unique continuation for sublinear elliptic equations based on carleman estimates abstract in this article we deal with different forms of the unique continuation property for second order elliptic equations with nonlinear potentials of sublinear growth under suitable regularity assumptions we prove the weak and the strong unique continuation property moreover we also discuss the unique continuation property from measurable sets which shows that nodal domains to these equations must have vanishing lebesgue measure our methods rely on suitable carleman estimates for which we include the sublinear potential into the main part of the operator

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

bounded and unbounded polynomials and multilinear forms characterizing continuity in this paper we prove a characterization of continuity for polynomials on a normed space namely we prove that a polynomial is continuous if and only if it maps compact sets into compact sets we also provide a partial answer to the question as to whether a polynomial is continuous if and only jilt transforms connected sets into connected sets these results motivate the natural question as to how many non continuous polynomials there are on an infinite dimensional normed space a problem on the lineability of the sets of non continuous polynomials and multilinear mappings on infinite dimensional normed spaces is answered

infinitary lambda calculus in a previous paper we have established the theory of transfinite reduction for orthogonal term rewriting systems in this paper we perform the same task for the lambda calculus from the viewpoint of infinitary rewriting the bohm model of the lambda calculus can be seen as an infinitary term model in contrast to term rewriting there are several different possible notions of infinite term which give rise to different bohm like models which embody different notions of lazy or eager computation

coalgebras in functional programming and type theory this is a survey article on the use of coalgebras in functional programming and type theory it presents the basic theory underlying the implementation of coinductive types families and predicates it gives an overview of the application of corecursive methods to the study of general recursion formal power series tabulations of functions on inductive data it also sketches some advanced topics in the study of the solutions to non guarded corecursive equations and the design of non standard type theory

carleman estimates and absence of embedded eigenvalues let l w be a schrodinger operator with a potential w in l frac n 1 2 mathbb r n n geq 2 we prove that there is no positive eigenvalue the main tool is an l p l p prime carleman type estimate which implies that eigenfunctions to positive eigenvalues must be compactly supported the carleman estimate builds on delicate dispersive estimates established in 7 we also consider extensions of the result to variable coefficient operators with long range and short range potentials and gradient potentials

a note on eigenvalue bounds for schrodinger operators abstract we obtain a new bound on the location of eigenvalues for a non self adjoint schrodinger operator with complex valued potentials by obtaining a weighted l 2 estimate for the resolvent of the laplacian

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

moduleability algebraic structures and nonlinear properties we show that some pathological phenomena occur more often than one could expect existing large algebraic structures infinite dimensional vector spaces algebras positive cones or infinitely generated modules enjoying certain special properties in particular we construct infinite dimensional vector spaces of non integrable measurable functions completing some recent results shown in garcia pacheco et al 2009 13 garcia pacheco and seoane sepulveda 2006 15 munoz fernandez et al 2008 20 we prove as well the existence of dense and not barrelled spaces of sequences every non zero element of which has a finite number of zero coordinates giving partial answers to a problem originally posed by r m aron and v i gurariy in 2003

highlights in infinitary rewriting and lambda calculus we present some highlights from the emerging theory of infinitary rewriting both for first order term rewriting systems and l calculus in the first section we introduce the framework of infinitary rewriting for first order rewrite systems so without bound variables we present a recent observation concerning the continuity of infinitary rewriting in the second section we present an excursion to the infinitary l calculus after the main definitions we mention a recent observation about infinite looping l terms that is terms that reduce in one step to themselves next we describe the fundamental trichotomy in the semantics of l calculus bohm trees levy longo trees and berarducci trees we conclude with a short description of a new refinement of bohm tree semantics called clocked semantics

unique continuation for δ and the c fefferman phong class we show that the strong unique continuation property holds for the inequality u v u where the potential v x satisfies the c fefferman phong condition in a certain range of p values we also deal with the situation of u x vanishing at infinity these are all consequences of appropriate carleman inequalities

