a new algebraization of the lameequation we develop a new way of writing the lamehamiltonian in lie algebraic form this yields in a natural way an explicit formula for both the lamepolynomials and the classical non meromorphic lame functions in terms of chebyshev polynomials and of a certain family of weakly orthogonal polynomials

coincidence of extendible vector valued ideals with their minimal kernel abstract we provide coincidence results for vector valued ideals of multilinear operators more precisely if a is an ideal of n linear mappings we give conditions for which the equality a e 1 e n f a min e 1 e n f holds isometrically as an application we obtain in many cases that the monomials form a schauder basis of the space a e 1 e n f several structural and geometric properties are also derived using this equality we apply our results to the particular case where a is the classical ideal of extendible or pietsch integral multilinear operators similar statements are given for ideals of vector valued homogeneous polynomials

a number of quasiexactly solvable n body problems we present several examples of quasiexactly solvable n body problems in one two and higher dimensions we study various aspects of these problems in some detail in particular we show that in some of these examples the corresponding polynomials form an orthogonal set and many of their properties are similar to those of the bender dunne polynomials we also discuss qes problems where the polynomials do not form an orthogonal set

influence of the anisotropy parameter δ on the spectrum of the generalized q symmetrized harper equation the 1 generalization of the q symmetrized harper equation is discussed in terms of wavefunctions expressed by laurent series proceeding by recursion leads to a nontrivial dependent generalization of the characteristic energy polynomial with a special emphasis on a continuous dependence on the commensurability parameter the multiplicity parameter which is responsible for the amount of coprime realizations of the commensurability parameter is also accounted for the present energies have been derived so as to reproduce particular ones obtained before as limiting cases

a new class of quasi exactly solvable potentials with a position dependent mass a new class of quasi exactly solvable potentials with a variable mass in the schrodinger equation is presented we have derived a general expression for the potentials including natanzon confluent potentials the general solution of the schrodinger equation is determined and the eigenstates are expressed in terms of the orthogonal polynomials

on the families of orthogonal polynomials associated to the razavy potential we show that there are two different families of weakly orthogonal polynomials associated to the quasi exactly solvable razavy potential v x cos 2x m 2 0 m one of these families encompasses the four sets of orthogonal polynomials recently found by khare and mandal while the other one is new these results are extended to the related periodic potential u x cos 2x m 2 for which we also construct two different families of weakly orthogonal polynomials we prove that either of these two families yields the ground state when m is odd and the lowest lying gaps in the energy spectrum of the latter periodic potential up to and including the m 1 th gap and having the same parity as m 1 moreover we show that the algebraic eigenfunctions obtained in this way are the well known finite solutions of the whittaker hill or hill s three term periodic differential equation thus the foregoing results provide a lie algebraic justification of the fact that the whittaker hill equation unlike for instance mathieu s equation admits finite solutions

the d dimensional softcore coulomb potential and the generalized confluent heun equation an analysis of the generalized confluent heun equation 2r2 1r y 2r2 1r 0 y e1r e0 y 0 in d dimensional space where i i ei are real parameters is presented with the aid of these general results the quasi exact solvability of the schrodinger eigenproblem generated by the softcore coulomb potential v r e2z r b b 0 is explicitly resolved necessary and sufficient conditions for polynomial solvability are given a three term recurrence relation is provided to generate the coefficients of polynomial solutions explicitly we prove that these polynomial solutions are sources of finite sequences of orthogonal polynomials properties such as the recurrence relations christoffel darboux formulas and moments of the weight function are discussed we also reveal a factorization property of these polynomials which permits the construction of other interesting related sequences of orthogonal polynomials an analysis of the generalized confluent heun equation 2r2 1r y 2r2 1r 0 y e1r e0 y 0 in d dimensional space where i i ei are real parameters is presented with the aid of these general results the quasi exact solvability of the schrodinger eigenproblem generated by the softcore coulomb potential v r e2z r b b 0 is explicitly resolved necessary and sufficient conditions for polynomial solvability are given a three term recurrence relation is provided to generate the coefficients of polynomial solutions explicitly we prove that these polynomial solutions are sources of finite sequences of orthogonal polynomials properties such as the recurrence relations christoffel darboux formulas and moments of the weight function are discussed we also reveal a factorization property of these polynomials which permits the construction of other interesting related sequences of orthogonal polynomials

