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

subsets of some families of real functions and their algebrability abstract we consider the following families of real valued functions darboux d quasi continuous q and wi tkowski functions s the aim of this paper is to study the level of algebrability for various intersections of families d q and s and their complements

clifford algebras of hyperbolic involutions for as a central simple algebra of even degree with hyperbolic orthogonal involution we describe the canonically induced involution underline sigma on the even clifford algebra c 0 a sigma underline sigma of a sigma when deg a equiv 0 mod 8 a cong m 2 b and the interesting part of underline sigma is isomorphic to the canonical involution on an exterior power algebra of b as a corollary we get some properties of the involution on the exterior power algebra

structure preserving stability and accuracy properties of the energy conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models summary the computational efficiency of a typical projection based nonlinear model reduction method hinges on the efficient approximation for explicit computations of the scalar projections onto a subspace of a residual vector for implicit computations it also hinges on the additional efficient approximation of similar projections of the jacobian of this residual with respect to the solution the computation of both approximations is often referred to in the literature as hyper reduction to this effect this paper focuses on the analysis and comparative performance study for nonlinear finite element reduced order models of solids and structures of the recently developed energy conserving mesh sampling and weighting ecsw hyper reduction method unlike most alternative approaches this method approximates the scalar projections of residuals and or jacobians directly instead of approximating first these vectors and matrices then projecting the resulting approximations onto the subspaces of interest in this paper it is shown that ecsw distinguishes itself furthermore from other hyper reduction methods through its preservation of the lagrangian structure associated with hamilton s principle for second order dynamical systems this enables it to also preserve the numerical stability properties of the discrete system to which it is applied it is also shown that for a fixed set of parameter values the approximation error committed online by ecsw is bounded by its counterpart error committed off line during the training of this method therefore this error can be estimated in this case a priori and controlled the performance of ecsw is demonstrated first for two academic but nevertheless interesting nonlinear dynamic response problems for both of them ecsw is shown to preserve numerical stability and deliver the desired level of accuracy whereas the discrete empirical interpolation method and its recently introduced unassembled variant are shown to be susceptible to failure because of numerical instability the potential of ecsw for enabling the effective reduction of nonlinear finite element dynamic models of solids and structures is also highlighted with the realistic simulation of the nonlinear transient dynamic response of a complete car engine to thermal and combustion pressure loads using an implicit scheme for this simulation ecsw is reported to enable the reduction of the cpu time required by the high dimensional nonlinear finite element dynamic analysis by more than four orders of magnitude while achieving a very good level of accuracy copyright 2015 john wiley sons ltd

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

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

spaceability of the set of continuous injections from blp into lp with nowhere continuous inverses abstract let p 1 we show that in the banach space of all bounded continuous mappings from b l p the open unit ball in l p into l p the subset consisting of all injections with nowhere continuous inverses is isometrically l p spaceable i e it contains an isometric copy of l p the proof uses some ideas of creswell s example

a posteriori error estimation for deim reduced nonlinear dynamical systems in this work an efficient approach for a posteriori error estimation for pod deim reduced nonlinear dynamical systems is introduced the considered nonlinear systems may also include time and parameter affine linear terms as well as parametrically dependent inputs and outputs the reduction process involves a galerkin projection of the full system and approximation of the system s nonlinearity by the deim method s chaturantabut and d c sorensen siam j sci comput 32 2010 pp 2737 2764 the proposed a posteriori error estimator can be efficiently decomposed in an offline online fashion and is obtained by a one dimensional auxiliary ode during reduced simulations key elements for efficient online computation are partial similarity transformations and matrix deim approximations of the nonlinearity jacobians the theoretical results are illustrated by application to an unsteady burgers equation and a cell apoptosis model

