What Is Variable Bandwidth?

We propose a new notion of variable bandwidth that is based on the spectral subspaces of an elliptic operator where p > 0 is a strictly positive function. Denote by the orthogonal projection of A_p corresponding to the spectrum of A_p in ; the range of this projection is the space of functions of variable bandwidth with spectral set in Λ. We will develop the basic theory of these function spaces. First, we derive (nonuniform) sampling theorems; second, we prove necessary density conditions in the style of Landau. Roughly, for a spectrum the main results say that, in a neighborhood of , a function of variable bandwidth behaves like a band‐limited function with local bandwidth . Although the formulation of the results is deceptively similar to the corresponding results for classical band‐limited functions, the methods of proof are much more involved. On the one hand, we use the oscillation method from sampling theory and frame‐theoretic methods; on the other hand, we need the precise spectral theory of Sturm‐Liouville operators and the scattering theory of one‐dimensional Schrödinger operators. © 2017 Wiley Periodicals, Inc.     

Global Existence of Weak Solutions of the Nematic Liquid Crystal Flow in Dimension Three

For any bounded smooth domain , we establish the global existence of a weak solution of the initial boundary value (or the Cauchy) problem of the simplified Ericksen‐Leslie system LLF modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary (or Cauchy) data , with (the upper hemisphere). Furthermore, (u,d) satisfies the global energy inequality.© 2016 Wiley Periodicals, Inc.     

A Rigidity Result for Overdetermined Elliptic Problems in the Plane

Let be a (locally) Lipschitz function and a domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problem we prove that Ω is a half‐plane. In particular, we obtain a partial answer to a question raised by H. Berestycki, L. Caffarelli, and L. Nirenberg in 1997.© 2017 Wiley Periodicals, Inc.     

Block‐Transitive and Point‐Primitive 2‐ Designs with Sporadic Socle

The purpose of this paper is to classify all pairs , where is a nontrivial 2‐ design, and acts transitively on the set of blocks of and primitively on the set of points of with sporadic socle. We prove that there exists only one such pair : is the unique 2‐(176,8,2) design and , the Higman–Sims simple group.     

On the Hamilton–Waterloo Problem with Odd Orders

Given nonnegative integers , the Hamilton–Waterloo problem asks for a factorization of the complete graph into α ‐factors and β ‐factors. Without loss of generality, we may assume that . Clearly, v odd, , , and are necessary conditions. To date results have only been found for specific values of m and n. In this paper, we show that for any integers , these necessary conditions are sufficient when v is a multiple of and , except possibly when or 3. For the case where we show sufficiency when with some possible exceptions. We also show that when are odd integers, the lexicographic product of with the empty graph of order n has a factorization into α ‐factors and β ‐factors for every , , with some possible exceptions.     

Almost‐spanning universality in random graphs

A graph G is said to be ‐universal if it contains every graph on at most n vertices with maximum degree at most Δ. It is known that for any and any natural number Δ there exists such that the random graph G(n, p) is asymptotically almost surely ‐universal for . Bypassing this natural boundary, we show that for the same conclusion holds when . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 380–393, 2017     

On Flag‐Transitive Symmetric Designs of Affine Type

It is shown that, if is a nontrivial 2‐ symmetric design, with , admitting a flag‐transitive automorphism group G of affine type, then , p an odd prime, and G is a point‐primitive, block‐primitive subgroup of . Moreover, acts flag‐transitively, point‐primitively on , and is isomorphic to the development of a difference set whose parameters and structure are also provided.     

List‐Coloring the Squares of Planar Graphs without 4‐Cycles and 5‐Cycles

Let G be a planar graph without 4‐cycles and 5‐cycles and with maximum degree . We prove that . For arbitrarily large maximum degree Δ, there exist planar graphs of girth 6 with . Thus, our bound is within 1 of being optimal. Further, our bound comes from coloring greedily in a good order, so the bound immediately extends to online list‐coloring. In addition, we prove bounds for ‐labeling. Specifically, and, more generally, , for positive integers p and q with . Again, these bounds come from a greedy coloring, so they immediately extend to the list‐coloring and online list‐coloring variants of this problem.     

Entanglement Thresholds for Random Induced States

For a random quantum state on obtained by partial tracing a random pure state on , we consider the question whether it is typically separable or typically entangled. For this problem, we show the existence of a sharp threshold of order roughly . More precisely, for any and for d large enough, such a random state is entangled with very large probability when , and separable with very large probability when . One consequence of this result is as follows: for a system of N identical particles in a random pure state, there is a threshold such that two subsystems of k particles each typically share entanglement if k > k₀, and typically do not share entanglement if k < k₀. Our methods also work for multipartite systems and for “unbalanced” systems such as , . The arguments rely on random matrices, classical convexity, high‐dimensional probability, and geometry of Banach spaces; some of the auxiliary results may be of reference value. © 2013 Wiley Periodicals, Inc.     

