Nontrivial Spaces Research Articles

On the K-divisibility constant for some special finite-dimensional Banach couples

We prove new estimates of the K -divisibility constants for some special Banach couples. In particular, we prove that the K -divisibility constant for a couple of the form ( U ⊕ V , U ) where U and V are non-trivial Hilbert spaces equals 2 / 3 . We also prove estimates for the K -divisibility constant of the two-dimensional version of the couple ( L 2 , L ∞ ) , proving in particular that this couple is not exactly K -divisible. There are also several auxiliary results, including some estimates for relative Calderón constants for finite-dimensional couples.

Irreducible weight modules over the twisted Schrödinger-Virasoro algebra

It is shown that the support of an irreducible weight module over the Schrodinger-Virasoro Lie algebra with an infinite-dimensional weight space coincides with the weight lattice, and all nontrivial weight spaces of such a module are infinite-dimensional. As a by-product, it is obtained that every simple weight module over Lie algebra of this type with a nontrivial finite-dimensional weight space is a Harish-Chandra module.

Classification of irreducible weight modules over W-algebra W(2,2)

In this paper, we show that the support of an irreducible weight module over the W-algebra W(2,2), which has an infinite dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the W-algebra W(2,2), having a nontrivial finite dimensional weight space, is a Harish–Chandra module (and hence is either an irreducible highest or a lowest weight module or an irreducible module of the intermediate series).

Generalized intersection bodies are not equivalent

In [A. Koldobsky, A functional analytic approach to intersection bodies, Geom. Funct. Anal. 10 (2000) 1507–1526], A. Koldobsky asked whether two types of generalizations of the notion of an intersection body are in fact equivalent. The structures of these two types of generalized intersection bodies have been studied by the author in [E. Milman, Generalized intersection bodies, J. Funct. Anal. 240 (2) (2006) 530–567], providing substantial evidence for a positive answer to this question. The purpose of this note is to construct a counter-example, which provides a surprising negative answer to this question in a strong sense. This implies the existence of non-trivial non-negative functions in the range of the spherical Radon transform, and the existence of non-trivial spaces which embed in L p for certain negative values of p.

Unitarity at infinity and topological holography

Recently it has been suggested that non-Gaussian inflationary perturbations can be usefully analysed in terms of a putative dual gauge theory defined on the future conformal infinity generated by an accelerating cosmology. The problem is that unitarity of this gauge theory implies a strong constraint (the “Strominger bound”) on the matter fields in the bulk. We argue that the bound is just a reflection of the equation of state of cosmological matter. The details motivate a discussion of the possible relevance of the “dS/CFT correspondence” to the resolution of the Big Bang singularity. It is argued that the correspondence may require the Universe to come into existence along a non-singular spacelike hypersurface, as in the theories of “creation from nothing” discussed by Firouzjahi, Sarangi, and Tye, and also by Ooguri et al. and others. The argument makes use of the unusual properties of gauge theories defined on topologically non-trivial spaces.

Classification of Irreducible Weight Modules with a Finite–dimensional Weight Space over Twisted Heisenberg–Virasoro Algebra

We show that the support of an irreducible weight module over the twisted Heisenberg– Virasoro algebra, which has an infinite–dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the twisted Heisenberg–Virasoro algebra, having a nontrivial finite–dimensional weight space, is a Harish–Chandra module (and hence is either an irreducible highest or lowest weight module or an irreducible module from the intermediate series).

Classification of simple weight Virasoro modules with a finite-dimensional weight space

We show that the support of a simple weight module over the Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all non-trivial weight spaces of such module are infinite-dimensional. As a corollary we obtain that every simple weight module over the Virasoro algebra, having a non-trivial finite-dimensional weight space, is a Harish-Chandra module (and hence is either a simple highest or lowest weight module or a simple module from the intermediate series). This implies positive answers to two conjectures about simple pointed and simple mixed modules over the Virasoro algebra.