weighted norm inequalities for the continuous square function we prove new weighted norm inequalities for real variable analogues of the lusin area function we apply our results to obtain new i weighted norm inequalities for singular integral operators ii weighted sobolev inequalities iii eigenvalue estimates for degenerate schrodinger operators

applications of infinitary lambda calculus we present an introduction to infinitary lambda calculus highlighting its main properties subsequently we give three applications of infinitary lambda calculus the first addresses the non definability of surjective pairing which was shown by the first author not to be definable in lambda calculus we show how this result follows easily as an application of berry s sequentiality theorem which itself can be proved in the setting of infinitary lambda calculus the second pertains to the notion of relative recursiveness of number theoretic functions the third application concerns an explanation of counterexamples to confluence of lambda calculus extended with non left linear reduction rules adding non left linear reduction rules such as dxx x or the reduction rules for surjective pairing to the lambda calculus yields non confluence as proved by the second author we discuss how an extension to the infinitary lambda calculus where bohm trees can be directly manipulated as infinite terms yields a more simple and intuitive explanation of the correctness of these church rosser counterexamples

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

on some sharp spectral inequalities for schrodinger operators on semiaxis in this paper we obtain sharp lieb thirring inequalities for a schrodinger operator on semiaxis with a matrix potential and show how they can be used to other related problems among them are spectral inequalities on star graphs and spectral inequalities for schrodinger operators on half spaces with robin boundary conditions

the weyl correspondence and time frequency analysis describes the weyl correspondence and its properties showing how it gives a window independent definition of time frequency concentration for use in models in signal detection the definition of concentration is justified by showing that it gives reasonable answers in certain intuitive cases the weyl correspondence expresses a linear transformation as a weighted superposition of time frequency shifts of the signal and then authors explain why this is not the same as transforming a signal into the time frequency domain multiplying by a weight in the transform domain and taking the inverse the investigation into time frequency concentration and the weyl correspondence is justified by a new result the authors show that convolving the wigner distribution with a general smoothing function is equivalent to evaluating a weighted sum of spectrograms this is a new interpretation of the process of smoothing the wigner distribution to reduce cross terms it relates smoothing of the wigner distribution to the multiple window technique pioneered by thomson 1982

carleman inequalities and the heat operator we prove a strong unique continuation theorem for functions u vanishing to infinite order in the space time variables at 0 0 and verifying the inequality 4u tu v x t u on rn 0 t when t1 n 2r 1 s v x t lt locl r x n 2r 1 s 1 n 2 r

the periodic schrodinger operators with potentials in the morrey class we consider the periodic schrodinger operator v x in rd d 3 with potential v in the morrey class let be a periodic cell for v we show that for p d 1 2 d 2 there exists a positive constant e depending only on the shape of p and d such that iflim supr 0supx r21 b x r b x r v y pdy1 p e then the spectrum of v is purely absolutely continuous we obtain this result as a consequence of certain weighted l2 sobolev inequalities on the d torus it improves an early result by the author for potentials in ld 2 or weak ld 2 space

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

spaceability in banach and quasi banach sequence spaces abstract let x be a banach space we prove that for a large class of banach or quasi banach spaces e of x valued sequences the sets e q l q x where is any subset of 0 and e c 0 x contain closed infinite dimensional subspaces of e if non empty of course this result is applied in several particular cases and it is also shown that the same technique can be used to improve a result on the existence of spaces formed by norm attaining linear operators

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

numerical linear algebra and solvability of partial differential equations it was observed long ago that the obstruction to the accurate computation of eigenvalues of large non self adjoint matrices is inherent in the problem the basic idea is that the resolvent of a highly non normal operator can be very large far away from the spectrum this leads to an easily observable fact that algorithms for locating eigenvalues will typically find some false eigenvalues these false eigenvalues also explain one of the most surprising phenomena in linear pdes namely the fact discovered by hans lewy in 1957 in berkeley that one cannot always locally solve the pde p u f almost immediately after that discovery hormander provided an explanation of lewy s example showing that almost all operators with non constanct complex valued coefficients are not locally solvable in modern language that was done by considering the essentially dual problem of existence of non propagating singularities the purpose of this article is to review this work in the context of almost eigenvalues and from the point of view of semi classical analysis