Minimum Degrees of Minimal Ramsey Graphs for Almost‐Cliques

For graphs F and H, we say F is Ramsey for H if every 2‐coloring of the edges of F contains a monochromatic copy of H. The graph F is Ramsey H‐minimal if F is Ramsey for H and there is no proper subgraph of F so that is Ramsey for H. Burr et al. defined to be the minimum degree of F over all Ramsey H‐minimal graphs F. Define to be a graph on vertices consisting of a complete graph on t vertices and one additional vertex of degree d. We show that for all values ; it was previously known that , so it is surprising that is much smaller. We also make some further progress on some sparser graphs. Fox and Lin observed that for all graphs H, where is the minimum degree of H; Szabó et al. investigated which graphs have this property and conjectured that all bipartite graphs H without isolated vertices satisfy . Fox et al. further conjectured that all connected triangle‐free graphs with at least two vertices satisfy this property. We show that d‐regular 3‐connected triangle‐free graphs H, with one extra technical constraint, satisfy ; the extra constraint is that H has a vertex v so that if one removes v and its neighborhood from H, the remainder is connected.     

Hierarchical Construction of Bounded Solutions in Critical Regularity Spaces

We construct uniformly bounded solutions for the equations div U = f and U = f in the critical cases and , respectively. Criticality in this context manifests itself by the lack of a linear solution operator mapping . Thus, the intriguing aspect here is that although the problems are linear, construction of their solutions is not. Our constructions are special cases of a general framework for solving linear equations of the form , where is a linear operator densely defined in Banach space with a closed range in a (proper subspace) of Lebesgue space , and with an injective dual . The solutions are realized in terms of a multiscale hierarchical representation, , interesting for its own sake. Here, u _j's are constructed recursively as minimizers of where the residuals are resolved in terms of a dyadic sequence of scales with large enough . The nonlinear aspect of this construction is a counterpart of the fact that one cannot linearly solve in critical regularity spaces.© 2016 Wiley Periodicals, Inc.     

On the Existence of

Let there is an . For or , has been determined by Hanani, and for or , has been determined by the first author. In this paper, we investigate the case . A necessary condition for is . It is known that , and that there is an for all with a possible exception . We need to consider the case . It is proved that there is an for all with an exception and a possible exception , thereby, .     

The Hamilton–Waterloo Problem for ‐Factors and ‐Factors

The Hamilton–Waterloo problem asks for a 2‐factorization of (for v odd) or minus a 1‐factor (for v even) into ‐factors and ‐factors. We completely solve the Hamilton–Waterloo problem in the case of C₃‐factors and ‐factors for .     

A Hypercube Variant with Small Diameter

This article introduces a new variant of hypercubes . The n‐dimensional twisted hypercube is obtained from two copies of the ‐dimensional twisted hypercube by adding a perfect matching between the vertices of these two copies of . We prove that the n‐dimensional twisted hypercube has diameter . This improves on the previous known variants of hypercube of dimension n and is optimal up to an error of order . Another type of hypercube variant that has similar structure and properties as is also discussed in the last section.     

Optimal Two‐Dimensional Optical Orthogonal Codes with the Best Cross‐Correlation Constraint

The study of optical orthogonal codes has been motivated by an application in an optical code‐division multiple access system. From a practical point of view, compared to one‐dimensional optical orthogonal codes, two‐dimensional optical orthogonal codes tend to require smaller code length. On the other hand, in some circumstances only with good cross‐correlation one can deal with both synchronization and user identification. These motivate the study of two‐dimensional optical orthogonal codes with better cross‐correlation than auto‐correlation. This paper focuses on optimal two‐dimensional optical orthogonal codes with the auto‐correlation and the best cross‐correlation 1. By examining the structures of w‐cyclic group divisible designs and semi‐cyclic incomplete holey group divisible designs, we present new combinatorial constructions for two‐dimensional ‐optical orthogonal codes. When and , the exact number of codewords of an optimal two‐dimensional ‐optical orthogonal code is determined for any positive integers n and .     

Subcubic Edge‐Chromatic Critical Graphs Have Many Edges

We consider graphs G with such that and for every edge e, so‐called critical graphs. Jakobsen noted that the Petersen graph with a vertex deleted, , is such a graph and has average degree only . He showed that every critical graph has average degree at least , and asked if is the only graph where equality holds. A result of Cariolaro and Cariolaro shows that this is true. We strengthen this average degree bound further. Our main result is that if G is a subcubic critical graph other than , then G has average degree at least . This bound is best possible, as shown by the Hajós join of two copies of .     

The Maximum Number of Complete Subgraphs of Fixed Size in a Graph with Given Maximum Degree

In this article, we make progress on a question related to one of Galvin that has attracted substantial attention recently. The question is that of determining among all graphs G with n vertices and , which has the most complete subgraphs of size t, for . The conjectured extremal graph is , where with . Gan et al. (Combin Probab Comput 24(3) (2015), 521–527) proved the conjecture when , and also reduced the general conjecture to the case . We prove the conjecture for and also establish a weaker form of the conjecture for all r.     

Cayley Graphs of Diameter Two from Difference Sets

Let and be the largest order of a Cayley graph and a Cayley graph based on an abelian group, respectively, of degree d and diameter k. When , it is well known that with equality if and only if the graph is a Moore graph. In the abelian case, we have . The best currently lower bound on is for all sufficiently large d. In this article, we consider the construction of large graphs of diameter 2 using generalized difference sets. We show that for sufficiently large d and if , and m is odd.     

On random k‐out subgraphs of large graphs

We consider random subgraphs of a fixed graph with large minimum degree. We fix a positive integer k and let G_k be the random subgraph where each independently chooses k random neighbors, making kn edges in all. When the minimum degree then G_k is k‐connected w.h.p. for ; Hamiltonian for k sufficiently large. When , then G_k has a cycle of length for . By w.h.p. we mean that the probability of non‐occurrence can be bounded by a function (or ) where . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 143–157, 2017