Regular Embedding Research Articles

CR regular embeddings and immersions of compact orientable 4-manifolds into ℂ3

We show that a compact orientable 4-manifold M has a CR regular immersion into ℂ3 if and only if both its first Pontryagin class p1(M) and its Euler characteristic χ(M) vanish, and has a CR regular embedding into ℂ3 if and only if in addition the second Stiefel–Whitney class w2(M) vanishes.

ON REDUCTIVE AUTOMORPHISM GROUPS OF REGULAR EMBEDDINGS

Let G be a connected reductive complex algebraic group acting on a smooth complete complex algebraic variety X. We assume that X is a regular embedding, a condition satisfied in particular by smooth toric varieties and flag varieties. For any set D of G-stable prime divisors, we study the action on X of the group Aut°(X, D), the connected automorphism group of X stabilizing all elements of D. We determine a Levi subgroup A(X, D) of Aut°(X, D), and also relevant invariants of X as a spherical A(X, D)-variety. As a byproduct, we obtain a complete description of the inclusion relation between closures of A(X, D)-orbits on X.

Overgroups of Subsystem Subgroups in Exceptional Groups: Levels

An embedding of root systems ∆ ⊆ Φ determines the corresponding regular embedding G(∆, R)≤ G(Φ, R) of Chevalley groups, over an arbitrary commutative ring R. Denote by E(∆, R) the elementary subgroup of G(∆, R). In the present paper we initiate the study of intermediate subgroups H, E∆, R) ≤ H ≤ G(Φ, R), provided that Φ=E6, E7, E8, F4 or G2, and there are no roots in Φ orthogonal to all of ∆. There are 72 such pairs (Φ, Δ)$. For F4 and G2 we assume, moreover, that 2 ∈ R * or 6 ∈ R *, respectively. For all such subsystems Δ we construct the levels of intermediate subgroups. We prove that these levels are determined by certain systems of ideals in R, one for each Δ-equivalence class of roots in Φ\∆, and calculate all relations among these ideals, in each case. Bibliography: 64 titles.

The structure of a local embedding and Chern classes of weighted blow-ups

For a proper local embedding between two Deligne–Mumford stacks Y and X , we find, under certain mild conditions, a new (possibly non-separated) Deligne–Mumford stack X' , with an etale, surjective and universally closed map to the target X , and whose fiber product with the image of the local embedding is a finite union of stacks with corresponding etale, surjective and universally closed maps to Y . Moreover, a natural set of weights on the substacks of X' allows the construction of a universally closed push-forward, and thus a comparison between the Chow groups of X' and X . We apply the construction above to the computation of the Chern classes of a weighted blow-up along a regular local embedding via deformation to a weighted normal cone and localization. We describe the stack X' in the case when X is the moduli space of stable maps with local embeddings at the boundary. We apply the methods above to find the Chern classes of the stable map spaces.

Classification of nonorientable regular embeddings of Hamming graphs

By a regular embedding of a graph K in a surface we mean a 2-cell embedding of K in a compact connected surface such that the automorphism group acts regularly on flags. In this paper, we classify the nonorientable regular embeddings of the Hamming graph H(d,n). We show that there exists such an embedding if and only if n=2 and d=2, or n=3 or 4 and d≥1, or n=6 and d=1 or 2. We also give constructions and descriptions of these embeddings.

On the orientable regular embeddings of complete multipartite graphs

Let Km[n] be the complete multipartite graph with m parts, while each part contains n vertices. The regular embeddings of complete graphs Km[1] have been determined by Biggs (1971) [1], James and Jones (1985) [12] and Wilson (1989) [23]. During the past twenty years, several papers such as Du et al. (2007, 2010) [6,7], Jones et al. (2007, 2008) [14,15], Kwak and Kwon (2005, 2008) [16,17] and Nedela et al. (2002) [20] contributed to the regular embeddings of complete bipartite graphs K2[n] and the final classification was given by Jones [13] in 2010. Since then, the classification for general cases m≥3 and n≥2 has become an attractive topic in this area. In this paper, we deal with the orientable regular embeddings of Km[n] for m≥3. We in fact give a reduction theorem for the general classification, namely, we show that if Km[n] has an orientable regular embedding M, then either m=p and n=pe for some prime p≥5 or m=3 and the normal subgroup Aut0+(M) of Aut+(M) preserving each part setwise is a direct product of a 3-subgroup Q and an abelian 3′-subgroup, where Q may be trivial. Moreover, we classify all the embeddings when m=3 and Aut0+(M) is abelian. We hope that our reduction theorem might be the first necessary approach leading to the general classification.

