riemann integrability and lebesgue measurability of the composite function abstract if f is continuous on the interval a b g is riemann integrable resp lebesgue measurable on the interval and g a b then f g is riemann integrable resp measurable on a well known fact on the other hand states that f g might not be riemann integrable resp measurable when f is riemann integrable resp measurable and g is continuous if c stands for the continuum in this paper we construct a 2 c dimensional space v and a c dimensional space w of respectively riemann integrable functions and continuous functions such that for every f v 0 and g w 0 f g is not riemann integrable showing that nice properties such as continuity or riemann integrability can be lost in a linear fashion via the composite function similarly we construct a c dimensional space w of continuous functions such that for every g w 0 there exists a c dimensional space v of measurable functions such that f g is not measurable for all f v 0

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 and integrability in the sense of riemann lebesgue denjoy and khintchine abstract in this paper we continue the ongoing research on lineability related questions on this occasion we shall consider among others the classes of integrable functions in the sense of riemann lebesgue denjoy and khintchine improving some already known results and expanding the study of lineability to other famous integrable classes never considered before

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

closed form nonlinear tracking controllers for quadrotors with model and input generator uncertainties the paper presents a new approach for trajectory tracking in a quadrotor type helicopter the proposed closed form control schemes apply smooth bounded hyperbolic functions of the state variables for achieving trajectory tracking in the fully nonlinear system the current approach accounts for main sources of uncertainties this is the first time explicit formulas are produced for solutions of the considered control problems a comparative study with existing controller is presented

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 question of rentschler and the dixmier problem the following theorem is the main result of the paper theorem let r and t be somewhat commutative algebras with the same holonomic number and let r t be an algebra homomorphism then every holonomic t module m is via a holonomic r module and has finite length as an r module when applied to the weyl algebra this result gives a positive answer to a question of rentschler in the important case where r d x and t d y are rings of differential operators on smooth irreducible algebraic affine varieties x and y of the same dimension the result means that holonomicity is preserved by twisting by an arbitrary algebra homomorphism a short proof is given of the well known fact that an affirmative solution to the dixmier problem whether every algebra endomorphism of the weyl algebra an is an automorphism implies the jacobian conjecture the dixmier problem has a positive answer if and only if the twisted from both sides a n bimodule an is simple for each to begin with it is not even clear whether it is finitely generated the theorem implies that it has finite length moreover the bimodule is holonomic thus simple subfactors of it have the least possible gelfand kirillov dimension

low level formation controls for a group of quadrotors with model uncertainties the article proposes low level control schemes for a group of quadrotors with highly nonlinear uncertain models in formation flights the paper suggests control strategies for formation flights in

dixmier s problem 5 for the weyl algebra abstract the purpose of this paper is to give a negative answer to problem 5 and a positive answer in the case of homogeneous elements to problem 4 of the article of j dixmier bull soc math france 96 1968 209 242 on the first weyl algebra

algebraic genericity within the class of sup measurable functions abstract it is well known that there exist real valued non measurable functions on the plane whose restrictions to the graph of all measurable functions on the real line are measurable in this paper it is shown that there is in fact a continuum dimensional vector space all of whose members except for zero are functions on the plane satisfying the mentioned property versions of this fact are proved in the baire category setting and in the case where the graphs are generated by softer functions

simple holonomic modules over the second weyl algebra a2 abstract for simple generalized weyl algebras of gelfand kirillov dimension 4 a class including the second weyl algebra a 2 some simple factor algebras of the universal enveloping algebra of the lie algebra sl 2 sl 2 and of u q sl 2 u q sl 2 etc the simple holonomic modules are classified up to pairs of irreducible elements of certain noncommutative euclidean ring

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

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

gelfand kirillov dimension of commutative subalgebras of simple inﬁnite dimensional algebras and their quotient division algebras

an analogue of the conjecture of dixmier is true for the algebra of polynomial integro differential operators abstract let a 1 k x d d x be the weyl algebra and i 1 k x d d x be the algebra of polynomial integro differential operators over a field k of characteristic zero the conjecture problem of dixmier 1968 still open is an algebra endomorphism of the weyl algebra a 1 an automorphism the aim of the paper is to prove that each algebra endomorphism of the algebra i 1 is an automorphism notice that in contrast to the weyl algebra a 1 the algebra i 1 is a non simple non noetherian algebra which is not a domain moreover it contains infinite direct sums of nonzero left and right ideals

an undecidable case of lineability in rr abstract recently it has been proved that assuming that there is an almost disjoint family of cardinality 2 c in c which is assured for instance by either martin s axiom or the continuum hypothesis or even 2 c c one has that the set of sierpinski zygmund functions is 2 c strongly algebrable and thus 2 c lineable here we prove that these two statements are actually equivalent and moreover that they both are undecidable this would be the first time in which one encounters an undecidable proposition in the recently coined theory of lineability and spaceability

krull gelfand kirillov and filter dimensions of simple affine algebras the aim of the paper is to prove the following two results 1 letabe a finitely partitive simple affine algebra with gk a then the krull dimension where gk and fil dim are the gelfand kirillov and the filter dimension respectively 2 letbbe an integral regular domain affine over a field of characteristic zero and let d b be its ring of differential operators the filter dimension fil dim d b 1

lineability and algebrability of pathological phenomena in analysis we show that in analysis many pathological phenomena occur more often than one could expect that is in a linear or algebraic way we show this by means of the construction of large algebraic structures infinite dimensional vector spaces or infinitely generated algebras enjoying some special or pathological properties

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