On the existence of MOLS with equal-sized holes

We consider a pair of MOLS (mutually orthogonal Latin squares) having holes, corresponding to missing sub-MOLS, which are disjoint and spanning It is shown that a pair of MOLS withn holes of sizeh exist forh 2 if and only ifn 4 For SOLS (self-orthogonal Latin squares) with holes, we have the same result, with two possible exceptions SOLS with 7 or 13 holes of size 6     

Perfect Mendelsohn designs with equal-sized holes and block size four

Let M = {m₁, m₂, …, m_h} and X be a v-set (of points). A holey perfect Mendelsohn designs (briefly (v, k, λ) - HPMD), is a triple (X, H, B), where H is a collection of subsets of X (called holes) with sizes M and which partition X, and B is a collection of cyclic k-tuples of X (called blocks) such that no block meets a hole in more than one point and every ordered pair of points not contained in a hole appears t-apart in exactly λ blocks, for 1 ≤ t ≤ k − 1. The vector (m₁, m₂, …, m_h) is called the type of the HPMD. If m₁ = m₂ = … = m_h = m, we write briefly m^h for the type. In this article, it is shown that the necessary condition for the existence of a (v, 4, λ) - HPMD of type m^h, namely, is also sufficient with the exception of types 2⁴ and 1⁸ with λ = 1, and type m⁴ for odd m with odd λ. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 203–213, 1997     

Remarks on global existence for the supercritical nonlinear Schrödinger equation with a harmonic potential

This paper is concerned with the nonlinear Schrödinger equation with a harmonic potential which describes the attractive Bose–Einstein condensate under the magnetic trap. By combining the best constant of Gagliardo–Nirenberg's inequality with the characteristic of this equation, we derive out a global existence condition for the supercritical equation which coincides with the critical case.     

Permutations, parenthesis words, and Schröder numbers

A different proof for the following result due to West is given: the Schröder number s[inn−1] equals the number of permutations on [1,2,] …, nþat avoid the pattern (3,1,4,2) and its dual (2,4,1,3).     

On the existence of positive least energy solutions for a coupled Schrödinger system with critical exponent

In this paper, we consider the following coupled Schrödinger system with critical exponent: where is a smooth bounded domain, λ > 0,μ≥0, and . Under certain conditions on λ and μ, we show that this problem has at least one positive least energy solution. Copyright © 2016 John Wiley & Sons, Ltd.     

The existence spectrum of golf designs

In this note, a golf design of order 41 is constructed. Combined Colbourn and Nonay's result, the existence spectrum of golf design of order υ is the set {υ: υ≡1 (mod 2), υ ≥ 3, υ ≠ 5}. © 2005 Wiley Periodicals, Inc. J Combin Designs 15: 84–89, 2007     

Kirkman covering designs with even-sized holes

A Kirkman holey covering design, denoted by KHCD(g^u), is a resolvable group-divisible covering design of type g^u. Each of its parallel class contains one block of size δ, while other blocks have size 3. Here δ is equal to 2, 3 and 4 when gu≡2, 3 and 4 (mod 3) in turn. In this paper, we study the existence problem of a KHCD(g^u) which has minimum possible number of parallel classes, and give a solution for most values of even g and u.     

Some identities on the Catalan, Motzkin and Schröder numbers

In this paper, some identities between the Catalan, Motzkin and Schröder numbers are obtained by using the Riordan group. We also present two combinatorial proofs for an identity related to the Catalan numbers with the Motzkin numbers and an identity related to the Schröder numbers with the Motzkin numbers, respectively.     

Generalized nonlinear Schrödinger equation with time-dependent dissipation

The generalized nonlinear Schrödinger equation with time-dependent dissipation [1] is considered using decomposition [2].     

Scattering for Schrödinger Operators with Magnetic Fields

We study the existence and completeness of the wave operators W_ω(A(b),-Δ) for general Schrodinger operators of the form is a magnetic potential.     

On Schrödinger operators with multipolar inverse-square potentials

Positivity, essential self-adjointness, and spectral properties of a class of Schrödinger operators with multipolar inverse-square potentials are discussed. In particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established.