On the well‐posedness of the Euler equations in the Triebel‐Lizorkin spaces

We prove local‐in‐time unique existence and a blowup criterion for solutions in the Triebel‐Lizorkin space for the Euler equations of inviscid incompressible fluid flows in ?ⁿ, n ≥ 2. As a corollary we obtain global persistence of the initial regularity characterized by the Triebel‐Lizorkin spaces for the solutions of two‐dimensional Euler equations. To prove the results, we establish the logarithmic inequality of the Beale‐Kato‐Majda type, the Moser type of inequality, as well as the commutator estimate in the Triebel‐Lizorkin spaces. The key methods of proof used are the Littlewood‐Paley decomposition and the paradifferential calculus by J. M. Bony. © 2002 John Wiley & Sons, Inc.     

A lower‐epiperimetric inequality for area‐minimizing surfaces

The epiperimetric inequality introduced by E. R. Reifenberg in [3] gives a rate of decay at a point for the decreasing k‐density of area of an area‐minimizing integral k‐cycle. While dilating the cycle at that point, this rate of decay holds once the configuration is close to a tangent cone configuration and above the limiting density corresponding to that configuration. This is why we propose to call the Reifenberg epiperimetric inequality an upper‐epiperimetric inequality. A direct consequence of this upper‐epiperimetric inequality is the statement that any point possesses a unique tangent cone. The upper‐epiperimetric inequality was proven by B. White in [5] for area‐minimizing 2‐cycles in ?ⁿ. In the present paper we introduce the notion of a lower‐epiperimetric inequality. This inequality gives this time a rate of decay for the decreasing k‐density of area of an area‐minimizing integral k‐cycle, while dilating the cycle at a point once the configuration is close to a tangent cone configuration and below the limiting density corresponding to that configuration. Our main result in the present paper is to prove the lower‐epiperimetric inequality for area‐minimizing 2‐cycles in ?ⁿ. As a consequence of this inequality we prove the “splitting before tilting” phenomenon for calibrated 2‐rectifiable cycles, which plays a crucial role in the proof of the regularity of 1‐1 integral currents in [4]. © 2004 Wiley Periodicals, Inc.     

Efficient algorithms for three‐dimensional axial and planar random assignment problems

Beautiful formulas are known for the expected cost of random two‐dimensional assignment problems, but in higher dimensions even the scaling is not known. In three dimensions and above, the problem has natural “Axial” and “Planar” versions, both of which are NP‐hard. For 3‐dimensional Axial random assignment instances of size n, the cost scales as Ω(1/ n), and a main result of the present paper is a linear‐time algorithm that, with high probability, finds a solution of cost O(n^–1+o(1)). For 3‐dimensional Planar assignment, the lower bound is Ω(n), and we give a new efficient matching‐based algorithm that with high probability returns a solution with cost O(n log n). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 160–196, 2015     

Partial Regularity for Singular Solutions to the Monge‐Ampère Equation

We prove that solutions to the Monge‐Ampère inequality in ?ⁿ are strictly convex away from a singular set of Hausdorff (n‐1)‐dimensional measure zero. Furthermore, we show this is optimal by constructing solutions to det D²u = 1 with singular set of Hausdorff dimension as close as we like to n‐1. As a consequence we obtain W^2,1 regularity for the Monge‐Ampère equation with bounded right‐hand side and unique continuation for the Monge‐Ampère equation with sufficiently regular right‐hand side. © 2015 Wiley Periodicals, Inc.     

Global Well‐Posedness of the Three‐Dimensional Primitive Equations with Only Horizontal Viscosity and Diffusion

In this paper, we consider the initial boundary value problem of the three‐dimensional primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well‐posedness of the strong solution is established for any H² initial data. An N‐dimensional logarithmic Sobolev embedding inequality, which bounds the L^∞‐norm in terms of the L^q‐norms up to a logarithm of the L^p‐norm for p > N of the first‐order derivatives, and a system version of the classic Grönwall inequality are exploited to establish the required a~priori H² estimates for global regularity.© 2016 Wiley Periodicals, Inc.     