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

preserving lagrangian structure in nonlinear model reduction with application to structural dynamics this work proposes a model reduction methodology that preserves lagrangian structure and achieves computational efficiency in the presence of high order nonlinearities and arbitrary parameter dependence as such the resulting reduced order model retains key properties such as energy conservation and symplectic time evolution maps we focus on parameterized simple mechanical systems subjected to rayleigh damping and external forces and consider an application to nonlinear structural dynamics to preserve structure the method first approximates the system s lagrangian ingredients the riemannian metric the potential energy function the dissipation function and the external force and subsequently derives reduced order equations of motion by applying the forced euler lagrange equation with these quantities from the algebraic perspective key contributions include two efficient techniques for approximating parameterized reduced matrices while preserving symmetry and positive definiteness matri

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

efficient non linear model reduction via a least squares petrov galerkin projection and compressive tensor approximations the first author acknowledges the partial support by a national science foundation graduate fellowship and the partial support by a national defense science and engineering graduate fellowship the second and third authors acknowledge the partial support by the motor sports division of the toyota motor corporation under agreement number 48737 and the partial support by a research grant from the academic excellence alliance program between king abdullah university of science and technology kaust and stanford university all authors also acknowledge the constructive comments received during the review process

a matrix deim technique for model reduction of nonlinear parametrized problems in cardiac mechanics abstract when relying on newton iterations to solve nonlinear problems in the context of reduced basis rb methods the assembling of the rb arrays during the online stage depends on the dimension of the underlying high fidelity approximation this is more of an issue when dealing with fully nonlinear problems for which the global jacobian matrix has to be entirely reassembled at each newton step in this paper the discrete empirical interpolation method deim and its matrix version mdeim are combined to evaluate both the residual vector and the jacobian matrix very efficiently in the case of complex parametrized nonlinear mechanical problems we compare this strategy with the classical deim approach and we derive a posteriori error estimates on the solution accounting for the contribution of deim mdeim errors the effectiveness of the proposed framework is assessed on quasi static nonlinear problems in particular we consider a nonlinear elasticity problem defined on a cube and a mechanical model describing heart contraction for an idealized left ventricle geometry the latter is a coupled problem in which the activation of the heart contraction is given by the solution of an electrophysiology model our numerical results show that mdeim is preferable to the classical deim both in terms of efficiency and accuracy

reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation we present a new approach to treating nonlinear operators in reduced basis approximations of parametrized evolution equations our approach is based on empirical interpolation of nonlinear differential operators and their frechet derivatives efficient offline online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals an a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically we introduce a new algorithm the podei greedy algorithm which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way the approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations we show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions in case of symmetries of the problem the approach realizes an automatic and intuitive space compression or even space dimensionality reduction we perform empirical investigations of the error convergence and run times in all cases we obtain a good run time acceleration

dimensional model reduction in non linear finite element dynamics of solids and structures a general approach to the dimensional reduction of non linear finite element models of solid dynamics is presented for the newmark implicit time discretization the computationally most expensive phase is the repeated solution of the system of linear equations for displacement increments to deal with this it is shown how the problem can be formulated in an approximation ritz basis of much smaller dimension similarly the explicit newmark algorithm can be also written in a reduced dimension basis and the computation time savings in that case follow from an increase in the stable time step length in addition the empirical eigenvectors are proposed as the basis in which to expand the incremental problem this basis achieves approximation optimality by using computational data for the response of the full model in time to construct a reduced basis which reproduces the full system in a statistical sense because of this global time viewpoint the basis need not be updated as with reduced bases computed from a linearization of the full finite element model if the dynamics of a finite element model is expressed in terms of a small number of basis vectors the asymptotic cost of the solution with the reduced model is lowered and optimal scalability of the computational algorithm with the size of the model is achieved at the same time numerical experiments indicate that by using reduced models substantial savings can be achieved even in the pre asymptotic range furthermore the algorithm parallelizes very efficiently the method we present is expected to become a useful tool in applications requiring a large number of repeated non linear solid dynamics simulations such as convergence studies design optimization and design of controllers of mechanical systems