The Deformation Spaces of Projective Structures on 3-Dimensional Coxeter Orbifolds

A real projective structure on a 3-orbifold is given by locally modeling the orbifold by real projective geometry. We present some methodology to study Coxeter groups which are fundamental groups of 3-orbifolds with representations in \({\rm SL}_{\pm} (4, \mathbb{R})\) and deformation spaces. There are related examples by Benoist. These examples give us nontrivial deformation spaces of projective structures.

Closed relativistic strings in geometrically nontrivial spaces

Closed relativistic strings in geometrically nontrivial spaces

N=1supersymmetricSU(2)rmoose theories

We study the quantum moduli spaces and dynamical superpotentials of four-dimensional ${\mathrm{SU}(2)}^{r}$ linear and ring moose theories with $\mathcal{N}=1$ supersymmetry and link chiral superfields in the fundamental representation. Nontrivial quantum moduli spaces and dynamical superpotentials are produced. When the moduli space is perturbed by a generic tree level superpotential, the vacuum space becomes discrete. The ring moose is in the Coulomb phase and we find two singular submanifolds with a nontrivial modulus that is a function of all the independent gauge invariants needed to parametrize the quantum moduli space. The massive theory near these singularities confines. The Seiberg-Witten elliptic curve that describes the quantum moduli space of the ring moose is produced.

Automorphisms of spine spaces

Automorphisms of non-trivial spine spaces are either, type-preserving, or type-exchanging, that is, they map stars onto stars or exchange them with tops. In the first case they are given by semi-linear bijections, in the latter, by sesqui-linear forms.

DUAL METRICS AND NONGENERIC SUPERSYMMETRIES FOR A CLASS OF SIKLOS SPACE–TIMES

The presence of Killing–Yano tensors implies the existence of nongeneric supercharges in spinning point particle theories on curved backgrounds. Dual metrics are defined through their associated nondegenerate Killing tensors of valence two. Siklos space–times, which are the only nontrivial Einstein spaces conformal to nonflat pp-waves are investigated with regard to the existence of their corresponding Killing and Killing–Yano tensors. It is found that a class of Siklos space–times admit dual metrics and nongeneric supercharges.

D-homogeneous and d-ultrahomogeneous linear spaces

A linear space S is d-homogeneous if, whenever the linear structures induced on two subsets S1 and S2 of cardinality at most d are isomorphic, there is at least one automorphism of S mapping S1 onto S2. S is called d-ultrahomogeneous if each isomorphism between the linear structures induced on two subsets of cardinality at most d can be extended into an automorphism of S. We have proved in [11;] (without any finiteness assumption) that every 6-homogeneous linear space is homogeneous (that is d-homogeneous for every positive integer d). Here we classify completely the finite nontrivial linear spaces that are d-homogeneous for d ≥ 4 or d-ultrahomogeneous for d ≥ 3. We also prove an existence theorem for infinite nontrivial 4-ultrahomogeneous linear spaces. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 321–329, 2000

Calibrated Embeddings in the Special Lagrangian and Coassociative Cases

Every closed, oriented, real analytic Riemannian3–manifold can be isometrically embedded as a specialLagrangian submanifold of a Calabi–Yau 3–fold, even as thereal locus of an antiholomorphic, isometric involution. Every closed,oriented, real analytic Riemannian 4–manifold whose bundle of self-dual2–forms is trivial can be isometrically embedded as a coassociativesubmanifold in a G2-manifold, even as the fixed locus of ananti-G2 involution. These results, when coupledwith McLean's analysis of the moduli spaces of such calibratedsubmanifolds, yield a plentiful supply of examples of compact calibratedsubmanifolds with nontrivial deformation spaces.

