Some Turán type results on the hypercube

For graphs H,G a classical problem in extremal graph theory asks what proportion of the edges of H a subgraph may contain without containing a copy of G. We prove some new results in the case where H is a hypercube. We use a supersaturation technique of Erd?s and Simonivits to give a characterization of a set of graphs such that asymptotically the answer is the same when G is a member of this set and when G is a hypercube of some fixed dimension. We apply these results to a specific set of subgraphs of the hypercube called Fibonacci cubes. Additionally, we use a coloring argument to prove new asymptotic bounds on this problem for a different set of graphs. Finally we prove a new asymptotic bound for the case where G is the cube of dimension 3.     

Hörmander-type multipliers on locally compact Vilenkin groups: theL 1(G)-case

In [3] and [4]Kitada presented Hörmander-type multiplier theorems for Lebesgue and Hardy spaces defined over a locally compact Vilenkin groupG. Like in the classical case, multipliers for the spaceL ¹(G) were not included in these results. In the present paper we discuss this particular case and we show how we need to modify the usual Hörmander multiplier condition to obtainL ¹ (G)-multipliers.     

On the fluid limit of the M/G/∞ queue

The paper deals with the fluid limits of some generalized M/G/∞ queues under heavy-traffic scaling. The target application is the modeling of Internet traffic at the flow level. Our paper first gives a simplified approach in the case of Poisson arrivals. Expressing the state process as a functional of some Poisson point process, an elementary proof for limit theorems based on martingales techniques and weak convergence results is given. The paper illustrates in the special Poisson arrivals case the classical heavy-traffic limit theorems for the G/G/∞ queue developed earlier by Borovkov and by Iglehart, and later by Krichagina and Puhalskii. In addition, asymptotics for the covariance of the limit Gaussian processes are obtained for some classes of service time distributions, which are useful to derive in practice the key parameters of these distributions.     

Integral properties of the classical warping function of a simply connected domain

Let u(z,G) be the classical warping function of a simply connected domain G. We prove that the L ^p -norms of the warping function with different exponents are related by a sharp isoperimetric inequality, including the functional u(G) = sup _x∈G u(x, G). A particular case of our result is the classical Payne inequality for the torsional rigidity of a domain. On the basis of the warping function, we construct a new physical functional possessing the isoperimetric monotonicity property. For a class of integrals depending on the warping function, we also obtain a priori estimates in terms of the L ^p -norms of the warping function as well as the functional u(G). In the proof, we use the estimation technique on level lines proposed by Payne.     

On the local case in the Aschbacher theorem for linear and unitary groups

We consider the subgroups H in a linear or a unitary group G over a finite field such that O _r (H) ? Z(G) for some odd prime r. We obtain a refinement of the well-known Aschbacher theorem on subgroups of classical groups for this case.     

A general Weyl-type integration formula for isometric group actions

We show that integration over a G-manifold M can be reduced to integration over a minimal section Σ with respect to an induced weighted measure and integration over a homogeneous space G/N. We relate our formula to integration formulæ for polar actions and calculate some weight functions. In the case of a compact Lie group acting on itself via conjugation, we obtain a classical result of Hermann Weyl.     

Classification and unfoldings of degenerate Hopf bifurcations

This paper initiates the classification, up to symmetry-covariant contact equivalence, of perturbations of local Hopf bifurcation problems which do not satisfy the classical non-degeneracy conditions. The only remaining hypothesis is that ±i should be simple eigenvalues of the linearized right-hand side at criticality. Then the Lyapunov-Schmidt method allows a reduction to a scalar equation G(x, λ) = 0, where G(?x, λ) = ?G(x, λ). A definition is given of the codimension of G, and a complete classification is obtained for all problems with codimension ?3, together with the corresponding universal unfoldings. The perturbed bifurcation diagrams are given for the cases with codimension ?2, and for one case with codimension 3; for this last case one of the unfolding parameters is a “modal” parameter, such that the topological codimension equals in fact 2. Formulas are given for the calculation of the Taylor coefficients needed for the application of the results, and finally the results are applied to two simple problems: a model of glycolytic oscillations and the Fitzhugh nerve equations.     

