An improvement of the Ivanov bound

Let be a distance-regular graph of diameterd, valencyk andr=max{i|(c _i ,b _i )=(c ₁,b ₁)}. In this paper, we prove that

Bijections Between Multiparking Functions, Dirichlet Configurations, and Descending R-Traversals

There are several combinatorial objects that are known to be in bijection with the spanning trees of a graph G. These objects include G-parking functions, critical configurations of G, and descending traversals of G. In this paper, we extend the bijections to generalizations of all three objects. Partially supported by NSF VIGRE grant # 9977354.     

A Non-Messing-Up Phenomenon for Posets

We classify finite posets with a particular sorting property, generalizing a result for rectangular arrays. Each poset is covered by two sets of disjoint saturated chains such that, for any original labeling, after sorting the labels along both sets of chains, the labels of the chains in the first set remain sorted. We also characterize posets with more restrictive sorting properties. Received October 19, 2005     

Parking Functions and Descent Algebras

We show that the notion of parkization of a word, a variant of the classical standardization, allows us to introduce an internal product on the Hopf algebra of parking functions. Its Catalan subalgebra is stable under this operation and contains the descent algebra as a left ideal. Received March 1, 2005     

Permutation statistics related to a class of noncommutative symmetric functions and generalizations of the Genocchi numbers

We prove conjectures of the third author [L. Tevlin, Proc. FPSAC’07, Tianjin] on two new bases of noncommutative symmetric functions: the transition matrices from the ribbon basis have nonnegative integral coefficients. This is done by means of two composition-valued statistics on permutations and packed words, which generalize the combinatorics of Genocchi numbers.      

Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions

We study asharpinterface model for phase transitions which incorporates the interaction of the phase boundaries with the walls of a container Ω. In this model, the interfaces move by their mean curvature and are normal to δΩ. We first establish local-in-time existence and uniqueness of smooth solutions for the mean curvature equation with a normal contact angle condition. We then discuss global solutions by interpreting the equation and the boundary condition in a weak (viscosity) sense. Finally, we investigate the relation of the aforementioned model with atransitionlayer model. We prove that if Ω isconvex, the transition-layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero. We conclude with a discussion of the difficulties that arise in establishing this result in nonconvex domains. Communicated by David Kinderlehrer     

Proof of a conjecture on the Sperner property of the subgroup lattice of an abelianp-group

Let (p) denote the subgroup lattice of the abelianp-group

Generalized cardinal B-splines: Stability, linear independence, and appropriate scaling matrices

Generalized cardinal B-splines are defined as convolution products of characteristic functions of self-affine lattice tiles with respect to a given integer scaling matrix. By construction, these generalized splines are refinable functions with respect to the scaling matrix and therefore they can be used to define a multiresolution analysis and to construct a wavelet basis. In this paper, we study the stability and linear independence properties of the integer translates of these generalized spline functions. Moreover, we give a characterization of the scaling matrices to which the construction of the generalized spline functions can be applied.     

Interpolating varieties for spaces of meromorphic functions

We consider in this paper interpolation problems for weighted spaces of entire and meromorphic functions. Various (analytic and geometric) conditions necessary and sufficient for multiplicity varieties to be interpolating varieties will be given. This research is supported in part by NSF Grants DMS-90-00616 and CDR 88-03012.     

The core and the Weber set for bicooperative games

This paper studies two classical solution concepts for the structure of bicooperative games. First, we define the core and the Weber set of a bicooperative game and prove that the core is always contained in the Weber set. Next, we introduce a special class of bicooperative games, the so-called bisupermodular games, and show that these games are the only ones in which the core and the Weber set coincide.      

Geometric conditions for interpolation in weighted spaces of entire functions

We use L² estimates for the equation to find geometric conditions on discrete interpolating varieties for weighted spaces A_p(ℂ) of entire functions such that |f(z)|≤Ae^Bp(z) for some A, B>0. In particular, we give a characterization when p(z)=e^|z| and more generally, when In p(e^r) is convex andIn p(r) is concave. Acknowledgements and Notes. The author wishes to thank X. Massaneda for useful talks and remarks.     