maximum or minimum entropy generation for open systems entropy generation is analysed and obtained from the entropy balance for open systems considering the lifetime of the natural real process the lagrangian approach is discussed in order to develop an analytical method to obtain the stationary states of the open irreversible systems the stationary conditions of the open systems are obtained in relation to the entropy generation and its maximum principle an analysis of both minimum and maximum entropy generation is proposed suggesting that they are two different viewpoints of the same aspect the first is related to the system while the second is related to the interaction between the system and the environment

n fold supersymmetry and quasi solvability associated with x2 laguerre polynomials we construct a new family of quasi solvable and n fold supersymmetric quantum systems where each hamiltonian preserves an exceptional polynomial subspace of codimension 2 we show that the family includes as a particular case the recently reported rational radial oscillator potential whose eigenfunctions are expressed in terms of the x2 laguerre polynomials of the second kind in addition we find that the two kinds of the x2 laguerre polynomials are ingeniously connected with each other by the n fold supercharge

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

anti isospectral transformations orthogonal polynomials and quasi exactly solvable problems we consider the double sinh gordon potential which is a quasi exactly solvable problem and show that in this case one has two sets of bender dunne orthogonal polynomials we study in some detail the various properties of these polynomials and the corresponding quotient polynomials in particular we show that the weight functions for these polynomials are not always positive we also study the orthogonal polynomials of the double sine gordon potential which is related to the double sinh gordon case by an anti isospectral transformation finally we discover a new quasi exactly solvable problem by making use of the anti isospectral transformation

geometry of 2 homogeneous polynomials on lp spaces 1 p abstract we characterise extreme 2 homogeneous polynomials of diagonal form on l p with p 2 and give a class of examples of smooth points we describe the unit ball of p 2 l p 2 for both the cases p 2 and p we apply the results thus found to study the geometry of the predual of p 2 l p 2 and to show that for p 2 an extreme 2 homogeneous polynomial on l p 2 remains extreme when regarded as a polynomial on l p

orthogonal polynomials from hermitian matrices a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions it can be considered as a matrix version of exactly solvable schrodinger equations the hermitian matrices factorizable hamiltonians are real symmetric tridiagonal jacobi matrices corresponding to second order difference equations by solving the eigenvalue problem in two different ways the duality relation of the eigenpolynomials and their dual polynomials is explicitly established through the techniques of exact heisenberg operator solution and shape invariance various quantities the two types of eigenvalues the eigenvalues and the sinusoidal coordinates the coefficients of the three term recurrence the normalization measures and the normalisation constants etc are determined explicitly

geometry of three homogeneous polynomials on real hilbert spaces let h be a two dimensional real hilbert space we give a characterisation of the extreme and the smooth points of the unit ball of the space of three homogeneous polynomials on h in terms of the coefficients of the polynomial we also determine the smooth points of the unit ball of the predual of the space of three homogeneous polynomials

centralisers of spaces of symmetric tensor products and applications we show that the centraliser of the space of n fold symmetric injective tensors n 2 on a real banach space is trivial with a geometric condition on the set of extreme points of its dual the space of integral polynomials we obtain the same result for complex banach spaces we give some applications of this results to centralisers of spaces of homogeneous polynomials and complex banach spaces in addition we derive a banach stone theorem for spaces of vector valued approximable polynomials

the physical character of information the mathematical theory of communication defines information in syntax without reference to its physical representation and semantic significance however in an everyday context information is tied to its representation and its content is valued the dichotomy between the formal definition and the practical perception of information is examined by the second law of thermodynamics that was recently formulated as an equation of motion thermodynamic entropy shows that the physical representation of information is not inconsequential in generation transmission and processing of information according to the principle of increasing entropy communication by dissipative transformations is a natural process among many other evolutionary phenomena that level energy density differences between components of a communication system and its surroundings in addition information guided processes direct down along descents on free energy landscapes the non integrable equation for irreversible processes reveals that there is no universal analytical algorithm to match source to channel noise infiltration is also regarded by the second law as an inevitable consequence of energy transduction between a communication system and its surroundings communication is invariably associated with misunderstanding because mechanisms and means of information processing at the receiver differ from those at the sender the significance of information is ascribed to the increase in thermodynamic entropy in the receiver system that results from execution of the received message