the spectrum of the schrodinger operator on decrit le spectre negatif de l operateur de schrodinger avec un potentiel singulier on determine la valeur exacte du fond du spectre et on etablit une double estimation

a sharper uncertainty principle abstract the work strengthens the result established by l cohen on uncertainty principle involving phase derivative we propose stronger uncertainty principles not only in the classical setting for fourier transform but also for self adjoint operators we also deduce the conditions that give rise to the equal relation of the uncertainty principle examples are provided to show that the new uncertainty principle is truly sharper than the existing ones in literature

parametrices and dispersive estimates for schrodinger operators with variable coefficients in this article we consider variable coefficient time dependent schr odinger evolutions in bbb r n using phase space methods we construct outgoing parametrices and prove strichartz type estimates globally in time this is done in the context of c 2 metrics which satisfy a weak asymptotic flatness condition at infinity

a strong unique continuation theorem for parabolic equations the unique continuation is best understood for second order elliptic operators the classical paper by carleman 7 established the strong unique continuation theorem for second order elliptic operators which need not have analytic coefficients this was remarkable since it indicates that even a non analytic solution of an elliptic equation can behave somewhat in an analytic manner the powerful technique he used the so called carleman weighted inequality has played a central role in later developments in 1950 s aronszajn 4 and cordes 11 generalized his result to higher dimensions in recent years this classical subject caught new attention from a great number of people much efforts have been made to relax the smoothness hypotheses on the coefficients see 6 17 43 1 22 a seminal paper by jerison and kenig 26 has shown the unique continuation property for operators of the form c x with c l 2 loc where n 3 is the space dimension further improvements have been carried out in considering other classes of coefficients fefferman phong class and kato class 42 8 12 29 in extending the result to operators with first derivative terms and variable leading coefficients 46 47 51 52 and etc see 25 15 16 28 5 27 and the references therein for second order linear parabolic operators with time independent coefficients the strong unique continuation property along with detailed estimates on the hausdorff measures of nodal sets was reduced in 32 33 to the previously established elliptic counterparts the reduction from time independent parabolic

behavioural differential equations a coinductive calculus of streams automata and power series we present a theory of streams infinite sequences automata and languages and formal power series in terms of the notions of homomorphism and bisimulation which are the cornerstones of the theory of universal coalgebra this coalgebraic perspective leads to a unified theory in which the observation that each of the aforementioned sets carries a so called final automaton structure plays a central role finality forms the basis for both definitions and proofs by coinduction the coalgebraic counterpart of induction coinductive definitions take the shape of what we have called behavioural differential equations after brzozowski s notion of input derivative a calculus is developed for coinductive reasoning about all of the afore mentioned structures closely resembling calculus from classical analysis

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

global unique continuation from a half space for the schrodinger equation abstract we obtain a global unique continuation result for the differential inequality i t u v x u in r n 1 this is the first result on global unique continuation for the schrodinger equation with time independent potentials v x in r n our method is based on a new type of carleman estimates for the operator i t on r n 1 as a corollary of the result we also obtain a new unique continuation result for some parabolic equations

on spectral stability of the nonlinear dirac equation abstract we study the point spectrum of the nonlinear dirac equation in any spatial dimension linearized at one of the solitary wave solutions we prove that in any dimension the linearized equation has no embedded eigenvalues in the part of the essential spectrum beyond the embedded thresholds we then prove that the birth of point eigenvalues with nonzero real part the ones which lead to linear instability from the essential spectrum is only possible from the embedded eigenvalues or thresholds and therefore can not take place beyond the embedded thresholds we also prove that in the nonrelativistic limit m the point eigenvalues can only accumulate to 0 and 2 m i