Missing Modules, the Gnome Lie Algebra, and E 10

We study the embedding of Kac-Moody algebras into Borcherds (or generalized Kac-Moody) algebras which can be explicitly realized as Lie algebras of physical states of some completely compactified bosonic string. The extra ``missing states'' can be decomposed into irreducible highest or lowest weight ``missing modules'' w.r.t. the relevant Kac-Moody subalgebra; the corresponding lowest weights are associated with imaginary simple roots whose multiplicities can be simply understood in terms of certain polarization states of the associated string. We analyse in detail two examples where the momentum lattice of the string is given by the unique even unimodular Lorentzian lattice $II_{1,1}$ or $II_{9,1}$, respectively. The former leads to the Borcherds algebra $g_{1,1}$, which we call ``gnome Lie algebra", with maximal Kac-Moody subalgebra $A_1$. By the use of the denominator formula a complete set of imaginary simple roots can be exhibited, whereas the DDF construction provides an explicit Lie algebra basis in terms of purely longitudinal states of the compactified string in two dimensions. The second example is the Borcherds algebra $g_{9,1}$, whose maximal Kac-Moody subalgebra is the hyperbolic algebra $E_{10}$. The imaginary simple roots at level 1, which give rise to irreducible lowest weight modules for $E_{10}$, can be completely characterized; furthermore, our explicit analysis of two non-trivial level-2 root spaces leads us to conjecture that these are in fact the only imaginary simple roots for $g_{9,1}$.

Semi-proximity continuous functions in digital images

Starting with the intuitive concept of “nearness” as a binary relation, semi-proximity spaces (sp-spaces) are defined. The restrictions on semi-proximity spaces are weaker than on topological proximity spaces. Thus, semi-proximity spaces generalize classical topological spaces. Moreover, it is possible to describe all digital pictures used in computer vision and computer graphics as non-trivial semi-proximity spaces. which is not possible in classical topology. Therefore, we use semi-proximity spaces to establish a formal relationship between the “topological” concepts of digital image processing and their continuous counterparts in ℝ n . Especially interesting are continuous functions in semi-proximity spaces which are called “semi-proximity” continuous functions. They can be used for characterizing well-behaved operations on digital images such as thinning. It will be shown that the deletion of a simple point can be treated as a semi-proximity continuous function. These properties and the fact that a variety of nearness relations can be defined on digital pictures indicate that semi-proximity continuous functions are a useful tool in the difficult task of shape description.

A classification of Riemannian 3-manifolds with constant principal ricci curvaturesρ1= ρ2≠ ρ3

This paper has been motivated by various problems and results in differential geometry. The main motivation is the study of curvature homogeneous Riemannian spaces initiated in 1960 by I.M. Singer (see Section 9—Appendix for the precise definitions and references). Up to recently, only sporadic classes of examples have been known of curvature homogeneous spaces which are not locally homogeneous. For instance, isoparametric hypersurfaces in space forms give nice examples of nontrivial curvature homogeneous spaces (see [FKM]). To study the topography of curvature homogeneous spaces more systematically, it is natural to start with the dimension n = 3. The following results and problems have been particularly inspiring.

Linear spaces on 16 points

AbstractThere are 398 nonisomorphic nontrivial linear spaces on 16 points having no lines of size two. In addition, there are 157,151 proper linear spaces on 17 points having one line of size five and 42 lines of size three. © 1993 John Wiley & Sons, Inc.

Normality of 𝜎-products

We prove that no σ \sigma -product of an uncountable family of nontrivial spaces is hereditarily normal or hereditarily ω 1 {\omega _1} -paracompact. Also, if every finite subproduct of a σ \sigma -product is (hereditarily) countably paracompact and (hereditarily) normal, then the σ \sigma -product is (hereditarily) countably paracompact iff it is (hereditarily) normal.

Quantum mechanics on topologically nontrivial spaces

The formulation of quantum mechanics on topologically nontrivial spaces is discussed. It is pointed out that the “obstacles” are represented by cohomology groups and not only byπ1(m) as usually stated. Some widespread errors and misunderstandings are cleared up.