Riemann surfaces with longest systole and an improved Voronoï algorithm

In this paper we introduce a new method in order to find the Riemann surface M of a fixed topological type with the longest systole; it is based on a cell decomposition of the Teichmüller space of M. The method also works in the Euclidean case and is similar to the so-called Voronoï algorithm for positive definite quadratic forms, or equivalently, for lattice sphere packings. In particular, we give a new proof of Rogers' theorem.     

L^p -uniqueness for Dirichlet operators with singular potentials

We study the problem of strong uniqueness in L^p for the Dirichlet operator perturbed by a singular complex-valued potential. First we construct the generator -H_p of a C₀-semigroup in L^p, with H_p extending the restriction of the perturbed Dirichlet operator to the set of smooth functions. The corresponding sesquilinear form in L² is not assumed to be sectorial. Then we reveal sufficient conditions on the logarithmic derivative # of the measure rdx \rho dx and the potential q which ensure that -H_p is the only extension of D+b·?-q \upharpoonright_C0^￥ \Delta +\beta \cdot \nabla -q \upharpoonright_{C_0^{\infty}} which generates a C₀-semigroup on L^p. The method of a priori estimates of solutions to corresponding differential equations is employed.     

Möbius Type Semigroups in SL(2,F)

In this work, we show that for an ordered field F with an order P, the semigroup S of elements in SL(2,F) having entries in P can be factored into upper triangular, diagonal and lower triangular matrices. Moreover, we show that the semigroup ±S is maximal in SL(2,F).     

A note on discrete heat conduction

If the longitudinal line method is applied to the Cauchy problem u_t = u_xx, u(0, x) = u₀(x) with a bounded function u₀, one is led to a linear initial value problem v￠(t)=A v(t), v(0)=wv'(t)=A v(t),\, v(0)=w in l^￥ (\Bbb Z)l^\infty (\Bbb Z). Using Banach limit techniques we study the asymptotic behaviour of the solutions of these problems as t tends to infinity.     

Variational analysis of non-Lipschitz spectral functions

We consider spectral functions f : 5 where f is any permutation-invariant mapping from Cⁿ to R, and 5 is the eigenvalue map from the set of n 2 n complex matrices to Cⁿ, ordering the eigenvalues lexicographically. For example, if f is the function "maximum real part", then f : 5 is the spectral abscissa, while if f is "maximum modulus", then f : 5 is the spectral radius. Both these spectral functions are continuous, but they are neither convex nor Lipschitz. For our analysis, we use the notion of subgradient extensively analyzed in Variational Analysis, R.T. Rockafellar and R. J.-B. Wets (Springer, 1998). We show that a necessary condition for Y to be a subgradient of an eigenvalue function f : 5 at X is that Y^* commutes with X. We also give a number of other necessary conditions for Y based on the Schur form and the Jordan form of X. In the case of the spectral abscissa, we refine these conditions, and we precisely identify the case where subdifferential regularity holds. We conclude by introducing the notion of a semistable program: maximize a linear function on the set of square matrices subject to linear equality constraints together with the constraint that the real parts of the eigenvalues of the solution matrix are non-positive. Semistable programming is a nonconvex generalization of semidefinite programming. Using our analysis, we derive a necessary condition for a local maximizer of a semistable program, and we give a generalization of the complementarity condition familiar from semidefinite programming.     

How often can a given distance occur in a finite set of integers?

Let n, a, d be natural numbers and A a set of integers of the closed interval [0, n] with | A | = a. Then we establish sharp lower and upper bounds for the number of pairs (x,y) ? A×A(x,y)\in A\times A for which y - x = d. Roughly spoken, we investigate how often a distance d can occur in A.     

On solutions to the Schroeder-Bernstein problem for Banach spaces

We investigate the geometry of the Banach spaces failing Schroeder-Bernstein Property (SBP). Initially we prove that every complex hereditarily indecomposable Banach space H is isomorphic to a complemented subspace of a Banach space S(H) that fails SBP in such a way that the only complemented hereditarily indecomposable subspaces of S(H) are those which are nearly isomorphic to H. Then we show that every Banach space having Mazur property is isomorphic to some complemented subspace of a Banach space which is not isomorphic to its square but isomorphic to its cube. Finally, we prove that if a Banach space X fails SBP then either it is not primary or the Grothendieck group K₀(L(X)) of the algebra of operators on X is not trivial.     

A note on the complexity of the transportation problem with a permutable demand vector

In this note we investigate the computational complexity of the transportation problem with a permutable demand vector, TP-PD for short. In the TP-PD, the goal is to permute the elements of the given integer demand vector b=(b₁,…,b_n) in order to minimize the overall transportation costs. Meusel and Burkard [6] recently proved that the TP-PD is strongly NP-hard. In their NP-hardness reduction, the used demand values b_j, j=1,…,n, are large integers. In this note we show that the TP-PD remains strongly NP-hard even for the case where b_j]{0,3} for j=1,…,n. As a positive result, we show that the TP-PD becomes strongly polynomial time solvable if b_j] {0,1,2} holds for j=1,…,n. This result can be extended to the case where b_j]{3,3+1,3+2} for an integer 3.     