characterizing polynomial time complexity of stream programs using interpretations this paper provides a criterion based on interpretation methods on term rewrite systems in order to characterize the polynomial time complexity of second order functionals for that purpose it introduces a first order functional stream language that allows the programmer to implement second order functionals this characterization is extended through the use of exp poly interpretations as an attempt to capture the class of basic feasible functionals bff moreover these results are adapted to provide a new characterization of polynomial time complexity in computable analysis these characterizations give a new insight on the relations between the complexity of functional stream programs and the classes of functions computed by oracle turing machine where oracles are treated as inputs

non lipschitz functions with bounded gradient and related problems let e be a topological vector space and let us consider a property p we say that the subset m of e formed by the vectors in e which satisfy p is lineable for certain cardinal finite or infinite if m 0 contains an infinite dimensional linear space of dimension in this note we prove that there exist uncountably infinite dimensional linear spaces of functions enjoying the following properties 1 being continuous on 0 1 a e differentiable with a e bounded derivative and not lipschitz 2 differentiable in r2 r and not enjoying the mean value theorem 3 real valued differentiable on an open connected and non convex set having bounded gradient non lipschitz and therefore not verifying the mean value theorem

transfinite reductions in orthogonal term rewriting systems strongly convergent reduction is the fundamental notion of reduction in infinitary orthogonal term rewriting systems otrss for these we prove the transfinite parallel moves lemma and the compressing lemma strongness is necessary as shown by counterexamples normal forms which we allow to be infinite are unique in contrast to normal forms strongly converging fair reductions result in normal forms

on quantitative uniqueness for elliptic equations we address the question of quantitative uniqueness for the equation delta u v u with either periodic or dirichlet boundary conditions in a disk we construct solutions u corresponding to potential functions v such that u vanishes of order mathrm const vert v vert l infty 2 3 the example also shows sharpness of recently obtained bounds in the case of a parabolic equation u t delta u v u

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

lazy productivity via termination abstract we present a procedure for transforming strongly sequential constructor based term rewriting systems trss into context sensitive trss in such a way that productivity of the input system is equivalent to termination of the output system thereby automated termination provers become available for proving productivity a trs is called productive if all its finite ground terms are constructor normalizing and all inductive constructor paths through the resulting possibly non wellfounded constructor normal form are finite to our knowledge this is the first complete transformation from productivity to termination the transformation proceeds in two steps i the strongly sequential trs is converted into a shallow trs where patterns do not have nested constructors ii the shallow trs is transformed into a context sensitive trs where rewriting below constructors and in arguments not consumed from is disallowed furthermore we show how lazy evaluation can be encoded by strong sequentiality thus extending our transformation to e g haskell programs finally we present a simple but fruitful extension of matrix interpretations to make them applicable for proving termination of context sensitive trss

on the role of quadratic oscillations in nonlinear schrodinger equations ii the l 2 critical case we consider a nonlinear semi classical schrodinger equation for which quadratic oscillations lead to focusing at one point described by a nonlinear scattering operator the relevance of the nonlinearity was discussed by r carles c fermanian kammerer and i gallagher for l 2 supercritical power like nonlinearities and more general initial data the present results concern the l 2 critical case in space dimensions 1 and 2 we describe the set of non linearizable data which is larger due to the conformal invariance as an application we precise a result by f merle and l vega concerning finite time blow up for the critical schrodinger equation the proof relies on linear and nonlinear profile decompositions

bialgebras for structural operational semantics bialgebras and distributive laws are an abstract categorical framework to study various flavors of structural operational semantics this paper aims to introduce the reader to the basics of bialgebras for operational semantics and to sketch the state of the art in this research area

restriction theorems for orthonormal functions strichartz inequalities and uniform sobolev estimates we generalize the theorems of stein tomas and strichartz about surface restrictions of fourier transforms to systems of orthonormal functions with an optimal dependence on the number of functions we deduce the corresponding strichartz bounds for solutions to schr odinger equations up to the endpoint thereby solving an open problem of frank lewin lieb and seiringer we also prove uniform sobolev estimates in schatten spaces extending the results of kenig ruiz and sogge we finally provide applications of these results to a limiting absorption principle in schatten spaces to the well posedness of the hartree equation in schatten spaces to lieb thirring bounds for eigenvalues of schr odinger operators with complex potentials and to schatten properties of the scattering matrix