Regular embeddings of Cartesian product graphs

It is proved in this paper that if a Cartesian power X n of a prime graph X (with respect to the Cartesian multiplication) has an orientable regular embedding, then X has an orientable regular embedding too; and if a graph with some extra conditions has orientable regular embeddings, then its Cartesian power also has. As an application of our main theorems, the regular embeddings of grid-like graphs are studied.

Classification and Galois conjugacy of Hamming maps

We show that for each d ≥ 1 the d -dimensional Hamming graph H ( d , q ) has an orientably regular surface embedding if and only if q is a prime power p e . If q > 2 there are up to isomorphism φ ( q − 1)/ e such maps, all constructed as Cayley maps for a d -dimensional vector space over the field F q . We show that for each such pair ( d , q ) the corresponding Belyi pairs are conjugate under the action of the absolute Galois group Gal \overline Q , and we determine their minimal field of definition. We also classify the orientably regular embedding of merged Hamming graphs for q > 3.

Classification of nonorientable regular embeddings of complete bipartite graphs

A 2-cell embedding of a graph G into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags – mutually incident vertex–edge–face triples. In this paper, we classify the regular embeddings of complete bipartite graphs K n , n into nonorientable surfaces. Such a regular embedding of K n , n exists only when n is of the form n = 2 p 1 a 1 p 2 a 2 ⋯ p k a k where the p i are primes congruent to ±1 mod 8. In this case, up to isomorphism the number of those regular embeddings of K n , n is 2 k .

On subgroups of the general linear group that contain a maximal nonsplit torus

The paper deals with the structure of intermediate subgroups of the general linear group GL(n, k) of degree n over a field k of odd characteristic that contain a nonsplit maximal torus related to a radical extension of degree n of the ground field k. The structure of ideal nets over a ring that determine the structure of intermediate subgroups containinga transvection is given. Let $$ K = k\left( {\sqrt[n]{d}} \right) $$ be a radical degree-n extension of a field k of odd characteristic, and let T =(d) be a nonsplit maximal torus, which is the image of the multiplicative group of the field K under the regular embedding in G =GL(n, k). In the paper, the structure of intermediate subgroups H, T ≤ H ≤ G, that contain a transvection is studied. The elements of the matrices in the torus T = T (d) generate a subring R(d) in the field k.Let R be an intermediate subring, R(d) ⊆ R ⊆ k, d ∈ R. Let σR denote the net in which the ideal dR stands on the principal diagonal and above it and all entries of which beneath the principal diagonal are equal to R. Let σR denote the net in which all positions on the principal diagonal and beneath it are occupied by R and all entries above the principal diagonal are equal to dR. Let E(σR) be the subgroup generated by all transvections from the net group G(σR). In the paper it is proved that the product TE(σR) is a group (and thus an intermediate subgroup). If the net σ associated with an intermediate subgroup H coincides with σR,then TE(σR) ≤ H ≤ N(σR),where N(σR) is the normalizer of the elementary net group E(σR) in G. For the normalizer N(σR),the formula N(σR)= TG(σR) holds. In particular, this result enables one to describe the maximal intermediate subgroups. Bibliography: 13 titles.

Advances in Modal Analysis Using a Robust and Multiscale Method

This paper presents a new approach to modal synthesis for rendering sounds of virtual objects. We propose a generic method that preserves sound variety across the surface of an object at different scales of resolution and for a variety of complex geometries. The technique performs automatic voxelization of a surface model and automatic tuning of the parameters of hexahedral finite elements, based on the distribution of material in each cell. The voxelization is performed using a sparse regular grid embedding of the object, which permits the construction of plausible lower resolution approximations of the modal model. We can compute the audible impulse response of a variety of objects. Our solution is robust and can handle nonmanifold geometries that include both volumetric and surface parts. We present a system which allows us to manipulate and tune sounding objects in an appropriate way for games, training simulations, and other interactive virtual environments.

