Arithmetic Laplacians

We develop an arithmetic analogue of elliptic partial differential equations. The role of the space coordinates is played by a family of primes, and that of the space derivatives along the various primes are played by corresponding Fermat quotient operators subjected to certain commutation relations. This leads to arithmetic linear partial differential equations on algebraic groups that are analogues of certain operators in analysis constructed from Laplacians. We classify all such equations on one-dimensional groups, and analyze their spaces of solutions.     

Interval hypergraphs and D-interval hypergraphs

A hypergraph $\text{H = (V,}\text{E}\text{)}$ is called an interval hypergraph if there exists a one-to-one function ? mapping the elements of V to points on the real line such that for each edge E, there is an interval I, containing the images of all elements of E, but not the images of any elements not in E₁. The difference hypergraph D(H) determined by H is formed by adding to $\text{E}$ all nonempty sets of the form E₁ ? E₁, where E₁ and E₁ are edges of HH is said to be a D-interval hypergraph if D(H) is an interval hypergraph. A forbidden subhypergraph characterization of D-interval hypergraphs is given. By relating D-interval hypergraphs to dimension theory for posets, we determine all 3-irreducible posets of length one.     

t-designs on hypergraphs

A $\text{t-}\text{design}\text{T=(X,}\text{B}\text{)}$, denoted by (λ; t, k, v), is a system $\text{B}$ of subsets of size k from a v-set X, such that each t-subset of X is contained in exactly λ elements of $\text{B}$. A hypergraph $\text{H=(Y,}\text{E}\text{)}$ is a finite set Y where $\text{E}\text{=(E}{}_{\mathrm{i}}\text{: i?I)}$ is a family of subsets (which we assume here are distinct) of Y such that E_i≠Ø, i?l, and ?E_i=Y. Let G be an automorphism group of $\text{H=(Y,}\text{E}\text{)}$ where O^l_i is the ith orbit of l-subsets of $\text{E}$. Let A(G; H; t, k)= (a_ij) be an m by n matrix, where a_ij is the number of copies of O^t_i that occur in the system of all t-subsets of all elements of O^k_j. Then there is a t-design $\text{T=(X,}\text{B}\text{)}$ with $\text{X=}\text{E}$, with parameters (λ; t, k, v), and with G an automorphism groupof T iff there is an m by s submatrix M of A(G; H; t, k) where M has uniform row sums λ. The calculus for applying this theorem is illustrated and numerous t-designs for 10?v?16 are found and presented. Using a theorem of Alltop on our (12; 4, 6, 13) and (60; 4, 7, 15) we obtain a (12; 5, 7, 14) and a (60; 5, 8, 16).     

Flows on hypergraphs

We consider the capacitated minimum cost flow problem on directed hypergraphs. We define spanning hypertrees so generalizing the spanning tree of a standard graph, and show that, like in the standard and in the generalized minimum cost flow problems, a correspondence exists between bases and spanning hypertrees. Then, we show that, like for the network simplex algorithms for the standard and for the generalized minimum cost flow problems, most of the computations performed at each pivot operation have direct hypergraph interpretations.     

Discrete connection Laplacians

To every Hermitian vector bundle with connection over a compact Riemannian manifold one can associate a corresponding connection Laplacian acting on the sections of the bundle. We define analogous combinatorial, metric dependent Laplacians associated to triangulations of and prove that their spectra converge, as the mesh of the triangulations approaches zero, to the spectrum of the connection Laplacian.

     

Higher Laplacians on pseudo-Hermitian symmetric spaces

One of the most fundamental operators studied in geometric analysis is the classical Laplace–Beltrami operator. On pseudo-Hermitian manifolds, higher Laplacians $L_m$ are defined for each positive integer m, where $L_1$ coincides with the Laplace–Beltrami operator. Despite their natural definition, these higher Laplacians have not yet been studied in detail. In this paper, we consider the setting of simple pseudo-Hermitian symmetric spaces, i.e., let $X=G/H$ be a symmetric space for a real simple Lie group G, equipped with a G-invariant complex structure. We show that the higher Laplacians $L_1,L_3,\dots ,L_{2r?1}$ form a set of algebraically independent generators for the algebra ${\mathbb{D}}_G(X)$ of G-invariant differential operators on X, where r denotes the rank of X. For higher rank, this is the first instance of a set of generators for ${\mathbb{D}}_G(X)$ defined explicitly in purely geometric terms, and confirms a conjecture of Engli? and Peetre, originally stated in 1996 for the class of Hermitian symmetric spaces.     

