Permutation polynomials of the form

Recently, several classes of permutation polynomials of the form (x²+x+δ)^s+x over have been discovered. They are related to Kloosterman sums. In this paper, the permutation behavior of polynomials of the form (x^p−x+δ)^s+L(x) over is investigated, where L(x) is a linearized polynomial with coefficients in . Six classes of permutation polynomials on are derived. Three classes of permutation polynomials over are also presented.     

On Ulyanov inequalities in Banach spaces and semigroups of linear operators

Let X,Y be Banach spaces and {T(t):t≥0} be a consistent, equibounded semigroup of linear operators on X as well as on Y; it is assumed that {T(t)} satisfies a Nikolskii type inequality with respect to X and Y:T(2t)f_Y(t)T(t)f_X. Then an abstract Ulyanov type inequality is derived between the (modified) K-functionals with respect to (X,D_X((-A)^α)) and (Y,D_Y((-A)^α)),α>0, where A is the infinitesimal generator of {T(t)}. Particular choices of X,Y are Lorentz–Zygmund spaces, of {T(t)} are those connected with orthogonal expansions such as Fourier, spherical harmonics, Jacobi, Laguerre, Hermite series. Known characterizations of the K-functionals lead to concrete Ulyanov type inequalities. In particular, results of Ditzian and Tikhonov in the case , are partly covered.     

Every 4-connected line graph of a quasi claw-free graph is hamiltonian connected

Let G be a graph. For u,vV(G) with dist_G(u,v)=2, denote J_G(u,v)={wN_G(u)∩N_G(v)|N_G(w)N_G(u)N_G(v){u,v}}. A graph G is called quasi claw-free if J_G(u,v)≠ for any u,vV(G) with dist_G(u,v)=2. In 1986, Thomassen conjectured that every 4-connected line graph is hamiltonian. In this paper we show that every 4-connected line graph of a quasi claw-free graph is hamiltonian connected.     

The rainbow number of matchings in regular bipartite graphs

Given a graph G and a subgraph H of G, let rb(G,H) be the minimum number r for which any edge-coloring of G with r colors has a rainbow subgraph H. The number rb(G,H) is called the rainbow number of H with respect to G. Denote as mK₂ a matching of size m and as B_n,k the set of all the k-regular bipartite graphs with bipartition (X,Y) such that X=Y=n and k≤n. Let k,m,n be given positive integers, where k≥3, m≥2 and n>3(m−1). We show that for every GB_n,k, rb(G,mK₂)=k(m−2)+2. We also determine the rainbow numbers of matchings in paths and cycles.     

Orthogonal double covers of Cayley graphs

Let X and G be graphs, such that G is isomorphic to a subgraph of X.An orthogonal double cover (ODC) of X by G is a collection of subgraphs of X, all isomorphic with G, such that (i) every edge of X occurs in exactly two members of and (ii) and share an edge if and only if x and y are adjacent in X. The main question is: given the pair (X,G), is there an ODC of X by G? An obvious necessary condition is that X is regular.A technique to construct ODCs for Cayley graphs is introduced. It is shown that for all (X,G) where X is a 3-regular Cayley graph on an abelian group there is an ODC, a few well known exceptions apart.     

The Diameter of Sparse Random Graphs

We consider the diameter of a random graph G(n, p) for various ranges of p close to the phase transition point for connectivity. For a disconnected graph G, we use the convention that the diameter of G is the maximum diameter of its connected components. We show that almost surely the diameter of random graph G(n, p) is close to if np → ∞. Moreover if , then the diameter of G(n, p) is concentrated on two values. In general, if , the diameter is concentrated on at most 21/c₀ + 4 values. We also proved that the diameter of G(n, p) is almost surely equal to the diameter of its giant component if np > 3.6.     

LOCAL TIME ANALYSIS OF ADDITIVE LEVY PROCESSES WITH DIFFERENT L(E)VY EXPONENTS

Let X1 XN be independent, classical Levy processes on R^d with Levy exponents ψ1,…, ψN, respectively. The corresponding additive Levy process is defined as the following N-parameter random field on R^d, X（t） △= X1（t1） ＋ ... ＋ XN（tN）, At∈N. Under mild regularity conditions on the ψi＇s, we derive estimate for the local and uniform moduli of continuity of local times of X = {X（t）; t ∈R^N}.     

On groups with conjugacy classes of distinct sizes

A finite group G is called an ah-group if any two distinct conjugacy classes of G have distinct cardinality. We show that if G is an ah-group, then the non-abelian socle of G is isomorphic to one of the following:

1. , for 1a5, a≠2.

2. A₈.

3. PSL(3,4)^e, for 1e10.

4. A₅×PSL(3,4)^e, for 1e10.

Based on this result, we virtually show that if G is an ah-group with π(G) 2,3,5,7 , then F(G)≠1, or equivalently, that G has an abelian normal subgroup.In addition, we show that if G is an ah-group of minimal size which is not isomorphic to S₃, then the non-abelian socle of G is either trivial or isomorphic to one of the following:

1. , for 3a5.

2. PSL(3,4)^e, for 1e10.

Our research lead us to interesting results related to transitivity and homogeneousity in permutation groups, and to subgroups of wreath products of form Z₂S_n. These results are of independent interest and are located in appendices for greater autonomy.     

Estimation de Yule–Walker d'un CAR(p) observé à temps discret

Let (X(lδ), l=0,n) be a discrete observation at mesh δ>0 of X, a CAR(p). Classical Yule–Walker estimation are biased and must be corrected. Resultant estimators converge if T=nδ→+∞, are asymptotically normal with rate , and efficient. The diffusion coefficient is also estimated, with rate .     

