On a question of Thas on partial 3‐(qn + 1, q + 1, 1) designs

In this note, we answer a question of JA Thas about partial $3$‐ $\left(q^n+1,q+1,1\right)$ designs. We then extend this answer to a result about the embedding of certain partial $3$‐ $\left(q^2+1,q+1,1\right)$ designs into Möbius planes.     

The crossing number of Cm × Cn is as conjectured for n ≥ m(m + 1)

It has been long conjectured that the crossing number of C_m × C_n is (m?2)n, for all m, n such that n ≥ m ≥ 3. In this paper, it is shown that if n ≥ m(m + 1) and m ≥ 3, then this conjecture holds. That is, the crossing number of C_m × C_n is as conjectured for all but finitely many n, for each m. The proof is largely based on techniques from the theory of arrangements, introduced by Adamsson and further developed by Adamsson and Richter. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 53–72, 2004     

Cycles passing through k + 1 vertices in k‐connected graphs

In this article, we prove the following theorem. Let k ≥ 3 be an integer, G be a k‐connected graph with minimum degree d and X be a set of k + 1 vertices on a cycle. Then G has a cycle of length at least min {2d,|V(G)|} passing through X. This result gives the positive answer to the Question posed by Locke [8]. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:179–190, 2008     

Group‐divisible designs with block size k having k + 1 groups,for k = 4, 5

We investigate the spectrum for k‐GDDs having k + 1 groups, where k = 4 or 5. We take advantage of new constructions introduced by R. S. Rees (Two new direct product‐type constructions for resolvable group‐divisible designs, J Combin Designs, 1 (1993), 15–26) to construct many new designs. For example, we show that a resolvable 4‐GDD of type g⁵ exists if and only if g ≡ 0 mod 12 and that a resolvable 5‐GDD of type g⁶ exists if and only if g ≡ 0 mod 20. We also show that a 4‐GDD of type g⁴m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 3 and 0 < m ≤ 3g/2, except possibly when (g,m) = (9,3) or (18,6), and that a 5‐GDD of type g⁵m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 4 and 0 < m ≤ 4g/3, with 32 possible exceptions. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 363–386, 2000     

Strong Disorder for a Certain Class of Directed Polymers in a Random Environment

We study a model of directed polymers in a random environment with a positive recurrent Markov chain, taking values in a countable space Σ. The random environment is a family () of independent and identically distributed real-valued variables. The asymptotic behaviour of the normalized partition function is characterized: when the common law of the g(·,·) is infinitely divisible and the Markov chain is exponentially recurrent we prove that the normalized partition function converges exponentially fast towards zero at all temperatures.     

More results on perfect (3,3,6k + 4; 6k − 2)‐threshold schemes

A (2,3)‐packing on X is a pair (X, ), where is a set of 3‐subsets (called blocks) of X, such that any pair of distinct points from X occurs together in at most one block. Its leave is a graph (X,E) such that E consists of all the pairs which do not appear in any block of . In this article, we shall construct a set of 6k ? 2 disjoint (2,3)‐packings of order 6k + 4 with K_1,3 ∪ 3kK₂ or G₁ ∪ (3k ? 1)K₂ as their common leave for any integer k ≥ 1 with a few possible exceptions (G₁ is a special graph of order 6). Such a system can be used to construct perfect threshold schemes as noted by Schellenberg and Stinson ( 22 ). © 2006 Wiley Periodicals, Inc. J Combin Designs     

On Nonnegative Solutions of the Inequality Δ u + uσ ≤ 0 on Riemannian Manifolds

We study the uniqueness of a nonnegative solution of the differential inequality on a complete Riemannian manifold, where σ > 1 is a parameter. We prove that if, for some x₀ ? M and all large enough r where , and B(x,r) is a geodesic ball, then the only nonnegative solution of (*) is identically 0. We also show the sharpness of the above values of the exponents p,q. © 2014 Wiley Periodicals, Inc.     

Quenched Free Energy and Large Deviations for Random Walks in Random Potentials

We study quenched distributions on random walks in a random potential on integer lattices of arbitrary dimension and with an arbitrary finite set of admissible steps. The potential can be unbounded and can depend on a few steps of the walk. Directed, undirected, and stretched polymers, as well as random walk in random environment, are covered. The restriction needed is on the moment of the potential, in relation to the degree of mixing of the ergodic environment. We derive two variational formulas for the limiting quenched free energy and prove a process‐level quenched large deviation principle (LDP) for the empirical measure. As a corollary we obtain LDPs for types of random walks in random environments not covered by earlier results. © 2012 Wiley Periodicals, Inc.     

A variety of distinct kinds of multiple soliton solutions for a ( 3 + 1)‐dimensional nonlinear evolution equation

In this work, a variety of distinct kinds of multiple soliton solutions is derived for a ( 3 + 1)‐dimensional nonlinear evolution equation. The simplified form of the Hirota's method is used to derive this set of distinct kinds of multiple soliton solutions. The coefficients of the spatial variables play a major role in the existence of this variety of multiple soliton solutions for the same equation. The resonance phenomenon is investigated as well. Copyright © 2012 John Wiley & Sons, Ltd.     

