On hyper‐torre isols

In this paper we present a contribution to a classical result of E. Ellentuck in the theory of regressive isols. E. Ellentuck introduced the concept of a hyper‐torre isol, established their existence for regressive isols, and then proved that associated with these isols a special kind of semi‐ring of isols is a model of the true universal‐recursive statements of arithmetic. This result took on an added significance when it was later shown that for regressive isols, the property of being hyper‐torre is equivalent to being hereditarily odd‐even. In this paper we present a simplification to the original proof for establishing that equivalence. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Conditional Resolvability of Honeycomb and Hexagonal Networks

Given a graph G = (V, E), a set W í V{W \subseteq V} is said to be a resolving set if for each pair of distinct vertices u, v ? V{u, v \in V} there is a vertex x in W such that d(u, x) 1 d(v, x){d(u, x) \neq d(v, x)} . The resolving number of G is the minimum cardinality of all resolving sets. In this paper, conditions are imposed on resolving sets and certain conditional resolving parameters are studied for honeycomb and hexagonal networks.     

On the point-stabiliser in a transitive permutation group

If V is a (possibly infinite) set, G a permutation group on V, v ? V{V, v\in V}, and Ω is an orbit of the stabiliser G _v , let G_v^W{G_v^{\Omega}} denote the permutation group induced by the action of G _v on Ω, and let N be the normaliser of G in Sym(V). In this article, we discuss a relationship between the structures of G _v and G_v^W{G_v^\Omega}. If G is primitive and G _v is finite, then by a theorem of Betten et al. (J Group Theory 6:415–420, 2003) we can conclude that every composition factor of the group G _v is also a composition factor of the group G_v^W(v){G_v^{\Omega(v)}}. In this paper we generalize this result to possibly imprimitive permutation groups G with infinite vertex-stabilisers, subject to certain restrictions that can be expressed in terms of the natural permutation topology on Sym(V). In particular, we show the following: If W = u^G_v{\Omega=u^{G_v}} is a suborbit of a transitive closed subgroup G of Sym(V) with a normalizing overgroup N ≤ N _Sym(V)(G) such that the N-orbital {(v^g,u^g) | u ? W, g ? N}{\{(v^g,u^g) \mid u\in \Omega, g\in N\}} is locally finite and strongly connected (when viewed as a digraph on V), then every closed simple section of G _v is also a section of G_v^W{G_v^\Omega}. To demonstrate that the topological assumptions on G and the simple sections of G _v cannot be omitted in this statement, we give an example of a group G acting arc-transitively on an infinite cubic tree, such that the vertex-stabiliser G _v is isomorphic to the modular group PSL(2,\mathbbZ) @ C₂*C₃{{\rm PSL}(2,\mathbb{Z}) \cong C_2*C_3}, which is known to have infinitely many finite simple groups among its sections.     

On signpost systems and connected graphs

By a signpost system we mean an ordered pair (W, P), where W is a finite nonempty set, P W × W × W and the following statements hold: if (u, v, w) P, then (v, u, u) P and (v, u, w) P, for all u, v, w W; if u v; then there exists r W such that (u, r, v) P, for all u, v W. We say that a signpost system (W, P) is smooth if the folowing statement holds for all u, v, x, y, z W: if (u, v, x), (u, v, z), (x, y, z) P, then (u, v, y) P. We say thay a signpost system (W, P) is simple if the following statement holds for all u, v, x, y W: if (u, v, x), (x, y, v) P, then (u, v, y), (x, y, u) P.By the underlying graph of a signpost system (W, P) we mean the graph G with V(G) = W and such that the following statement holds for all distinct u, v W: u and v are adjacent in G if and only if (u, v, v) P. The main result of this paper is as follows: If G is a graph, then the following three statements are equivalent: G is connected; G is the underlying graph of a simple smooth signpost system; G is the underlying graph of a smooth signpost system.Research was supported by Grant Agency of the Czech Republic, grant No. 401/01/0218.     

