Counting Split Interval Orders

A poset (X,) is a split interval order (a.k.a. unit bitolerance order, proper bitolerance order) if a real interval and a distinguished point in that interval can be assigned to each xX so that xy precisely when x's distinguished point precedes y's interval, and x's interval precedes y's distinguished point. For each |X|9, we count the split interval orders and identify all posets that are minimal forbidden posets for split interval orders. The paper is a companion to Counting Split Semiorders by Fishburn and Reeds (this issue).     

Cycle Orders

Let X, T and C be, respectively, a finite set with at least three points, a set of ordered triples of distinct points from X, and a cyclic ordering of the points in X. Define T C to mean that, for every a b c T, the elements a, b, c occur in that cyclic order in C, and let C(T) denote the set of cyclic orderings of X for which T C. We say that T is noncyclic if C(T) is empty, cyclic if C(T) is nonempty, uniquely cyclic if | C(T) | = 1, a partial cycle order if it is cyclic and T ={a b c :{a b c} C for all C C(T)}, and a total cycle order if it is a uniquely cyclic partial cycle order. Many years ago E. V. Huntington axiomatized total cycle orders by independent necessary and sufficient conditions on T. The present paper studies the more relaxed structures of cyclic T sets and partial cycle orders. We focus on conditions for cyclicity, a theory of cycle dimension of partial cycle orders, and extremal problems that address combinatorial structures of T sets.     

Every Finite Topology is Generated by a Partial Pseudometric

Given any preorder on a finite set , we present an algorithm to construct a partial pseudometric on which generates in the sense that if and only if . The specialization topology generated by agrees with the topology generated by the partial pseudometric -balls, and consequently any topology on is generated by a partial pseudometric.     

Théorie du potentiel pour des opérateurs elliptiques non linéaires du second ordre à coefficients discontinus

Let a open subset of ⁿ , n3, and an open. Existence and unicity are proved for the Dirichlet problem

Markov subshifts and partial representation of $$\mathbb{F}_{\text n}$$

In this paper we fix a set * of positive elements of the free group (e. g. the set of finite words occurring in a Markov subshift) as well as n partial isometries on a Hilbert space H. Based on these we define a map S : which we prove to be a partial representation of on H under certain conditions studied by Matsumoto.*Supported by Capes.     

Asymptotic behavior of solutions of A 2nth order nonlinear differential equation

In this paper we prove two results. The first is an extension of the result of G. D. Jones [4[:Every nontrivial solution for

The Cauchy-Neumann problem for a viscous Hamilton-Jacobi equation

We study the weak solvability of viscous Hamilton-Jacobi equation: \,0,\,x\,\in\,\Omega,$" align="middle" border="0"> with Neumann boundary condition and irregular initial data ₀. The domain is a bounded open set and p > 0. The last part deals with the case a convex set and the initial data ₀ = in a open set D such that and      

Characterizing cryptogroups with a finite number of $${\mathcal{H}}$$ -classes in each $${\mathcal{D}}$$ -class by their subsemigroups

We construct a family of completely regular semigroups with the property that each completely regular semigroup S with a finite number of -classes in each -class is non-cryptic if and only if S contains an isomorphic image of a member of . Each member F of is an ideal extension of a Rees matrix semigroup J by a cyclic group B with a zero adjoined and the identity of B is the identity of F. Here with I and Λ finite, G is given by generators and relations, and P is given explicitly. Within completely regular semigroups, the cryptic property is equivalent to where is the natural partial order and a if and only if a ² = ab = ba. Hence the above result can be formulated in terms of and .      

A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ) ^m u  =  u ^q in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .      

Nonparametric Inference for a Class of Stochastic Partial Differential Equations II

Consider the stochastic partial differential equationdu (t,x) = (t)u (t, x)dt + dW _Q(t,x), 0 t T where = ₂/x ₂, and is a class of positive valued functions. We obtain an estimator for the linear multiplier (t) and establish the consistency, rate of convergence and asymptotic normality of this estimator as 0.     