THIRD-REGULAR BI-EMBEDDINGS OF LATIN SQUARES

AbstractFor each positive integer n ≥ 2, there is a well-known regular orientable Hamiltonian embedding of Kn, n, and this generates a regular face 2-colourable triangular embedding of Kn, n, n. In the case n ≡ 0 (mod 8), and only in this case, there is a second regular orientable Hamiltonian embedding of Kn, n. This paper presents an analysis of the face 2-colourable triangular embedding of Kn, n, n that results from this. The corresponding Latin squares of side n are determined, together with the full automorphism group of the embedding.

Quotients of Boolean algebras and regular subalgebras

Let \({\mathbb{B}}\) and \({\mathbb{C}}\) be Boolean algebras and \({e: \mathbb{B}\rightarrow \mathbb{C}}\) an embedding. We examine the hierarchy of ideals on \({\mathbb{C}}\) for which \({ \bar{e}: \mathbb{B}\rightarrow \mathbb{C} / \fancyscript{I}}\) is a regular (i.e. complete) embedding. As an application we deal with the interrelationship between \({\fancyscript{P}(\omega)/{{\rm fin}}}\) in the ground model and in its extension. If M is an extension of V containing a new subset of ω, then in M there is an almost disjoint refinement of the family ([ω]ω)V. Moreover, there is, in M, exactly one ideal \({\fancyscript{I}}\) on ω such that \({(\fancyscript{P}(\omega)/{{\rm fin}})^V}\) is a dense subalgebra of \({(\fancyscript{P}(\omega)/\fancyscript{I})^M}\) if and only if M does not contain an independent (splitting) real. We show that for a generic extension V[G], the canonical embedding $$\fancyscript{P}^V(\omega){/}{\rm fin}\hookrightarrow \fancyscript{P}(\omega){/}(U(Os)(\mathbb{B}))^G$$ is a regular one, where \({U(Os)(\mathbb{B})}\) is the Urysohn closure of the zero-convergence structure on \({\mathbb{B}}\).

Nonorientable regular embeddings of graphs of order [formula omitted

A map is called regular if its automorphism group acts regularly on the set of all flags (incident vertex–edge–face triples). An orientable map is called orientably regular if the group of all orientation-preserving automorphisms is regular on the set of all arcs (incident vertex–edge pairs). If an orientably regular map admits also orientation-reversing automorphisms, then it is regular, and is called reflexible. A regular embedding and orientably regular embedding of a graph G are, respectively, 2-cell embeddings of G as a regular map and orientably regular map on some closed surface. In Du et al. (2004) [7], the orientably regular embeddings of graphs of order p q for two primes p and q ( p may be equal to q ) have been classified, where all the reflexible maps can be easily read from the classification theorem. In [11], Du and Wang (2007) classified the nonorientable regular embeddings of these graphs for p ≠ q . In this paper, we shall classify the nonorientable regular embeddings of graphs of order p 2 where p is a prime so that a complete classification of regular embeddings of graphs of order p q for two primes p and q is obtained. All graphs in this paper are connected and simple.

Finite and infinite-dimensional symmetries of pure 𝒩 = 2 supergravity inD= 4

We study the symmetries of pure = 2 supergravity in D = 4. As is known, this theory reduced on one Killing vector is characterised by a non-linearly realised symmetry SU(2,1) which is a non-split real form of SL(3,). We consider the BPS brane solutions of the theory preserving half of the supersymmetry and the action of SU(2,1) on them. Furthermore we provide evidence that the theory exhibits an underlying algebraic structure described by the Lorentzian Kac-Moody group SU(2,1)+++. This evidence arises both from the correspondence between the bosonic space-time fields of = 2 supergravity in D = 4 and a one-parameter sigma-model based on the hyperbolic group SU(2,1)++, as well as from the fact that the structure of BPS brane solutions is neatly encoded in SU(2,1)+++. As a nice by-product of our analysis, we obtain a regular embedding of the Kac-Moody algebra (2,1)+++ in 11 based on brane physics.

Preserving topology and elasticity for embedded deformable models

In this paper we introduce a new approach for the embedding of linear elastic deformable models. Our technique results in significant improvements in the efficient physically based simulation of highly detailed objects. First, our embedding takes into account topological details, that is, disconnected parts that fall into the same coarse element are simulated independently. Second, we account for the varying material properties by computing stiffness and interpolation functions for coarse elements which accurately approximate the behaviour of the embedded material. Finally, we also take into account empty space in the coarse embeddings, which provides a better simulation of the boundary. The result is a straightforward approach to simulating complex deformable models with the ease and speed associated with a coarse regular embedding, and with a quality of detail that would only be possible at much finer resolution.

