rank equalities related to the generalized inverses at s 2 bt 1 s1 2 of two matrices a and b for given matrices a and b by applying the group inverse expressions of generalized inverses a t s 2 and b t 1 s 1 2 we present some rank equalities related to a t s 2 and b t 1 s 1 2 as applications we obtain a variety of rank equalities related to the moore penrose inverses the drazin inverses the group inverses and the weighted moore penrose inverses of a and b

the block independence in the generalized inverse at s 2 for some ordered matrices and applications abstract in this paper the definition of block independence in the generalized inverse a t s 2 is firstly given and then a necessary and sufficient condition for some ordered matrices to be block independent in the generalized inverse a t s 2 is derived as an application a necessary and sufficient condition for a 1 a 2 a k t s 2 a 1 t 1 s 1 2 a 2 t 2 s 2 2 a k t k s k 2 is proved moreover some results are shown with respect to the moore penrose inverse the weighted moore penrose inverse and the drazin inverse respectively

on g inverses of a bordered matrix revisited abstract in this article we study the structure and properties of g inverse m of the bordered matrix m a b c 0 which is partitioned conforming with m m d 1 d 2 d 3 d 4 by applying the multiple quotient singular value decomposition qq svd from the structure of m we discuss the relations among d1 d2 d3 and d4 and derive equivalent conditions for m to be a bordered matrix by studying d1 in m we obtain some equivalent conditions for d1 to be a g inverse of a

diophantine definability of infinite discrete nonarchimedean sets and diophantine models over large subrings of number fields we prove that some infinite p adically discrete sets have diophantine definitions in large subrings of number fields first if k is a totally real number field or a totally complex degree 2 extension of a totally real number field then for every prime p of k there exists a set of k primes s of density arbitrarily close to 1 such that there is an infinite p adically discrete set that is diophantine over the ring ok s of s integers in k second if k is a number field over which there exists an elliptic curve of rank 1 then there exists a set of k primes s of density 1 and an infinite diophantine subset of ok s that is v adically discrete for every place v of k third if k is a number field over which there exists an elliptic curve of rank 1 then there exists a set of k primes s of density 1 such that there exists a diophantine model of z over ok s this line of research is motivated by a question of mazur concerning the distribution of rational points on varieties in a nonarchimedean topology and questions concerning extensions of hilbert s tenth problem to subrings of number fields

the reverse order law for the generalized inverse a 2 t s by using the ranks of matrices the necessary and sufficient condition is given for the reverse order law of the generalized inverse a 2 t s of matrix product a a 1a 2 a n to hold as the special cases similar results for moore penrose inverse a the weighted moore penrose inverse a m n drazin inverse a d and group inverse a g are also derived

extension of hilbert s tenth problem to some algebraic number fields we extend the solution of hilbert s tenth problem to algebraic number fields having one pair of complex conjugated embeddings the proof is based on the extended method of j denef used for totally real algebraic number fields

elliptic curves retaining their rank in finite extensions and hilbert s tenth problem for rings of algebraic numbers using poonen s version of the weak vertical method we produce new examples of large and small rings of algebraic numbers including rings of integers where z and or the ring of integers of a subfield are existentially definable and or where the ring version of mazur s conjecture on the topology of rational points does not hold

the inverse of a block circulant matrix the inverse a 1 of a block circulant matrix bcm a is given in a closed form by using the fact that a bcm is a combination of permutation matrices whose eigenvalues and eigenvectors are found with the help of the complex roots of unity special results are also given when a is block symmetric or symmetric

hilbert s tenth problem and mazur s conjecture for large subrings of mathbb q we give the first examples of infinite sets of primes s such that hilbert s tenth problem over z s 1 has a negative answer in fact we can take s to be a density 1 set of primes we show also that for some such s there is a punctured elliptic curve e over z s 1 such that the topological closure of e z s 1 in e r has infinitely many connected components

diophantine definability over some rings of algebraic numbers with infinite number of primes allowed in the denominator let k be a number field let w be a set of non archimedean primes of k let o k w x k ord p x 0 p w then if k is a totally real non trivial cyclic extension of there exists an infinite set w of finite primes of k such that and the ring of algebraic integers of k have a diophantine definition over o k w thus the diophantine problem of o k w is undecidable

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

diophantine classes of holomorphy rings of global fields abstract this paper introduces the notions of diophantine generation and diophantine equivalence and explores the resulting diophantine classes of holomorphy rings of number fields and of one variable function fields over finite fields of constants

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

ranks of twists of elliptic curves and hilbert s tenth problem in this paper we investigate the 2 selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields we give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2 selmer rank and we give lower bounds for the number of twists with bounded conductor that have a given 2 selmer rank as a consequence under appropriate hypotheses we can find many twists with trivial mordell weil group and assuming the shafarevich tate conjecture many others with infinite cyclic mordell weil group using work of poonen and shlapentokh it follows from our results that if the shafarevich tate conjecture holds then hilbert s tenth problem has a negative answer over the ring of integers of every number field

the moore penrose inverses of m n block matrices and their applications abstract this article gives the expressions for the moore penrose inverses of m n block matrices when they satisfy rank additivity conditions and presents some of their special cases and applications

hilbert s tenth problem for fields of rational functions over finite fields we prove that there is no algorithm to solve arbitrary polynomial equations over a field of rational functions in one letter with constants in a finite field of characteristic other than 2 and hence hilbert s tenth problem for any such field is undecidable

rank equalities for submatrices in generalized inverse mt s 2 of m by applying the group inverse expressions of the generalized inverse m t s 2 of the matrix m we present some rank equalities for submatrices in m t s 2 as applications we derive rank equalities for submatrices in the moore penrose inverse and the drazin inverse

on the p q outer generalized inverse in banach algebra abstract in this note the p q outer generalized inverse in banach algebra is considered a new representation based on the 1 5 inverse is presented also we construct an iterative method for computing the p q outer generalized inverse in banach algebra

periodic solutions for a rayleigh system of p laplacian type abstract by using the generalized borsuk theorem in coincidence degree theory some criteria for guaranteeing the existence of periodic solutions for a rayleigh system of p laplacian type are derived