Exact travelling wave solutions of the generalized Bretherton equation

We search for exact travelling wave solutions of the generalized Bretherton equation for integer, greater than one, values of the exponent m of the nonlinear term by two methods: the truncated Painlevé expansion method and an algebraic method. We find periodic solutions for m=2 and m=5, to add to those already known for m=3; in all three cases these solutions exist for finite intervals of the wave velocity. We also find a “kink” shaped solitary wave for m=5 and a family of elementary unbounded solutions for arbitrary m.     

The maximal eigenvalue of 0–1 matrices with prescribed number of ones

We determine the maximum spectral radius for 0–1 matrices with m² + l ones for l=2m, 2m ? 3 for all m and for a fixed l and m> M(l). Similar results are obtained for symmetric 0–1 matrix with zero diagonal. In all cases, equality is characterized.     

Upper bounds for the formula size of symmetric Boolean functions

We prove that the complexity of the implementation of the counting function of n Boolean variables by binary formulas is at most n ^3.03, and it is at most n ^4.47 for DeMorgan formulas. Hence, the same bounds are valid for the formula size of any threshold symmetric function of n variables, particularly, for the majority function. The following bounds are proved for the formula size of any symmetric Boolean function of n variables: n ^3.04 for binary formulas and n ^4.48 for DeMorgan ones. The proof is based on the modular arithmetic.     

An Excursion into Nonlinear Ramsey Theory

Formulas for two-color Rado numbers have been established for many families of linear equations. However, there are no explicit formulas for two-color Rado numbers for any nonlinear equations. In this paper, we will establish formulas for the two-color Rado numbers for three families of equations: x + y ⁿ = z, x + y ² + c = z, and x +  y ² = az , where c and a are positive integers.     

Minimum cutsets for an element of a boolean lattice

An informative new proof is given for the theorem of Nowakowski that determines for all n and k the minimum size of a cutset for an element A with |A|=k of the Boolean algebra B _n of all subsets of {1,...,n}, ordered by inclusion. An inequality is obtained for cutsets for A that is reminiscent of Lubell's inequality for antichains in B _n. A new result that is provided by this approach is a list of all minimum cutsets for A.Research supported in part by NSF Grant DMS 87-01475.Research supported in part by NSF Grant DMS 86-06225 and Air Force OSR-86-0076.     

On (<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">m</Emphasis>)-semigroups

We generalize Green’s lemma and Green’s theorem for usual binary semigroups to (n,m)-semigroups, define and describe the regularity for an element of an (n,m)-semigroup, give some criteria for an element of an (n,m)-semigroup to be invertible, and further apply the invertibility for (n,m)-semigroups to (n,m)-groups and give some equivalent characterizations for (n,m)-groups. We establish Hosszú-Gluskin theorems for (n,m)-semigroups in two cases, as generalizations of the corresponding theorems for n-groups.     

Representations of Central Matrix-Valued Carathéodory Functions in Both Nondegenerate and Degenerate Cases

We derive left and right quotient representations for central q × q matrix-valued Carathéodory functions. Moreover, we obtain recurrent formulas for the matrix polynomials involved in the quotient representations. These formulas are the starting point for getting recurrent formulas for those matrix polynomials which occur in the Arov-Krein resolvent matrix for the nondegenerate matricial Carathéodory problem.     

The isogeny conjecture for <Emphasis Type="Italic">A</Emphasis>-motives

We prove the isogeny conjecture for A-motives over finitely generated fields K of transcendence degree ≤1. This conjecture says that for any semisimple A-motive M over K, there exist only finitely many isomorphism classes of A-motives M′ over K for which there exists a separable isogeny M′→M. The result is in precise analogy to known results for abelian varieties and for Drinfeld modules and will have strong consequences for the \mathfrak p{\mathfrak {p}}-adic and adelic Galois representations associated to M. The method makes essential use of the Harder–Narasimhan filtration for locally free coherent sheaves on an algebraic curve.     

Hypoelliptic heat kernel inequalities on the Heisenberg group

We study the existence of “L^p-type” gradient estimates for the heat kernel of the natural hypoelliptic “Laplacian” on the real three-dimensional Heisenberg Lie group. Using Malliavin calculus methods, we verify that these estimates hold in the case p>1. The gradient estimate for p=2 implies a corresponding Poincaré inequality for the heat kernel. The gradient estimate for p=1 is still open; if proved, this estimate would imply a logarithmic Sobolev inequality for the heat kernel.     

A-optimal designs for an additive cubic model

Chan et al. (1998a) obtained A-optimal designs for an additive quadratic mixture model for q≥3 mixture components. In this paper, we obtain the A-optimal designs for an additive cubic model for q≥3 mixture components using the class of symmetric weighted centroid designs based on barycentres of various depths. We observe that barycentres of depths 0 and 2 are possible support points for an A-optimal design. We have also given the optimal weights of A-optimal designs for 3≤q≤17.