Minimal non-$${\mathcal {F}}$$ -groups

Let be a saturated formation. We describe minimal non- -, minimal non- -, and minimal non-metabelian groups. Dedicated to L. A. Shemetkov on the occasion of his seventieth birthday.     

Fixed points in {\mathcal{M}}-fuzzy metric spaces

In this paper, we give some common fixed point theorems for five mappings satisfying some conditions in -fuzzy metric spaces.     

On the Intertwining of \partial{\mathcal{D}}-Isometries

Let be a strictly pseudoconvex bounded domain in with C ² boundary . If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on , then T is referred to as a -isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those -isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of . Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) -isometry T (of which the multiplication tuple on the Hardy space of the unit ball in is a rather special example). Submitted: September 9, 2007. Revised: October 10, 2007. Accepted: October 24, 2007.     

Pseudomonotone$${_{\ast}}$$ maps and the cutting plane property

Pseudomonotone maps are a generalization of paramonotone maps which is very closely related to the cutting plane property in variational inequality problems (VIP). In this paper, we first generalize the so-called minimum principle sufficiency and the maximum principle sufficiency for VIP with multivalued maps. Then we show that pseudomonotonicity of the map implies the “maximum principle sufficiency” and, in fact, is equivalent to it in a sense. We then present two applications of pseudomonotone maps. First we show that pseudomonotone maps can be used instead of the much more restricted class of pseudomonotone₊ maps in a cutting plane method. Finally, an application to a proximal point method is given.      

The geometric invariants $${\Omega^{n}}$$ of a product of groups

We compute the geometric invariants of a product G × H of groups in terms of and . This gives a sufficient condition in terms of and for a normal subgroup of G × H with abelian quotient to be of type F _n . We give an example involving the direct product of the Baumslag–Solitar group BS_1,2 with itself.      

Compact solutions to the equation Tx = y in a weakly closed $$ \mathcal{T}(\mathcal{N}) $$-module

Given two vectors x, y in a Hilbert space and a weakly closed -module , we provide a necessary and sufficient condition for the existence of a compact operator T in satisfying Tx = y.     

CR-Admissible $$\mathbb{{Z}}_{2}$$-gradations and CR-symmetries

Complementing the results of (Lotta and Nacinovich, Adv. Math. 191(1): 114–146, 2005), we show that the minimal orbit M of a real form G of a semisimple complex Lie group in a flag manifold is CR-symmetric (see (Kaup and Zaitsev Adv. Math. 149(2):145–181, 2000)) if and only if the corresponding CR algebra admits a gradation compatible with the CR structure.      

Compactness and local compactness for {\mathbb{R}}-trees

We establish (geometric) criteria for an -tree to be compact and to be locally compact. It follows that locally compact -trees are separable. Received: 10 September 2007     

Failure of hypoellipticity for the canonical solution to $${\bar\partial_b}$$ on some nilpotent Lie groups

In this paper we show that Kohn’s solution to the \({\bar\partial_b}\) problem fails to be hypoelliptic on some class of high codimension submanifolds of \({\mathbb{C}^n}\). The examples presented here carry a Lie group structure which generalize the one-dimensional Heisenberg group.     

Fundamentals for symplectic $$ \mathcal{A} $$-modules. Affine Darboux theorem

In his [9–11], the first author shows that the sheaf-theoreti-cally based Abstract Differential Geometry incorporates and generalizes classical differential geometry. Here, we undertake to explore the implications of Abstract Differential Geometry to classical symplectic geometry. The full investigation will be presented elsewhere.      

The rational cohomology of $${\overline{\mathcal{M}_4}}$$

We present two approaches to the study of the cohomology of moduli spaces of curves. Together, they allow us to compute the rational cohomology of the moduli space of stable complex curves of genus 4, with its Hodge structure.     

$${\mathcal{C}}_{0}$$ ( X)-algebras,stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras

We study permanence properties of the classes of stable and so-called -stable -algebras, respectively. More precisely, we show that a (X)-algebra A is stable if all its fibres are, provided that the underlying compact metrizable space X has finite covering dimension or that the Cuntz semigroup of A is almost unperforated (a condition which is automatically satisfied for -algebras absorbing the Jiang–Su algebra tensorially). Furthermore, we prove that if is a K ₁-injective strongly self-absorbing -algebra, then A absorbs tensorially if and only if all its fibres do, again provided that X is finite-dimensional. This latter statement generalizes results of Blanchard and Kirchberg. We also show that the condition on the dimension of X cannot be dropped. Along the way, we obtain a useful characterization of when a -algebra with weakly unperforated Cuntz semigroup is stable, which allows us to show that stability passes to extensions of -absorbing -algebras. Research supported by: Deutsche Forschungsgemeinschaft (through the SFB 478), by the EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280), and by the Center for Advanced Studies in Mathematics at Ben-Gurion University     

On ideals and quotients of A \mathcal{T} -algebras

Some results on A -algebras are given. We study the problem when ideals, quotients and hereditary subalgebras of A -algebras are A -algebras or A -algebras, and give a necessary and sufficient condition of a hereditary subalgebra of an A -algebra being an A -algebra.     

