Oscillations in some nonlinear economic relationships

In this paper we consider a simple family of nonlinear dynamical systems generated by smooth functions. Some theorems for the existence and the uniqueness of the limit cycles of the systems are presented. If f and g are generating functions with unique limit cycles and xf(x) < xg(x), for all x ≠ 0, then according to the ‘bounding theorem’ proved in the paper, the limit cycle of the system generated by f is bounded by the limit cycle of the system generated by g. This gives us a method to estimate the amplitude of the oscillations also for systems for which we do not know the generating function exactly. As an application we extend the nonlinear business cycle model proposed by Tönu Puu (1989).     

Some results on characterizing the edges of connected graphs with a given domination number

A dominating set for a graph G = (V, E) is a subset of vertices V′ V such that for all v ε V − V′ there exists some u ε V′ for which {v, u} ε E. The domination number of G is the size of its smallest dominating set(s). For a given graph G with minimum size dominating set D, let m₁ (G, D) denote the number of edges that have neither endpoint in D, and let m₂ (G, D) denote the number of edges that have at least one endpoint in D. We characterize the possible values that the pair (m₁ (G, D), m₂ (G, D)) can attain for connected graphs having a given domination number.     

Mathematical analysis of the Tyson model of the regulation of the cell division cycle

In this paper, we study the mathematical properties of a family of models of Eukaryotic cell cycle, which extend the qualitative model proposed by Tyson [Proc. Natl. Acad. Sci. 88 (1991) 7328–7332]. By means of some recent results in the theory of Lienard's systems, conditions for the uniqueness of the limit cycle and on the global asymptotic stability of the unique equilibrium (idest of the arrest of the cell division) are given.     

All trees are 1-embeddable and all except stars are 2-embeddable

A graph G with n vertices is said to be embeddable (in its complement) if there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))=. It is known that all trees T with n (≥2) vertices and T K_1,n−1 are embeddable. We say that G is 1-embeddable if, for every edge e, there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))={e};and that it is 2-embeddable if,for every pair e₁, e₂ of edges, there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))={e₁, e₂}. We prove here that all trees with n (3) vertices are 1-embeddable; and that all trees T with n (4) vertices and T K_1,n−1 are 2-embeddable. In a certain sense, this result is sharp.     

An extended result of Kleitman and Saks concerning binary trees

In a recent paper, D.J. Kleitman and M.E. Saks gave a proof of Huang's conjecture on alphabetic binary trees.

Given a set E = {e_i}, I = 0, 1, 2, …, m and assigned positive weights to its elements and supposing the elements are indexed such that w(e₀) ≤ w(e₁) ≤ … ≤w (e_m), where w(e_i) is the weight of e_i, we call the following sequence E^* a ‘saw-tooth’ sequence

E^*=(e₀,e_m,e₁,…,e_j,e_m−j,…).

Huang's conjecture is: E^* is the most expensive sequence for alphabetic binary trees. This paper shows that this property is true for the L-restricted alphabetic binary trees, where L is the maximum length of the leaves and log₂(m + 1) ≤L≤m.     

Harary's conjectures on integral sum graphs

Let N denote the set of positive integers and Z denote all integers. The (integral) sum graph of a finite subset S N(Z) is the graph (S, E) with uv ε E if and only if u + v ε S. A graph G is said to be an (integral) sum graph if it is isomorphic to the (integral) sum graph of some S N(Z). The (integral) sum number of a given graph G is the smallest number of isolated nodes which when added to G result in an (integral) sum graph.

We show that the integral sum number of a complete graph with n 4 nodes equals 2n − 3, which proves a conjecture of Harary. And we disprove another conjecture of Harary by showing that there are infinitely many trees which are not caterpillars but are integral sum graphs.     

Some results on integral sum graphs

Let Z denote the set of all integers. The integral sum graph of a finite subset S of Z is the graph (S,E) with vertex set S and edge set E such that for u,vS, uvE if and only if u+vS. A graph G is called an integral sum graph if it is isomorphic to the integral sum graph of some finite subset S of Z. The integral sum number of a given graph G, denoted by ζ(G), is the smallest number of isolated vertices which when added to G result in an integral sum graph. Let x denote the least integer not less than the real x. In this paper, we (i) determine the value of ζ(K_n−E(K_r)) for r2n/3−1, (ii) obtain a lower bound for ζ(K_n−E(K_r)) when 2r<2n/3−1 and n5, showing by construction that the bound is sharp when r=2, and (iii) determine the value of ζ(K_r,r) for r2. These results provide partial solutions to two problems posed by Harary (Discrete Math. 124 (1994) 101–108). Finally, we furnish a counterexample to a result on the sum number of K_r,s given by Hartsfiedl and Smyth (Graphs and Matrices, R. Rees (Ed.), Marcel, Dekker, New York, 1992, pp. 205–211).     

Subgraph distances in graphs defined by edge transfers

For two edge-induced subgraphs F and H of the same size in a graph G, the subgraph H can be obtained from F by an edge jump if there exist four distinct vertices u, v, w, and x in G such that uv ε E(F), wx ε E(G) - E(F), and H = F - uv + wx. The subgraph F is j-transformed into H if H can be obtained from F by a sequence of edge jumps. Necessary and sufficient conditions are presented for a graph G to have the property that every edge-induced subgraph of a fixed size in G can be j-transformed into every other edge-induced subgraph of that size. The minimum number of edge jumps required to transform one subgraph into another is called the jump distance. This distance is a metric and can be modeled by a graph. The jump graph J(G) of a graph G is defined as that graph whose vertices are the edges of G and where two vertices of J(G) are adjacent if and only if the corresponding edges of G are independent. For a given graph G, we consider the sequence {{J^k(G)}} of iterated jump graphs and classify each graph as having a convergent, divergent, or terminating sequence.     

