Affine congruence by dissection of discs - appropriate groups and optimal dissections

Let be a group of affine transformations of the Euclidean plane . Two topological discs D, are called congruent by dissection with respect to if D can be dissected into a finite number of subdiscs that can be rearranged by maps from to a dissection of E. Our main result says in particular that admits congruence by dissection of any circular disc C with any square S if and only if contains a contractive map and all orbits , , are dense in . In this case any two discs D and E are congruent by dissection with respect to and every disc D is congruent by dissection with n copies of D for every n ≥ 2. Moreover, we give estimates on minimal numbers of pieces that are needed to realize congruences by dissection. Dedicated to Irmtraud Stephani on the occasion of her 70th birthday     

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.     

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     

Sampling Theory Associated with a Symmetric Operator with Compact Resolvent and De Branges Spaces

Let be a symmetric operator with compact resolvent defined in a Hilbert space For any fixed we consider an entire function K_a which involves the resolvent of Associated with K_a we obtain, by duality in a Hilbert space of entire functions which becomes a De Branges space of entire functions. This property provides a characterization of regardless of the anti-linear mapping which has as its range space. There exists also a sampling formula allowing to recover any function in from its samples at the sequence of eigenvalues of This work has been supported by the grant BFM2003–01034 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología.     

{\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     

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.     

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.     

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.     

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.     

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.     

Metric Geometry in Homogeneous Spaces of the Unitary Group of a C*-Algebra. Part II. Geodesics Joining Fixed Endpoints

This article focuses on the study of the metric geometry of homogeneous spaces (the unitary group of a C*-algebra modulo the unitary group of a C*-subalgebra ) where the invariant Finsler metric in is induced by the quotient norm of Under the assumption that is of compact type, i.e. when the unitary group is relatively compact in the strong operator topology, this work presents local and global versions of Hopf-Rinow-like theorems: given points there exists a minimal uniparametric group curve joining ρ₀ and ρ₁.     

The Automorphism Group of Plane Algebraic Curves with Singer Automorphisms

The main result in Cossidente and Siciliano (J. Number Theory, Vol. 99 (2003) pp. 373–382) states that if a Singer subgroup of PGL(3,q) is an automorphism group of a projective, geometric irreducible, non-singular plane algebraic curve then either or . In the former case is projectively equivalent to the curve with equation X^q+1Y+Y^q+1+X=0 studied by Pellikaan. Furthermore, the curve has a very nice property from Finite Geometry point of view: apart from the three distinguished points fixed by the Singer subgroup, the set of its -rational points can be partitioned into finite projective planes . In this paper, the full automorphism group of such curves is determined. It turns out that is the normalizer of a Singer group in .     

\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.     

C*-Algebras of Integral Operators with Piecewise Slowly Oscillating Coefficients and Shifts Acting Freely

We establish a symbol calculus for the C*-subalgebra of generated by the operators of multiplication by slowly oscillating and piecewise continuous functions and the operators where is the Cauchy singular integral operator and The C*-algebra is invariant under the transformations

The Topology on the Space of δ-psh Functions in the Cegrell Classes

The aim of the present paper is to introduce a metric locally convex topology on the space of δ-psh functions in the Cegrell class . We prove that with this topology is a non-separable and non-reflexive Fréchet space. At the same time, we extend the Monge–Ampère operator from the class to .     

A class number formula for the p-cyclotomic field

Let p be an odd prime number and . Let be the classical Stickelberger ideal of the group ring . Iwasawa [6] proved that the index equals the relative class number of . In [2], [4] we defined for each subgroup H of G a Stickelberger ideal of , and studied some of its properties. In this note, we prove that when mod 4 and [G : H] = 2, the index equals the quotient . Received: 13 January 2006     

A Topological View of Ramsey Families of Finite Subsets of Positive Integers

If is an initially hereditary family of finite subsets of positive integers (i.e., if and G is initial segment of F then ) and M an infinite subset of positive integers then we define an ordinal index . We prove that if is a family of finite subsets of positive integers such that for every the characteristic function χ_F is isolated point of the subspace