Fixed point property for Banach algebras associated to locally compact groups

In this paper we investigate when various Banach algebras associated to a locally compact group G have the weak or weak^∗ fixed point property for left reversible semigroups. We proved, for example, that if G is a separable locally compact group with a compact neighborhood of the identity invariant under inner automorphisms, then the Fourier-Stieltjes algebra of G has the weak^∗ fixed point property for left reversible semigroups if and only if G is compact. This generalizes a classical result of T.C. Lim for the case when G is the circle group T.     

INVARIANT HILBERT SCHEMES AND DESINGULARIZATIONS OF QUOTIENTS BY CLASSICAL GROUPS

Let W be a finite-dimensional representation of a reductive algebraic group G. The invariant Hilbert scheme $ \mathcal{H} $ is a moduli space that classifies the G-stable closed subschemes Z of W such that the affine algebra k[Z] is the direct sum of simple G-modules with prescribed multiplicities. In this article, we consider the case where G is a classical group acting on a classical representation W and k[Z] is isomorphic to the regular representation of G as a G-module. We obtain families of examples where $ \mathcal{H} $ is a smooth variety, and thus for which the Hilbert–Chow morphism $ \gamma :\mathcal{H}\to W//G $ is a canonical desingularization of the categorical quotient.     

Additive bases in groups

We study the problem of removing an element from an additive basis in a general abelian group. We introduce analogues of the classical functions X, S and E (defined in the case of ?) and obtain bounds on them. Our estimates on the functions S _G and E _G are valid for general abelian groups G while in the case of X _G we show that distinct types of behaviours may occur depending on G.     

The local structure of tangent G-vector fields

We introduce a notion of equivariant index in order to describe the behavior of tangent G-vector fields on smooth G-manifolds near isolated zeros. Our methods result in a calculation of the monoid of G-homotopy classes of self-maps of the unit sphere S(V) in a real orthogonal (finite dimensional) G-module V, this being the unstable analogue of a classical result of Segal. During the course of our calculation, we prove general position results on tangent G-vector fields and obtain canonical local structures for such fields.     

Graphic matroids,shellability and the Poincare Conjecture

In this paper we introduce a theory of edge shelling of graphs. Whereas the standard notion of shelling a simplicial complex involves a sequential removal of maximal simplexes, edge shelling involves a sequential removal of the edges of a graph. A necessary and sufficient condition for edge shellability is given in the case of 3-colored graphs, and it is conjectured that the result holds in general. Questions about shelling, and the dual notion of closure, are motivated by topological problems. The connection between graph theory and topology is by way of a complex ΔG associated with a graph G. In particular, every closed 2- or 3-manifold can be realized in this way. If ΔG is shellable, then G is edge shellable, but not conversely. Nevertheless, the condition that G is edge shellable is strong enough to imply that a manifold ΔG must be a sphere. This leads to completely graph-theoretic generalizations of the classical Poincaré Conjecture.     

Glider Representations of Group Algebra Filtrations of Nilpotent Groups

We continue the study of glider representations of finite groups G with given structure chain of subgroups e ? G ₁ ?… ? G _d = G. We give a characterization of irreducible gliders of essential length e ≤ d which in the case of p-groups allows to prove some results about classical representation theory. The paper also contains an introduction to generalized character theory for glider representations and an extension of the decomposition groups in the Clifford theory. Furthermore, we study irreducible glider representations for products of groups and nilpotent groups.     

An NP-Completeness Result of Edge Search in Graphs

Consider the following generalization of the classical sequential group testing problem for two defective items: suppose a graph G contains n vertices two of which are defective and adjacent. Find the defective vertices by testing whether a subset of vertices of cardinality at most p contains at least one defective vertex or not. What is then the minimum number c _p (G) of tests, which are needed in the worst case to find all defective vertices? In Gerzen (Discrete Math 309(20):5932–5942, 2009), this problem was partly solved by deriving lower and sharp upper bounds for c _p (G). In the present paper we show that the computation of c _p (G) is an NP-complete problem. In addition, we establish some results on c _p (G) for random graphs.     