on lineability of sets of continuous functions we study the existence of vector spaces of dimension at least two of continuous functions on subsets of r every non zero element of which admits one and only one absolute maximum

a faber krahn inequality for solutions of schrodinger s equation abstract we consider nontrivial solutions of u x v x u x where u 0 on the boundary of a bounded open region d r n and v x l d we prove a sharp relationship between v and the measure of d which generalizes the well known faber krahn theorem we also prove some geometric properties of the zero sets of the solution of the schrodinger equation u x v x u x

dispersive estimates for the three dimensional schrodinger equation with rough potentials the three dimensional schrodinger propogator e ith h v is a bounded map from l 1 to l with norm controlled by t 3 2 provided the potential satisfies two conditions an integrability condition limiting the singularities and decay of v and a zero energy spectral condition on h this is shown by expressing the spectral measure of h in terms of its resolvents and proving a family of l p mapping estimates for the resolvents previous results in this direction had required v to satisfy explicit pointwise bounds

sharp counterexamples in unique continuation for second order elliptic equations we construct nontrivial solutions with compact support for the elliptic equation u v u with v lp p n 2 or v l w for n 3 and with v l1 for n 2 the same method also yields nontrivial solutions with compact support for the elliptic equation u w u with w lq p n or w lw for n 2 for the second order elliptic equations u v u in r 1 respectively u w u in r 2 we define the weak unique continuation property ucp respectively the strong unique continuation property sucp as follows let u be a solution to 1 which vanishes in an open set then u 0 ucp let u be a solution to 1 which vanishes of infinite order at some point x0 r then u 0 sucp research partially supported by the dfg and daad research partially supported by the nsf grants dms 9970297 int 9815286

weighted doubling properties and unique continuation theorems for the degenerate schrodinger equations with singular potentials abstract let u be the weak solution to the degenerate schrodinger equation with singular coefficients in lipschitz domain as following div w x a x u x v x u x w x 0 where a x is a real symmetric matrix function satisfying the elliptic condition and the lipschitz continuity w x is an a 2 weight function of muckenhoupt class and v x is the fefferman phong s potential the weighted doubling properties and unique continuations for the weak solution u in the interior of any domains as well as at the boundary of some lipschitz domains are derived in this paper

remark on unique continuation for higher powers of the laplace operator abstract there are unique continuation results 2 5 for the differential inequality u x v x u x n when v l loc d 2 r d d 2 in this paper we obtain a result for the case d 2

spaceability of strict order integrability abstract in this paper sharp conditions on a measure space are provided in order that the subset of functions in the corresponding lebesgue space l p which are in no other l q contains except for zero an infinite dimensional closed subspace this completes earlier work of several authors the main result is derived from a new general criterion for spaceability on function spaces a further application to spaceability of the class of non absolutely continuous functions is supplied

strictly pseudoconvex domains in c n the theory which we have developed so far will be expanded in two ways 1 in this chapter we shall use the cauchy fantappie calculus to transform the bochner martinelli kernel into a kernel adapted to strictly pseudoconvex domains in n and prove a number of finiteness and vanishing theorems for these domains 2 in the next chapter we shall pass from n to arbitrary hermitian manifolds and establish the analogue of the bmk formula in that context obtaining along the way the hodge decomposition theorem on compact manifolds both lines of thought will merge in the subsequent chapters and lead to a solution of the levi problem and finiteness theorems for strictly pseudoconvex manifolds as well as to a solution of the overline partial neumann problem to be formulated later

characterization of multiplier spaces by wavelets and logarithmic morrey spaces in this paper we consider multipliers from sobolev spaces to lebesgue spaces we establish some wavelet characterization of multiplier spaces without using capacity further we give a sharp logarithmic morrey space condition for multipliers which lessens fefferman s morrey space condition to the logarithm level and generalizes lemarie s counter example to non integer cases and expresses his results in a more precise way