a priori hyperreduction method an adaptive approach model reduction methods are usually based on preliminary computations to build the shape function of the reduced order model rom before the computation of the reduced state variables they are a posteriori approaches most of the time these preliminary computations are as complex as the simulation which we want to simplify by the rom the reduction method we propose avoids such preliminary computations it is an a priori approach based on the analysis of some state evolutions such that all the state evolutions needed to perform the model reduction are described by an approximate rom the rom and the state evolution are simultaneously improved by the method thanks to an adaptive strategy obviously an initial set of known shape functions can be used to define the rom to adapt but it is not necessary the adaptive procedure includes extensions of the subspace spanned by the shape functions of the rom and selections of the most relevant shape functions in order to represent the state evolution the hyperreduction is achieved by selecting a part of the integration points of the finite element model to forecast the evolution of the reduced state variables hence both the number of degrees of freedom and the number of integration points are reduced to perform the adaptive procedure different computational strategies can be developed in this paper we propose an incremental algorithm involving adaptive periods during these adaptive periods the incremental computation is restarted until a quality criterion is satisfied this approach is compatible with classical formulations of the equations

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

a note on latticeability and algebrability abstract suppose a is a subset of a banach lattice banach algebra x we look for large sublattices resp subalgebras of a if x is a banach lattice we prove 1 if y is a closed subspace of x of codimension at least n then x y 0 contains a sublattice of dimension n 2 if y is a closed infinite codimensional ideal in x then x y 0 contains a closed infinite dimensional sublattice 3 if the order in x is induced by a 1 unconditional basis and y is a closed infinite codimensional subspace of x then x y 0 contains a closed infinite dimensional ideal further we show that 4 l p q p l q 0 contains a sublattice which is dense in l p and that 5 the sets l 1 t p 1 l p t 0 and s p s p 0 contain a dense subalgebra with a continuum of free generators here s p denotes the schatten p space

pod deim reduced order strategies for efficient four dimensional variational data assimilation this work studies reduced order modeling rom approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models it is shown that a key requirement for a successful reduced order solution is that reduced order karush kuhn tucker conditions accurately represent their full order counterparts in particular accurate reduced order approximations are needed for the forward and adjoint dynamical models as well as for the reduced gradient new strategies to construct reduced order based are developed for proper orthogonal decomposition pod rom data assimilation using both galerkin and petrov galerkin projections for the first time pod tensorial pod and discrete empirical interpolation method deim are employed to develop reduced data assimilation systems for a geophysical flow model namely the two dimensional shallow water equations numerical experiments confirm the theoretical framework for galerkin projection in the case of petrov galerkin projection stabilization strategies must be considered for the reduced order models the new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time

a method for interpolating on manifolds structural dynamics reduced order models a rigorous method for interpolating a set of parameterized linear structural dynamics reduced order models roms is presented by design this method does not operate on the underlying set of parameterized full order models hence it is amenable to an online real time implementation it is based on mapping appropriately the rom data onto a tangent space to the manifold of symmetric positive definite matrices interpolating the mapped data in this space and mapping back the result to the aforementioned manifold algorithms for computing the forward and backward mappings are offered for the case where the roms are derived from a general galerkin projection method and the case where they are constructed from modal reduction the proposed interpolation method is illustrated with applications ranging from the fast dynamic characterization of a parameterized structural model to the fast evaluation of its response to a given input in all cases good accuracy is demonstrated at real time processing speeds copyright 2009 john wiley sons ltd

efficient model reduction of parametrized systems by matrix discrete empirical interpolation in this work we apply a matrix version of the so called discrete empirical interpolation mdeim for the efficient reduction of nonaffine parametrized systems arising from the discretization of linear partial differential equations dealing with affinely parametrized operators is crucial in order to enhance the online solution of reduced order models roms however in many cases such an affine decomposition is not readily available and must be recovered through often intrusive procedures such as the empirical interpolation method eim and its discrete variant deim in this paper we show that mdeim represents a very efficient approach to deal with complex physical and geometrical parametrizations in a non intrusive efficient and purely algebraic way we propose different strategies to combine mdeim with a state approximation resulting either from a reduced basis greedy approach or proper orthogonal decomposition a posteriori error estimates accounting for the mdeim error are also developed in the case of parametrized elliptic and parabolic equations finally the capability of mdeim to generate accurate and efficient roms is demonstrated on the solution of two computationally intensive classes of problems occurring in engineering contexts namely pde constrained shape optimization and parametrized coupled problems

reduced basis approximation and a posteriori error estimation for stress intensity factors we develop reduced basis approximations rigorous a posteriori error bounds and offline online computational procedures for the accurate fast and reliable prediction of stress intensity factors or strain energy release rates for mode i linear elastic crack problems relevant to fracture mechanics we demonstrate the efficiency and rigor of our numerical method in several examples