Finding induced trees

Let G=(V(G),E(G)) be a finite connected undirected graph and WV(G) a subset of vertices. We are searching for a subset XV(G) such that WX and the subgraph induced on X is a tree. -completeness results and polynomial time algorithms are given assuming that the cardinality of W is fixed or not. Besides we give complexity results when X has to induce a path or when G is a weighted graph. We also consider problems where the cardinality of X has to be minimized.     

Le lemme de Schur pour les représentations orthogonales

Let σ be an orthogonal representation of a group G on a real Hilbert space. We show that σ is irreducible if and only if its commutant σ(G)' is isomorphic to , or . This result is an analogue of the classical Schur lemma for unitary representations. In both cases (orthogonal and unitary), a representation is irreducible if and only if its commutant is a field. If σ is irreducible, we show that there exists a unitary irreducible representation π of G such that the complexification σ is unitarily equivalent to π if σ(G)' , to π π̄ if σ(G)' , and to π π if σ(G)' (here π̄ denotes the contragredient representation of π). These results are classical for a finite-dimensional σ, but seem to be new in the general case.     

Symplectic geometry of semisimple orbits

Let G be a complex semisimple group, T G a maximal torus and B a Borel subgroup of G containing T. Let Ω be the Kostant-Kirillov holomorphic symplectic structure on the adjoint orbit O = Ad(G)c G/Z(c), where c Lie(T), and Z(c) is the centralizer of c in G. We prove that the real symplectic form Re Ω (respectively, Im Ω) on O is exact if and only if all the eigenvalues ad(c) are real (respectively, purely imaginary). Furthermore, each of these real symplectic manifolds is symplectomorphic to the cotangent bundle of the partial flag manifold G/Z(cc)B, equipped with the Liouville symplectic form. The latter result is generalized to hyperbolic adjoint orbits in a real semisimple Lie algebra.     

Small subsets of groups

Given an infinite group G and an infinite cardinal κ|G|, we say that a subset A of G is κ-large (κ-small) if there exists F[G]^<κ such that G=FA (GFA is κ-large for each F[G]^<κ). The subject of the paper is the family of all κ-small subsets. We describe the left ideal of the right topological semigroup βG determined by . We study interrelations between κ-small and other (P_κ-small and κ-thin) subsets of groups, and prove that G can be generated by some 2-thin subsets. We partition G in countable many subsets which are κ-small for each κω. We show that [G]^<κ is dual to provided that either κ is regular and κ=|G|, or G is Abelian and κ is a limit cardinal, or G is a divisible Abelian group.     

On the Norm of the Metric Projections

LetXbe a Banach space. GivenMa subspace ofXwe denote withP_Mthe metric projection ontoM. We defineπ(X) sup{P_M: Ma proximinal subspace ofX}. In this paper we give a bound forπ(X). In particular, whenX=L_p, we obtain the inequality P_M2^|2/p−1|, for every subspaceMofL_p. We also show thatπ(X)=π(X*).     

Equivalence relation groupoids associated with certain linearly ordered dimension groups

We consider linearly ordered, Archimedean dimension groups (G,G⁺,u) for which the group G/u is torsion-free. It will be shown that if, in addition, G/u is generated by a single element (i.e., ), then (G,G⁺,u) is isomorphic to for some irrational number τ(0,1). This amounts to an extension of related results where dimension groups for which G/u is torsion were considered. We will prove, in the case of the Fibonacci dimension group, that these results can be used to directly construct an equivalence relation groupoid whose C^*-algebra is the Fibonacci C^*-algebra.     

About equivalent interval colorings of weighted graphs

Given a graph G=(V,E) with strictly positive integer weights ω_i on the vertices iV, a k-interval coloring of G is a function I that assigns an interval I(i){1,…,k} of ω_i consecutive integers (called colors) to each vertex iV. If two adjacent vertices x and y have common colors, i.e. I(i)∩I(j)≠0/ for an edge [i,j] in G, then the edge [i,j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χ_int(G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ω_i=1 for all vertices iV).Two k-interval colorings I₁ and I₂ are said equivalent if there is a permutation π of the integers 1,…,k such that ℓI₁(i) if and only if π(ℓ)I₂(i) for all vertices iV. As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions.     

On non-solvable Camina pairs

In this paper we study non-solvable and non-Frobenius Camina pairs (G,N). It is known [D. Chillag, A. Mann, C. Scoppola, Generalized Frobenius groups II, Israel J. Math. 62 (1988) 269–282] that in this case N is a p-group. Our first result (Theorem 1.3) shows that the solvable residual of G/O_p(G) is isomorphic either to SL(2,p^e),p is a prime or to SL(2,5), SL(2,13) with p=3, or to SL(2,5) with p7.Our second result provides an example of a non-solvable and non-Frobenius Camina pair (G,N) with |O_p(G)|=5⁵ and G/O_p(G)SL(2,5). Note that G has a character which is zero everywhere except on two conjugacy classes. Groups of this type were studies by S.M. Gagola [S.M. Gagola, Characters vanishing on all but two conjugacy classes, Pacific J. Math. 109 (1983) 363–385]. To our knowledge this group is the first example of a Gagola group which is non-solvable and non-Frobenius.     

Characterisations of Chebyshev sets in c0

A subset MX of a normed linear space X is a Chebyshev set if, for every xX, the set of all nearest points from M to x is a singleton. We obtain a geometrical characterisation of approximatively compact Chebyshev sets in c₀. Also, given an approximatively compact Chebyshev set M in c₀ and a coordinate affine subspace Hc₀ of finite codimension, if M∩H≠, then M∩H is a Chebyshev set in H, where the norm on H is induced from c₀.