A constructive arbitrary‐degree Kronecker product decomposition of tensors

We generalize the matrix Kronecker product to tensors and propose the tensor Kronecker product singular value decomposition that decomposes a real k‐way tensor into a linear combination of tensor Kronecker products with an arbitrary number of d factors. We show how to construct , where each factor is also a k‐way tensor, thus including matrices (k=2) as a special case. This problem is readily solved by reshaping and permuting into a d‐way tensor, followed by a orthogonal polyadic decomposition. Moreover, we introduce the new notion of general symmetric tensors (encompassing symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors, etc.) and prove that when is structured then its factors will also inherit this structure.     

Set Systems with L‐Intersections and k‐Wise L‐Intersecting Families

In this paper, by employing linear algebra methods we obtain the following main results:

• (i) Let and be two disjoint subsets of such that Suppose that is a family of subsets of such that for every pair and for every i. Then Furthermore, we extend this theorem to k‐wise L‐intersecting and obtain the corresponding result on two cross L‐intersecting families. These results show that Snevily's conjectures proposed by Snevily (2003) are true under some restricted conditions. This result also gets an improvement of a theorem of Liu and Hwang (2013).
• (ii) Let p be a prime and let and be two subsets of such that or and Suppose that is a family of subsets of [n] such that (1) for every pair (2) for every i. Then This result improves the existing upper bound substantially.

     

Partitions of V(n,q) into 2‐ and s‐Dimensional Subspaces

Let denote a vector space of dimension n over the field with q elements. A set of subspaces of V is a (vector space) partition of V if every nonzero element of V is contained in exactly one subspace in . Suppose that is a partition of V with subspaces of dimension for . Then we call the type of the partition . Which possible types correspond to actual partitions is in general an open question. We prove that for any odd integer and for any integer , the existence of partitions of across a suitable range of types guarantees the existence of partitions of of essentially all the types for any integer . We then apply this result to construct new classes of partitions of V. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 467‐482, 2012     

k‐Kings in k‐Quasitransitive Digraphs

Let D be a digraph with vertex set and arc set . A vertex x is a k‐king of D, if for every , there is an ‐path of length at most k. A subset N of is k‐independent if for every pair of vertices , we have and ; it is l‐absorbent if for every there exists such that . A ‐kernel of D is a k‐independent and l‐absorbent subset of . A k‐kernel is a ‐kernel. A digraph D is k‐quasitransitive, if for any path of length k, x₀ and are adjacent. In this article, we will prove that a k‐quasitransitive digraph with has a k‐king if and only if it has a unique initial strong component and the unique initial strong component is not isomorphic to an extended ‐cycle where each has at least two vertices. Using this fact, we show that every strong k‐quasitransitive digraph has a ‐kernel.     

Directional ‐matrix compression for high‐frequency problems

Standard numerical algorithms, such as the fast multipole method or ‐matrix schemes, rely on low‐rank approximations of the underlying kernel function. For high‐frequency problems, the ranks grow rapidly as the mesh is refined, and standard techniques are no longer attractive. Directional compression techniques solve this problem by using decompositions based on plane waves. Taking advantage of hierarchical relations between these waves' directions, an efficient approximation is obtained. This paper is dedicated to directional ‐matrices that employ local low‐rank approximations to handle directional representations efficiently. The key result is an algorithm that takes an arbitrary matrix and finds a quasi‐optimal approximation of this matrix as a directional ‐matrix using a prescribed block tree. The algorithm can reach any given accuracy, and the approximation requires only units of storage, where n is the matrix dimension, κ is the wave number, and k is the local rank. In particular, we have a complexity of if κ is constant and for high‐frequency problems characterized by κ²～n. Because the algorithm can be applied to arbitrary matrices, it can serve as the foundation of fast techniques for constructing preconditioners.     

Structure Theorem for U5‐free Tournaments

Let U₅ be the tournament with vertices v₁, …, v₅ such that , and if , and . In this article, we describe the tournaments that do not have U₅ as a subtournament. Specifically, we show that if a tournament G is “prime”—that is, if there is no subset , , such that for all , either for all or for all —then G is U₅‐free if and only if either G is a specific tournament or can be partitioned into sets X, Y, Z such that , , and are transitive. From the prime U₅‐free tournaments we can construct all the U₅‐free tournaments. We use the theorem to show that every U₅‐free tournament with n vertices has a transitive subtournament with at least vertices, and that this bound is tight.     