Homogeneous polynomials and extensions of Hardy‐Hilbert's inequality

If L is a continuous symmetric n‐linear form on a real or complex Hilbert space and $\widehat{L}$ is the associated continuous n‐homogeneous polynomial, then $\Vert L\Vert =\big \Vert \widehat{L}\big \Vert$. We give a simple proof of this well‐known result, which works for both real and complex Hilbert spaces, by using a classical inequality due to S. Bernstein for trigonometric polynomials. As an application, an open problem for the optimal lower bound of the norm of a homogeneous polynomial, which is a product of linear forms, is related to the so‐called permanent function of an n × n positive definite Hermitian matrix. We have also derived generalizations of Hardy‐Hilbert's inequality.     

Homological connectivity of random k‐dimensional complexes

Let Δ_n?1 denote the (n ? 1)‐dimensional simplex. Let Y be a random k‐dimensional subcomplex of Δ_n?1 obtained by starting with the full (k ? 1)‐dimensional skeleton of Δ_n?1 and then adding each k‐simplex independently with probability p. Let H_k?1(Y; R) denote the (k ? 1)‐dimensional reduced homology group of Y with coefficients in a finite abelian group R. It is shown that for any fixed R and k ≥ 1 and for any function ω(n) that tends to infinity © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Inequalities for the first‐fit chromatic number

The First‐Fit (or Grundy) chromatic number of G, written as χ_FF(G), is defined as the maximum number of classes in an ordered partition of V(G) into independent sets so that each vertex has a neighbor in each set earlier than its own. The well‐known Nordhaus‐‐Gaddum inequality states that the sum of the ordinary chromatic numbers of an n‐vertex graph and its complement is at most n + 1. Zaker suggested finding the analogous inequality for the First‐Fit chromatic number. We show for n ≥ 10 that ?(5n + 2)/4? is an upper bound, and this is sharp. We extend the problem for multicolorings as well and prove asymptotic results for infinitely many cases. We also show that the smallest order of C₄‐free bipartite graphs with χ_FF(G) = k is asymptotically 2k² (the upper bound answers a problem of Zaker [9]). © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 75–88, 2008     

Inexpensive d‐dimensional matchings

Suppose that independent U(0, 1) weights are assigned to the ${d\choose 2}n^{2}$ edges of the complete d‐partite graph with n vertices in each of the d maximal independent sets. Then the expected weight of the minimum‐weight perfect d‐dimensional matching is at least $\frac{3}{16}n^{1-(2/d)}$. We describe a randomized algorithm that finds a perfect d‐dimensional matching whose expected weight is at most 5d³n^1?(2/d)+d¹⁵ for all d≥3 and n≥1. © 2002 John Wiley & Sons, Inc. Random Struct. Alg., 20, 50–58, 2002     

Maximal σ‐polynomials of connected 3‐chromatic graphs

In the set of graphs of order n and chromatic number k the following partial order relation is defined. One says that a graph G is less than a graph H if c_i(G) ≤ c_i(H) holds for every i, k ≤ i ≤ n and at least one inequality is strict, where c_i(G) denotes the number of i‐color partitions of G. In this paper the first ? n/2 ? levels of the diagram of the partially ordered set of connected 3‐chromatic graphs of order n are described. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 210–222, 2003     

Carleson measures and conformal self‐mappings in the real unit ball

Bounded and compact Carleson measures in the unit ball B of R ⁿ , n ≥ 2, are characterized by means of global Dirichlet integrals of the conformal self‐map T_a taking a ∈ B to the origin. The same proof applies in the unit ball of C ⁿ . It is also proved that the powers of the Jacobian of T_a satisfy the weak Harnack inequality and even Harnack's inequality with a constant independent of a. As an application of these results it is shown that the two different definitions for Carleson measures in the existing literature are equivalent for a certain range of parameter values. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On cubic non‐Cayley vertex‐transitive graphs

