the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

nonseparable spaceability and strong algebrability of sets of continuous singular functions abstract let cbv denote the banach algebra of all continuous real valued functions of bounded variation defined in 0 1 we show that the set of strongly singular functions in cbv is nonseparably spaceable we also prove that certain families of singular functions constitute strongly c algebrable sets the argument is based on a criterion of strong c algebrability using compositions with exponential like functions

lineability spaceability and additivity cardinals for darboux like functions we introduce the concept of maximal lineability cardinal number ml m of a subset m of a topological vector space and study its relation to the cardinal numbers known as additivity a m homogeneous lineability hl m and lineability l m of m in particular we will describe in terms of l the lineability and spaceability of the families of the following darboux like functions on r n n 1 extendable jones and almost continuous functions

the tortoise and the hare restart gmres when solving large nonsymmetric systems of linear equations with the restarted gmres algorithm one is inclined to select a relatively large restart parameter in the hope of mimicking the full gmres process surprisingly cases exist where small values of the restart parameter yield convergence in fewer iterations than larger values here two simple examples are presented where gmres 1 converges exactly in three iterations while gmres 2 stagnates one of these examples reveals that gmres 1 convergence can be extremely sensitive to small changes in the initial residual

the set of space filling curves topological and algebraic structure abstract in this paper a study of topological and algebraic properties of two families of functions from the unit interval i into the plane r 2 is performed the first family is the collection of all peano curves that is of those continuous mappings onto the unit square the second one is the bigger set of all space filling curves i e of those continuous functions i r 2 whose images have the positive jordan content emphasis is put on the size of these families in both topological and algebraic senses when endowed with natural structures

smoothness for strong solutions of a hessian quotient equation we obtain the local c 1 1 estimates for the w 2 p strong solutions of a hessian quotient equations with pn n 1 2 and then prove these solutions are smooth there is a couaterexample to show that this regularity result is optimal for n 3

on the zeros of a polynomial and its derivatives if p z is univariate polynomial with complex coefficients having all its zeros inside the closed unit disk then the gauss lucas theorem states that all zeros of p z lie in the same disk we study the following question what is the maximum distance from the arithmetic mean of all zeros of p z to a nearest zero of p z we obtain bounds for this distance depending on degree we also show that this distance is equal to 1 for polynomials of degree 3 and polynomials with real zeros

strong c algebrability of strong sierpinski zygmund smooth nowhere analytic and other sets of functions abstract we present a useful technique of proving strong c algebrability as an outcome we obtain the strong c algebrability of the following sets of functions strong sierpinski zygmund nowhere holder bruckner garg nowhere monotone differentiable a certain baire class smooth and nowhere analytic functions

structural aspects of the non uniformly continuous functions and the unbounded functions within c x abstract we prove in this paper that if a metric space supports a real continuous function which is not uniformly continuous then under appropriate mild assumptions there exists in fact a plethora of such functions in both topological and algebraical senses corresponding results are also obtained concerning unbounded continuous functions on a non compact metrizable space

lineability criteria with applications lineability is a property enjoyed by some subsets within a vector space x a subset a of x is called lineable whenever a contains except for zero an infinite dimensional vector subspace if additionally x is endowed with richer structures then the more stringent notions of dense lineability maximal dense lineability and spaceability arise naturally in this paper several lineability criteria are provided and applied to specific topological vector spaces mainly function spaces sometimes such criteria furnish unified proofs of a number of scattered results in the related literature families of strict order integrable functions hypercyclic vectors non extendable holomorphic mappings riemann non lebesgue integrable functions sequences not satisfying the lebesgue dominated convergence theorem nowhere analytic functions bounded variation functions entire functions with fast growth and peano curves among others are analyzed from the point of view of lineability

algebraic structures in the sets of surjective functions abstract in the paper we construct several algebraic structures vector spaces algebras and free algebras inside sets of different types of surjective functions among many results we prove that the set of everywhere but not strongly everywhere surjective complex functions is strongly c algebrable and that its 2 c algebrability is consistent with zfc under ch the set of everywhere surjective complex functions which are sierpinski zygmund in the sense of continuous but not borel functions is strongly c algebrable the set of jones complex functions is strongly 2 c algebrable

