system analysis via integral quadratic constraints this paper introduces a unified approach to robustness analysis with respect to nonlinearities time variations and uncertain parameters from an original idea by yakubovich 1967 the approach has been developed under a combination of influences from the western and russian traditions of control theory it is shown how a complex system can be described using integral quadratic constraints iqc for its elementary components a stability theorem for systems described by iqcs is presented that covers classical passivity dissipativity arguments but simplifies the use of multipliers and the treatment of causality a systematic computational approach is described and relations to other methods of stability analysis are discussed last but not least the paper contains a summarizing list of iqcs for important types of system components

c0 is isometrically isomorphic to a subspace of cantor lebesgue functions abstract we prove what the title of this note states i e c 0 that is the set of all sequences converging to 0 is isometrically isomorphic to a subspace of continuous almost everywhere differentiable and non lipschitz functions on 0 1 having almost everywhere null derivative this in particular improves a recent result jimenez rodriguez et al 2012 9 on the lineability of this set of functions

stability analysis with popov multipliers and integral quadratic constraints it is shown that a general form of popov multipliers can be used in stability analysis based on integral quadratic constraints iqc the popov multiplier is nonproper and a condition that the nominal plant is strictly proper will be imposed in order to ensure boundedness of the iqc corresponding to the popov multiplier a consequence of our main result is that the classical popov criterion can be combined with a stability criterion for slope restricted nonlinearities developed by zames and falb an example shows that the combination of these two criteria is useful in applications

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

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

the proximal augmented lagrangian method for nonsmooth composite optimization we study a class of optimization problems in which the objective function is given by the sum of a differentiable but possibly nonconvex component and a nondifferentiable convex regularization term we introduce an auxiliary variable to separate the objective function components and utilize the moreau envelope of the regularization term to derive the proximal augmented lagrangian a continuously differentiable function obtained by constraining the augmented lagrangian to the manifold that corresponds to the explicit minimization over the variable in the nonsmooth term the continuous differentiability of this function with respect to both primal and dual variables allows us to leverage the method of multipliers mm to compute optimal primal dual pairs by solving a sequence of differentiable problems the mm algorithm is applicable to a broader class of problems than proximal gradient methods and it has stronger convergence guarantees and a more refined step size update rules than the alternating direction method of multipliers admm these features make it an attractive option for solving structured optimal control problems we also develop an algorithm based on the primal descent dual ascent gradient method and prove global exponential asymptotic stability when the differentiable component of the objective function is strongly convex and the regularization term is convex finally we identify classes of problems for which the primal dual gradient flow dynamics are convenient for distributed implementation and compare contrast our framework to the existing approaches

on some natural complete operators abstract an operator is a mapping from integer functions to integer functions it is known from a result by baker gill and solovay 1975 that there exists an operator computable in polynomial time by a nondeterministic oracle turing machine called an np operator but not computable in polynomial time by any deterministic oracle turing machine we investigate several natural operators which share similar properties we use the concept of completeness to give a precise classification of the complexity of these operators for example the question of finding maximum values of polynomial time computable functions can be formulated as an operator complete for the class of np operators

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

complexity classes without machines on complete languages for up this paper develops techniques for studying complexity classes that are not covered by known recursive enumerations of machines often counting classes probabilistic classes and intersection classes lack such enumerations concentrating on the counting class up we show that there are relativizations for which upa has no complete languages and other relativizations for which pb upb npb and upb has complete languages among other results we show that p up if and only if there exists a set s in p of boolean formulas with at most one satisfying assignment such that s sat is not in p p up coup if and only if there exists a set s in p of uniquely satisfiable boolean formulas such that no polynomial time machine can compute the solutions for the formulas in s if up has complete languages then there exists a set r in p of boolean formulas with at most one satisfying assignment so that sat r is complete for up finally we indicate the wide applicability of our techniques to counting and probabilistic classes by using them to examine the probabilistic class bpp there is a relativized world where bppa has no complete languages if bpp has complete languages then it has a complete language of the form b majority where b p and majority f f is true for at least half of all assignments is the canonical pp complete set

robust stability analysis for feedback interconnections of time varying linear systems feedback interconnections of causal linear systems are studied in a continuous time setting the developments include a linear time varying ltv generalization of vinnicombe s nu gap metric and an integral quadratic constraint based robust l2 stability theorem for uncertain feedback interconnections of potentially open loop unstable systems these main results are established in terms of toeplitz wiener hopf and hankel operators and the fredholm index for a class of causal linear systems with the following attributes i a system graph i e subspace of l2 input output pairs admits normalized strong right i e image and left i e kernel representations and ii the corresponding hankel operators are compact these properties are first verified for stabilizable and detectable ltv state space models to initially motivate the abstract formulation and subsequently verified for frequency domain multiplication by constantly proper callier desoer transfer functions in analysis that confirms consistency of the developments with the time invariant theory to conclude the aforementioned robust stability theorem is applied in an illustrative example concerning the feedback interconnection of distributed parameter systems over a network with time varying gains less