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

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

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

discriminant and clifford algebras the centralizer of a square central skew symmetric unit in a central simple algebra with orthogonal involution carries a unitary involution the discriminant algebra of this unitary involution is shown to be an orthogonal summand in one of the components of the clifford algebra of the orthogonal involution as an application structure theorems for orthogonal involutions on central simple algebras of degree 8 are obtained

non linear model reduction for the navier stokes equations using residual deim method this article presents a new reduced order model based upon proper orthogonal decomposition pod for solving the navier stokes equations the novelty of the method lies in its treatment of the equation s non linear operator for which a new method is proposed that provides accurate simulations within an efficient framework the method itself is a hybrid of two existing approaches namely the quadratic expansion method and the discrete empirical interpolation method deim that have already been developed to treat non linear operators within reduced order models the method proposed applies the quadratic expansion to provide a first approximation of the non linear operator and deim is then used as a corrector to improve its representation in addition to the treatment of the non linear operator the pod model is stabilized using a petrov galerkin method this adds artificial dissipation to the solution of the reduced order model which is necessary to avoid spurious oscillations and unstable solutions a demonstration of the capabilities of this new approach is provided by solving the incompressible navier stokes equations for simulating a flow past a cylinder and gyre problems comparisons are made with other treatments of non linear operators and these show the new method to provide significant improvements in the solution s accuracy

the reduced model multiscale method r3m for the non linear homogenization of hyperelastic media at finite strains this paper presents a new multi scale method for the homogenization analysis of hyperelastic solids undergoing finite strains the key contribution is to use an incremental nonlinear homogenization technique in tandem with a model reduction method in order to alleviate the complexity of multiscale procedures which usually involve a large number of nonlinear nested problems to be solved the problem associated with the representative volume element rve is solved via a model reduction method proper orthogonal decomposition the reduced basis is obtained through pre computations on the rve the technique coined as reduced model multiscale method r3m allows reducing significantly the computation times as no large matrix needs to be inverted and as the convergence of both macro and micro problems is enhanced furthermore the r3m drastically reduces the size of the data base describing the history of the micro problems in order to validate the technique in the context of porous elastomers at finite strains a comparison between a full and a reduced multiscale analysis is performed through numerical examples involving different micro and macro structures as well as different nonlinear models neo hookean mooney rivlin it is shown that the r3m gives good agreement with the full simulations at lower computational and data storage requirements

orthogonal involutions on central simple algebras and function fields of severi brauer varieties abstract an orthogonal involution on a central simple algebra a after scalar extension to the function field f a of the severi brauer variety of a is adjoint to a quadratic form q over f a which is uniquely defined up to a scalar factor some properties of the involution such as hyperbolicity and isotropy up to an odd degree extension of the base field are encoded in this quadratic form meaning that they hold for the involution if and only if they hold for q as opposed to this we prove that there exists non totally decomposable orthogonal involutions that become totally decomposable over f a so that the associated form q is a pfister form we also provide examples of nonisomorphic involutions on an index 2 algebra that yield similar quadratic forms thus proving that the form q does not determine the isomorphism class of even when the underlying algebra has index 2 as a consequence we show that the e 3 invariant for orthogonal involutions is not classifying in degree 12 and does not detect totally decomposable involutions in degree 16 as opposed to what happens for quadratic forms