ritz values of normal matrices and ceva s theorem abstract this paper investigates the behavior of ritz values of normal matrices we apply ceva s theorem a classical geometric result to understand the geometric relationship between pairs of ritz values for 3 3 normal non hermitian matrices and then analyze the implications for larger matrices we find that in the case of normal non hermitian matrices the geometric constraints on the placement of ritz values provide less freedom than the cauchy interlacing theorem in the hermitian case using our results we analyze the restarted arnoldi method with exact shifts applied to a 3 3 normal non hermitian matrix

linear structure of sets of divergent sequences and series abstract we show that there exist infinite dimensional spaces of series every non zero element of which enjoys certain pathological property some of these properties consist on being i conditional convergent ii divergent or iii being a subspace of l of divergent series we also show that the space l 1 x of all weakly unconditionally cauchy series in x has an infinite dimensional vector space of non weakly convergent series and that the set of unconditionally convergent series on x contains a vector space e of infinite dimension so that if x e 0 then i xi

basic sequences and spaceability in lp spaces let x be a sequence space and denote by z x the subset of x formed by sequences having only a finite number of zero coordinates we study algebraic properties of z x and show among other results that for p is an element of 1 infinity z l p does not contain infinite dimensional closed subspaces this solves an open question originally posed by r m aron and v i gurariy in 2003 on the linear structure of z l infinity in addition to this we also give a thorough analysis of the existing algebraic structures within the sets z x and x z x and their algebraic genericities

a bernstein problem for special lagrangian equations where is are the eigenvalues of the hessian d2u namely any global convex solution to 1 1 in rn must be a quadratic polynomial recall the classical result any global convex solution in rn to the laplace equation u 1 n c or the monge ampere equation log det d2u log 1 log n c must be quadratic equation 1 1 originates from special lagrangian geometry hl the lagrangian graph x u x rn rn is called special when the argument of the complex number 1 1 1 1 1 n is constant c or u satisfies 1 1 and it is special if and only if x u x is a minimal surface in rn rn hl theorem 2 3 proposition 2 17 in terms of minimal surface our result is the following

lineability and additivity in r r we give a condition for a family of functions to be lineable by means of its additivity the novelty presented here is that we solve the lineability problem using a technique that is not constructive as are most approaches to this problem we relate the notions of additivity and lineability and use this relation to give a general method to find the lineability of large families of functions we also study more examples of pathologically behaving functions in particular the class of jones functions which is a highly pathological subclass of perfectly everywhere surjective functions we work on the additivity lineability and main properties of this class

large free linear algebras of real and complex functions let x be a set of cardinality such that we prove that the linear algebra rx or cx contains a free linear algebra with 2 generators using this we prove several algebrability results for spaces cc and rr in particular we show that the set of all perfectly everywhere surjective functions f c c is strongly 2c algebrable we also show that the set of all functions f r r whose sets of continuity points equals some fixed g set g is strongly 2c algebrable if and only if r g is c dense in itself

differentiators and the geometry of polynomials in 1959 davis introduced the concept of a differentiator of an operator on a finite dimensional hilbert space we prove that every such operator possesses a differentiator we also use the theory of differentiators to solve several problems in the geometry of polynomials for instance we answer in the affirmative a twenty year old unsolved conjecture of schoenberg a related conjecture of katsoprinakis and a fifty year old unsolved conjecture of de bruijn and springer