quasi exactly solvable systems and orthogonal polynomials this paper shows that there is a correspondence between quasi exactly solvable models in quantum mechanics and sets of orthogonal polynomials pn the quantum mechanical wave function is the generating function for the pn e which are polynomials in the energy e the condition of quasi exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value j the zeros of the critical polynomial pj e are the quasi exact energy eigenvalues of the system

natural selection for least action the second law of thermodynamics is a powerful imperative that has acquired several expressions during the past centuries connections between two of its most prominent forms i e the evolutionary principle by natural selection and the principle of least action are examined although no fundamentally new findings are provided it is illuminating to see how the two principles rationalizing natural motions reconcile to one law the second law when written as a differential equation of motion describes evolution along the steepest descents in energy and when it is given in its integral form the motion is pictured to take place along the shortest paths in energy in general evolution is a noneuclidian energy density landscape in flattening motion

isometries between spaces of homogeneous polynomials abstract we derive banach stone theorems for spaces of homogeneous polynomials we show that every isometric isomorphism between the spaces of homogeneous approximable polynomials on real banach spaces e and f is induced by an isometric isomorphism of e onto f with an additional geometric condition we obtain the analogous result in the complex case isometries between spaces of homogeneous integral polynomials and between the spaces of all n homogeneous polynomials are also investigated

on one dimensional schrodinger problems allowing polynomial solutions we discuss the explicit construction of the schrodinger equations admitting a representation through some family of general polynomials almost all solvable quantum potentials are shown to be generated by this approach some generalization has also been performed in higher dimensional problems

representation reduction and solution space contraction in quasi exactly solvable systems in quasi exactly solvable problems partial analytic solutions energy spectrum and associated wavefunctions are obtained if some potential parameters are assigned specific values we introduce a new class in which exact solutions are obtained at a given energy for a special set of values of the potential parameters to obtain a larger solution space one varies the energy over a discrete set the spectrum by simply changing the value of a given integer a unified treatment that includes the standard as well as the new class of quasi exactly solvable problems is presented and a few examples are given the solution space is spanned by discrete square integrable basis functions in which the matrix representation of the hamiltonian is tridiagonal consequently the wave equation gives a three term recursion relation for the expansion coefficients of the wavefunction imposing quasi exact solvability constraints results in a complete reduction of the representation to the direct sum of a finite and an infinite component the finite is real and exactly solvable whereas the infinite is complex and associated with zero norm states consequently the whole physical space contracts to a finite dimensional subspace with normalizable states

properties of q reflexive banach spaces abstract we investigate whether the q reflexive banach spaces have the following properties containment of l p reflexivity dunford pettis property etc

factorization of q p summing polynomials through lorentz spaces abstract we present a vector valued duality between factorable q p summing polynomials and q p summing linear operators on symmetric tensor products of banach spaces several applications are provided first we prove a polynomial characterization of cotype of banach spaces we also give a variant of pisier s factorization through lorentz spaces of factorable q p summing polynomials from c k spaces finally we show a coincidence result for q p concave polynomials

extended jaynes cummings models and quasi exact solvability the original jaynes cummings model is described by a hamiltonian which is hermitian and exactly solvable here we extend this model by several types of interactions leading to a non hermitian operator which does not satisfy the physical condition of spacetime reflection symmetry pt symmetry the new hamiltonians are either exactly solvable admitting an entirely real spectrum or quasi exactly solvable with a real algebraic part of their spectrum