porous sets and lineability of continuous functions on locally compact groups abstract let g be a non compact locally compact group in this paper we study the size of the set f g a b f g is well defined on g where a and b are normed spaces of continuous functions on g we also consider the problem of the spaceability of the set c 0 g c 0 g c 0 g c 00 g and among other results we show that for g r n the above set is strongly c algebrable and therefore algebrable and lineable with respect to the convolution product

logical reformulation of quantum mechanics i foundations the basic rules of quantum mechanics are reformulated they deal primarily with individual systems and do not assume that every ket may represent a physical state the customary kinematic and dynamic rules then allow to construct consistent boolean logics describing the history of a system following essentially griffiths proposal logical implication is defined within these logics the multiplicity of which reflects the complementary principle only one interpretative rule of quantum mechanics is necessary in such a framework it states that these logics providebona fide foundations for the description of a quantum system and for reasoning about it one attempts to build up classical physics including classical logic on these quantum foundations the resulting theory of measurement needs not to statea priori that the eigenvalues of an observable have to be the results of individual measurements nor to assume wave packet reduction both these properties can be obtained as consequences of the basic rules one also needs not to postulate that every observable is measurable even in principle a proposition calculus is obtained allowing in principle the replacement of the discussion of problems concerned with the practical interpretation of experiments by due calculations

lineability in subsets of measure and function spaces we show among other results that if lambda denotes the lebesgue measure on the borel sets in 0 1 and x is an infinite dimensional banach space then the set of measures whose range is neither closed nor convex is lineable in ca lambda x we also show that in certain situations we have lineability of the set of x valued and non sigma finite measures with relatively compact range the lineability of sets of the type l p i l q i is studied and some open questions are proposed some classical techniques together with the converse of the lyapunov convexity theorem are used

large structures in certain subsets of orlicz spaces abstract assume that and n n 1 are young functions in this paper we prove that under sharp conditions on a measure space and the young functions and n s the subset of functions in the corresponding orlicz space l which are not in l n for all n n admits various large linear structures

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

on minimal support properties of solutions of schrodinger equations abstract in this paper we obtain minimal support properties of solutions of schrodinger equations we improve previously known conditions on the potential for which the measure of the support of solutions cannot be too small we also use these properties to obtain some new results on unique continuation for the schrodinger operator

schrodinger operators and unique continuation towards an optimal result abstract in this article we prove the property of unique continuation also known for c functions as quasianalyticity for solutions of the differential inequality u v u for v from a wide class of potentials including l loc d 2 r d class and u in a space of solutions y v containing all eigenfunctions of the corresponding self adjoint schrodinger operator motivating question is it true that for potentials v for which self adjoint schrodinger operator is well defined the property of unique continuation holds

bounded linear non absolutely summing operators abstract we show that in certain situations we have lineability in the set of bounded linear and non absolutely summing operators examples on lineability of the set p e f i p e f are also presented and some open questions are proposed

lp 0 1 q plq 0 1 is spaceable for every p 0 abstract in this short note we prove the result stated in the title that is for every p 0 there exists an infinite dimensional closed linear subspace of l p 0 1 every nonzero element of which does not belong to q p l q 0 1 this answers in the positive a question raised in 2010 by r m aron on the spaceability of the above sets for both the banach and quasi banach cases we also complete some recent results from botelho et al 2011 3 for subsets of sequence spaces

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

productivity of stream definitions we give an algorithm for deciding productivity of a large and natural class of recursive stream definitions a stream definition is called productive if it can be evaluated continually in such a way that a uniquely determined stream in constructor normal form is obtained as the limit whereas productivity is undecidable for stream definitions in general we show that it can be decided for pure stream definitions for every pure stream definition the process of its evaluation can be modelled by the dataflow of abstract stream elements called pebbles in a finite pebbleflow net work and the production of a pebbleflow net associated with a pure stream definition that is the amount of pebbles the net is able to produce at its output port can be calculated by reducing nets to trivial nets