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

convex searches for discrete time zames falb multipliers in this article we develop and analyze convex searches for zames falb multipliers we present two different approaches infinite impulse response iir and finite impulse response fir multipliers the set of fir multipliers is complete in that any iir multipliers can be phase substituted by an arbitrarily large order fir multiplier we show that searches in discrete time for fir multipliers are effective even for large orders as expected the numerical results provide the best ell 2 stability results in the literature for slope restricted nonlinearities in particular we establish the equivalence between the state of the art lyapunov results for slope restricted nonlinearities and a subset of the fir multipliers finally we demonstrate that the discrete time search can provide an effective method to find suitable continuous time multipliers

lineability of the set of bounded linear non absolutely summing operators in this note we solve except for extremely pathological cases a question posed by puglisi and seoane sepulveda on the lineability of the set of bounded linear non absolutely summing operators we also show how the idea of the proof can be adapted to several related situations

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

rolewicz type chaotic operators abstract in this article we introduce a new class of rolewicz type operators in l p 1 p we exhibit a collection f having the cardinality of the continuum consisting of operators of this type which are chaotic and remain so under almost all finite linear combinations provided that the linear combination has sufficiently large norm as a corollary to our main result we also obtain that there exists a countable collection of such operators whose all finite linear combinations are chaotic provided that they have sufficiently large norm

zames falb multipliers for convergence rate motivating example and convex searches the bound of convergence rates for discrete time lurye sometimes written lur e systems has recently attracted much attention the contributions of this technical note are twofold firstly we sho

s linear and convex structures in function families abstract in this paper the notion of s lineability originally coined by vladimir i gurariy is introduced and developed in a general abstract setting this new notion is then applied to specific situations as for instance classes of differentiable nowhere monotone functions as well as families of vectors having dense orbit with respect to an operator large convex structures are also shown to exist inside the family of topologically mixing continuous selfmaps of a real compact interval

operator ranges and spaceability recent contributions on spaceability have overlooked the applicability of results on operator range subspaces of banach spaces or frechet spaces here we consider general results on spaceability of the complement of an operator range some of which we extend to the complement of a union of countable chains of operator ranges applications we give include spaceability of the non absolutely convergent power series in the disk algebra and of the non absolutely p summing operators between certain pairs of banach spaces another application is to ascent and descent of countably generated sets of continuous linear operators where we establish some closed range properties of sets with finite ascent and descent

stability analysis with dissipation inequalities and integral quadratic constraints this technical note considers the stability of a feedback connection of a known linear time invariant system and a perturbation the input output behavior of the perturbation is described by an integral quadratic constraint iqc iqc stability theorems can be formulated in the frequency domain or with a time domain dissipation inequality the two approaches are connected by a non unique factorization of the frequency domain iqc multiplier the factorization must satisfy two properties for the dissipation inequality to be valid first the factorization must ensure the time domain iqc holds for all finite times second the factorization must ensure that a related matrix inequality when feasible has a positive semidefinite solution this technical note shows that a class of frequency domain iqc multipliers has a factorization satisfying these two properties thus the dissipation inequality test with an appropriate factorization can be used with no additional conservatism

exponential decay rate conditions for uncertain linear systems using integral quadratic constraints this technical note develops linear matrix inequality lmi conditions to test whether an uncertain linear system is exponentially stable with a given decay rate alpha these new alpha exponential stability tests are derived for an uncertain system described by an interconnection of a nominal linear time invariant system and a troublesome perturbation the perturbation can contain uncertain parameters time delays or nonlinearities this technical note presents two key contributions first alpha exponential stability of the uncertain lti system is shown to be equivalent to internal linear stability of a related scaled system this enables derivation of alpha exponential stability tests from linear stability tests using integral quadratic constraints iqcs this connection requires iqcs to be constructed for a scaled perturbation operator the second contribution is a list of iqcs derived for the scaled perturbation using the detailed structure of the original perturbation finally connections between the proposed approach and related work are discussed

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

algebrability of some families of darboux like functions abstract we prove that the class of all almost continuous in the sense of stallings functions possessing the perfect road property and the class of all almost continuous functions without the perfect road property contain a 2 c generated free algebras we prove also that the family of all extendable functions contains a 2 c dimensional linear space

