Generalized subdifferentials of the rank function

We give an explicit formula for the generalized subdifferentials; i.e. the proximal subdifferential, the Fréchet subdifferential, the limitting subdifferential and the Clarke subdifferential of the counting function. Then, thanks to theorems of A.S. Lewis and H.S. Sendov, we obtain the corresponding generalized subdifferentials of the rank function.     

A variational approach of the rank function

In the same spirit as the one of the paper (Hiriart-Urruty and Malick in J. Optim. Theory Appl. 153(3):551–577, 2012) on positive semidefinite matrices, we survey several useful properties of the rank function (of a matrix) and add some new ones. Since the so-called rank minimization problems are the subject of intense studies, we adopt the viewpoint of variational analysis, that is the one considering all the properties useful for optimizing, approximating or regularizing the rank function.     

A note on the rank parity function

We provide some further theorems on the partitions generated by the rank parity function. New Bailey pairs are established, which are of independent interest.     

On the rank of a spectral function

Let P(x), 0 x 1, be an absolutely continuous spectral function in the separable Hilbert spacesS. If the vectors h_j, j=1, 2, ..., s; s are such that the set P(x)h_j is complete inS, then the rank of the function P(x) equals the general rank of the matrix-function d/dxP(x)h_i,h_js₁.Translated from Matematicheskie Zametki, Vol. 5, No. 4, pp. 457–460, April, 1969.     

On the rank of abelian varieties over function fields

Let be a smooth projective curve defined over a number field k, A/k() an abelian variety and (τ, B) the k()/k-trace of A. We estimate how the rank of A(k())/τB(k) varies when we take a finite geometrically abelian cover defined over k. This work was partially supported by CNPq research grant 304424/2003-0, Pronex 41.96.0830.00 and CNPq Edital Universal 470099/2003-8. I would like to thank Douglas Ulmer for comments on how to treat the case of arbitrary ramification, but the conductor prime to the ramification locus, in the case of elliptic fibrations. I would also like to thank Marc Hindry for comments on the inequality comparing the conductors of A and A'. Finally, I also thank the referee for his comments and criticisms.     

Modularity of proof-nets

When we cut a multiplicative proof-net of linear logic in two parts we get two modules with a certain border. We call pretype of a module the set of partitions over its border induced by Danos-Regnier switchings. The type of a module is then defined as the double orthogonal of its pretype. This is an optimal notion describing the behaviour of a module: two modules behave in the same way precisely if they have the same type.In this paper we define a procedure which allows to characterize (and calculate) the type of a module only exploiting its intrinsic geometrical properties and without any explicit mention to the notion of orthogonality. This procedure is simply based on elementary graph rewriting steps, corresponding to the associativity, commutativity and weak-distributivity of the multiplicative connectives of linear logic.This work was carried out at the University Roma Tre, in the framework of the European TMR Research Programme Linear Logic in Computer Science. The authors are grateful to Paul Ruet and to the anonymous referee for their useful comments and remarks on a previous version of this paper.     

Higher rank subgroups in the class groups of imaginary function fields

Let F be a finite field and T a transcendental element over F. In this paper, we construct, for integers m and n relatively prime to the characteristic of F(T), infinitely many imaginary function fields K of degree m over F(T) whose class groups contain subgroups isomorphic to (Z/nZ)^m. This increases the previous rank of m−1 found by the authors in [Y. Lee, A. Pacelli, Class groups of imaginary function fields: The inert case, Proc. Amer. Math. Soc. 133 (2005) 2883-2889].     

An interesting q-series related to the 4th symmetrized rank function

This paper presents the method to utilizing the $s$-fold extension of Bailey’s lemma to obtain $spt$-type functions related to the symmetrized rank function ${\eta }_{2k}\left(n\right).$ We provide the $k=2$ example, but clearly illustrate how deep connections between higher-order spt functions exist for any integer $k>1,$ and provide several directions for possible research. In particular, we present why the function $sp{t}_M^1\left(n\right),$ the total number of appearances of the smallest parts of partitions where parts greater than the smallest plus $M$ do not occur, is an $spt$ function that appears to have central importance. We also make note about extending $spt$-type functions to partition pairs.     