Abelian Kernels of Some Monoids of Injective Partial Transformations and an Application

In this paper we compute the abelian kernels of the monoids POI_n and POPI_n of all injective order preserving and respectively, orientation preserving, partial transformations on a chain with n elements. As an application, we show that the pseudovariety POPI generated by the monoids POPI_n (n epsilon N) is not contained in the Mal'cev product of the pseudovariety POI generated by the monoids POI_n (n epsilon N) with the pseudovariety Ab of all finite abelian groups.     

Orbit nonproper actions on Lorentz manifolds

If a topological group G acts on a topological space X, then we say that the action is orbit nonproper provided that, for some x ? X x \in X , the orbit map g ? gx : G ? X g \mapsto gx : G \to X is nonproper. We consider the problem of classifying the connected, simply connected real Lie groups G such that G admits a locally faithful, orbit nonproper action on a connected Lorentz manifold. In this paper, we describe three collections of groups such that G admits such an action iff G is in one of the three collections. In an earlier paper, we effectively described the first collection. In yet another paper, we describe effectively those groups in the second collection which are not contained in the union of the first and third. Finally, in another paper, we describe effectively the third collection.     

Internal injectivity of Boolean algebras in MSet

This paper deals with the internal notion of injectivity for Boolean algebras in the topos of M-sets. Given that, for ordinary Boolean algebraas, injectivity is the same as completeness (Sikorski's theorem) and the injective hull is the same as normal completion, we investigate here how the internal notion of completeness relates to internal injectivity. Further, we consider the internal injectivity of the initial Boolean algebra 2 which is equivalent to the prime ideal theorem for Boolean algebras in this topos. Before we turn specificially to Boolean algebras, we develop the bassic general facts concerning internal injectivity in MSet for arbitrary equational classes of algebras.     

Auto-equivalences of derived categories acting on cohomology

Let A be a k-algebra which is projective as a k-module, let M be an A-module whose endomorphisms are given by multiplication by central elements of A, and let TrPic_k(A) be the group of standard self-equivalences of the derived category of bounded complexes of A-modules. Then we define an action of the stabilizer of M in TrPic_k(A) on the Ext-algebra of M. In case M is the trivial module for the group algebra kG = A, this defines an action on the cohomology ring of G which extends the well-known action of the automorphism group of G on the cohomology group.     

Sur les tours localement cyclotomiques

For a given number field K and any prime l\ell we construct an increasing sequence of l\ell -extensions K_n of K which are locally cyclotomic over K. We give various criterious of finiteness or non-finiteness of this l\ell -tower and we characterise the number fields which have such a finite tower in terms of Iwasawa theory.     

Four Questions on Birkhoff Polytope

We ask several questions on the structure of the polytope P_n of doubly stochastic n 2 n matrices, known as a Birkhoff polytope. We discuss the volume of P_n, the work of the simplex method, and the mixing of random walks on P_n.     

Power residues on Abelian varieties

Given a prime l and an elliptic curve E defined over a number field k, we show that a non-zero point P] E(k) lies in lE(k) if and only if P lies in lE(k)(mod ) for almost all finite primes  of k. We give conditions on l under which analogous results hold for Abelian varieties and with one point replaced by a finite number of points. We also construct examples to show that these conditions are essential.     

Lie p-algebras of finite p-subalgebra rank

We say that a Lie p-algebra L has finite p-subalgebra rank if the minimal number of generators required to generate every finitely generated p-subalgebra is uniformly bounded by some integer r. This paper is concerned with the following problem: does L being of finite p-subalgebra rank force ad(L) to be finite-dimensional? Although this seems unlikely in general, we show that this is indeed the case for Lie p-algebras in a large class including all locally, residually, and virtually soluble Lie p-algebras.     

The Spectral Mapping Theorem for Evlution Semigroups on L p Associated with Strongly Continuous Cocycles

In this note we prove the spectral mapping theorem for certain evolution semigroups. Specifically, we study the evolution semigroup on L^p(Theta,mu;X), 1≤p相似文献   

Profinite groups with many commuting pairs or involutions

We prove that if the set of commuting pairs of a profinite group G has positive Haar measure then G is abelian by finite. Using this we show that the set I of involutions has positive measure exactly if I contains a nonempty open subset of G.     

On Cohn's conjecture concerning the diophantine equation¶x2+ 2m = yn

In the past fifty years and more, there are many papers concerned with the solutions (x,y,m,n) of the exponential diophantine equation $ x^2 + 2^m = y^n, x, y, m, n \in \mathbb{N}, 2 \not|\, y, n > 2 $ x^2 + 2^m = y^n, x, y, m, n \in \mathbb{N}, 2 \not|\, y, n > 2 , written by Ljunggren, Nagell, Brown, Toyoizumi, Cohn and the others. In 1992, Cohn conjectured that the equation has no solutions (x, y, m, n) with m > 2 and 2 | m 2 \mid m . In this paper, using a quantitative result of Laurent, Mignotte and Nesterenko on linear forms in the logarithms of two algebraic numbers, we verify Cohn's conjecture. Thus, according to known results, we prove that the equation has only three solutions (x, y, m, n) = (5, 3, 1, 3), (7, 3, 5, 4) and (11, 5, 2, 3).     

The Cartan matrix of the Schur algebra S(2, r)

Let k be an infinite field of prime characteristic and let r be a positive integer. Using admissible decompositions, we determine explicitly the entries of the decomposition matrix of the Schur algebra S(2, r) over k and prove that any two blocks with the same number of simple modules have the same decomposition matrix and hence the same Cartan matrix.