duality in spaces of polynomials of degree at most n abstract we study spaces of continuous polynomials of degree at most n between banach spaces using symmetric tensor products we show that any polynomial of degree at most n has a natural linearisation and that the space of all scalar valued polynomials of degree at most n has an isometric predual we introduce the spaces of integral and nuclear polynomials of degree at most n and endow them with norms that allows us to develop a duality theory for spaces of polynomials of degree at most n

the symmetric radon nikodým property for tensor norms abstract we introduce the symmetric radon nikod m property srn property for finitely generated s tensor norms of order n and prove a lewis type theorem for s tensor norms with this property as a consequence if is a projective s tensor norm with the srn property then for every asplund space e the canonical mapping n s e n s e is a metric surjection this can be rephrased as the isometric isomorphism q min e q e for some polynomial ideal q we also relate the srn property of an s tensor norm with the asplund or radon nikod m properties of different tensor products as an application results concerning the ideal of n homogeneous extendible polynomials are obtained as well as a new proof of the well known isometric isomorphism between nuclear and integral polynomials on asplund spaces an analogous study is carried out for full tensor products

atomic decompositions for tensor products and polynomial spaces abstract we study the existence of atomic decompositions for tensor products of banach spaces and spaces of homogeneous polynomials if a banach space x admits an atomic decomposition of a certain kind we show that the symmetrized tensor product of the elements of the atomic decomposition provides an atomic decomposition for the symmetric tensor product s n x for any symmetric tensor norm in addition the reciprocal statement is investigated and analogous consequences for the full tensor product are obtained finally we apply the previous results to establish the existence of monomial atomic decompositions for certain ideals of polynomials on x

continuous hahn polynomials and the heisenberg algebra continuous hahn polynomials have surfaced in a number of somewhat obscure physical applications for example they have emerged in the description of two photon processes in hydrogen hard hexagon statistical mechanical models and clebsch gordan expansions for unitary representations of the lorentz group so 3 1 in this paper it is shown that there is a simple and elegant way to construct these polynomials using the heisenberg algebra

the intersection of vibronic levels the basic one dimensional vibronic problem has been solved exactly in an analytical form for degenerate configurations the solution using the intermediate p representation and the heun functions expansion on the gauss basis set has indicated the existence of dynamic symmetry the direct solution in x representation has specified an effective method for the calculation of vibronic problems conical intersection parameters have been obtained and analyzed for a few lower levels

entropy generation minimum inside and maximum outside abstract the extremum of entropy generation is evaluated for both maximum and minimum cases using a thermodynamic approach which is usually applied in engineering to design energy transduction systems a new result in the thermodynamic analysis of the entropy generation extremum theorem is proved by the engineering approach it follows from the proof that the entropy generation results as a maximum when it is evaluated by the exterior surroundings of the system and a minimum when it is evaluated within the system the bernoulli equation is analyzed as an example in order to evaluate the internal and external dissipations in accordance with the theoretical results obtained

eigenvalue integrodifferential equations for orthogonal polynomials on the real line the one dimensional harmonic oscillator wave functions are solutions to a sturm liouville problem posed on the whole real line this problem generates the hermite polynomials however no other set of orthogonal polynomials can be obtained from a sturm liouville problem on the whole real line in this paper it is shown how to characterize an arbitrary set of polynomials orthogonal on in terms of a system of integrodifferential equations of the hartree fock type this system replaces and generalizes the linear differential equation associated with a sturm liouville problem our results are demonstrated for the special case of continuous hahn polynomials

geometry of integral polynomials m ideals and unique norm preserving extensions we use the aron berner extension to prove that the set of extreme points of the unit ball of the space of integral k homogeneous polynomials over a real banach space x is k x 1 with this description we show that for real banach spaces x and y if x is a nontrivial m ideal in y then ek sk sx the k th symmetric tensor product of x endowed with the injective symmetric tensor norm is never an m ideal in ek sk sy this result marks up a difference with the behavior of nonsymmetric tensors since when x is an m ideal in y it is known that ekkx the k th tensor product of x endowed with the injective tensor norm is an m ideal in ekky nevertheless if x is also asplund we prove that every integral k homogeneous polynomial in x has a unique extension to y that preserves the integral norm other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given