Exponential convergence in ‐Wasserstein distance for diffusion processes without uniformly dissipative drift

By adopting the coupling by reflection and choosing an auxiliary function which is convex near infinity, we establish the exponential convergence of diffusion semigroups with respect to the standard ‐Wasserstein distance for all . In particular, we show that for the Itô stochastic differential equation if the drift term b is such that for any , holds with some positive constants K₁, K₂ and , then there is a constant such that for all , and , where is a positive constant. This improves the main result in 14 where the exponential convergence is only proved for the L¹‐Wasserstein distance.     

Numerical solution to a linear equation with tensor product structure

We consider the numerical solution of a c‐stable linear equation in the tensor product space , arising from a discretized elliptic partial differential equation in . Utilizing the stability, we produce an equivalent d‐stable generalized Stein‐like equation, which can be solved iteratively. For large‐scale problems defined by sparse and structured matrices, the methods can be modified for further efficiency, producing algorithms of computational complexity, under appropriate assumptions (with n_s being the flop count for solving a linear system associated with ). Illustrative numerical examples will be presented.     

Maximal regularity for non‐autonomous Robin boundary conditions

We consider a non‐autonomous Cauchy problem where is associated with the form , where V and H are Hilbert spaces such that V is continuously and densely embedded in H. We prove H‐maximal regularity, i.e., the weak solution u is actually in (if and ) under a new regularity condition on the form with respect to time; namely Hölder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.     

Cycle Extensions in BIBD Block‐Intersection Graphs

A cycle C in a graph G is extendable if there is some other cycle in G that contains each vertex of C plus one additional vertex. A graph is cycle extendable if every non‐Hamilton cycle in the graph is extendable. A balanced incomplete block design, BIBD, consists of a set V of v elements and a block set of k‐subsets of V such that each 2‐subset of V occurs in exactly λ of the blocks of . The block‐intersection graph of a design is the graph having as its vertex set and such that two vertices of are adjacent if and only if their corresponding blocks have nonempty intersection. In this paper, we prove that the block‐intersection graph of any BIBD is cycle extendable. Furthermore, we present a polynomial time algorithm for constructing cycles of all possible lengths in a block‐intersection graph.     

Detachments of Amalgamated 3‐Uniform Hypergraphs Factorization Consequences

A is a hypergraph obtained from by splitting some or all of its vertices into more than one vertex. Amalgamating a hypergraph can be thought of as taking , partitioning its vertices, then for each element of the partition squashing the vertices to form a single vertex in the amalgamated hypergraph . In this paper, we use Nash‐Williams lemma on laminar families to prove a detachment theorem for amalgamated 3‐uniform hypergraphs, which yields a substantial generalization of previous amalgamation theorems by Hilton, Rodger, and Nash‐Williams. To demonstrate the power of our detachment theorem, we show that the complete 3‐uniform n‐partite multihypergraph can be expressed as the union of k edge‐disjoint factors, where for , is ‐regular, if and only if:

1. for all ,
2. for each i, , and
3. .

     

On 5‐Cycles and 6‐Cycles in Regular n‐Tournaments

Let T be a tournament of order n and be the number of cycles of length m in T. For and odd n, the maximum of is achieved for any regular tournament of order n (M. G. Kendall and B. Babington Smith, 1940), and in the case it is attained only for the unique regular locally transitive tournament RLT_n of order n (U. Colombo, 1964). A lower bound was also obtained for in the class of regular tournaments of order n (A. Kotzig, 1968). This bound is attained if and only if T is doubly regular (when ) or nearly doubly regular (when ) (B. Alspach and C. Tabib, 1982). In the present article, we show that for any regular tournament T of order n, the equality holds. This allows us to reduce the case to the case In turn, the pure spectral expression for obtained in the class implies that for a regular tournament T of order the inequality holds, with equality if and only if T is doubly regular or T is the unique regular tournament of order 7 that is neither doubly regular nor locally transitive. We also determine the value of c₆(RLT_n) and conjecture that this value coincides with the minimum number of 6‐cycles in the class for each odd      