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

reduced basis method for linear elasticity problems with many parameters he reduced basis rb methods are proposed here for the solution of parametrized equations in linear elasticity problems the fundamental idea underlying rb methods is to decouple the generation and projection stages offline online computational procedures of the approximation process in order to solve parametrized equations in a rapid inexpensive and reliable way the method allows important computational savings with respect to the classical galerkin finite element method ill suited to a repetitive environment like the parametrized contexts of optimization many queries and sensitivity analysis we consider different parametrization for the systems either physical quantities to model the materials and loads and geometrical parameters to model different geometrical configurations then we describe three different applications of the method in problems with isotropic and orthotropic materials working in plane stress and plane strain approximation and subject to harmonic loads

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

localized discrete empirical interpolation method this paper presents a new approach to construct more efficient reduced order models for nonlinear partial differential equations with proper orthogonal decomposition and the discrete empirical interpolation method deim whereas deim projects the nonlinear term onto one global subspace our localized discrete empirical interpolation method ldeim computes several local subspaces each tailored to a particular region of characteristic system behavior then depending on the current state of the system ldeim selects an appropriate local subspace for the approximation of the nonlinear term in this way the dimensions of the local deim subspaces and thus the computational costs remain low even though the system might exhibit a wide range of behaviors as it passes through different regimes ldeim uses machine learning methods in the offline computational phase to discover these regions via clustering local deim approximations are then computed for each cluster in the online computational phase machine

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

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

structure preserving model reduction for mechanical systems this paper focuses on methods of constructing of reduced order models of mechanical systems which preserve the lagrangian structure of the original system these methods may be used in combination with standard spatial decomposition methods such as the karhunen loeve expansion balancing and wavelet decompositions the model reduction procedure is implemented for three dimensional finite element models of elasticity and we show that using the standard newmark implicit integrator significant savings are obtained in the computational costs of simulation in particular simulation of the reduced model scales linearly in the number of degrees of freedom and parallelizes well

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

bridging proper orthogonal decomposition methods and augmented newton krylov algorithms an adaptive model order reduction for highly nonlinear mechanical problems this article describes a bridge between pod based model order reduction techniques and the classical newton krylov solvers this bridge is used to derive an efficient algorithm to correct on the fly the reduced order modelling of highly nonlinear problems undergoing strong topological changes damage initiation problems tackled via a corrected hyperreduction method are used as an example it is shown that the relevancy of reduced order model can be significantly improved with reasonable additional costs when using this algorithm even when strong topological changes are involved

the generalized even clifford algebra abstract for a central simple algebra a with an orthogonal involution over a field f of characteristic not equal to two we associate to it the generalized even clifford algebrac0 a we will present results on the structure of this algebra c0 a some of which are due to jacobson in j algebra1 1964 288 300 using a generic method we then give a formula for computing c0 a when a is decomposable as algebras with involution using these results a necessary condition for decomposability of an algebra with orthogonal involution compatible with the involution is then obtained

an efficient reduced order modeling approach for non linear parametrized partial differential equations summary for general non linear parametrized partial differential equations pdes the standard galerkin projection is no longer efficient to generate reduced order models this is because the evaluation of the integrals involving the non linear terms has a high computational complexity and cannot be pre computed this situation also occurs for linear equations when the parametric dependence is nonaffine in this paper we propose an efficient approach to generate reduced order models for large scale systems derived from pdes which may involve non linear terms and nonaffine parametric dependence the main idea is to replace the non linear and nonaffine terms with a coefficient function approximation consisting of a linear combination of pre computed basis functions with parameter dependent coefficients the coefficients are determined efficiently by an inexpensive and stable interpolation at some pre computed points the efficiency and accuracy of this method are demonstrated on several test cases which show significant computational savings relative to the standard galerkin projection reduced order approach copyright q 2008 john wiley sons ltd

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

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

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 the size of certain subsets of invariant banach sequence spaces abstract the essence of the notion of lineability and spaceability is to find linear structures in somewhat chaotic environments the existing methods in general use ad hoc arguments and few general techniques are known motivated by the search of general methods in this paper we formally extend recent results of g botelho and v v favaro on invariant sequence spaces to a more general setting our main results show that some subsets of invariant sequence spaces contain up to the null vector a closed infinite dimensional subspace