A Korovkin type approximation theorems via $$\mathcal{I}$$ -convergence

Using the concept of -convergence we provide a Korovkin type approximation theorem by means of positive linear operators defined on an appropriate weighted space given with any interval of the real line. We also study rates of convergence by means of the modulus of continuity and the elements of the Lipschitz class.     

The Abelian category of quotients of $$\mathcal{L}\mathfrak{F}$$ -spaces

We construct the category of quotients of -spaces and we show that it is Abelian. This answers a question of L. Waelbroeck from 1990.     

Counting SL2（F2^s） Representations of Torus Knot Groups

In this paper,we count the number of SL2(F2^s)-representations of torus knot groups up to a conjugacy.For the finite field F2^s with character 2,the counting method is similar to that in out previous work[1].Explicit formulae of the effective counting are given in this paper.Twisted Alexander polynomials related to those reprsentations are discussed.     

Eden-Staudacher and Beisert-Eden-Staudacher equations in the $$\mathcal{N} = 4$$ supersymmetric gauge theory

We investigate the Eden-Staudacher and Beisert-Eden-Staudacher equations for the anomalous dimension of twist-2 operators at a large spin s in the supersymmetric gauge theory. We reduce these equations to a set of linear algebraic equations and calculate their kernels analytically. We demonstrate that in the perturbation theory, the anomalous dimension is a sum of products of the Euler functions ζ(k) having the maximum transcendentality property. We also show that at a large coupling, the “singular” solution of the Beisert-Eden-Staudacher equation reproduces the anomalous dimension constants predicted from the string side of the AdS/CFT correspondence. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 117–129, April, 2008.     

Quasilinearization and curvature of Aleksandrov spaces

We present a new distance characterization of Aleksandrov spaces of non-positive curvature. By introducing a quasilinearization for abstract metric spaces we draw an analogy between characterization of Aleksandrov spaces and inner product spaces; the quasi-inner product is defined by means of the quadrilateral cosine—a metric substitute for the angular measure between two directions at different points. Our main result states that a geodesically connected metric space is an Aleksandrov domain (also known as a CAT(0) space) if and only if the quadrilateral cosine does not exceed one for every two pairs of distinct points in . We also observe that a geodesically connected metric space is an domain if and only if, for every quadruple of points in , the quadrilateral inequality (known as Euler’s inequality in ) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces of non-positive curvature.      

Almost Sure Convergence of the General Jamison Weighted Sum of $$ {\user1{B}} $$-Valued Random Variables

In this paper．two new functions are introduced to depict the Jamison weighted sum of random variables instead using the common methods．their properties and relationships are systematically discussed．We also analysed the implication of the conditions in previous papers．Then we apply these consequences to B-valued random variables．and greatly improve the original results of the strong convergence of the general Janfison weighted sum．Furthermore,our discussions are useful to the corresponding questions of real-valued random variables．     

A New \mathcal{N}\mathcal{P} -Complete Problem and Public-Key Identification

The appearance of the theory of zero-knowledge, presented by Goldwasser, Micali and Rackoff in 1985, opened a way to secure identification schemes. The first application was the famous Fiat-Shamir scheme based on the problem of modular square roots extraction. In the following years, many other schemes have been proposed, some Fiat-Shamir extensions but also new discrete logarithm based schemes. Therefore, all of them were based on problems from number theory. Their main common drawback is high computational load because of arithmetical operations modulo large integers. Implementation on low-cost smart cards was made difficult and inefficient.With the Permuted Kernels Problem (PKP), Shamir proposed the first efficient scheme allowing for an implementation on such low-cost smart cards, but very few others have afterwards been suggested.In this paper, we present an efficient identification scheme based on a combinatorial -complete problem: the Permuted Perceptrons Problem (PPP). This problem seems hard enough to be unsolvable even with very small parameters, and some recent cryptanalysis studies confirm that position. Furthermore, it admits efficient zero-knowledge proofs of knowledge and so it is well-suited for cryptographic purposes. An actual implementation completes the optimistic opinion about efficiency and practicability on low-cost smart cards, and namely with less than 2KB of EEPROM and just 100 Bytes of RAM and 6.4 KB of communication.