A distance between elliptical distributions based in an embedding into the Siegel group

This paper describes two different embeddings of the manifolds corresponding to many elliptical probability distributions with the informative geometry into the manifold of positive-definite matrices with the Siegel metric, generalizing a result published previously elsewhere. These new general embeddings are applicable to a wide class of elliptical probability distributions, in which the normal, t-Student and Cauchy are specific examples. A lower bound for the Rao distance is obtained, which is itself a distance, and, through these embeddings, a number of statistical tests of hypothesis are derived.     

Total embedding distributions of Ringel ladders

The total embedding distributions of a graph consists of the orientable embeddings and non-orientable embeddings and are known for only a few classes of graphs. The orientable genus distribution of Ringel ladders is determined in [E.H. Tesar, Genus distribution of Ringel ladders, Discrete Mathematics 216 (2000) 235–252] by E.H. Tesar. In this paper, using the overlap matrix, we obtain nonhomogeneous recurrence relation for rank distribution polynomial, which can be solved by the Chebyshev polynomials of the second kind. The explicit formula for the number of non-orientable embeddings of Ringel ladders is obtained. Also, the orientable genus distribution of Ringel ladders is re-derived.     

An Euler‐genus approach to the calculation of the crosscap‐number polynomial

In 1994, J. Chen, J. Gross, and R. Rieper demonstrated how to use the rank of Mohar's overlap matrix to calculate the crosscap‐number distribution, that is, the distribution of the embeddings of a graph in the nonorientable surfaces. That has ever since been by far the most frequent way that these distributions have been calculated. This article introduces a way to calculate the Euler‐genus polynomial of a graph, which combines the orientable and the nonorientable embeddings, without using the overlap matrix. The crosscap‐number polynomial for the nonorientable embeddings is then easily calculated from the Euler‐genus polynomial and the genus polynomial.     

Absolutely Continuous Embeddings of Rearrangement-Invariant Spaces

Compactness of embeddings between rearrangement invariant spaces is closely related to absolute continuity of these embeddings. We study absolutely continuous embeddings between rearrangement invariant spaces. In particular it is shown that an absolutely continuous embedding is never optimal. We give sufficient (and under additional hypotheses necessary) conditions for absolute continuity of these embeddings. We also provide quantitative estimates of absolutely continuous embeddings.     

Embedding of circulant graphs and generalized Petersen graphs on projective plane

Both the circulant graph and the generalized Petersen graph are important types of graphs in graph theory. In this paper, the structures of embeddings of circulant graph C(2n + 1; {1, n}) on the projective plane are described, the number of embeddings of C(2n + 1; {1, n}) on the projective plane follows, then the number of embeddings of the generalized Petersen graph P(2n +1, n) on the projective plane is deduced from that of C(2n +1; {1, n}), because C(2n + 1;{1, n}) is a minor of P(2n + 1, n), their structures of embeddings have relations. In the same way, the number of embeddings of the generalized Petersen graph P(2n, 2) on the projective plane is also obtained.     

Radial subspaces of Besov and Lizorkin-Triebel classes: Extended strauss lemma and compactness of embeddings

In this article, subspaces of radial distributions of Besov-Lizorkin-Triebel type are investigated. We give sufficient and necessary conditions for the compactness of the Sobolev-type embeddings. It is also proved that smoothness of the radial function implies decay of the function at infinity. This extends the classical Strauss lemma. The main tool in our investigations consists of an adapted atomic decomposition.     

Orthogonal sequences and regularity of embeddings into finite-dimensional spaces

This paper focuses on the regularity of linear embeddings of finite-dimensional subsets of Hilbert and Banach spaces into Euclidean spaces. We study orthogonal sequences in a Hilbert space H, whose elements tend to zero, and similar sequences in the space c₀ of null sequences. The examples studied show that the results due to Hunt and Kaloshin (Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity 12 (1999) 1263-1275) and Robinson (Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces, Nonlinearity 22 (2009) 711-728) for subsets of Hilbert and Banach spaces with finite box-counting dimension are asymptotically sharp. An analogous argument allows us to obtain a lower bound for the power of the logarithmic correction term in an embedding theorem proved by Olson and Robinson (Almost bi-Lipschitz embeddings and almost homogeneous sets, Trans. Amer. Math. Soc. 362 (1) (2010) 145-168) for subsets X of Hilbert spaces when X−X has finite Assouad dimension.     

Forest embeddings in regular graphs of large girth

This paper deals with the enumeration of distinct embeddings (both induced and partial) of arbitrary graphs in regular graphs of large girth. A simple explicit recurrence formula is presented for the number of embeddings of an arbitrary forest F in an arbitrary regular graph G of sufficiently large girth. This formula (and hence the number of embeddings) depends only on the order and degree of regularity of G, and the degree sequence and component structure (multiset of component orders) of F. A concept called c-subgraph regularity is introduced which generalizes the familiar notion of regularity in graphs. (Informally, a graph is c-subgraph regular if its vertices cannot be distinguished on the basis of embeddings of graphs of order less than or equal to c.) A central result of this paper is that if G is regular and has girth g, then G is (g ? 1)-subgraph regular.     

Limits of discrete distributions and Gibbs measures on random graphs

Building upon the theory of graph limits and the Aldous–Hoover representation and inspired by Panchenko’s work on asymptotic Gibbs measures [Annals of Probability 2013], we construct continuous embeddings of discrete probability distributions. We show that the theory of graph limits induces a meaningful notion of convergence and derive a corresponding version of the Szemerédi regularity lemma. Moreover, complementing recent work Bapst et al. (2015), we apply these results to Gibbs measures induced by sparse random factor graphs and verify the “replica symmetric solution” predicted in the physics literature under the assumption of non-reconstruction.     

