Minimal embeddings in the projective plane

We show that if G is a graph embedded on the projective plane in such a way that each noncontractible cycle intersects G at least n times and the embedding is minimal with respect to this property (i.e., the representativity of the embedding is n), then G can be reduced by a series of reduction operations to an n × n × n projective grid. The reduction operations consist of changing a triangle of G to a triad, changing a triad of G to a triangle, and several others. We also show that if every proper minor of the embedding has representativity < n (i.e., the embedding is minimal), then G can be obtained from an n × n × n projective grid by a series of the two reduction operations described above. Hence every minimal embedding has the same number of edges. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 153–163, 1997     

An explicit solution to the Skorokhod embedding problem for functionals of excursions of Markov processes

We develop an explicit non-randomized solution to the Skorokhod embedding problem in an abstract setup of signed functionals of excursions of Markov processes. Our setting allows us to solve the Skorokhod embedding problem, in particular, for the age process of excursions of a Markov process, for diffusions and their signed age processes, for Azéma’s martingale and for Bessel processes of dimension smaller than 2.This work is a continuation and an important generalization of Obłój and Yor [J. Obłój, M. Yor, An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale, Stochastic Process. Appl. 110 (1) (2004) 83–110]. Our methodology is based on excursion theory and the solution to the Skorokhod embedding problem is described in terms of the Itô measure of the functional. We also derive an embedding for positive functionals and we correct a mistake in the formula of Obłój and Yor [J. Obłój, M. Yor, An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale, Stochastic Process. Appl. 110 (1) (2004) 83–110] for measures with atoms.     

Segre embeddings and the canonical image of a curve

We prove that there is no g for which the canonical embedding of a general curve of genus g lies on the Segre embedding of any product of three or more projective spaces.     

Minimal p-extensions and the embedding problem

We investigate the ultrasolvability problem for minimal p-group extensions of odd order: for the factorgroup of such extension, there exists a Galois extension of number fields such as corresponding embedding problem is ultrasolvable (i.e. this embedding problem is solvable and all its solutions are fields).     

A relative maximum genus graph embedding and its local maximum genus

A relative embedding of a connected graph is an embedding of the graph in some surface with respect to some closed walks, each of which bounds a face of the embedding. The relative maximum genus of a connected graph is the maximum of integerk with the property that the graph has a relative embedding in the orientable surface withk handles. A polynomial algorithm is provided for constructing relative maximum genus embedding of a graph if the relative tree of the graph is planar. Under this condition, just like maximum genus embedding, a graph does not have any locally strict maximum genus.     

Embedding theorems for classes of convex sets

Rådström's embedding theorem states that the nonempty compact convex subsets of a normed vector space can be identified with points of another normed vector space such that the embedding map is additive, positively homogeneous, and isometric. In the present paper, extensions of Rådström's embedding theorem are proven which provide additional information on the embedding space. These results include those of Hörmander who proved a similar embedding theorem for the nonempty closed bounded convex subsets of a Hausdorff locally convex vector space. In contrast to Hörmander's approach via support functionals, all embedding theorems of the present paper are proven by a refinement of Rådström's original method which is constructive and does not rely on Zorn's lemma. This paper also includes a brief discussion of some actual or potential applications of embedding theorems for classes of convex sets in probability theory, mathematical economics, interval mathematics, and related areas.     

On the sphericity testing of single source digraphs

A digraph D=(V,A) is called spherical, if it has an upward embedding on the round sphere which is an embedding of D on the round sphere so that all edges are monotonic arcs and all point to a fixed direction, say to the north pole. It is easy to see that [S.M. Hashemi, Digraph embedding, Discrete Math. 233 (2001) 321-328] for upward embedding, plane and sphere are not equivalent, which is in contrast with the fact that they are equivalent for undirected graphs. On the other hand, it has been proved that sphericity testing for digraphs is an NP-complete problem [S.M. Hashemi, A. Kisielewicz, I. Rival, The complexity of upward drawings on spheres, Order 14 (1998) 327-363]. In this work we study sphericity testing of the single source digraphs. In particular, we shall present a polynomial time algorithm for sphericity testing of three connected single source digraphs.     

Embedding problems with cyclic kernel of order 4

We consider Galois theoretical embedding problems with kernelC ₄, and prove that such an embedding problem can be ‘constructively’ reduced to two embedding problems, where the kernels are groups of roots of unity.     