A characterization of the duality mapping for convex bodies

We characterize the duality of convex bodies in d-dimensional Euclidean vector space, viewed as a mapping from the space of convex bodies containing the origin in the interior into the same space. The question for such a characterization was posed by Vitali Milman. The property that the duality interchanges pairwise intersections and convex hulls of unions is sufficient for a characterization, up to a trivial exception and the composition with a linear transformation. Received: March 2007, Accepted: April 2007     

Mutually Independent Hamiltonian Connectivity of (n, k)-Star Graphs

A graph G is hamiltonian connected if there exists a hamiltonian path joining any two distinct nodes of G. Two hamiltonian paths and of G from u to v are independent if u = u ₁ = v ₁, v = u _v(G) = v _v(G) , and u _i ≠ v _i for every 1 < i < v(G). A set of hamiltonian paths, {P ₁, P ₂, . . . , P _k }, of G from u to v are mutually independent if any two different hamiltonian paths are independent from u to v. A graph is k mutually independent hamiltonian connected if for any two distinct nodes u and v, there are k mutually independent hamiltonian paths from u to v. The mutually independent hamiltonian connectivity of a graph G, IHP(G), is the maximum integer k such that G is k mutually independent hamiltonian connected. Let n and k be any two distinct positive integers with n–k ≥ 2. We use S _n,k to denote the (n, k)-star graph. In this paper, we prove that IHP(S _n,k ) = n–2 except for S _4,2 such that IHP(S _4,2) = 1.      

Unitary invariants of spectral measures with the CGS-property

A «CGS-property» for the spectral measures is introduced and the classical results of determining complete systems of unitary invariants for self-adjoint and bounded normal operators on separable Hilbert spaces are extended to the class of spectral measures with this property. As a consequence, the above mentioned results are extended to unbounded normal operators on separable Hilbert spaces. Moreover, three different kinds of multiplicity are defined and it is shown that for the measures with the «CGS-property» they all coincide. In the last section some analogues of the multiplicity functions defined by Stone [14] are related to the total multiplicity.     

Random points associated with rectangles

The study of the distribution and moments of the distance between random points within a rectangle or in two coplanar rectangles is required in a wide variety of fields. Formulae for the distributions and arbitrary moments of the distance between two random points associated with one or two rectangles in various situations are given here explicitly. These explicit formulae will be helpful to those who work in various applied areas for the computations required in their problems. The third Author has partially been supported by C.N.R..     

A Characterization of Mixed Branching Greedoids

Branching greedoids have been defined and characterized for both directed and undirected rooted graphs. Such greedoids can be extended to rooted mixed graphs – graphs with both directed and undirected edges. These greedoids are characterized by a list of forbidden minors. If Ω is a rooted mixed graph, its mixed branching greedoid has the edges of Ω as its ground set and the collection of arborescences as its feasible sets. The set of mixed branching greedoids is exactly the set of local forest greedoids without

Generalized symmetric functions and invariants of matrices

We generalize the classical isomorphism between symmetric functions and invariants of a matrix. In particular, we show that the invariants over several matrices are given by the abelianization of the symmetric tensors over the free associative algebra. The main result is proved by finding a characteristic free presentation of the algebra of symmetric tensors over a free algebra. The author is supported by research grant Politecnico di Torino n.119, 2004.     

Rational approximation in the real Hardy spaceH 2 and Stieltjes integrals: A uniquencess theorem

The paper deals with rational approximation over the real Hardy spaceH _{2, R}(V), whereV is the complement of the closed unit disk. The results concern Stieltjes functions

Local and improper Daniell-Loomis integrals

In this paper we start from previous results obtained in [7] on the abstract space of Daniell-Loomis integrable functionsL, which is constructed like to the Daniell extension process, but without continuity assumptions on the elementary integral. The localized integral is used to prove thatL consists of those functions whose local upper and lower integrals are equal and finite, or thatL is closed with respect to improper integration. Our results are also holded in integration with respect to finitely additive measures.