Asymptotic formulas for the triple Gamma function Γ3 by means of its integral representation

There is an abundant literature on inequalities for the Gamma function Γ and its various related functions as well as their approximations. Only very recently, several authors began to investigate various inequalities for the double Gamma function Γ₂ and its approximation. Here, in this sequel to some of these recent works, we aim at presenting an integral representation of the triple Gamma function Γ₃, which is then used to derive an asymptotic formula for Γ₃. As a by-product of the results presented here, integral representations and asymptotic formulas for the Gamma function Γ and the double Gamma function Γ₂ are also given. The methods and techniques used in this paper can easily be extended to derive the corresponding integral representations and asymptotic formulas for the multiple Gamma functions Γ_n (n ≧ 4).     

On certain generalized incomplete gamma functions

Recently, Chaudhry and Zubair have introduced a generalized incomplete gamma function Γ(v,x;z) which reduces to the incomplete gamma function Γ(v,x) when its variable z vanishes. We show that Γ(v,x;z) may be written essentially as a single Kampé de Fériet function which in turn may be expressed as a linear combination of two incomplete Weber integrals. Then by using properties of the latter integrals we deduce additional representations for Γ(v,x;z). In particular, we show that Γ(v,x;z) is essentially completely determined by a finite number of modified Bessel functions for all v ≠ 0 provided we know the values of the two incomplete Weber integrals when 0 < Re v ⩽ 1. When v = 0 we derive connections between the generalized incomplete gamma function and incomplete Lipschitz-Hankel integrals, and indicate that there exist connections with other special functions.     

On G-locally primitive graphs of locally Twisted Wreath type and a conjecture of Weiss

Let Γ be a connected G-vertex-transitive graph and let v be a vertex of Γ. The graph Γ is said to be G-locally primitive if the action of the vertex-stabiliser Gv on the neighbourhood Γ(v) of v is primitive. Furthermore, Γ is said to be of locally Twisted Wreath type if Gv is a primitive group of Twisted Wreath type in its action on Γ(v).Richard Weiss conjectured in 1978 that, there exists a function f:N→N such that if Γ is a connected G-vertex-transitive locally primitive graph of valency d and v is a vertex of Γ, then |Gv|?f(d). In this paper we prove this conjecture when Γ is of locally Twisted Wreath type.     

The trace class conjecture without the K-finiteness assumption

Let G be the group of real points of a reductive algebraic ℚ-group satisfying the same assumptions as in [5], Chapter I, and let Γ be a discrete subgroup of G. Let R_Γ be the right regular representation of G in L²(Γ\G). We prove in this Note that, for any integrable rapidly decreasing function ƒ on G, the restriction of R_Γ(ƒ) to the discrete spectrum of R_Γ is a trace class operator.     

Extensions of the Cauchy-Goursat Integral Theorem

It is natural to conjecture that if a function f is continuous on the closed region determined by a rectifiable 1-cycle Γ and complex-differentiable on the open region then Γ_∫f=0. The main result is an extension of the classical Cauchy-Goursat Theorem: the equality conjectured holds (with no boundary condition on f^′) under the additional hypothesis that the winding numbers of Γ define an Lp function and f satisfies a matching Hölder continuity condition near the image of Γ. (In particular, continuity suffices if p=∞.) The proof uses approximations of a rectifiable path by piecewise linear paths.     

Grundy number and products of graphs

The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedyk-colouring, that is a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic and cartesian products of two graphs in terms of the Grundy numbers of these graphs.Regarding the lexicographic product, we show that Γ(G)×Γ(H)≤Γ(G[H])≤2^Γ(G)−1(Γ(H)−1)+Γ(G). In addition, we show that if G is a tree or Γ(G)=Δ(G)+1, then Γ(G[H])=Γ(G)×Γ(H). We then deduce that for every fixed c≥1, given a graph G, it is CoNP-Complete to decide if Γ(G)≤c×χ(G) and it is CoNP-Complete to decide if Γ(G)≤c×ω(G).Regarding the cartesian product, we show that there is no upper bound of Γ(G□H) as a function of Γ(G) and Γ(H). Nevertheless, we prove that Γ(G□H)≤Δ(G)⋅2^Γ(H)−1+Γ(H).     