In 1983, the second author [D. Maru?i?, Ars Combinatoria 16B (1983), 297–302] asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. (The term non‐Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265–269] asked to determine the smallest valency ?(n) among valencies of non‐Cayley vertex‐transitive graphs of order n. As cycles are clearly Cayley graphs, ?(n)?3 for any non‐Cayley number n. In this paper a goal is set to determine those non‐Cayley numbers n for which ?(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n. It is known that for a prime p every vertex‐transitive graph of order p, p² or p³ is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2p, 4p or 2p² is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4p², p>7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2p^k, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012     

Strengthened Cauchy–Bunyakowski–Schwarz inequality for a three‐dimensional elasticity system

The constant γ of the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality plays a fundamental role in the convergence rate of multilevel iterative methods. The main purpose of this work is to give an estimate of the constant γ for a three‐dimensional elasticity system. The theoretical results obtained are practically important for the successful implementation of the finite element method to large‐scale modelling of complicated structures as they allow us to construct optimal order algebraic multilevel iterative solvers for a wide class of real‐life elasticity problems. Copyright © 2001 John Wiley & Sons, Ltd.     

Single‐cell compact finite‐difference discretization of order two and four for multidimensional triharmonic problems

In this article, we discuss finite‐difference methods of order two and four for the solution of two‐and three‐dimensional triharmonic equations, where the values of u,(?²u/?n²) and (?⁴u/?n⁴) are prescribed on the boundary. For 2D case, we use 9‐ and for 3D case, we use 19‐ uniform grid points and a single computational cell. We introduce new ideas to handle the boundary conditions and do not require to discretize the boundary conditions at the boundary. The Laplacian and the biharmonic of the solution are obtained as byproduct of the methods. The resulting matrix system is solved by using the appropriate block iterative methods. Computational results are provided to demonstrate the fourth‐order accuracy of the proposed methods. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Anomalous scaling for three‐dimensional Cahn‐Hilliard fronts

We prove the stability of the one‐dimensional kink solution of the Cahn‐Hilliard equation under d‐dimensional perturbations for d ≥ 3. We also establish a novel scaling behavior of the large‐time asymptotics of the solution. The leading asymptotics of the solution is characterized by a length scale proportional to t^1/3 instead of the usual t^1/2 scaling typical to parabolic problems. © 2004 Wiley Periodicals, Inc.     

Analyticity for Kuramoto–Sivashinsky‐type equations in two spatial dimensions

I. Stratis In this work, we investigate the analyticity properties of solutions of Kuramoto–Sivashinsky‐type equations in two spatial dimensions, with periodic initial data. In order to do this, we explore the applicability in three‐dimensional models of a spectral method, which was developed by the authors for the one‐dimensional Kuramoto–Sivashinsky equation. We introduce a criterion, which provides a sufficient condition for analyticity of a periodic function u∈C^∞, involving the rate of growth of ?ⁿu, in suitable norms, as n tends to infinity. This criterion allows us to establish spatial analyticity for the solutions of a variety of systems, including Topper–Kawahara, Frenkel–Indireshkumar, and Coward–Hall equations and their dispersively modified versions, once we assume that these systems possess global attractors. Copyright © 2015 John Wiley & Sons, Ltd.     

