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.     

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 .     

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     

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.     

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.     

On the Range of the Aluthge Transform

Let be the algebra of all bounded linear operators on a complex separable Hilbert space For an operator let be the Aluthge transform of T and we define for all where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range of Δ. We prove that R(Δ) is neither closed nor dense in However R(Δ) is strongly dense if is infinite dimensional. An erratum to this article is available at .     

Symmetrization of Closure Operators and Visibility

For an arbitrary set E and a given closure operator , we want to construct a symmetric closure operator via some – possibly infinite – iteration process. If E is finite, the corresponding symmetric closure operator . defines a matroid. If and is the convex closure operator, turns out to be the affine closure operator. Moreover, we apply the symmetrization process to closure operators induced by visibility. Received March 9, 2005     

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.     

Entropy and Decay of Correlations for Real Analytic Semi-Flows

We consider real analytic suspension semi-flows over uniformly expanding real-analytic map of the interval. We show that for any -invariant equilibrium measure related to an analytic potential g, there exists a Banach space of test functions such that for generic observables in , the corresponding correlation functions cannot decay faster than , where h_g is the measure theoretic entropy of . This statement implies the existence of essential spectrum for the Perron-Frobenius operator associated to the semi-flow, when acting on any reasonable Banach space. Submitted: September 16, 2008. Accepted: March 30, 2009.     

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     

Operator Matrices: SVEP and Weyl’s Theorem

It is known that if and are Banach space operators with the single-valued extension property, SVEP, then the matrix operator has SVEP for every operator and hence obeys Browder’s theorem. This paper considers conditions on operators A, B, and M₀ ensuring Weyls theorem for operators M_C.     

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.     

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

Exponential Dichotomy on the Real Line and Admissibility of Function Spaces

The purpose of this paper is to give characterizations for uniform exponential dichotomy of evolution families on the real line. We consider a general class of Banach function spaces denoted and we prove that if with and the pair is admissible for an evolution family then is uniformly exponentially dichotomic. By an example we show that the admissibility of the pair for an evolution family is not a sufficient condition for uniform exponential dichotomy. As applications, we deduce necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pairs and with      

Multipliers of Fractional Cauchy Transforms

Let B_n denote the unit ball of , n ≥ 2. Given an α > 0, let denote the class of functions defined for by integrating the kernel against a complex-valued measure on the sphere . Let denote the space of holomorphic functions in the ball. A function is called a multiplier of provided that for every . In the present paper, we obtain explicit analytic conditions on which imply that g is a multiplier of . Also, we discuss the sharpness of the results obtained. This research was supported by RFBR (grant no. 08-01-00358-a), by the Russian Science Support Foundation and by the programme “Key scientific schools NS 2409.2008.1”.     

Non-existence of regular algebraic minimal surfaces of spheres of degree 3

In this paper we prove that if is a closed minimal surface, then, , for any homogeneous polynomial f of degree 3 with 0 a regular value of the function .     

Levi Classes Generated by Nilpotent Groups

Let be a class of all groups G for which the normal closure (x) ^G of every element x belongs to a class . is a Levi class generated by . Let and ₀ be classes of finitely generated nilpotent groups and of torsion-free, finitely generated, nilpotent groups, respectively. We prove that and , and so and . It is shown that quasivarieties and are closed under free products, and that each contains at most one maximal proper subquasivariety. It is also proved that is closed under free products if so is .     

Variationally harmonic maps with general boundary conditions: Boundary regularity

Let and be Riemannian manifolds, compact without boundary. We develop a definition of a variationally harmonic map with respect to a general boundary condition of the kind u(x)∊Γ(x) for a.e. , where are given submanifolds depending smoothly on x. The given definition of variationally harmonic maps is slightly more restrictive, but also more natural than the usual definition of stationary harmonic maps. After deducing an energy monotonicity formula, it is possible to derive a regularity theory for variationally harmonic maps with general boundary data. The results include full boundary regularity in the Dirichlet boundary case Γ(x) = {g(x)} for if does not carry a nonconstant harmonic 2-sphere.     

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.