A Turán-type inequality for the gamma function

We prove that the following Turán-type inequality holds for Euler's gamma function. For all odd integers n?1 and real numbers x>0 we have

α?Γ(n−1)(x)Γ(n+1)(x)−Γ(n)²(x),     

A generalized characteristic polynomial of a graph having a semifree action

For an abelian group Γ, a formula to compute the characteristic polynomial of a Γ-graph has been obtained by Lee and Kim [Characteristic polynomials of graphs having a semi-free action, Linear algebra Appl. 307 (2005) 35-46]. As a continuation of this work, we give a computational formula for generalized characteristic polynomial of a Γ-graph when Γ is a finite group. Moreover, after showing that the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, we compute the reciprocals of the Bartholdi zeta functions of wheels and complete bipartite graphs as an application of our formula.     

The price of anarchy for non-atomic congestion games with symmetric cost maps and elastic demands

By showing that there is an upper bound for the price of anarchyρ(Γ) for a non-atomic congestion game Γ with only separable cost maps and fixed demands, Roughgarden and Tardos show that the cost of forgoing centralized control is mild. This letter shows that there is an upper bound for ρ(Γ) in Γ for fixed demands with symmetric cost maps. It also shows that there is a weaker bound for ρ(Γ) in Γ with elastic demands.     

On the divergence of polynomial interpolation

Consider a triangular interpolation scheme on a continuous piecewise C¹ curve of the complex plane, and let Γ be the closure of this triangular scheme. Given a meromorphic function f with no singularities on Γ, we are interested in the region of convergence of the sequence of interpolating polynomials to the function f. In particular, we focus on the case in which Γ is not fully contained in the interior of the region of convergence defined by the standard logarithmic potential. Let us call Γ_out the subset of Γ outside of the convergence region.In the paper we show that the sequence of interpolating polynomials, {P_n}_n, is divergent on all the points of Γ_out, except on a set of zero Lebesgue measure. Moreover, the structure of the set of divergence is also discussed: the subset of values z for which there exists a partial sequence of {P_n(z)}_n that converges to f(z) has zero Hausdorff dimension (so it also has zero Lebesgue measure), while the subset of values for which all the partials are divergent has full Lebesgue measure.The classical Runge example is also considered. In this case we show that, for all z in the part of the interval (−5,5) outside the region of convergence, the sequence {P_n(z)}_n is divergent.     

Principal Γ-cone for a tree

For any tree Γ, we introduce Γ-cones consisting of chambers and enumerate the number of chambers contained in two particular (called principal) Γ-cones. The problem is equivalent to the combinatorial problem of the enumeration of linear extensions of two bipartite orderings on a tree Γ. We characterize the principal Γ-cones among other Γ-cones by the strict maximality of the number of their chambers, and give a formula for this maximal (called principal) number by a finite sum of hook length formulae. We explain the formula through the simplicial block decomposition of principal Γ-cones. The results have their origin and application in the study of the topology related to Coxeter groups and Artin groups.     

Lattice paths with given number of turns and semimodules over numerical semigroups

Let Γ=〈α,β〉 be a numerical semigroup. In this article we consider several relations between the so-called Γ-semimodules and lattice paths from (0,α) to (β,0): we investigate isomorphism classes of Γ-semimodules as well as certain subsets of the set of gaps of Γ, and finally syzygies of Γ-semimodules. In particular we compute the number of Γ-semimodules which are isomorphic with their k-th syzygy for some k.     

Robust Bayesian prediction and estimation under a squared log error loss function

Robust Bayesian analysis is concerned with the problem of making decisions about some future observation or an unknown parameter, when the prior distribution belongs to a class Γ instead of being specified exactly. In this paper, the problem of robust Bayesian prediction and estimation under a squared log error loss function is considered. We find the posterior regret Γ-minimax predictor and estimator in a general class of distributions. Furthermore, we construct the conditional Γ-minimax, most stable and least sensitive prediction and estimation in a gamma model. A prequential analysis is carried out by using a simulation study to compare these predictors.     

