h) is the (expected) error term in the asymptotic formula for Σ _N<n≤2N d _k (n)d _k (n + h), and d _k (n) is the divisor function generated by ζ(s) ^k . When k = 3, the result improves, for H ≥ N ^1/2, the bound given in a recent work of Baier, Browning, Marasingha and Zhao, who dealt with the case k = 3.     

Finite element methods for semilinear parabolic interface problems

In this exposition we study the finite element methods for second-order semilinear parabolic interface problems in two dimensional convex polygonal domains with smooth interface. Both semidiscrete and fully discrete schemes are analyzed. Optimal order error estimates in the L²(0, T; H¹(Ω))-norm are established. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

and Norms Error Estimates in Finite Element Method for Linear Parabolic Interface Problems

We derive new a priori error estimates for linear parabolic equations with discontinuous coefficients. Due to low global regularity of the solutions the error analysis of the standard finite element method for parabolic problems is difficult to adopt for parabolic interface problems. A finite element procedure is, therefore, proposed and analyzed in this paper. We are able to show that the standard energy technique of finite element method for non-interface parabolic problems can be extended to parabolic interface problems if we allow interface triangles to be curved triangles. Optimal pointwise-in-time error estimates in the L ²(Ω) and H ¹(Ω) norms are shown to hold for the semidiscrete scheme. A fully discrete scheme based on backward Euler method is analyzed and pointwise-in-time error estimates are derived. The interfaces are assumed to be arbitrary shape but smooth for our purpose.     

Total weight choosability of graphs

A graph G = (V, E) is called (k, k′)‐total weight choosable if the following holds: For any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k′ real numbers, there is a mapping f: V∪E→? such that f(y)∈L(y) for any y∈V∪Eand for any two adjacent vertices x, x′, . We conjecture that every graph is (2, 2)‐total weight choosable and every graph without isolated edges is (1, 3)‐total weight choosable. It follows from results in [7] that complete graphs, complete bipartite graphs, trees other than K₂ are (1, 3)‐total weight choosable. Also a graph G obtained from an arbitrary graph H by subdividing each edge with at least three vertices is (1, 3)‐total weight choosable. This article proves that complete graphs, trees, generalized theta graphs are (2, 2)‐total weight choosable. We also prove that for any graph H, a graph G obtained from H by subdividing each edge with at least two vertices is (2, 2)‐total weight choosable as well as (1, 3)‐total weight choosable. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:198‐212, 2011     

Cocycle d'Euler etK2

In this paper we construct a canonical 2-cocycle on the groupP SL(2,k) with values in the Witt groupW(k) of the fieldk. This allows us to produce anatural homomorphism :H ₂(SL(2,k); Z)I ²(k), whereI ²(k) is the square of the fundamental ideal. We prove that this homomorphism is in fact a lift of Milnor's symbol.

     

The links of 3‐dimensional singularities defined by Brieskorn polynomials

In this paper, we completely determine the diffeomorphism types of the 5‐dimensional links of 3‐dimensional log‐canonical singularities defined by Brieskorn polynomials. Moreover, we show that if k is an integer with 1 ≤ k < 611, then there is no link K defined by a Brieskorn polynomial in ?⁴ such that the order of H₂(K) is 6k. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)