Transformation de Fourier des distributions de type positif sur un groupe de Lie unimodulaire

Call a locally compact group G, C¹-unique, if L¹(G) has exactly one (separating) C¹-norm. It is easy to see that a ¹-regular group G is C¹-unique and that a C¹-unique group is amenable. For connected groups G it is proved that G is C¹-unique, if the interior R(G)⁰ of a certain part R(G) of Prim(G), called the regular part of Prim(G), is dense in Prim(G), and that C¹-uniqueness of G implies the density of R(G) in Prim(G). From this it is derived that a connected group of type I is C¹-unique if and only if R(G)⁰ is dense in Prim(G). For exponential G, a quite explicit version of this result in terms of the Lie algebra of G is given. As an easy consequence, examples of amenable groups, which are not C¹-unique, and C¹-unique groups, which are not ¹-regular are obtained. Furthermore it is shown that a connected locally compact group G is amenable if and only if L¹(G) has exactly one C¹-norm, which is invariant under the isometric ¹-automorphisms of L¹(G).     

A geometric Gordan-Stiemke theorem

Let C and K be closed cones in Rⁿ. Denote by φ (K∩C) the face of C generated by K ∩ C, by φ(K ∩ D)^D the dual face of φ(K ∩ C) in C¹, and by φ(-K¹ ∩ C¹) the face of C¹ generated by -K¹ ∩ C¹. It is proved that φ(K ∩ C¹) if and only if -C¹ ∩ [span(K ∩ C)] ⊥ ? C¹ + K¹. In particular, the closedness of C¹ + K¹ is a sufficient condition. Our result contains a generalization of the Gordon-Stiemke theorem which appeared in a recent paper of Saunders and Schneider.     

Completely inverse AG ∗∗-groupoids

A completely inverse AG ^??-groupoid is a groupoid satisfying the identities (xy)z=(zy)x, x(yz)=y(xz) and xx ^?1=x ^?1 x, where x ^?1 is a unique inverse of x, that is, x=(xx ^?1)x and x ^?1=(x ^?1 x)x ^?1. First we study some fundamental properties of such groupoids. Then we determine certain fundamental congruences on a completely inverse AG ^??-groupoid; namely: the maximum idempotent-separating congruence, the least AG-group congruence and the least E-unitary congruence. Finally, we investigate the complete lattice of congruences of a completely inverse AG ^??-groupoids. In particular, we describe congruences on completely inverse AG ^??-groupoids by their kernel and trace.     

On pseudoinverses of operator products

The analytical structure of the Moore-Penrose pseudoinverse of the product ab of any two operators over finite-dimensional unitary spaces is studied. The existence of the unique representation of the form (ab)⁺=b⁺(h+g)a⁺ is proved. Here h:= (a⁺abb⁺)⁺ is an (oblique) projector and g is an operator with a number of special properties. In particular, h+g is a projector, g is orthogonal to h in some metric, and g³=0. A necessary and sufficient condition for the case (ab)⁺=b⁺ha⁺ is established. This case contains the classical one (ab)⁺=b⁺a⁺ (the reverse-order law). For the latter a new necessary and sufficient condition is given.     

On the size of the Durfee square of a random integer partition

We prove a local limit theorem for the length of the side of the Durfee square in a random partition of a positive integer n as n→∞. We rely our asymptotic analysis on the power series expansion of x^m2∏_j=1^m(1−x^j)⁻² whose coefficient of xⁿ equals the number of partitions of n in which the Durfee square is m².     

Dirichlet Heat Kernel Estimates for Stable Processes with Singular Drift In Unbounded C 1,1 Open Sets

Suppose d ≥ 2 and α ∈ (1, 2). Let D be a (not necessarily bounded) C ^1,1 open set in ? ^d and μ = (μ ¹, . . . , μ ^d ) where each μ ^j is a signed measure on ? ^d belonging to a certain Kato class of the rotationally symmetric α-stable process X. Let X ^μ be an α-stable process with drift μ in ? ^d and let X ^μ,D be the subprocess of X ^μ in D. In this paper, we derive sharp two-sided estimates for the transition density of X ^μ,D .     

On simultaneous representations of primes by binary quadratic forms

Let p ≡ ± 1 (mod 8) be a prime which is a quadratic residue modulo 7. Then p = M² + 7N², and knowing M and N makes it possible to “predict” whether p = A² + 14B² is solvable or p = 7C² + 2D² is solvable. More generally, let q and r be distinct primes, and let an integral solution of H²p = M² + qN² be known. Under appropriate assumptions, this information can be used to restrict the possible values of K for which K²q = A² + qrB² is solvable and the possible values of K′ for which K′²p = qC² + rD² is solvable. These restrictions exclude some of the binary quadratic forms in the principal genus of discriminant ?4qr from representing p.     

On sesquilinear forms over fields with involution

Let k be a field of characteristic ≠2 with an involution σ. A matrix A is split if there is a change of variables Q such that (Q^σ)^TAQ consists of two complementary diagonal blocks. We classify all matrices that do not split. As a consequence we obtain a new proof for the following result. Given a square matrix A there is a matrix S such that (S^σ)^TAS=A^T and S^σS=I.     

Classifying simple closed curve pairs in the 2-sphere and a generalization of the Schoenflies theorem

A classification theory is developed for pairs of simple closed curves (A,B) in the sphere S², assuming that A∩B has finitely many components. Such a pair of simple closed curves is called an SCC-pair, and two SCC-pairs (A,B) and (A^′,B^′) are equivalent if there is a homeomorphism from S² to itself sending A to A^′ and B to B^′. The simple cases where A and B coincide or A and B are disjoint are easily handled. The component code is defined to provide a classification of all of the other possibilities. The component code is not uniquely determined for a given SCC-pair, but it is straightforward that it is an invariant; i.e., that if (A,B) and (A^′,B^′) are equivalent and C is a component code for (A,B), then C is a component code for (A^′,B^′) as well. It is proved that the component code is a classifying invariant in the sense that if two SCC-pairs have a component code in common, then the SCC-pairs are equivalent. Furthermore code transformations on component codes are defined so that if one component code is known for a particular SCC-pair, then all other component codes for the SCC-pair can be determined via code transformations. This provides a notion of equivalence for component codes; specifically, two component codes are equivalent if there is a code transformation mapping one to the other. The main result of the paper asserts that if C and C^′ are component codes for SCC-pairs (A,B) and (A^′,B^′), respectively, then (A,B) and (A^′,B^′) are equivalent if and only if C and C^′ are equivalent. Finally, a generalization of the Schoenflies theorem to SCC-pairs is presented.     

Regular Hadamard Matrices Generating Infinite Families of Symmetric Designs

If H is a regular Hadamard matrix with row sum 2h, m is a positive integer, and q = (2h ? 1)², then (4h ²(q ^{m + 1} ? 1)/(q ?1),(2h ² ? h)q ^m ,(h ²-h)q ^m ) are feasible parameters of a symmetric designs. If q is a prime power, then a balanced generalized weighing matrix BGW((q ^{m +1} ? 1)/(q?1),q ^m ,q ^m ?q ^{m ?1}) can be applied to construct such a design if H satisfies certain structural conditions. We describe such conditions and show that if H satisfies these conditions and B is a regular Hadamard matrix of Bush type, then B×H satisfies these structural conditions. This allows us to construct parametrically new infinite families of symmetric designs.