On permutation characters of wreath products

It is known that the character rings of symmetric groups S_n and the character rings of hyperoctahedral groups S₂?S_n are generated by (transitive) permutation characters. These results of Young are generalized to wreath products G?H (G a finite group, H a permutation group acting on a finite set). It is shown that the character ring of G?H is generated by permutation characters if this holds for G, H and certain subgroups of H. This result can be sharpened for wreath products G?S_n;if the character ring of G has a basis of transitive permutation characters, then the same holds for the character ring of G?S_n.     

Some generalizations of torsion-free Crawley groups

In this paper we investigate two new classes of torsion-free Abelian groups which arise in a natural way from the notion of a torsion-free Crawley group. A group G is said to be an Erd?s group if for any pair of isomorphic pure subgroups H,K with G/H ? G/K, there is an automorphism of G mapping H onto K; it is said to be a weak Crawley group if for any pair H,K of isomorphic dense maximal pure subgroups, there is an automorphism mapping H onto K. We show that these classes are extensive and pay attention to the relationship of the Baer-Specker group to these classes. In particular, we show that the class of Crawley groups is strictly contained in the class of weak Crawley groups and that the class of Erd?s groups is strictly contained in the class of weak Crawley groups.     

A generalization of the Weierstrass semigroup

The Weierstrass semigroup H(P) is well known and has been studied. Recently there has been a renewed interest in these semigroups because of applications in coding theory. Generalizations of the Weierstrass semigroup H(P) to n-tuples P₁,…,P_n have been made and studied. We will state and study another possible generalization.     

Polynomial cases of graph decomposition: A complete solution of Holyer’s problem

Let H be a fixed graph. A graph G has an H-decomposition if the edge set of G can be partitioned into subsets inducing graphs isomorphic to H. Let P_H denote the following decision problem: “Does an instance graph G admit H-decomposition?” In this paper we prove that the problem P_H is polynomial time solvable if H is a graph whose every component has at most 2 edges. This way we complete a solution of Holyer’s problem which is the problem of classifying the problems P_H according to their computational complexities.     

Subdivisions of graphs: A generalization of paths and cycles

One of the basic results in graph theory is Dirac's theorem, that every graph of order n?3 and minimum degree ?n/2 is Hamiltonian. This may be restated as: if a graph of order n and minimum degree ?n/2 contains a cycle C then it contains a spanning cycle, which is just a spanning subdivision of C. We show that the same conclusion is true if instead of C, we choose any graph H such that every connected component of H is non-trivial and contains at most one cycle. The degree bound can be improved to (n-t)/2 if H has t components that are trees.We attempt a similar generalization of the Corrádi-Hajnal theorem that every graph of order ?3k and minimum degree ?2k contains k disjoint cycles. Again, this may be restated as: every graph of order ?3k and minimum degree ?2k contains a subdivision of kK₃. We show that if H is any graph of order n with k components, each of which is a cycle or a non-trivial tree, then every graph of order ?n and minimum degree ?n-k contains a subdivision of H.     

The connected Ramsey number

A graph G is totally connected if both G and ? (its complement) are connected. The connected Ramsey number r_c(F, H) is the smallest integer k ? 4 so that if G is a totally connected graph of order k then either F ? G or H ? ?. We show that if neither of F nor H contains a bridge, then r_c = r(F, H), the usual generalized Ramsey number of F and H. We compute r_c (P_mP_m), the connected Ramsey number for paths.     

Extremals in some classes of Carnot groups

Let G be a Carnot group and D={e 1,e 2 } be a bracket generating left invariant distribution on G.In this paper,we obtain two main results.We first prove that there only exist normal minimizers in G if the type of D is (2,1,...,1) or (2,1,...,1,2).This immediately leads to the fact that there are only normal minimizers in the Goursat manifolds.As one corollary,we also obtain that there are only normal minimizers when dim G 5.We construct a class of Carnot groups such as that of type (2,1,...,1,2,n 0,...,n a) with n 0 1,n i 0,i=1,...,a,in which there exist strictly abnormal extremals.This implies that,for any given manifold of dimension n 6,we can find a class of n-dimensional Carnot groups having strictly abnormal minimizers.We conclude that the dimension n=5 is the border line for the existence and nonexistence of strictly abnormal extremals.Our main technique is based on the equations for the normal and abnormal extremals.     