A subadditive property of the gamma function

Let Δ=min_x?0Γ(2x)/Γ(x) and . We prove that the function x?(Γ(x))^α is subadditive on (0,∞) if and only if α∗?α?0.     

Tetravalent Edge-transitive Cayley Graphs of PGL （2, p）

t Let F = Cay（G, S）, R（G） be the right regular representation of G. The graph Г is called normal with respect to G, if R（G） is normal in the full automorphism group Aut（F） of F. Г is called a bi-normal with respect to G if R（G） is not normal in Aut（Г）, but R（G） contains a subgroup of index 2 which is normal in Aut（F）. In this paper, we prove that connected tetravalent edge-transitive Cayley graphs on PGL（2,p） are either normal or bi-normal when p ≠ 11 is a prime.     

Multiplicity of limit cycles and analytic m-solutions for planar differential systems

This work deals with limit cycles of real planar analytic vector fields. It is well known that given any limit cycle Γ of an analytic vector field it always exists a real analytic function f₀(x,y), defined in a neighborhood of Γ, and such that Γ is contained in its zero level set. In this work we introduce the notion of f₀(x,y) being an m-solution, which is a merely analytic concept. Our main result is that a limit cycle Γ is of multiplicity m if and only if f₀(x,y) is an m-solution of the vector field. We apply it to study in some examples the stability and the bifurcation of periodic orbits from some non-hyperbolic limit cycles.     

Construction of Green functions for weighted biharmonic operators

A constructive proof is given of the existence of the weighted biharmonic Green function Γ_α for α?0. The method is used to derive the explicit formula for Γ₁ previously stated by Hedenmalm. In addition, a formula for Γ₂ is found, which is then shown to take both positive and negative values in the bidisk .     

Homogeneous factorisations of graph products

A homogeneous factorisation of a digraph Γ consists of a partition P={P₁,…,P_k} of the arc set AΓ and two vertex-transitive subgroups M?G?Aut(Γ) such that M fixes each P_i setwise while G leaves P invariant and permutes its parts transitively. Given two graphs Γ₁ and Γ₂ we consider several ways of taking a product of Γ₁ and Γ₂ to form a larger graph, namely the direct product, cartesian product and lexicographic product. We provide many constructions which enable us to lift homogeneous factorisations or certain arc partitions of Γ₁ and Γ₂, to homogeneous factorisations of the various products.     

An explicit spine for the Picard modular group over the Gaussian integers

Let Γ\D be an arithmetic quotient of a symmetric space of non-compact type. A spine D₀ is a Γ-equivariant deformation retraction of D with dimension equal to the virtual cohomological dimension of Γ. We explicitly construct a spine for the case of Γ=SU(2,1;Z[i]). The spine is then used to compute the cohomology of Γ\D with various local coefficients.     

Vanishing viscosity in the plane for vorticity in borderline spaces of Besov type

The existence and uniqueness of solutions to the Euler equations for initial vorticity in BΓ∩Lp₀∩Lp₁ was proved by Misha Vishik, where BΓ is a borderline Besov space parameterized by the function Γ and 1<p₀<2<p₁. Vishik established short time existence and uniqueness when Γ(n)=O(logn) and global existence and uniqueness when . For initial vorticity in BΓ∩L², we establish the vanishing viscosity limit in L²(R²) of solutions of the Navier-Stokes equations to a solution of the Euler equations in the plane, convergence being uniform over short time when Γ(n)=O(logn) and uniform over any finite time when Γ(n)=O(logκn), 0?κ<1, and we give a bound on the rate of convergence. This allows us to extend the class of initial vorticities for which both global existence and uniqueness of solutions to the Euler equations can be established to include BΓ∩L² when Γ(n)=O(logκn) for 0<κ<1.