Cycle Frames of Complete Multipartite Multigraphs ‐ III

For two graphs G and H their wreath product has vertex set in which two vertices and are adjacent whenever or and . Clearly, , where is an independent set on n vertices, is isomorphic to the complete m‐partite graph in which each partite set has exactly n vertices. A 2‐regular subgraph of the complete multipartite graph containing vertices of all but one partite set is called partial 2‐factor. For an integer λ, denotes a graph G with uniform edge multiplicity λ. Let J be a set of integers. If can be partitioned into edge‐disjoint partial 2‐factors consisting cycles of lengths from J, then we say that has a ‐cycle frame. In this paper, we show that for and , there exists a ‐cycle frame of if and only if and . In fact our results completely solve the existence of a ‐cycle frame of .     

Classification of Graeco‐Latin Cubes

A q‐ary code of length n, size M, and minimum distance d is called an code. An code with is said to be maximum distance separable (MDS). Here one‐error‐correcting () MDS codes are classified for small alphabets. In particular, it is shown that there are unique (5, 5³, 3)₅ and (5, 7³, 3)₇ codes and equivalence classes of (5, 8³, 3)₈ codes. The codes are equivalent to certain pairs of mutually orthogonal Latin cubes of order q, called Graeco‐Latin cubes.     

Perfect Matchings Avoiding Several Independent Edges in a Star‐Free Graph

In Aldred and Plummer (Discrete Math 197/198 (1999) 29–40) proved that every m‐connected ‐free graph of even order has a perfect matching M with and , where F₁ and F₂ are prescribed disjoint sets of independent edges with and . It is known that if l satisfies , then the star‐free condition in the above result is best possible. In this paper, for , we prove a refinement of the result in which the condition is replaced by the weaker condition that G is ‐free (note that the new condition does not depend on l). We also show that if m is even and either or , then for m‐connected graphs G with sufficiently large order, one can replace the condition by the still weaker condition that G is ‐free. The star‐free conditions in our results are best possible.     

Pre‐Hilbert spaces with anomalous splitting and orthogonally‐closed subspace structures

Four classes of closed subspaces of an inner product space S that can naturally replace the lattice of projections in a Hilbert space are: the complete/cocomplete subspaces , the splitting subspaces , the quasi‐splitting subspaces and the orthogonally‐closed subspaces . It is well‐known that in general the algebraic structure of these families differ remarkably and they coalesce if and only if S is a Hilbert space. It is also known that when S is a hyperplane in its completion i.e. then and . On the other extreme, when i.e. then and . Motivated by this and in contrast to it, we show that in general the codimension of S in bears very little relation to the properties of these families. In particular, we show that the equalities and can hold for inner product spaces with arbitrary codimension in . At the end we also contribute to the study of the algebraic structure of by testing it for the Riesz interpolation property. We show that may fail to enjoy the Riesz interpolation property in both extreme situations when S is “very small” (i.e. and when S is ‘very big’ (i.e. .     

Hyperhypersimple sets and Q1‐reducibility

We prove that the c.e. Q₁‐degrees are not dense, and there exists a c.e. Q₁‐degree with no minimal c.e. predecessors. It is proved that if M₁ and M₂ are maximal sets such that then or . We also show that there exist infinite collections of Q₁‐degrees and such that the following hold: (i) for every , , , and , (ii) each consists entirely of maximal sets; and (iii) each consists entirely of non‐maximal hyperhypersimple sets.     

The Large‐Time Solution of Burgers' Equation with Time‐Dependent Coefficients. I. The Coefficients Are Exponential Functions

In this paper, we consider an initial‐value problem for Burgers' equation with variable coefficients where x and t represent dimensionless distance and time, respectively, and , are given functions of t. In particular, we consider the case when the initial data have algebraic decay as , with as and as . The constant states and are problem parameters. Two specific initial‐value problems are considered. In initial‐value problem 1 we consider the case when and , while in initial‐value problem 2 we consider the case when and . The method of matched asymptotic coordinate expansions is used to obtain the large‐t asymptotic structure of the solution to both initial‐value problems over all parameter values.