The structure of certain rigid tensor categories

We consider rigid tensor categories over a field of characteristic zero in which some exterior power of each object is zero. To cite this article: P. O'Sullivan, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Singular values of a real rectangular tensor

Real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we systematically study properties of singular values of a real rectangular tensor, and give an algorithm to find the largest singular value of a nonnegative rectangular tensor. Numerical results show that the algorithm is efficient.     

Estimation of Ricci tensor on certain Fano manifolds

We give time existence to some Monge-Ampère equations on certain Fano manifolds which cannot carry Einstein-K?hler metrics. These solutions allow us to obtain an estimation of Ricci tensors on our manifolds.

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Résumé. On met en évidence un intervalle de temps pour lequel des équations de Monge-Ampère sur certaines variétés de Fano, qui ne possèdent pas de métriques d'Einstein-K?hler, admettent des solutions. Ces solutions permettent d'obtenir des métriques dont on sait minorer le tenseur de Ricci.

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Received June 22, 1998; in final form January 25, 1999     

On special values of certain Dirichlet L-functions

Let r _k (n) denote the number of ways n can be expressed as a sum of k squares. Recently, S. Cooper (Ramanujan J. 6:469–490, [2002]), conjectured a formula for r ₉(t), t≡5 (mod 8), r ₁₁(t), t≡7 (mod 8), where t is a square-free positive integer. In this note we observe that these conjectures follow from the works of Lomadze (Akad. Nauk Gruz. Tr. Tbil. Mat. Inst. Razmadze 17:281–314, [1949]; Acta Arith. 68(3):245–253, [1994]). Further we express r ₉(t), r ₁₁(t) in terms of certain special values of Dirichlet L-functions. Combining these two results we get expressions for these special values of Dirichlet L-functions involving Jacobi symbols.      

Critical values of autonomous Lagrangian systems

Let M be a closed manifold and a convex superlinear Lagrangian. We consider critical values of Lagrangians as defined by R. Ma?é in [5]. Let c _u (L) denote the critical value of the lift of L to the universal covering of M and let c _a (L) denote the critical value of the lift of L to the abelian covering of M. It is easy to see that in general, . Let c ₀ (L) denote the strict critical value of L defined as the smallest critical value of where ranges among all possible closed 1-forms. We show that c _a (L) = c ₀ (L). We also show that if there exists k such that the Euler-Lagrange flow of L on the energy level k' is Anosov for all , then . Afterwards, we exhibit a Lagrangian on a compact surface of genus two which possesses Anosov energy levels with energy , thus answering in the negative a question raised by Ma?é. This example also shows that the inequality could be strict. Moreover, by a result of M.J. Dias Carneiro [4] these Anosov energy levels do not have minimizing measures. Finally, we describe a large class of Lagrangians for which c _u (L) is strictly bigger than the maximum of the energy restricted to the zero section of TM. Received: October 2, 1996     

Algebraic independence of values of certain Fourier series

The algebraic independence of values of certain Fourier series with algebraic coefficients and their derivatives at algebraic points is proved.     

Exact values of widths for certain functional classes

For classes of functions analytic in a unit circle in a Hardy-Banach space, we obtain exact values of Kolmogorov widths in the case where the metrics of the class and the space do not coincide.     

Critical points and values of complex polynomials

This paper is primarily concerned with complex polynomials which have critical points which are also fixed points. We show that certain perturbations of a critical fixed point satisfy an inequality. This inequality permits us to prove a local version of Smale's mean value conjecture. We also use Thurston's topological characterization of critically finite rational mappings to enumerate explicitly as branched mappings the set of complex polynomials which have all their critical points fixed.     

Anisotropic tensor spaces and functions,the mean values of tensor quantities and the limiting criteria

The concept of an anisotropic vector space with a tensor basis which is invariant under a symmetry transformations of a three-dimensional Euclidean vector space is introduced using the example of symmetric second- and fourth-rank Euclidean tensors. In addition to the traditional operation of summation, the operation of multiplication in a fixed tensor basis is introduced for the elements of this space, that is, the axioms of a ring with an identity element and zero divisors, which enable one to carry out algebraic and functional operations. The possibilities of the proposed mathematical procedure are illustrated using examples of anisotropic tensor functions of a tensor argument, by the general solution of the classical problem of calculating the mean value of the tensor of the moduli of elasticity of a single-phase grain-oriented polycrystalline material and the construction of the strength surfaces of anisotropic composite materials.     

Critical points for certain statistical measures of agreement

This paper further explores the analytic connections between three commonly used statistical measures of agreement for 2?×?2 contingency tables: sensitivity, specificity, and the kappa coefficient, which are often employed in evaluating diagnostic tests. In particular, for a given fixed kappa the corresponding locus of minimum values of sensitivity and specificity is rigorously derived.     

Optimal cubature formulas for tensor products of certain classes of functions

We study the problem of constructing an optimal formula of approximate integration along a d-dimensional parallelepiped. Our construction utilizes mean values along intersections of the integration domain with n hyperplanes of dimension (d−1), each of which is perpendicular to some coordinate axis. We find an optimal cubature formula of this type for two classes of functions. The first class controls the moduli of continuity with respect to all variables, whereas the second class is the intersection of certain periodic multivariate Sobolev classes. We prove that all node hyperplanes of the optimal formula in each case are perpendicular to a certain coordinate axis and are equally spaced and the weights are equal. For specific moduli of continuity and for sufficiently large n, the formula remains optimal for the first class among cubature formulas with arbitrary positions of hyperplanes.     

On the distribution of values of certain divisor functions

Let {?_d} be a sequence of nonnegative numbers and f(n) = Σ?_d, the sum being over divisors d of n. We say that f has the distribution function F if for all c ≥ 0, the number of integers n ≤ x for which f(n) > c is asymptotic to xF(c), and we investigate when F exists and when it is continuous.     

Classification of certain commutator ideals and the tensor product closure property

Research partially supported by NSF Grant DMS 8503390.