Base sizes for S-actions of finite classical groups

Let G be a permutation group on a set Ω. A subset B of Ω is a base for G if the pointwise stabilizer of B in G is trivial; the base size of G is the minimal cardinality of a base for G, denoted by b(G). In this paper we calculate the base size of every primitive almost simple classical group with point stabilizer in Aschbacher’s collection S of irreducibly embedded almost simple subgroups. In this situation we also establish strong asymptotic results on the probability that randomly chosen subsets of Ω form a base for G. Indeed, with some specific exceptions, we show that almost all pairs of points in Ω are bases.     

Tensor products as induced representations: The case of finite GL(3)

We describe the tensor products of two irreducible linear complex representations of the group G = GL(3, $\mathbb{F}_q $ ) in terms of induced representations by linear characters of maximal tori and also in terms of Gelfand-Graev representations. Our results includeMacDonald’s conjectures for G and are extensions to G of finite counterparts to classical results on tensor products of principal series as well as holomorphic and antiholomorphic representations of the group SL(2,?); besides, they provide an easy way to decompose these tensor products with the help of Frobenius reciprocity. We also state some conjectures for the general case of GL(n, $\mathbb{F}_q $ ).     

Lévy Processes in a Step 3 Nilpotent Lie Group

The infinitesimal generators of Lévy processes in Euclidean space are pseudodifferential operators with symbols given by the Lévy-Khintchine formula. This classical analysis relies heavily on Fourier analysis which, in the case when the state space is a Lie group, becomes much more subtle. Still the notion of pseudo-differential operators can be extended to connected, simply connected nilpotent Lie groups by employing the Weyl functional calculus. With respect to this definition, the generators of Lévy processes in the simplest step 3 nilpotent Lie group G are pseudodifferential operators which admit C _c (G) as its core.     

Sandwiching random graphs: universality between random graph models

The goal of this paper is to establish a connection between two classical models of random graphs: the random graph G(n,p) and the random regular graph G_d(n). This connection appears to be very useful in deriving properties of one model from the other and explains why many graph invariants are universal. In particular, one obtains one-line proofs of several highly non-trivial and recent results on G_d(n).     

Backbone colorings of graphs with bounded degree

We study backbone colorings, a variation on classical vertex colorings: Given a graph G and a subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex k-coloring of G in which the colors assigned to adjacent vertices in H differ by at least 2. The minimal k∈N for which such a coloring exists is called the backbone chromatic number of G. We show that for a graph G of maximum degree Δ where the backbone graph is a d-degenerated subgraph of G, the backbone chromatic number is at most Δ+d+1 and moreover, in the case when the backbone graph being a matching we prove that the backbone chromatic number is at most Δ+1. We also present examples where these bounds are attained.Finally, the asymptotic behavior of the backbone chromatic number is studied regarding the degrees of G and H. We prove for any sparse graph G that if the maximum degree of a backbone graph is small compared to the maximum degree of G, then the backbone chromatic number is at most .     

Generalizations of the image conjecture and the Mathieu conjecture

We first propose a generalization of the image conjecture Zhao (submitted for publication) [31] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent to a variation of the Mathieu conjecture Mathieu (1997) [21] from integrals of G-finite functions over reductive Lie groups G to integrals of polynomials over open subsets of Rⁿ with any positive measures. Via this equivalence, the generalized image conjecture can also be viewed as a natural variation of the Duistermaat and van der Kallen theorem Duistermaat and van der Kallen (1998) [14] on Laurent polynomials with no constant terms. To put all the conjectures above in a common setting, we introduce what we call the Mathieu subspaces of associative algebras. We also discuss some examples of Mathieu subspaces from other sources and derive some general results on this newly introduced notion.