Combinatorial Isols and the Arithmetic of Dekker Semirings

In his long and illuminating paper [1] Joe Barback defined and showed to be non‐vacuous a class of infinite regressive isols he has termed “complete y torre” (CT) isols. These particular isols a enjoy a property that Barback has since labelled combinatoriality. In [2], he provides a list of properties characterizing the combinatoria isols. In Section 2 of our paper, we extend this list of characterizations to include the fact that an infinite regressive isol X is combinatorial if and only if its associated Dekker semiring D (X) satisfies all those Π₂ sentences of the anguage L_N for isol theory that are true in the set ω of natural numbers. (Moreover, with X combinatorial, the interpretations in D(X)of the various function and relation symbols of L_N via the “lifting ” to D(X) of their Σ₁ definitions in ω coincide with their interpretations via isolic extension.) We also note in Section 2 that Π₂(L)‐correctness, for semirings D(X), cannot be improved to Π ₃(L)‐correctness, no matter how many additional properties we succeed in attaching to a combinatoria isol; there is a fixed (L) sentence that blocks such extension. (Here L is the usual basic first‐order language for arithmetic.) In Section 3, we provide a proof of the existence of combinatorial isols that does not involve verification of the extremely strong properties that characterize Barback's CT isols.     

Multiple solutions for a superlinear and periodic elliptic system on \mathbbRN{\mathbb{R}^N}

This paper is concerned with the following periodic Hamiltonian elliptic system

Nonexistence of certain symmetric spherical codes

A spherical 1-codeW is any finite subset of the unit sphere inn dimensionsS ^n–1, for whichd(u, v)1 for everyu, v fromW, uv. A spherical 1-code is symmetric ifuW implies –uW. The best upper bounds in the size of symmetric spherical codes onS ^n–1 were obtained in [1]. Here we obtain the same bounds by a similar method and improve these bounds forn=5, 10, 14 and 22.     

Global Existence and Blow-Up for a Class of Degenerate Parabolic Systems with Localized Source

This paper deals with a class of localized and degenerate quasilinear parabolic systems

Pan-connectedness of graphs with large neighborhood unions

Let G be a simple graph with n vertices. For any v ? V(G){v \in V(G)} , let N(v)={u ? V(G): uv ? E(G)}{N(v)=\{u \in V(G): uv \in E(G)\}} , NC(G) = min{|N(u) èN(v)|: u, v ? V(G){NC(G)= \min \{|N(u) \cup N(v)|: u, v \in V(G)} and uv \not ? E(G)}{uv \not \in E(G)\}} , and NC₂(G) = min{|N(u) èN(v)|: u, v ? V(G){NC_2(G)= \min\{|N(u) \cup N(v)|: u, v \in V(G)} and u and v has distance 2 in E(G)}. Let l ≥ 1 be an integer. A graph G on n ≥ l vertices is [l, n]-pan-connected if for any u, v ? V(G){u, v \in V(G)} , and any integer m with l ≤ m ≤ n, G has a (u, v)-path of length m. In 1998, Wei and Zhu (Graphs Combinatorics 14:263–274, 1998) proved that for a three-connected graph on n ≥ 7 vertices, if NC(G) ≥ n − δ(G) + 1, then G is [6, n]-pan-connected. They conjectured that such graphs should be [5, n]-pan-connected. In this paper, we prove that for a three-connected graph on n ≥ 7 vertices, if NC ₂(G) ≥ n − δ(G) + 1, then G is [5, n]-pan-connected. Consequently, the conjecture of Wei and Zhu is proved as NC ₂(G) ≥ NC(G). Furthermore, we show that the lower bound is best possible and characterize all 2-connected graphs with NC ₂(G) ≥ n − δ(G) + 1 which are not [4, n]-pan-connected.     