Some nilpotent and locally nilpotent matrix groups

Say a division ring D is special if for every finite subset X of D there is a homomorphism of the subring of D generated by X into a division ring of finite Schur index a power of its positive characteristic. (D is not assumed to have positive characteristic.) We make a detailed study of nilpotent and locally nilpotent matrix groups over special division rings.

This has been done previously for a number of ‘special’ division rings arising from group algebras and Lie algebras, particularly by A.I. Lichtman. The present paper therefore presents single proofs of all these results. It also covers many division rings not considered before and produces some new results for those that have been considered before.

In view of the definition of ‘special’ it is not surprising that the proofs depend on a detailed analysis of the finite-dimensional case.     

Two theorems on Euclidean distance matrices and Gale transform

We present a characterization of those Euclidean distance matrices (EDMs) D which can be expressed as D=λ(E−C) for some nonnegative scalar λ and some correlation matrix C, where E is the matrix of all ones. This shows that the cones

where is the elliptope (set of correlation matrices) and is the (closed convex) cone of EDMs.

The characterization is given using the Gale transform of the points generating D. We also show that given points , for any scalars λ₁,λ₂,…,λ_n such that

∑_j=1ⁿλ_jp^j=0, ∑_j=1ⁿλ_j=0,

we have

∑_j=1ⁿλ_jpⁱ−p^j2= forall i=1,…,n,

for some scalar independent of i.     

On optimal search plans to detect a target moving randomly on the real line

A target, whose initial position is unknown, is performing a random walk on the integers. A searcher, starting at the origin, wants to follow a search plan for which E[τ^k] is finite, where k ≥ 1 and τ is the time to capture. The searcher, who has a prior distribution over the target's initial position, can move only to adjacent positions, and cannot travel faster than the target. Necessary and sufficient conditions are given for the existence of search plans for which E[τ^k] is finite and a minimum.     

Graph rigidity via Euclidean distance matrices

Let G=(V,E,ω) be an incomplete graph with node set V, edge set E, and nonnegative weights ω_ij's on the edges. Let each edge (v_i,v_j) be viewed as a rigid bar, of length ω_ij, which can rotate freely around its end nodes. A realization of a graph G is an assignment of coordinates, in some Euclidean space, to each node of G. In this paper, we consider the problem of determining whether or not a given realization of a graph G is rigid. We show that each realization of G can be epresented as a point in a compact convex set ; and that a generic realization of G is rigid if and only if its corresponding point is a vertex of Ω, i.e., an extreme point with full-dimensional normal cone.     

On the primitive idempotents of distance-regular graphs

Let Γ denote a distance-regular graph with diameter d3. Let E, F denote nontrivial primitive idempotents of Γ such that F corresponds to the second largest or the least eigenvalue. We investigate the situation that there exists a primitive idempotent H of Γ such that EF is a linear combination of F and H. Our main purpose is to obtain the inequalities involving the cosines of E, and to show that equality is closely related to Γ being Q-polynomial with respect to E. This generalizes a result of Lang on bipartite graphs and a result of Pascasio on tight graphs.     

On the limit of a product of matrix exponentials

The existence of the limit as ε→0 exp[(A+B/ε)t] exp[-Bt/ε]is studied for n×n matrices A,B. Necessary and sufficient conditions on B that the limit exist for all A are given.     

Compatibilité entre structures d''intervalles et relations d''orde

Compatibility between interval structures and partial orderings.

If H=(X,E) is a hypergraph, n the cardinality of X,I_n the ordered set {1..n} and < an order relation on X, we call F(X,<) the set of the one-to-one functions from X to I_n which are compatible with <. If AI_n we denote by (A) the length of the smallest interval of I_n which contains A.

We first deal with the following problem: Find ƒF(X,<) which minimise . The a_e, eR are positive coefficients.

This problem can be understood as a scheduling problem and is checked to be NP-complete. We learn how to recognize in polynomial time those hypergraphs H=(X,E) which induce an optimal value of z min equal to .

Next we work on a dual question which arises about interval graphs, when some partial orderings on the vertex set of these graphs intend to represent inclusion, overlapping or anteriority relations between closed intervals of the real line.     

The spatial diffusion of diseases

Using the linear theory of semigroups, existence and uniqueness of the solution are studied for an S → E → I model which takes into account spatial inhomogeneity, nonlocal interactions, and an open population.     

A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model

For the pth-order linear ARCH model,

, where ₀ > 0, _i 0, I = 1, 2, …, p, {_t} is an i.i.d. normal white noise with E_t = 0, E_t² = 1, and _t is independent of {X_s, s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, ₁ + ₂ + ··· + _p < 1. In this note, we assume that _t has the probability density function p(t) which is positive and lower-semicontinuous over the real line, but not necessarily Gaussian, then the geometric ergodicity of the ARCH(p) process is proved under E_t² = 1. When _t has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.     

Maximal chordal subgraphs

An algorithm for finding maximal chordal subgraphs is developed that has worst-case time complexity of O(|E|Δ), where |E| is the number of edges in G and Δ is the maximum vertex degree in G. The study of maximal chordal subgraphs is motivated by their usefulness as computationally efficient structures with which to approximate a general graph. Two examples are given that illustrate potential applications of maximal chordal subgraphs. One provides an alternative formulation to the maximum independent set problem on a graph. The other involves a novel splitting scheme for solving large sparse systems of linear equations.