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

additivity and lineability in vector spaces abstract gamez merino munoz fernandez and seoane sepulveda proved that if additivity a f c then f is a f lineable where f r r they asked if a f c can be weakened we answer this question in negative moreover we introduce and study the notions of homogeneous lineability number and lineability number of subsets of linear spaces

feasible computability and resource bounded topology townsend introduced a resource bounded extension of polynomial time computable functions on strings to type two functionals and studied a type two version of the meyer stockmeyer polynomial hierarchy which is founded on polynomial time computable functionals of type two a functional of type two is polynomial time computable if it is computed by a deterministic function oracle turing machine whose runtime is bounded by a polynomial that does not depend on the choice of oracle functions townsend also introduced a boldface polynomial hierarchy of type two by a relativization method and gave a topological characterization of the first level of this hierarchy we reformulate townsend s topological notions associated with time bounded computations of function oracle turing machines and further extend his topological characterization to all levels of the boldface polynomial hierarchy of type two this leads to a polynomialized version of descriptive set theory

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

locally recurrent functions density topologies and algebraic genericity we study the algebraic and topological genericity of certain subsets of locally recurrent functions obtaining among other results algebrability and spaceability within these classes

computational complexity of real functions in this chapter we introduce the formal definitions of computational complexity of real functions we first review the definitions of computable real numbers and computable real functions and their basic properties then the oracle turing machine is introduced as the formal model for computing real functions this model allows us to define the complexity measures for computing real functions in a natural way the class of polynomial time computable real functions is then defined and several characterizations of this class will be given other complexity classes of real numbers and real functions such as np real functions and log space computable real functions will be defined in later chapters

stability analysis of systems with uncertain time varying delays stability of systems in the presence of bounded uncertain time varying delays in the feedback loop is studied the delay parameter is assumed to be an unknown time varying function for which the upper bounds on the magnitude and the variation are given the stability problem is treated in the integral quadratic constraint iqc framework criteria for verifying robust stability are formulated as feasibility problems over a set of frequency dependent linear matrix inequalities the criteria can be equivalently formulated as semi definite programs sdp using kalman yakubovich popov lemma as such checking robust stability can be performed in a computationally efficient fashion

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

universality of sequences of operators related to taylor series abstract in this note the universality of a sequence of operators associated to the partial sums of the taylor series of a holomorphic function is investigated the emphasis is put on the fact that the taylor series are evaluated at a prescribed point and the variable is the center of the expansion the dynamics of the sequence of operators linked to the partial sums of a power series that is not generated by an entire function is also studied

holomorphic functions on non runge domains and related problems let u be a domain in cn that is not a runge domain we study the topological and algebraic properties of the family of holomorphic functions on u which cannot be approximated by polynomials

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

on lineability and additivity of real functions with finite preimages abstract we study lineability of real functions with finite preimages in particular we prove that the class of n to one functions contains a vector subspace of dimension n but not of dimension n 1 additionally we give examples of star like families of functions closed under multiplication by a non zero scalar for which lineability is less than additivity

nonlinear subsets of function spaces and spaceability abstract in this paper we study the existence of infinite dimensional closed linear subspaces of a rearrangement invariant space on 0 1 every nonzero element of which does not belong to any included rearrangement invariant space of the same class such that the inclusion operator is disjointly strictly singular we consider lorentz marcinkiewicz and orlicz spaces the answer is affirmative for marcinkiewicz spaces and negative for lorentz and orlicz spaces also the same problem is studied for nakano spaces assuming different hypothesis

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

a tight relationship between generic oracles and type 2 complexity theory we show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type 2 classes are distinct

integral quadratic constraints for delayed nonlinear and parameter varying systems the stability and performance of nonlinear and linear parameter varying lpv time delayed systems are analyzed first the input output behavior of the time delay operator is bounded in the frequency domain by integral quadratic constraints iqcs a simple geometric interpretation is used to derive new iqcs for both constant and varying delays second the performance of nonlinear and lpv delayed systems is bounded using dissipation inequalities that incorporate iqcs the nonlinear or lpv part of the system is treated directly in the analysis and not bounded by iqcs this step makes use of recent results that show under mild technical conditions that an iqc has an equivalent representation as a finite horizon time domain constraint a numerical example with a nonlinear delayed system is provided to demonstrate the effectiveness of the method

almost overcomplete and almost overtotal sequences in banach spaces abstract we introduce the new concepts of almost overcomplete sequence in a banach space and almost overtotal sequence in a dual space we prove that any of such sequences is relatively norm compact and we present several applications of this fact

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

convolution functions that are nowhere differentiable abstract in 1951 v jarnik constructed two continuous functions whose volterra convolution is nowhere differentiable we generalize jarnik s results by proving that the set of such functions is maximal lineable this would shed some light on a question posed in 1973 on the structure of the set of continuous functions whose volterra convolution is nowhere differentiable

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