periodic potentials and supersymmetry we review the current status of one dimensional periodic potentials and also present several new results it is shown that using the formalism of supersymmetric quantum mechanics one can considerably enlarge the limited class of analytically solvable one dimensional periodic potentials further using the landen transformations as well as cyclic identities for jacobi elliptic functions discovered by us recently it is shown that a linear superposition of lame as well as associated lame potentials is also analytically solvable finally using anti isospectral transformations we also obtain a class of analytically solvable complex pt invariant periodic potentials having real band spectra

some spectral equivalences between schrodinger operators spectral equivalences of the quasi exactly solvable sectors of two classes of schrodinger operators are established using gaudin type bethe ansatz equations in some instances the results can be extended leading to full isospectrality in this manner we obtain equivalences between pt symmetric problems and hermitian problems we also find equivalences between some classes of hermitian operators

econophysics and bio chemical engineering thermodynamics the exergetic analysis of a municipality abstract exergy is a fundamental quantity because it allows us to obtain information on the useful work obtainable in a process the analyses of irreversibility are important not only in the design and development of the industrial devices but also in fundamental thermodynamics and in the socio economic analysis of municipality consequently the link between entropy and exergy is discussed in order to link econophysics to the bio chemical engineering thermodynamics last this link holds to the fundamental role of fluxes and to the exergy exchanged in the interaction between the system and its environment the result consists in a thermodynamic approach to the analysis of the unavailability of the economic productive or social systems the unavailability is what the system cannot use in relation to its internal processes this quantity result is interesting also as a support to public manager for economic decisions here the alessandria municipality is analyzed in order to highlight the application of the theoretical results

weakly open sets in the unit ball of some banach spaces and the centralizer abstract we show that every banach space x whose centralizer is infinite dimensional satisfies that every non empty weakly open set in b y has diameter 2 where y n s x n fold symmetric projective tensor product of x endowed with the symmetric projective norm for every natural number n we provide examples where the above conclusion holds that includes some spaces of operators and infinite dimensional c algebras we also prove that every non empty weak open set in the unit ball of the space of n homogeneous and integral polynomials on x has diameter two for every natural number n whenever the cunningham algebra of x is infinite dimensional here we consider the space of n homogeneous integral polynomials as the dual of the space n s e x n fold symmetric injective tensor product of x endowed with the symmetric injective norm for instance every infinite dimensional l 1 satisfies that its cunningham algebra is infinite dimensional we obtain the same result for every non reflexive l embedded space and so for every predual of an infinite dimensional von neumann algebra

new quasi exactly solvable sextic polynomial potentials a hamiltonian is said to be quasi exactly solvable qes if some of the energy levels and the corresponding eigenfunctions can be calculated exactly and in a closed form an entirely new class of qes hamiltonians having sextic polynomial potentials is constructed these new hamiltonians are different from the sextic qes hamiltonians in the literature because their eigenfunctions obey symmetry rather than hermitian boundary conditions these new hamiltonians present a novel problem that is not encountered when the hamiltonian is hermitian it is necessary to distinguish between the parametric region of unbroken symmetry in which all of the eigenvalues are real and the region of broken symmetry in which some of the eigenvalues are complex the precise location of the boundary between these two regions is determined numerically using the extrapolation techniques and analytically using the wkb analysis

quasi exactly solvable potentials on the line and orthogonal polynomials in this paper we show that a quasi exactly solvable normalizable or periodic one dimensional hamiltonian satisfying very mild conditions defines a family of weakly orthogonal polynomials which obey a three term recursion relation in particular we prove that normalizable exactly solvable one dimensional systems are characterized by the fact that their associated polynomials satisfy a two term recursion relation we study the properties of the family of weakly orthogonal polynomials defined by an arbitrary one dimensional quasi exactly solvable hamiltonian showing in particular that its associated stieltjes measure is supported on a finite set from this we deduce that the corresponding moment problem is determined and that the kth moment grows like the kth power of a constant as k tends to infinity we also show that the moments satisfy a constant coefficient linear difference equation and that this property actually characterizes weakly orthogonal polynomial systems

geometry of spaces of polynomials abstract connections between the shape of the unit ball of a banach space and analytic properties of the banach space have been studied for many years in this article some geometric properties of spaces related ton homogeneous polynomials are considered in particular the rotundity and smoothness of spaces of continuousn homogeneous polynomials and its preduals are studied furthermore an inequality relating the product of the norms of linear functionals on a banach space with the norm of the continuousn homogeneous polynomial determined by the product of the linear functionals is derived this inequality is used to study the strongly exposed points of the predual of the space of continuous 2 homogeneous polynomials