Invariant metrics and Laplacians on Siegel-Jacobi space

In this paper, we compute Riemannian metrics on the Siegel-Jacobi space which are invariant under the natural action of the Jacobi group explicitly and also provide the Laplacians of these invariant metrics. These are expressed in terms of the trace form.     

Spectral asymptotics for Laplacians on self-similar sets

Given a self-similar Dirichlet form on a self-similar set, we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a ‘geometric counting function’ defined through a family of coverings of the self-similar set naturally associated with the Dirichlet space. Secondly, under (sub-)Gaussian heat kernel upper bound, we prove a detailed short time asymptotic behavior of the partition function, which is the Laplace-Stieltjes transform of the eigenvalue counting function associated with the Dirichlet form. This result can be applicable to a class of infinitely ramified self-similar sets including generalized Sierpinski carpets, and is an extension of the result given recently by B.M. Hambly for the Brownian motion on generalized Sierpinski carpets. Moreover, we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function.     

Invariant Metrics and Laplacians on Siegel-Jacobi Disk

Let Dn be the generalized unit disk of degree n. In this paper, Riemannian metrics on the Siegel-Jacobi disk Dn × C（^m,n） which are invariant under the natural action of the Jacobi group are found explicitly and the Laplacians of these invariant metrics are computed explicitly. These are expressed in terms of the trace form.     

无界拉普拉斯算子下图上的泛函不等式

本文探讨图上的泛函不等式,并且在无界拉普拉斯算子的意义下,利用图的完备性和图上超压缩性的性质,证明了图上对数Sobolev不等式的成立,以及超压缩性与图上Nash不等式的等价关系.     

Threshold hypergraphs

This paper deals with three generalizations of threshold graphs to hypergraphs proposed by M. Ch. Golumbic. Answering a question of M. Ch. Golumbic we show that these three definitions are not equivalent. The main results of the paper are Theorems 2.5 and 2.6 which characterize hypergraphs satisfying the most general of above definitions.     

Maximal quasi-accretive Laplacians on finite metric graphs

For a finite not necessarily compact metric graph, one considers the differential expression ${-\frac{d^2}{d x^2}}$ on each edge. The boundary conditions at the vertices of the graph yielding quasi-m-accretive as well as m-accretive operators are completely characterised.     

Searching for perfection on hypergraphs

In this paper we study the family of oriented transitive 3-hypergraphs that arise from cyclic permutations and intervals in the circle, in order to search for the notion of perfection on hypergraphs.     

Partitive hypergraphs

We define a new class of hypergraphs (partitive hypergraphs) which generalizes both, the set of all externally related subsets of a graph and the set of all committees of an hypergraph.We give a characterization of the partitive hypergraphs and moreover of those which are associated with hypergraphs or graphs.     

Quasi-random hypergraphs

We introduce an equivalence class of varied properties for hypergraphs. Any hypergraph possessing any one of these properties must of necessity possess them all. Since almost all random hypergraphs share these properties, we term these properties quasi-random. With these results, it becomes quite easy to show that many natural explicit constructions result in hypergraphs which imitate random hypergraphs in a variety of ways.     

Circular representation problem on hypergraphs

A hypergraph J=(X,E) is said to be circular representable, if its vertices can be placed on a circle, in such way that every edge of H induces an interval. This concept is a translation into the vocabulary of hypergraphs of the circular one's property for the (0, 1) matrices [6] studied by Tucker [9, 10]. We give here a characterization of the hypergraphs which are circular representable. We study when the associated representation is unique, and we characterize the possible transformations of a representation into another, a kind of problem which has already been treated from the algorithmic point of view by Booth and Lueker [1] or Duchet [2] in the case of the interval representable hypergraphs.Finally, we establish a connection between circular graphs and circular representable hypergraphs of the type of the Fulkerson-Gross connection between interval graphs and matrices having the consecutive one's property [5], in some special cases.