Relations of Borel Type for Generalizations of Exponential Series

We prove that the condition is necessary and sufficient for the validity of the relation ln F() ln (, F), +, outside a certain set for every function from the class . Here, H(, f) is the class of series that converge for all 0 and have a form

Malliavin Calculus for White Noise Driven Parabolic SPDEs

We consider the parabolic SPDE

Closed polynomials and saturated subalgebras of polynomial algebras

The behavior of closed polynomials, i.e., polynomials such that the subalgebra is integrally closed in , is studied under extensions of the ground field. Using some properties of closed polynomials, we prove that, after shifting by constants, every polynomial can be factorized into a product of irreducible polynomials of the same degree. We consider some types of saturated subalgebras , i.e., subalgebras such that, for any , a generative polynomial of f is contained in A. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1587–1593, December, 2007.     

A Graph Theoretic Approach to Strong Monotonicity with respect to Polyhedral Cones

Consider the flow _t for the system of differential equations , x, ⁿ, open. Let K(t) be an expanding polyhedral cone of constant dimension, k be a unit vector in K(0), and x ₀. A sufficient condition for K(t) for t0 is that there exists an l so that Df(_t(x₀))+lI leaves K(t) invariant for all t0. If in addition (Df(_t(x₀))+lI)^n-1 takes k into the relative interior of K(t) for all t>0 then is in the relative interior of K(t) for all t>0. The latter condition for strong monotonicity may be cumbersome to check; a graph theoretic condition which can replace it is presented in this paper. Knowledge of the facial structure of K(t) is required. The results contained in this paper are extensions of the Kamke-Müller theorem and Hirsch's theorem for strong monotone flows. Applications from chemical kinetics and epidemiology are considered.     

Multiple Wick Product Chaos Processes

Let u(x) xR ^q be a symmetric nonnegative definite function which is bounded outside of all neighborhoods of zero but which may have u(0)=. Let p _x, (·) be the density of an R ^q valued canonical normal random variable with mean x and variance and let {G _x, ; (x, )R ^q ×[0,1 ]} be the mean zero Gaussian process with covariance

Completion of an additive measure to a σ-additive measure by an extension of the space

For an algebra of subsets of a set X there is constructed a set and an algebra of its subsets so that the mapping is a one-to-one correspondence between and and for each additive measure on the measure on defined by the equation is countably additive.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 71–76, January, 1968.The author wishes to express his deep appreciation to S. V. Fomin, under whose guidance this paper was written.     

On extendable planes,M.D.S. codes and hyperovals in PG(2, q), q=2 t

The question of the existence of finite planes seems to be beyond the range of present techniques. In this paper we skirmish with an easier question: extendable planes. We show how extendable planes arise as special cases of certain maximum distance separable codes (M.D.S. codes). A synthetic characterization of extendable planes is obtained. A different characterization is obtained in terms of hyperoval systems. Moreover, since =PG(2, q), q=2 ^t , is extendable this leads to new insights concerning the subtle and marvellous structure of certain hyperoval systems in . A priori, it seems somewhat surprising that very much can be said about hyperovals in , as they have certainly not been classified. In particular, we obtain a partial generalization of the famous even intersection property of hyperovals in PG(2, 4). We conclude with a discussion of hyperoval spreads and packings in along with some open questions.     

Interpolation theorems for a family of spanning subgraphs

Let G be a graph with order p, size q and component number . For each i between p – and q, let be the family of spanning i-edge subgraphs of G with exactly components. For an integer-valued graphical invariant if H H is an adjacent edge transformation (AET) implies |(H)-(H^')|1 then is said to be continuous with respect to AET. Similarly define the continuity of with respect to simple edge transformation (SET). Let M _j() and m _j() be the invariants defined by . It is proved that both M _p–() and m _p–(;) interpolate over , if is continuous with respect to AET, and that M _j() and m _j() interpolate over , if is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.