The acyclic edge chromatic number of a random d‐regular graph is d + 1

We prove the theorem from the title: the acyclic edge chromatic number of a random d‐regular graph is asymptotically almost surely equal to d + 1. This improves a result of Alon, Sudakov, and Zaks and presents further support for a conjecture that Δ(G) + 2 is the bound for the acyclic edge chromatic number of any graph G. It also represents an analog of a result of Robinson and the second author on edge chromatic number. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 69–74, 2005     

A study on the (2 + 1)‐dimensional KdV4 equation derived by using the KdV recursion operator

We derive a new ( 2 + 1)‐dimensional Korteweg–de Vries 4 (KdV4) equation by using the recursion operator of the KdV equation. This study shows that the new KdV4 equation possess multiple soliton solutions the same as the multiple soliton solutions of the KdV hierarchy, but differ only in the dispersion relations. We also derive other traveling wave solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

Directed B(K, 1; v) with K = {4, 5} and {4, 6} related to deletion/insertion‐correcting codes

Motivated by the construction of t‐deletion/insertion‐correcting codes, we consider the existence of directed PBDs with block sizes from K = {4, 5} and {4, 6}. The spectra of such designs are determined completely in this paper. For any integer {υ ≥ 4, a DB({4,5} ,1; υ) exists if and only if υ∉{6, 8, 9, 12, 14}, and a DB({4, 6}, 1; υ) exists if and only if υ ≡ 0,1 mod 3 and υ∉{9,15}. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 147–156, 2001     

The decomposability of simple orthogonal arrays on 3 symbols having t + 1 rows and strength t

It is well‐known that all orthogonal arrays of the form OA(N, t + 1, 2, t) are decomposable into λ orthogonal arrays of strength t and index 1. While the same is not generally true when s = 3, we will show that all simple orthogonal arrays of the form OA(N, t + 1, 3, t) are also decomposable into orthogonal arrays of strength t and index 1. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 442–458, 2000     

A study of integrability and symmetry for the (p + 1)th Boltzmann equation via Painlevé analysis and Lie‐group method

In this paper,we applied the Painlevé property test on Krook‐Wu model of the nonlinear Boltzmann equation (p = 1). As a result, by using Bäcklund transformation, we obtained three solutions two of them were known earlier, while the third one is new and more general, we have also two reductions one of them is Abel's equation. Also, Lie‐group method is applied to the (p + 1)th Boltzmann equation. The complete Lie algebra of infinitesimal symmetries is established. Three nonequivalent sub‐algebraic of the complete Lie algebra are used to investigate similarity solutions and similarity reductions in the form of nonlinear ordinary equations for (p + 1)th Boltzmann equation; we obtained two general solutions for (p + 1)th Boltzmann equation and new solutions for Krook‐Wu model of Boltzmann equation (p = 1). Copyright © 2014 John Wiley & Sons, Ltd.     

A classification result on weighted {δ (p3 + 1), δ; 3, p3}‐minihypers

We classify all {δ (p³ + 1), δ; 3, p³}‐minihypers, , for a prime number p₀ ≥ 7, with excess e ≤ p³. Such a minihyper is a sum of lines and of possibly one projected subgeometry PG(5, p), or a sum of lines and a minihyper which is a projected subgeometry PG(5, p) minus one line. When p is a square, also (possibly projected) Baer subgeometries PG(3, p^3/2) can occur. © 2004 Wiley Periodicals, Inc.     

Cyclic bi‐embeddings of Steiner triple systems on 12s + 7 points

A cyclic face 2‐colourable triangulation of the complete graph K_n in an orientable surface exists for n ≡ 7 (mod 12). Such a triangulation corresponds to a cyclic bi‐embedding of a pair of Steiner triple systems of order n, the triples being defined by the faces in each of the two colour classes. We investigate in the general case the production of such bi‐embeddings from solutions to Heffter's first difference problem and appropriately labelled current graphs. For n = 19 and n = 31 we give a complete explanation for those pairs of Steiner triple systems which do not admit a cyclic bi‐embedding and we show how all non‐isomorphic solutions may be identified. For n = 43 we describe the structures of all possible current graphs and give a more detailed analysis in the case of the Heawood graph. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 92–110, 2002; DOI 10.1002/jcd.10001     

Pointwise convergence along a tangential curve for the fractional Schrödinger equation with 0 < m < 1

In this article, we study the pointwise convergence problem about solution to the fractional Schrödinger equation with 0 < m < 1 along a tangential curve and estimate the capacitary dimension of the divergence set. We extend the results of Cho and Shiraki (2021) for the case m > 1 to the case 0 < m < 1, which is sharp up to the endpoint.     

Optimal (r,N)‐policy for discrete‐time Geo ∕ G ∕ 1 queue with different input rate and setup time

This paper presents a queue‐length analysis of Geo ∕ G ∕ 1 queue with ( r , N )‐policy and different input rate. Using a different method, the recursive expressions of queue‐length distribution at different epochs are obtained. Furthermore, some performance measures are also investigated. Finally, the Tabu search algorithm is used to search the joint optimum value of ( r , N ), which minimizes the state‐dependent operating cost. Copyright © 2014 John Wiley & Sons, Ltd.     

Two constructions of (v, (v − 1)/2, (v − 3)/2) difference families

In this article, two constructions of (v, (v ? 1)/2, (v ? 3)/2) difference families are presented. The first construction produces both cyclic and noncyclic difference families, while the second one gives only cyclic difference families. The parameters of the second construction are new. The difference families presented in this article can be used to construct Hadamard matrices. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 164–171, 2008