The behaviour of microstructures with small shears of the austenite-martensite interface in martensitic phase transformations

Let W ì \BbbR²\Omega \subset \Bbb{R}^2 denote a bounded domain whose boundary ?W\partial \Omega is Lipschitz and contains a segment G₀\Gamma_0 representing the austenite-twinned martensite interface. We prove inf_{u ? W(W)} ò_W j(?u(x,y))dxdy=0\displaystyle{\inf_{{u\in \cal W}(\Omega)} \int_\Omega \varphi(\nabla u(x,y))dxdy=0}     

Embeddings of resolvable group divisible designs with block size 3

It is proved in this paper that an RGD(3, g;v) can be embedded in an RGD(3, g;u) if and only if , , , v ≥ 3g, u ≥ 3v, and (g,v) ≠ (2,6),(2,12),(6,18).     

Graph Connectivity After Path Removal

Let G be a graph and u, v be two distinct vertices of G. A u—v path P is called nonseparating if G—V(P) is connected. The purpose of this paper is to study the number of nonseparating u—v path for two arbitrary vertices u and v of a given graph. For a positive integer k, we will show that there is a minimum integer (k) so that if G is an (k)-connected graph and u and v are two arbitrary vertices in G, then there exist k vertex disjoint paths P ₁[u,v], P ₂[u,v], . . ., P _k [u,v], such that G—V (P _i [u,v]) is connected for every i (i = 1, 2, ..., k). In fact, we will prove that (k) 22k+2. It is known that (1) = 3.. A result of Tutte showed that (2) = 3. We show that (3) = 6. In addition, we prove that if G is a 5-connected graph, then for every pair of vertices u and v there exists a path P[u, v] such that G—V(P[u, v]) is 2-connected.* Supported by NSF grant No. DMS-0070059 Supported by ONR grant N00014-97-1-0499 Supported by NSF grant No. 9531824     

The zero dispersion limit of the korteweg-de vries equation for initial potentials with non-trivial reflection coefficient

The inverse scattering method is used to determine the distribution limit as ? → 0 of the solution u(x, t, ?) of the initial value problem. U_t ? 6uu_x + ?²u_xxx = 0, u(x, 0) = v(x), where v(x) is a positive bump which decays sufficiently fast as x x→±α. The case v(x) ? 0 has been solved by Peter D. Lax and C. David Levermore [8], [9], [10]. The computation of the distribution limit of u(x, t, ?) as ? → 0 is reduced to a quadratic maximization problem, which is then solved.     

On a functional equation related to income inequality measures

The functional equationg(u, x)+g(v, y)=g(u, y)+g(v, x) for allu, v, x, y>0 withu+v=x+y is initiated by F. A. Cowell and A. F. Shorrocks in their research on the aggregation of inequality indices. We solve the equation by extension theorems.Dedicated to Professor Janos Aczél on his 60th birthday     

n-cubes and median graphs

The n-cube is characterized as a connected regular graph in which for any three vertices u, v, and w there is a unique vertex that lies simultaneously on a shortest (u, v)-path, a shortest (v, w)-path, and a shortest (w, u)-path.     

Global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth

The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}The aim of this study is to prove global existence of classical solutions for systems of the form \frac?u?t -a Du=-f(u,v){\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)} , \frac?v?t -b Dv=g(u,v){\frac{\partial v}{\partial t} -b \Delta v=g(u,v)} in (0, +∞) × Ω where Ω is an open bounded domain of class C ¹ in \mathbbRⁿ{\mathbb{R}^n}, a > 0, b > 0 and f, g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(x,h) ￡ C j(x)e^ahb{g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}}} and g(ξ, η) ≤ ψ(η)f(ξ, η) for some constants C > 0, α > 0 and β ≥ 1 where j{\varphi} and ψ are any nonnegative continuously differentiable functions on [0, +∞) such that j(0)=0{\varphi(0)=0} and lim_{h? +￥}h^b-1y(h) = l{ \lim_{\eta \rightarrow +\infty}\eta^{\beta -1}\psi(\eta)= \ell} where ℓ is a nonnegative constant. The asymptotic behavior of the global solutions as t goes to +∞ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates and Lyapunov functional methods.     