Vertex pancyclic in‐tournaments

An in‐tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. The topic of this paper is to investigate vertex k‐pancyclicity of in‐tournaments of order n, where for some 3 ≤ k ≤ n, every vertex belongs to a cycle of length p for every k ≤ p ≤ n. We give sharp lower bounds for the minimum degree such that a strong in‐tournament is vertex k‐pancyclic for k ≤ 5 and k ≥ n − 3. In the latter case, we even show that the in‐tournaments in consideration are fully (n − 3)‐extendable which means that every vertex belongs to a cycle of length n − 3 and that the vertex set of every cycle of length at least n − 3 is contained in a cycle of length one greater. In accordance with these results, we state the conjecture that every strong in‐tournament of order n with minimum degree greater than is vertex k‐pancyclic for 5 < k < n − 3, and we present a family of examples showing that this bound would be best possible. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 84–104, 2001     

Preconditioning and a posteriori error estimates using h‐ and p‐hierarchical finite elements with rectangular supports

We show some of the properties of the algebraic multilevel iterative methods when the hierarchical bases of finite elements (FEs) with rectangular supports are used for solving the elliptic boundary value problems. In particular, we study two types of hierarchies; the so‐called h‐ and p‐hierarchical FE spaces on a two‐dimensional domain. We compute uniform estimates of the strengthened Cauchy–Bunyakowski–Schwarz inequality constants for these spaces. Moreover, for diagonal blocks of the stiffness matrices corresponding to the fine spaces, the optimal preconditioning matrices can be found, which have tri‐ or five‐diagonal forms for h‐ or p‐refinements, respectively, after a certain reordering of the elements. As another use of the hierarchical bases, the a posteriori error estimates can be computed. We compare them in test examples for h‐ and p‐hierarchical FEs with rectangular supports. Copyright © 2008 John Wiley & Sons, Ltd.     

Groups and C*‐Algebras from a 4‐Dimensional Anzai Flow

The rotation flow on the circle T gives a concrete representation of the irrational rotation algebra, which is an in finite dimensional simple quotient of the group C*‐algebra of the discrete Heisenberg group H₃ analogously certain 2‐ and 3‐dimensional Anzai flows on T ² and T ³are known to give concrete representations of the corresponding quotients of the group C*‐algebras of the groups H₄ and H_5,5. Considered here is the (minimal, effective) 4‐dimensional Anzai flow F = (ℤ, T ⁴) generated by the homeomorphism (y, x, w, v) ↦ (λy, yx, xw, wv); a group H_6,10 is determined by F the faithful in finite dimensional simple quotients of whose group C*‐algebra C*‐(H_6,10 have concrete representations given by F. Furthermore, the rest of the infinite dimensional simple quotients of C*‐(H_6,10 are identified and displayed as C*‐crossed products generated by minimal effective actions and also as matrix algebras over simple C*‐algebras from groups of lower dimension; these lower dimensional groups are H₃ and subgroups of H₄ and H_5,5.     

The existence of a 2‐factor in K1, n‐free graphs with large connectivity and large edge‐connectivity

In this article, we study the existence of a 2‐factor in a K_{1, n}‐free graph. Sumner [J London Math Soc 13 (1976), 351–359] proved that for n?4, an (n?1)‐connected K_{1, n}‐free graph of even order has a 1‐factor. On the other hand, for every pair of integers m and n with m?n?4, there exist infinitely many (n?2)‐connected K_{1, n}‐free graphs of even order and minimum degree at least m which have no 1‐factor. This implies that the connectivity condition of Sumner's result is sharp, and we cannot guarantee the existence of a 1‐factor by imposing a large minimum degree. On the other hand, Ota and Tokuda [J Graph Theory 22 (1996), 59–64] proved that for n?3, every K_{1, n}‐free graph of minimum degree at least 2n?2 has a 2‐factor, regardless of its connectivity. They also gave examples showing that their minimum degree condition is sharp. But all of them have bridges. These suggest that the effects of connectivity, edge‐connectivity and minimum degree to the existence of a 2‐factor in a K_{1, n}‐free graph are more complicated than those to the existence of a 1‐factor. In this article, we clarify these effects by giving sharp minimum degree conditions for a K_{1, n}‐free graph with a given connectivity or edge‐connectivity to have a 2‐factor. Copyright © 2010 Wiley Periodicals, Inc. J Graph Theory 68:77‐89, 2011