Ramsey-minimal graphs for star-forests

It is shown that if G and H are star-forests with no single edge stars, then (G, H) is Ramsey-finite if and only if both G and H are single stars with an odd number of edges. Further (S_m ∪kS₁, S_n ∪tS₁) is Ramsey-finite when m and n are odd, where S_i denotes a star with i edges. In general, for G and H star-forests, (G∪kS₁, H∪tS₁) can be shown to be Ramsey-finite or Ramsey-infinite depending on the choice of G, H, k, and l with the general case unsettled. This disproves the conjecture given in [2] where it is suggested that the pair of graphs (L, M) is Ramsey-finite if and only if (1) either L or M is a matching, or (2) both L and M are star-forests of the type S_m ∪kS₁, m odd and k ? 0.     

On characterizing equilibrium points in two person strictly competitive games

A game is strictly competitive if all possible outcomes are Pareto optimal. It has been known that ifs′ = (s′ ₁,s′ ₂) ands″ = (s″ ₁ s″ ₂) are equilibrium points in a two person strictly competitive game, then payoffs are the same ats′ as ats″ and that (s′ ₁,s″ ₂) and (s″ ₁ s′ ₂) are equilibrium points as well. It is proved in this paper that, for 0 ?k?1,ks′ + (1?k)s″ is also an equilibrium point.     

Join of two graphs admits a nowhere-zero 3-flow

Let G be a graph, and λ the smallest integer for which G has a nowherezero λ-flow, i.e., an integer λ for which G admits a nowhere-zero λ-flow, but it does not admit a (λ ? 1)-flow. We denote the minimum flow number of G by Λ(G). In this paper we show that if G and H are two arbitrary graphs and G has no isolated vertex, then Λ(G ∨ H) ? 3 except two cases: (i) One of the graphs G and H is K ₂ and the other is 1-regular. (ii) H = K ₁ and G is a graph with at least one isolated vertex or a component whose every block is an odd cycle. Among other results, we prove that for every two graphs G and H with at least 4 vertices, Λ(G ∨ H) ? 3.     

Weierstrass points and Z2 homology

In a recent paper (1993), Lustig established a beautiful connection between the six Weierstrass points on a Riemann surface M₂ of genus 2 and intersection points of closed geodesics for the associated hyperbolic metric. As a consequence, he was able to construct an action of the mapping class group Out(π₁M₂) of M₂ on the set of Weierstrass points of M₂ and a virtual splitting of the natural homomorphism Aut(π₁M₂) → Out(π₁M₂). Our discussion in this paper begins with the observation that these two results of Lustig's are direct consequences of the work of Birman and Hilden (1973) on equivariant homotopies for surface homeomorphisms.It is well known that Γ₂ acts naturally on the Z₂ symplectic vector space of rank 4, H₁(M₂, Z₂). We identify this action with Lustig's action by constructing a natural correspondence between pairs of distinct Weierstrass points on M₂ and nonzero elements in H₁(M₂,Z₂). In this manner, the well-known exceptional isomorphism of finite group theory, S₆ ≅ Sp(4, Z₂), arises from a natural isomorphism of Γ₂ spaces.     

On the unique continuation problem for CR mappings into nonminimal hypersurfaces

We prove the following results on the unique continuation problem for CR mappings between real smooth hypersurfaces in ? ⁿ . If the CR mappingH extends holomorphically to one side of the source manifoldM near the pointp ₀ εM, the target manifoldM′ contains a holomorphic hypersurface σ′ throughp′₀ =H(p ₀ (i.e.,M′ is nonminimal atp′ ₀), andH(M) ? Σ′ (forcingM to be nonminimal atp ₀), then the transversal component ofH is not flat atp ₀. Furthermore, we show that the assumption thatH extends holomorphically to one side ofM cannot be removed in general. Indeed, we give an example of a smooth CR mappingH, withM, M′ ? ?², real analytic and of infinite type atp ₀ andp′₀ respectively (without being Levi flat), such thatH(M) ? Σ′ but the transversal component ofH is flat atp ₀ (in particular,H is not real analytic!). However, we show that ifM andM′ are assumed to be real analytic, and if the sourceM is “sufficiently far from being Levi flat” in a certain sense (so as to exclude the above mentioned counterexample) then the assumption thatH extends holomorphically to one side ofM can be dropped. Also, in the general case, we prove that the rate of vanishing of the transversal component cannot be too rapid (unlessH(M) ? Σ′), and we relate the possible rate of vanishing to the order of vanishing of the Levi form on a certain holomorphic submanifold ofM.     