Approximation of rank function and its application to the nearest low-rank correlation matrix

The rank function rank(.) is neither continuous nor convex which brings much difficulty to the solution of rank minimization problems. In this paper, we provide a unified framework to construct the approximation functions of rank(.), and study their favorable properties. Particularly, with two families of approximation functions, we propose a convex relaxation method for the rank minimization problems with positive semidefinite cone constraints, and illustrate its application by computing the nearest low-rank correlation matrix. Numerical results indicate that this convex relaxation method is comparable with the sequential semismooth Newton method (Li and Qi in SIAM J Optim 21:1641–1666, 2011) and the majorized penalty approach (Gao and Sun, 2010) in terms of the quality of solutions.     

Modularity and monotonicity of games

The purpose of this paper is twofold. First, we generalize Kajii et al. (J Math Econ 43:218–230, 2007) and provide a condition under which for a game \(v\) , its Möbius inverse is equal to zero within the framework of the \(k\) -modularity of \(v\) for \(k \ge 2\) . This condition is more general than that in Kajii et al. (J Math Econ 43:218–230, 2007). Second, we provide a condition under which for a game \(v\) , its Möbius inverse takes non-negative values, and not just zero. This paper relates the study of totally monotone games to that of \(k\) -monotone games. Furthermore, this paper shows that the modularity of a game is related to \(k\) -additive capacities proposed by Grabisch (Fuzzy Sets Syst 92:167–189, 1997). To illustrate its application in the field of economics, we use these results to characterize a Gini index representation of Ben-Porath and Gilboa (J Econ Theory 64:443–467, 1994). Our results can also be applied to potential functions proposed by Hart and Mas-Colell (Econometrica 57:589–614, 1989) and further analyzed by Ui et al. (Math Methods Oper Res 74:427–443, 2011).     

Chapter 2:rank and tensor rank

Let Vbe a vector space of matrices over a field and ka fixed positive integer. In this chapter we first survey results concerning linear maps on certain types of Vthat preserve one of the following:(a) the set of rank kmatrices, (b) the set of matrices of rank less than k. We next survey results concerning linear maps on certain symmetry classes of tensors that preserve nonzero decomposable elements.     

The prime at infinity and the rank of the class group in global function fields

In this paper we construct, for any integers m and n, and 2?g?m-1, infinitely many function fields K of degree m over F(T) such that the prime at infinity splits into exactly g primes in K and the ideal class group of K contains a subgroup isomorphic to (Z/nZ)^m-g. This extends previous results of the author and Lee for the cases g=1 and g=m.     

Relation between the constant rank and the relaxed constant rank constraint qualifications

This article discusses a relation between the constant rank constraint qualification (CRCQ) and the recently proposed relaxed constant rank constraint qualification (RCRCQ). We show that a parametric constraint system satisfying the RCRCQ is locally diffeomorphic to a system satisfying the CRCQ. We use this result to extend some existing results for the CRCQ to the RCRCQ, establish a relation between the RCRCQ and the Mangasarian–Fromovitz constraint qualification, and obtain a weakened version of the Aubin property under the RCRCQ.     

Modularity of tangential k-blocks

This paper examines the role of modularity in tangential k-blocks over GF(q). It is shown that if M is a tangential k-block over GF(q) and F is a modular flat of M which is affine over GF(q) then the simple matroid associated with the complete Brown truncation of M by F is also a tangential k-block over GF(q). This enables us to construct tangetial k-blocks over GF(q) of all ranks r where q^k − q + 2⩽r ⩽ q^k. We also consider tangential k-blocks which have modular hyperplanes; bounds are placed on the rank of members of this class and some of their minors are exhibited.