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 .     

Representation Rings of Lie Superalgebras

Given a Lie superalgebra , we introduce several variants of the representation ring, built as subrings and quotients of the ring of virtual -supermodules, up to (even) isomorphisms. In particular, we consider the ideal of virtual -supermodules isomorphic to their own parity reversals, as well as an equivariant K-theoretic super representation ring on which the parity reversal operator takes the class of a virtual -supermodule to its negative. We also construct representation groups built from ungraded -modules, as well as degree-shifted representation groups using Clifford modules. The full super representation ring , including all degree shifts, is then a -graded ring in the complex case and a -graded ring in the real case. Our primary result is a six-term periodic exact sequence relating the rings , and . We first establish a version of it working over an arbitrary (not necessarily algebraically closed) field of characteristic 0. In the complex case, this six-term periodic long exact sequence splits into two three-term sequences, which gives us additional insight into the structure of the complex super representation ring . In the real case, we obtain the expected 24-term version, as well as a surprising six-term version, of this periodic exact sequence. (Received: October 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.     

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.     

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     

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     

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     

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.     

K-Theory for the Banach Algebra of Operators on James''s Quasi-reflexive Banach Spaces

We prove that the K-groups of the Banach algebra of bounded, linear operators on the pth James space , where 1 < p < , are given by and . Moreover, for each Banach space and each non-zero, closed ideal contained in the ideal of inessential operators, we show that and . This enables us to calculate the K-groups of for each Banach space which is a direct sum of finitely many James spaces and -spaces.     

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.     

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.     

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.     

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     

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.     

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.     

Equivariant K-Homology and Restriction to Finite Cyclic Subgroups

For a discrete group G, we prove that a G-map between proper G–CW-complexes induces an isomorphism in G-equivariant K-homology if it induces an isomorphism in C-equivariant K-homology for every finite cyclic subgroup C of G. As an application, we show that the source of the Baum–Connes assembly map, namely K _* ^G (E(G, in)), is isomorphic to K _* ^G (E(G, )), where E(G, ) denotes the classifying space for the family of finite cyclic subgroups of G. Letting be the family of virtually cyclic subgroups of G, we also establish that and related results.     

On graded generalized local cohomology

Let be a homogeneous Noetherian ring with local base ring (R₀,m₀) and let M,N be two finitely generated graded R-modules. Let denote the i-th graded generalized local cohomology of N relative to M with support in . We study the vanishing, tameness and asymptotical stability of the homogeneous components of . Received: 22 March 2005; revised: 25 June 2005     

Supersaturation For Ramsey-Turán Problems

For an l-graph , the Turán number is the maximum number of edges in an n-vertex l-graph containing no copy of . The limit is known to exist [8]. The Ramsey–Turán density is defined similarly to except that we restrict to only those with independence number o(n). A result of Erdős and Sós [3] states that as long as for every edge E of there is another edge E′of for which |E∩E′|≥2. Therefore a natural question is whether there exists for which . Another variant proposed in [3] requires the stronger condition that every set of vertices of of size at least εn (0<ε<1) has density bounded below by some threshold. By definition, for every . However, even is not known for very many l-graphs when l>2. We prove the existence of a phenomenon similar to supersaturation for Turán problems for hypergraphs. As a consequence, we construct, for each l≥3, infinitely many l-graphs for which . We also prove that the 3-graph with triples 12a, 12b, 12c, 13a, 13b, 13c, 23a, 23b, 23c, abc, satisfies . The existence of a hypergraph satisfying was conjectured by Erdős and Sós [3], proved by Frankl and R?dl [6], and later by Sidorenko [14]. Our short proof is based on different ideas and is simpler than these earlier proofs. * Research supported in part by the National Science Foundation under grants DMS-9970325 and DMS-0400812, and an Alfred P. Sloan Research Fellowship. † Research supported in part by the National Science Foundation under grants DMS-0071261 and DMS-0300529.     

Polar Graphs and Corresponding Involution Sets, Loops and Steiner Triple Systems

A 1-factorization (or parallelism) of the complete graph with loops is called polar if each 1-factor (parallel class) contains exactly one loop and for any three distinct vertices x₁, x₂, x₃, if {x₁} and {x₂, x₃} belong to a 1-factor then the same holds for any permutation of the set {1, 2, 3}. To a polar graph there corresponds a polar involution set , an idempotent totally symmetric quasigroup (P, *), a commutative, weak inverse property loop (P, + ) of exponent 3 and a Steiner triple system . We have: satisfies the trapezium axiom is self-distributive ⇔ (P, + ) is a Moufang loop is an affine triple system; and: satisfies the quadrangle axiom is a group is an affine space.     

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