Embeddings of anisotropic Besov spaces into Sobolev spaces

We study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix A into Sobolev spaces, with a focus on the influence of A on the embedding behavior. For a large range of parameters, we derive sharp characterizations of embeddings.     

Exponentially many maximum genus embeddings and genus embeddings for complete graphs

There are many results on the maximum genus, among which most are written for the existence of values of such embeddings, and few attention has been paid to the estimation of such embeddings and their applications. In this paper we study the number of maximum genus embeddings for a graph and find an exponential lower bound for such numbers. Our results show that in general case, a simple connected graph has exponentially many distinct maximum genus embeddings. In particular, a connected cubic graph G of order n always has at least distinct maximum genus embeddings, where α and m denote, respectively, the number of inner vertices and odd components of an optimal tree T. What surprise us most is that such two extremal embeddings (i.e., the maximum genus embeddings and the genus embeddings) are sometimes closely related with each other. In fact, as applications, we show that for a sufficient large natural number n, there are at least many genus embeddings for complete graph K _n with n ≡ 4, 7, 10 (mod12), where C is a constance depending on the value of n of residue 12. These results improve the bounds obtained by Korzhik and Voss and the methods used here are much simpler and straight. This work was supported by the National Natural Science Foundation of China (Grant No. 10671073), Science and Technology commission of Shanghai Municipality (Grant No. 07XD14011) and Shanghai Leading Academic Discipline Project (Grant No. B407)     

Genera of Cayley maps

The genus distribution of a graph G is defined to be the sequence {gm}, where gm is the number of different embeddings of G in the closed orientable surface of genus m. In this paper, we examine the genus distributions of Cayley maps for several Cayley graphs. It will be shown that the genus distribution of Cayley maps has many different properties from its usual genus distribution.     

Highest weight modules and polarized embeddings of shadow spaces

The present paper was inspired by the work on polarized embeddings by Cardinali et al. (J. Algebr. Comb. 25(1):7–23, 2007) although some of our results in it date back to 1999. They study polarized embeddings of certain dual polar spaces, and identify the minimal polarized embeddings for several such geometries. We extend some of their results to arbitrary shadow spaces of spherical buildings, and make a connection to work of Burgoyne, Wong, Verma, and Humphreys on highest weight representations for Chevalley groups.     

A duality theorem for graph embeddings

A generalized type of graph covering, called a “Wrapped quasicovering” (wqc) is defined. If K, L are graphs dually embedded in an orientable surface S, then we may lift these embeddings to embeddings of dual graphs K?,L? in orientable surfaces S?, such that S? are branched covers of S and the restrictions of the branched coverings to K?,L? are wqc's of K, L. the theory is applied to obtain genus embeddings of composition graphs G[nK₁] from embeddings of “quotient” graphs G.     

Triangular embeddings of complete graphs (neighborly maps) with 12 and 13 vertices

In this paper, we describe the generation of all nonorientable triangular embeddings of the complete graphs K₁₂ and K₁₃. (The 59 nonisomorphic orientable triangular embeddings of K₁₂ were found in 1996 by Altshuler, Bokowski, and Schuchert, and K₁₃ has no orientable triangular embeddings.) There are 182,200 nonisomorphic nonorientable triangular embeddings for K₁₂, and 243,088,286 for K₁₃. Triangular embeddings of complete graphs are also known as neighborly maps and are a type of twofold triple system. We also use methods of Wilson to provide an upper bound on the number of simple twofold triple systems of order n, and thereby on the number of triangular embeddings of K_n. We mention an application of our results to flexibility of embedded graphs. © 2005 Wiley Periodicals, Inc. J Combin Designs     

On the genus distributions of wheels and of related graphs

We are concerned with families of graphs in which there is a single root-vertex ofunbounded valence, and in which, however, there is a uniform upper bound for the valences of all the other vertices. Using a result of Zagier, we obtain formulas and recursions for the genus distributions of several such families, including the wheel graphs. We show that the region distribution of a wheel graph is approximately proportional to the sequence of Stirling numbers of the first kind. Stahl has previously obtained such a result for embeddings in surfaces whose genus is relatively near to the maximum genus. Here, we generalize Stahl’s result to the entire genus distributions of wheels. Moreover, we derive the genus distributions for four other graph families that have some similarities to wheels.     

Embeddings of SL(2; ?) into the cremona group

Geometric and dynamic properties of embeddings of SL(2; ℤ) into the Cremona group are studied. Infinitely many nonconjugate embeddings that preserve the type (i.e., that send elliptic, parabolic and hyperbolic elements onto elements of the same type) are provided. The existence of infinitely many nonconjugate elliptic, parabolic and hyperbolic embeddings is also shown. In particular, a group G of automorphisms of a smooth surface S obtained by blowing up 10 points of the complex projective plane is given. The group G is isomorphic to SL(2; ℤ), preserves an elliptic curve and all its elements of infinite order are hyperbolic.     

New regular embeddings of n‐cubes Qn

We investigate highly symmetrical embeddings of the n‐dimensional cube Q_n into orientable compact surfaces which we call regular embeddings of Q_n. We derive some general results and construct some new families of regular embeddings of Q_n. In particular, for n = 1, 2,3(mod 4), we classify the regular embeddings of Q_n which can be expressed as balanced Cayley maps. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 297–312, 2004     

An extension theorem in symplectic geometry

 We extend the ``Extension after Restriction Principle' for symplectic embeddings of bounded starlike domains to a large class of symplectic embeddings of unbounded starlike domains. Received: 21 January 2002 / Revised version: 5 July 2002 Mathematics Subject Classification (2000): Primary 53D35, Secondary 54C20