Modified Alternating $$\vec{k}$$–generators

This paper deals with a class of pseudorandom bit generators – modified alternating –generators. This class is constructed similarly to the class of alternating step generators. Three subclasses of are distinguished, namely linear, mixed and nonlinear generators. The main attention is devoted to the subclass of linear and mixed generators generating periodic sequences with maximal period lengths. A necessary and sufficient condition for all sequences generated by the linear generators of to be with maximal period lengths is formulated. Such sequences have good statistical properties, such as distribution of zeroes and ones, and large linear complexity. Two methods of cryptanalysis of the proposed generators are given. Finally, three new classes of modified alternating –generators, designed especially to be more secure, are presented.     

Line bundles over a moduli space of logarithmic connections on a Riemann surface

We consider logarithmic connections, on rank n and degree d vector bundles over a compact Riemann surface X, singular over a fixed point x₀ ∈ X with residue in the center of the integers n and d are assumed to be mutually coprime. A necessary and sufficient condition is given for a vector bundle to admit such a logarithmic connection. We also compute the Picard group of the moduli space of all such logarithmic connections. Let denote the moduli space of all such logarithmic connections, with the underlying vector bundle being of fixed determinant L, and inducing a fixed logarithmic connection on the determinant line L. Let be the Zariski open dense subset parametrizing all connections such that the underlying vector bundle is stable. The space of all global sections of certain line bundles on are computed. In particular, there are no nonconstant algebraic functions on Therefore, there are no nonconstant algebraic functions on although is biholomorphic to a representation space which admits nonconstant algebraic functions. The moduli space admits a natural compactification by a smooth divisor. We investigate numerically effectiveness of this divisor at infinity. It turns out that the divisor is not numerically effective in general. Received: March 2004 Revision: May 2004 Accepted: May 2004     

Locally finite varieties of Heyting algebras

We show that for a variety of Heyting algebras the following conditions are equivalent: (1) is locally finite; (2) the -coproduct of any two finite -algebras is finite; (3) either coincides with the variety of Boolean algebras or finite -copowers of the three element chain are finite. We also show that a variety of Heyting algebras is generated by its finite members if, and only if, is generated by a locally finite -algebra. Finally, to the two existing criteria for varieties of Heyting algebras to be finitely generated we add the following one: is finitely generated if, and only if, is residually finite. Received November 11, 2001; accepted in final form July 25, 2005.     

{\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pairs}}

Let R be a unital associative ring and two classes of left R-modules. In this paper we introduce the notion of a In analogy to classical cotorsion pairs as defined by Salce [10], a pair of subclasses and is called a if it is maximal with respect to the classes and the condition for all and Basic properties of are stated and several examples in the category of abelian groups are studied. Received: 17 March 2005     

A Helly-type transversal theorem for n-dimensional unit balls

Let be a family of unit balls in with the property that the mutual distances of the centers are at least . If any n² members of have a common line transversal, then has a line transversal too. Received: 27 January 2005; revised: 17 October 2005     

Lefschetz decompositions and categorical resolutions of singularities

Let Y be a singular algebraic variety and let be a resolution of singularities of Y. Assume that the exceptional locus of over Y is an irreducible divisor in . For every Lefschetz decomposition of the bounded derived category of coherent sheaves on we construct a triangulated subcategory ) which gives a desingularization of . If the Lefschetz decomposition is generated by a vector bundle tilting over Y then is a noncommutative resolution, and if the Lefschetz decomposition is rectangular, then is a crepant resolution.     

Remarks on balance in generalized Tate cohomology

We consider two pairs of complete hereditary cotorsion theories on the category of left R-modules, such that We prove that for any left R-modules M, N and for any n ≧ 1, the generalized Tate cohomology modules can be computed either using a left of M and a left of M or using a right a right of N. Received: 17 December 2004     

\mathcal{P}\mathcal{A}{\text{-isomorphisms}} of inverse semigroups

A partial automorphism of a semigroup S is any isomorphism between its subsemigroups, and the set all partial automorphisms of S with respect to composition is an inverse monoid called the partial automorphism monoid of S. Two semigroups are said to be if their partial automorphism monoids are isomorphic. A class of semigroups is called if it contains every semigroup to some semigroup from Although the class of all inverse semigroups is not we prove that the class of inverse semigroups, in which no maximal isolated subgroup is a direct product of an involution-free periodic group and the two-element cyclic group, is It follows that the class of all combinatorial inverse semigroups (those with no nontrivial subgroups) is A semigroup is called if it is isomorphic or antiisomorphic to any semigroup that is to it. We show that combinatorial inverse semigroups which are either shortly connected [5] or quasi-archimedean [10] are To Ralph McKenzieReceived April 15, 2004; accepted in final form October 7, 2004.     

Local Schur’s Lemma and Commutative Semifields

A set of linear maps , V a finite vector space over a field K, is regular if to each there corresponds a unique element such that R(x)=y. In this context, Schur’s lemma implies that is a field if (and only if) it consists of pairwise commuting elements. We consider when is locally commutative: at some μ ∈V*, AB(μ)=BA(μ) for all , and has been normalized to contain the identity. We show that such locally commutative are equivalent to commutative semifields, generalizing a result of Ganley, and hence characterizing commutative semifield spreads within the class of translation planes. This enables the determination of the orders |V| for which all locally commutative on V are (globally) commutative. Similarly, we determine a sharp upperbound for the maximum size of the Schur kernel associated with strictly locally commutative . We apply our main result to demonstrate the existence of a partial spread of degree 5, with nominated shears axis, that cannot be extend to a commutative semifield spread. Finally, we note that although local commutativity for a regular linear set implies that the set of Lie products consists entirely of singular maps, the converse is false.     