On Petersen's graph theorem

In this paper we prove the following: let G be a graph with e_G edges, which is (k ? 1)-edge- connected, and with all valences ?k. Let 1?r?k be an integer, then G contains a spanning subgraph H, so that all valences in H are ?r, with no more than ?re_G?k? edges. The proof is based on a useful extension of Tutte's factor theorem [4,5], due to Lovász [3]. For other extensions of Petersen's theorem, see [6,7,8].     

Note on the uniquely colorable graphs

For n ≥ 3, if there exists a uniquely n colorable graph which contains no subgraph isomorphic to K₃, then the number of points in the graph must be strictly greater than n²+n?1.     

H∞ estimation for fuzzy membership function optimization

Given a fuzzy logic system, how can we determine the membership functions that will result in the best performance? If we constrain the membership functions to a specific shape (e.g., triangles or trapezoids) then each membership function can be parameterized by a few variables and the membership optimization problem can be reduced to a parameter optimization problem. The parameter optimization problem can then be formulated as a nonlinear filtering problem. In this paper we solve the nonlinear filtering problem using H_∞ state estimation theory. However, the membership functions that result from this approach are not (in general) sum normal. That is, the membership function values do not add up to one at each point in the domain. We therefore modify the H_∞ filter with the addition of state constraints so that the resulting membership functions are sum normal. Sum normality may be desirable not only for its intuitive appeal but also for computational reasons in the real time implementation of fuzzy logic systems. The methods proposed in this paper are illustrated on a fuzzy automotive cruise controller and compared to Kalman filtering based optimization.     

The global dimension of ω-smash coproducts

We mainly study the global dimension of ω-smash coproducts. We show that if H is a Hopf algebra with a bijective antipode S _H , and C _ω ? H denotes the ω-smash coproduct, then gl.dim(C _ω ? H) ≤ gl.dim(C) + gl.dim(H), where gl.dim(H) denotes the global dimension of H as a coalgebra.     

On the continuity of Mackey's extension process

If N is a closed normal subgroup of a locally compact group H, and if L is an H-invariant unitary irreducible representation of N, then it is known that there exists an irreducible unitary representation ? of H whose restriction to N is a multiple of L. The representation ? is important in Mackey's theory, but it is not constructively defined and it is often difficult to determine ? explicitly. In this paper the continuity properties of the assignment L to ? are studied. The kinds of continuity conditions possible are investigated and a theorem for a special case is proved.     

An existence theorem for antichains

Let k₁, k₂,…, k_n be given integers, 1 ? k₁ ? k₂ ? … ? k_n, and let S be the set of vectors x = (x₁,…, x_n) with integral coefficients satisfying 0 ? x_i ? k_i, i = 1, 2, 3,…, n. A subset H of S is an antichain (or Sperner family or clutter) if and only if for each pair of distinct vectors x and y in H the inequalities x_i ? y_i, i = 1, 2,…, n, do not all hold. Let |H| denote the number of vectors in H, let K = k₁ + k₂ + … + k_n and for 0 ? l ? K let (l)H denote the subset of H consisting of vectors h = (h₁, h₂,…, h_n) which satisfy h₁ + h₂ + … + h_n = l. In this paper we show that if H is an antichain in S, then there exists an antichain H′ in S for which |(l)H′| = 0 if $\text{l <}\text{K}\text{2}$, $\text{|(}\text{K}\text{2}\text{)H′| = |(}\text{K}\text{2}\text{)H|}$ if K is even and |(l)H′| = |(l)H| + |(K ? l)H| if $\text{l>}\text{K}\text{2}$.     

Nonlinear reaction-diffusion models for interacting populations

This paper proves that several initial-boundary value problems for a wide class of nonlinear reaction-diffusion equations have solutions c_i(x, t), 1 ? i ? N (with c_i(x, t) representing the concentration of the ith species at position x in a set $\text{Ω}$ at time t ? 0), which exist for all t ? 0 and are unique, smooth, nonnegative, and strictly positive for t > 0. The Volterra-Lotka predator-prey model with diffusion (to which the results above are proved to apply) is then studied in more detail. It is proved that any bounded solution of this model loses its spatial dependence and behaves like a periodic function of time alone as t → ∞. It is proved that if the spatial dimension is one or if the diffusion coefficients of the two species are equal, then all solutions are bounded.