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

imbedding conditions for normal matrices abstract when can an n k n k normal matrix b be imbedded in an n n normal matrix a this question was studied for the first time 50 years ago by ky fan and gordon pall who gave the complete answer in the case k 1 since then a few authors obtained additional results in this note we show how an approach inspired by the hermitian case can throw some light on the problem

interaction between hermitian and normal imbeddings abstract for several applications it is highly desirable to understand how the eigenvalues of an imbeddable matrix v av where v c n k is an isometry are distributed throughout the numerical range of a c n n there has been extensive study for a hermitian while a geometric description for the eigenvalues of imbeddings in non hermitian matrices remains a challenging problem toward this direction a subspace is introduced wherein all n n complex diagonal matrices for which a given isometry v c n k generates diagonal imbeddings are defined in particular conditions upon which a real diagonal matrix may be imbeddable in some normal are obtained including an application for higher rank numerical ranges finally a procedure determining whether two given sets of complex numbers may be realized as spectra of a pair of imbeddable normal matrices is established

on viscosity solutions of fully nonlinear equations with measurable ingredients we study fully nonlinear uniformly elliptic equations with measurable ingredients caffarelli s recent work on w2 p estimates for viscosity solutions has led to significant progress in this area here we present a unified treatment of this theory based on an appropriate notion of viscosity solution for instance it is shown that strong solutions are viscosity solutions that viscosity solutions are twice differentiable a e and that the pointwise derivatives satisfy the equation a e an important consequence of our approach is the possibility of passage to various kinds of limits in fully nonlinear equations this extends results of this type due to evans and krylov our work is to some extent expository the main purpose being to provide an easily accessible set of tools and techniques to study equations with measurable ingredients 1996 john wiley sons inc

the gauss lucas theorem and jensen polynomials a characterization is given of the sequences fyj o vvith the property that for any complex polynomial z 1akzk and convex region kcontaining the origin and the zeros of the zeros of 2 yi 0 for all n k

peano curves on topological vector spaces the starting point of this paper is the existence of peano curves that is continuous surjections mapping the unit interval onto the unit square from this fact one can easily construct of a continuous surjection from the real line r to any euclidean space rn the algebraic structure of the set of these functions as well as extensions to spaces with higher dimensions is analyzed from the modern point of view of lineability and large algebras are found within the families studied we also investigate topological vector spaces that are continuous image of the real line providing an optimal lineability result

inverse spectral problem for normal matrices and the gauss lucas theorem we establish an analog of the cauchy poincare interlacing theorem for normal matrices in terms of majorization and we provide a solution to the corresponding inverse spectral problem using this solution we generalize and extend the gauss lucas theorem and prove the old conjecture of de bruijn springer on the location of the roots of a complex polynomial and its derivative and an analog of rolle s theorem conjectured by schoenberg

maximal polynomials and the ilieff sendov conjecture in this paper we consider those complex polynomials which have all their roots in the unit disk one fixed root and all the roots of their first derivatives as far as possible from a fixed point we conjecture that any such polynomial has all the roots of its derivative on a circle centered at the fixed point and as many of its own roots as possible on the unit circle we prove a part of this conjecture and use it to define an algorithm for constructing some of these polynomials with this algorithm we investigate the 1962 conjecture of sendov and the 1969 conjecture of goodman rahman and ratti and independently schmeisser obtaining counterexamples of degrees 6 8 10 and 12 for the latter

algebraic structure of continuous unbounded and integrable functions abstract in this paper we study the large linear and algebraic size of the family of unbounded continuous and integrable functions in 0 and of the family of sequences of these functions converging to zero uniformly on compacta and in l 1 norm in addition we concentrate on the speed at which these functions grow their smoothness and the strength of their convergence to zero

some results and open questions on spaceability in function spaces a subset m of a topological vector space x is called lineable respectively spaceable in x if there exists an infinite dimensional linear space respectively an infinite dimensional closed linear space y subset of m boolean or 0 in this article we prove that for every infinite dimensional closed subspace x of c 0 1 the set of functions in x having infinitely many zeros in 0 1 is spaceable in x we discuss problems related to these concepts for certain subsets of some important classes of banach spaces such as c 0 1 or muntz spaces we also propose several open questions in the field and study the properties of a new concept that we call the oscillating spectrum of subspaces of c 0 1 as well as oscillating and annulling properties of subspaces of c 0 1