The Reduced Minimum Modulus in <Emphasis>C</Emphasis><Superscript>*</Superscript>-Algebras

We define the reduced minimum modulus of a nonzero element a in a unital C ^*-algebra by . We prove that . Applying this result to and its closed two side ideal , we get that dist , and for any if RR = 0, where and is the quotient homomorphism and . These results generalize corresponding results in Hilbert spaces.     

Congruence properties in congruence permutable and in ideal determined varieties, with applications.

We define a weak version of EDPC (equationally definable principal congruences), called EDPC*, that is shown to be preserved under varietal closure in congruence permutable varieties. We show that if is a congruence permutable variety generated by a class then has EDPC iff has EDPC* iff has EDPC*. An equational condition is given which, if satisfied by implies that has the CEP (congruence extension property). Similar results are proved for ideal determined varieties. These results are applied to the variety of residuated lattices, with examples.Received January 15, 2004; accepted in final form October 8, 2004.     

The geometry of rational parameterized representations

We study the projective space of univariate rational parameterized equations of degree d or less in real projective space The parameterized equations of degree less than d form a special algebraic variety We investigate the subspaces on and their relation to rational curves in give a geometric characterization of the automorphism group of and outline applications of the theory to projective kinematics.     

Elliptic Curves Suitable for Pairing Based Cryptography

For pairing based cryptography we need elliptic curves defined over finite fields whose group order is divisible by some prime with where k is relatively small. In Barreto et al. and Dupont et al. [Proceedings of the Third Workshop on Security in Communication Networks (SCN 2002), LNCS, 2576, 2003; Building curves with arbitrary small Mov degree over finite fields, Preprint, 2002], algorithms for the construction of ordinary elliptic curves over prime fields with arbitrary embedding degree k are given. Unfortunately, p is of size .We give a method to generate ordinary elliptic curves over prime fields with p significantly less than which also works for arbitrary k. For a fixed embedding degree k, the new algorithm yields curves with where or depending on k. For special values of k even better results are obtained.We present several examples. In particular, we found some curves where is a prime of small Hamming weight resp. with a small addition chain.AMS classification: 14H52, 14G50     

Hyperbolic Function Theory in the Clifford Algebra {\mathcal {C}}\ell_{n+1, 0}

The aim of this paper is to give the basic principles of hyperbolic function theory on the Clifford algebra . The structure of the theory is quite similar to the case of Clifford algebras with negative generators, but the proofs are not obvious. The (real) Clifford algebra is generated by unit vectors with positive squares e²_i = + 1. The hyperbolic Dirac operator is of the form where Q₀f is represented by the composition . If is a solution of H_kf = 0, then f is called k-hypergenic in Ω, where is an open set. We introduce some basic results of hyperbolic function theory and give some representation theorems on . Received: October, 2007. Accepted: February, 2008.     

Elements of Prescribed Order,Prescribed Traces and Systems of Rational Functions Over Finite Fields

Let k 1 and be a system of rational functions forming a strongly linearly independent set over a finite field . Let be arbitrarily prescribed elements. We prove that for all sufficiently large extensions , there is an element of prescribed order such that is the relative trace map from onto We give some applications to BCH codes, finite field arithmetic and ordered orthogonal arrays. We also solve a question of Helleseth et~al. (Hypercubic 4 and 5-designs from Double-Error-Correcting codes, Des. Codes. Cryptgr. 28(2003). pp. 265–282) completely.classification 11T30, 11G20, 05B15     

The Polynomial Degree of the Grassmannian {\mathcal G_{\bf 1,}{\bf n,}{\bf 2}}

For a subset ψ of PG(N, 2) a known result states that ψ has polynomial degree ≤ r, r≤ N, if and only if ψ intersects every r-flat of PG(N, 2) in an odd number of points. Certain refinements of this result are considered, and are then applied in the case when ψ is the Grassmannian , to show that for n <8 the polynomial degree of is .     

Forbidden (0,1)-vectors in Hyperplanes of $$\mathbb{R}^{n}$$: The unrestricted case

In this paper, we continue our investigation on “Extremal problems under dimension constraints” introduced [1]. The general problem we deal with in this paper can be formulated as follows. Let be an affine plane of dimension k in . Given determine or estimate .Here we consider and solve the problem in the special case where is a hyperplane in and the “forbidden set” . The same problem is considered for the case, where is a hyperplane passing through the origin, which surprisingly turns out to be more difficult. For this case we have only partial results.AMS Classification: 05C35, 05B30, 52C99     

Convexities of partial monounary algebras

In the present paper we prove that the collection of all convexities of partial monounary algebras is finite; namely, it has exactly 23 elements. Further, we show that for each element there exists a subset of such that is generated by and card . This work was supported by the Science and Technology Assistance Agency under the contract No. APVT-20-004104. Supported by Grant VEGA 1/3003/06.     

On the best bound of the minimal twisted height of linear subspaces

Let V be a vector space over a global field k, g an element of the adele group and H_g the twisted height defined on the k-subspaces of V . We show that the square root of the generalized Hermite-Rankin constant for k gives the best upper bound of the function , where runs over all m-dimensional k-subspaces of V and runs over all n-dimensional k-subspaces of . Received: 17 June 2005