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

geometric theory of spaces of integral polynomials and symmetric tensor products abstract we investigate the nature of extreme weak exposed and weak strongly exposed points of the unit ball of spaces of n homogeneous integral polynomials on and n fold symmetric products of a banach space e for the space of integral polynomials we show the set of extreme points is contained in the set n e 1 we give smul yan type theorems for spaces of n homogeneous polynomials and n fold symmetric tensors that characterise weak exposed resp weak strongly exposed points in terms of g teaux resp frechet differentiability of the norm on various spaces of tensor products and polynomials our study of the geometry of these spaces has many applications when e has the radon nikod m property we show that the spaces of n homogeneous integral and nuclear polynomials are isomerically isomorphic for each integer n when the dimensions of e and n are both at least 2 then the space of n homogeneous polynomials on e is neither smooth nor rotund for a certain class of reflexive banach space the space of n homogeneous approximable polynomials on e is either reflexive or is not isometric to a dual banach space we conclude with a choquet theorem for a space of homogeneous polynomials

duality in spaces of nuclear and integral polynomials abstract we study the spaces of nuclear and integral vector valued polynomials and their duals we prove that if e is asplund p n ne f p i ne f isometrically for any banach space f we describe p n ne f as a subspace of p ne f in terms of weak star continuous polynomials we show that p w ne f p c ne f when e has the approximation property

continued fraction as a discrete nonlinear transform the connection between a taylor series and a continued fraction involves a nonlinear relation between the taylor coefficients an and the continued fraction coefficients bn in many instances it turns out that this nonlinear relation transforms a complicated sequence an into a very simple one bn this simplification is illustrated in the context of graph combinatorics

a moments analysis of quasi exactly solvable systems a new perspective on the sextic potential gx 6 bx 4 mx 2 beta over x 2 a quantum system is quasi exactly solvable qes when a subset of its eigenvalues can be obtained in closed form or they are the roots of closed form expressions in one dimension qes states assume the form involving a known positive reference function and a polynomial p x so that the taylor expansion of truncates at a finite order for non qes states this truncation procedure for suitable reference functions can provide approximate values for the eigenvalues it corresponds to the hill determinant method which is known to be unstable when assumes the controlling asymptotic form of the physical state for this reason it cannot be used to simultaneously generate exact qes states while approximating non qes states to address these limitations the orthogonal polynomial projection quantization oppq method was developed handy and vrinceanu 2013 j phys a math theor 46 135202 2013 j phys b atom mol opt phys 46 115002 it is demonstrated in this paper that oppq can directly and transparently provide at the same time both algebraic solutions for qes states and convergent numerically stable approximations for non qes states within the oppq analysis for the wavefunction representation the bender dunne energy orthogonal polynomials correspond exactly up to a numerical factor to the energy dependent power moments p dx xp x within this perspective the existence of qes states is associated with an anomalous kink behavior in the order of the finite difference moment equation corresponding to the s suggesting a change in the number of degrees of freedom this was first noted through the implementation of the eigenvalue moment method the first application of semidefinite programming analysis to quantum operators handy and bessis 1985 phys rev lett 55 931 this moments perspective also reveals additional properties for the non qes states of the same symmetry as the qes states their lower order moments must be zero we demonstrate our results in the context of the two sextic potentials vsa x x6 mx2 bx4 and vss x x6 mx2 b x2

bases in spaces of multilinear forms over banach spaces fil dimant veronica universidad de san andres departamento de matematica y ciencias argentina