Regulation of Podosome Formation in Macrophages by a Splice Variant of the Sodium Channel SCN8A

Voltage-gated sodium channels initiate electrical signaling in excitable cells such as muscle and neurons. They also are expressed in non-excitable cells such as macrophages and neoplastic cells. Previously, in macrophages, we demonstrated expression of SCN8A, the gene that encodes the channel NaV1.6, and intracellular localization of NaV1.6 to regions near F-actin bundles, particularly at areas of cell attachment. Here we show that a splice variant of NaV1.6 regulates cellular invasion through its effects on podosome and invadopodia formation in macrophages and melanoma cells. cDNA sequence analysis of SCN8A from THP-1 cells, a human monocyte-macrophage cell line, confirmed the expression of a full-length splice variant that lacks exon 18. Immunoelectron microscopy demonstrated NaV1.6-positive staining within the electron dense podosome rosette structure. Pharmacologic antagonism with tetrodotoxin (TTX) in differentiated THP-1 cells or absence of functional NaV1.6 through a naturally occurring mutation (med) in mouse peritoneal macrophages inhibited podosome formation. Agonist-mediated activation of the channel with veratridine caused release of sodium from cationic vesicular compartments, uptake by mitochondria, and mitochondrial calcium release through the Na/Ca exchanger. Invasion by differentiated THP-1 and HTB-66 cells, an invasive melanoma cell line, through extracellular matrix was inhibited by TTX. THP-1 invasion also was inhibited by small hairpin RNA knockdown of SCN8A. These results demonstrate that a variant of NaV1.6 participates in the control of podosome and invadopodia formation and suggest that intracellular sodium release mediated by NaV1.6 may regulate cellular invasion of macrophages and melanoma cells.

N= 2 Einstein-Yang-Mills's BPS solutions

We find the general form of all the supersymmetric configurations and solutions of N=2,d=4 Einstein-Yang-Mills theories. In the timelike case, which we study in great detail, giving many examples, the solutions to the full supergravity equations can be constructed from known flat spacetime solutions of the Bogomol'nyi equations. This allows the regular supersymmetric embedding in supergravity of regular monopole solutions ('t Hooft-Poyakov's, Weinberg's, Wilkinson and Bais's) but also embeddings of irregular solutions to the Bogomol'nyi equations which turn out to be regular black holes with different forms of non-Abelian hair once the non-triviality of the spacetime metric is taken into account. The attractor mechanism is realized in a gauge-covariant way. In the null case we determine the general equations that supersymmetric configurations and solutions must satisfy but we do not find relevant new supersymmetric solutions.

On the extension of real regular embedding

Let X be a real nonsingular affine algebraic variety of dimension k. It is proved that any two regular (algebraic) embeddings X → ℝn are regularly equivalent, provided that n ⩾ 4k + 2.

Kernel methods for learning languages

This paper studies a novel paradigm for learning formal languages from positive and negative examples which consists of mapping strings to an appropriate high-dimensional feature space and learning a separating hyperplane in that space. Such mappings can often be represented flexibly with string kernels, with the additional benefit of computational efficiency. The paradigm inspected can thus be viewed as that of using kernel methods for learning languages. We initiate the study of the linear separability of automata and languages by examining the rich class of piecewise-testable languages. We introduce a subsequence feature mapping to a Hilbert space and prove that piecewise-testable languages are linearly separable in that space. The proof makes use of word combinatorial results relating to subsequences. We also show that the positive definite symmetric kernel associated to this embedding is a rational kernel and show that it can be computed in quadratic time using general-purpose weighted automata algorithms. Our examination of the linear separability of piecewise-testable languages leads us to study the general problem of separability with other finite regular covers. We show that all languages linearly separable under a regular finite cover embedding, a generalization of the subsequence embedding we use, are regular. We give a general analysis of the use of support vector machines in combination with kernels to determine a separating hyperplane for languages and study the corresponding learning guarantees. Our analysis includes several additional linear separability results in abstract settings and partial characterizations for the linear separability of the family of all regular languages.