Global Lipschitz regularity for almost minimizers of asymptotically convex variational problems

We prove global, up to the boundary of a domain ${{\it \Omega}\subset\mathbb {R}^n}We prove global, up to the boundary of a domain W ì \mathbb Rⁿ{{\it \Omega}\subset\mathbb {R}^n}, Lipschitz regularity results for almost minimizers of functionals of the form

u ? òW g(x, u(x), ?u(x)) dx.u \mapsto \int \limits_{\Omega} g(x, u(x), \nabla u(x))\,dx.      18. On the Approximation of Irrationals by Rationals      Carstenelsner 《Mathematische Nachrichten》1998,189(1):243-256 Let ξ be a real irrational number, and a ≧ 0, b ≧ 0, s > 1 be integers. A theorem of S. UCHIYAMA states that there are infinitely many pairs of integers u and v ≠ 0 such that OVBARξ?u/vOVBAR ≤ s2/4v2 and u ? a, v ? b mod s, provided that it is not a ? 6 ? 0 mod s. It is shown that this result is best-possible for all integers s > 1.      19. One-sided continuous dependence of maximal solutions      Eric Schechter 《Journal of Differential Equations》1981,39(3):413-425 We consider the maximal solution of $\text{u′(s) = ?(s, u(s))}$, where ? satisfies a one-sided variant of Carathéodory's conditions. A best-possible condition is proved for the dependence of u on ?. Also we show that a function v satisfies $\text{v(t) ? v(r) ? ?}{}_{\mathrm{r}}{}^{\mathrm{t}}\text{?(s, v(s))ds}$ if and only if v is dominated by the maximal solution u.      20. Ground state alternative for <Emphasis Type="Italic">p</Emphasis>-Laplacian with potential term      Yehuda Pinchover  Kyril Tintarev 《Calculus of Variations and Partial Differential Equations》2007,28(2):179-201 Let Ω be a domain in , d ≥ 2, and 1 < p < ∞. Fix . Consider the functional Q and its Gateaux derivative Q′ given by If Q ≥ 0 on, then either there is a positive continuous function W such that for all, or there is a sequence and a function v > 0 satisfying Q′ (v) = 0, such that Q(u k ) → 0, and in . In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every satisfying there exists a constant C > 0 such that . As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.

On the Approximation of Irrationals by Rationals

Let ξ be a real irrational number, and a ≧ 0, b ≧ 0, s > 1 be integers. A theorem of S. UCHIYAMA states that there are infinitely many pairs of integers u and v ≠ 0 such that OVBARξ?u/vOVBAR ≤ s²/4v² and u ? a, v ? b mod s, provided that it is not a ? 6 ? 0 mod s. It is shown that this result is best-possible for all integers s > 1.     

One-sided continuous dependence of maximal solutions

We consider the maximal solution of $\text{u′(s) = ?(s, u(s))}$, where ? satisfies a one-sided variant of Carathéodory's conditions. A best-possible condition is proved for the dependence of u on ?. Also we show that a function v satisfies $\text{v(t) ? v(r) ? ?}{}_{\mathrm{r}}{}^{\mathrm{t}}\text{?(s, v(s))ds}$ if and only if v is dominated by the maximal solution u.     

Ground state alternative for <Emphasis Type="Italic">p</Emphasis>-Laplacian with potential term

Let Ω be a domain in , d ≥ 2, and 1 < p < ∞. Fix . Consider the functional Q and its Gateaux derivative Q′ given by If Q ≥ 0 on, then either there is a positive continuous function W such that for all, or there is a sequence and a function v > 0 satisfying Q′ (v) = 0, such that Q(u _k ) → 0, and in . In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every satisfying there exists a constant C > 0 such that . As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.