Advances in metric embedding theory

Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The mathematical theory of metric embedding is well studied in both pure and applied analysis and has more recently been a source of interest for computer scientists as well. Most of this work is focused on the development of bi-Lipschitz mappings between metric spaces. In this paper we present new concepts in metric embeddings as well as new embedding methods for metric spaces. We focus on finite metric spaces, however some of the concepts and methods are applicable in other settings as well.One of the main cornerstones in finite metric embedding theory is a celebrated theorem of Bourgain which states that every finite metric space on n points embeds in Euclidean space with distortion. Bourgain?s result is best possible when considering the worst case distortion over all pairs of points in the metric space. Yet, it is natural to ask: can an embedding do much better in terms of the average distortion? Indeed, in most practical applications of metric embedding the main criteria for the quality of an embedding is its average distortion over all pairs.In this paper we provide an embedding with constant average distortion for arbitrary metric spaces, while maintaining the same worst case bound provided by Bourgain?s theorem. In fact, our embedding possesses a much stronger property. We define the ?q-distortion of a uniformly distributed pair of points. Our embedding achieves the best possible ?q-distortion for all 1?q?∞simultaneously.The results are based on novel embedding methods which improve on previous methods in another important aspect: the dimension of the host space. The dimension of an embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. However, no previous result improved the bound on the dimension which can be derived from Bourgain?s embedding. Our embedding methods achieve better dimension, and in fact, shed new light on another fundamental question in metric embedding, which is: whether the embedding dimension of a metric space is related to its intrinsic dimension? I.e., whether the dimension in which it can be embedded in some real normed space is related to the intrinsic dimension which is reflected by the inherent geometry of the space, measured by the space?s doubling dimension. The existence of such an embedding was conjectured by Assouad,⁴and was later posed as an open problem in several papers. Our embeddings give the first positive result of this type showing any finite metric space obtains a low distortion (and constant average distortion) embedding in Euclidean space in dimension proportional to its doubling dimension.Underlying our results is a novel embedding method. Probabilistic metric decomposition techniques have played a central role in the field of finite metric embedding in recent years. Here we introduce a novel notion of probabilistic metric decompositions which comes particularly natural in the context of embedding. Our new methodology provides a unified approach to all known results on embedding of arbitrary finite metric spaces. Moreover, as described above, with some additional ideas they allow to get far stronger results.The results presented in this paper⁵have been the basis for further developments both within the field of metric embedding and in other areas such as graph theory, distributed computing and algorithms. We present a comprehensive study of the notions and concepts introduced here and provide additional extensions, related results and some examples of algorithmic applications.     

Explicit representation of roots on <Emphasis Type="Italic">p</Emphasis>-adic solenoids and non-uniqueness of embeddability into rational one-parameter subgroups

This note generalizes known results concerning the existence of roots and embedding one-parameter subgroups on p-adic solenoids. An explicit representation of the roots leads to the construction of two distinct rational embedding one-parameter subgroups. The results contribute to enlighten the group structure of solenoids and to point out difficulties arising in the context of the embedding problem in probability theory. As a consequence, the uniqueness of embedding of infinitely divisible probability measures on p-adic solenoids is solved under a certain natural condition.     

Equations for the Jacobian of a hyperelliptic curve

We give an explicit embedding of the Jacobian of a hyperelliptic curve, , into projective space such that the image is isomorphic to the Jacobian over the splitting field of . The embedding is a modification of the usual embedding by theta functions with half integer characteristics.

     

The maximal coarse Baum–Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space

We introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromov?s notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum–Connes conjecture holds for metric spaces with bounded geometry which admit a fibred coarse embedding into Hilbert space.     

Embeddings Determined by Universal Words in the Rank 2 Free Group

We consider a specific method for embedding a countable group that is given by generators and relations into some 2-generated group. This embedding enables us to express the images of generators of the countable group in the 2-generated group and explicitly deduce from the defining relations of the latter those of the former which inherit some special properties. The method can be used to construct the explicit embeddings of recursive groups into finitely presented groups.

     

A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems

We make a contribution to the theory of embeddings of anisotropic Sobolev spaces into L ^p -spaces (Sobolev case) and spaces of Hölder continuous functions (Morrey case). In the case of bounded domains the generalized embedding theorems published so far pose quite restrictive conditions on the domain’s geometry (in fact, the domain must be “almost rectangular”). Motivated by the study of some evolutionary PDEs, we introduce the so-called “semirectangular setting”, where the geometry of the domain is compatible with the vector of integrability exponents of the various partial derivatives, and show that the validity of the embedding theorems can be extended to this case. Second, we discuss the a priori integrability requirement of the Sobolev anisotropic embedding theorem and show that under a purely algebraic condition on the vector of exponents, this requirement can be weakened. Lastly, we present a counterexample showing that for domains with general shapes the embeddings indeed do not hold.     

Extrinsic estimates for eigenvalues of the Dirac operator

For a compact spin manifold M isometrically embedded into Euclidean space, we derive the extrinsic estimates from above and below for eigenvalues of the square of the Dirac operator, which depend on the second fundamental form of the embedding. We also show the bounds of the ratio of the eigenvalues. Since the unit sphere and the projective spaces admit the standard embedding into Euclidean spaces, we also obtain the corresponding results for their compact spin submanifolds.     

一个关于球面和环面上嵌入图的近-三角剖分嵌入的内插定理(I)

一个近-三角剖分嵌入是指一个曲面上的嵌入图使得几乎所有的面都是三角形,至多只有一个可能的例外.文中作者证明了如下结论:如果一个图G 在球面S₀(或环面S₁)上有近-三角剖分嵌入,那么G在每一个可定向曲面S_k有近-三角剖分嵌入,其中k=h,h+1,\cdots ,\lfloor\frac{\beta(G)}{2}\rfloor$, 而h=0(或1)并且β(G)是图G的Betti数.特别地,G是上可嵌入的.     

On the cover relationship for local interior <Emphasis Type="Italic">G</Emphasis>-algebras

In this paper, we generalize the notion of cover relationship on modules over finite groups and blocks of modular group algebras to the case of local interior G-algebras by the embedding method. We get some properties on the embedding method and establish a relationship between the principal block correspondence and the cover relationship.     

Embedding constants for periodic Sobolev spaces of fractional order

We obtain an explicit expression for the norms of the embedding operators of the periodic Sobolev spaces into the space of continuous functions (the norms of this type are usually called embedding constants). The corresponding formulas for the embedding constants express them in terms of the values of the well-known Epstein zeta function which depends on the smoothness exponent s of the spaces under study and the dimension n of the space of independent variables. We establish that the embeddings under consideration have the embedding functions coinciding up to an additive constant and a scalar factor with the values of the corresponding Epstein zeta function. We find the asymptotics of the embedding constants as s → n/2.