tightness and distinguished frechet spaces in this chapter we apply the concept of tightness to study distinguished frechet spaces we show that a frechet space is distinguished if and only if its strong dual has countable tightness this approach to studying distinguished frechet spaces leads to a rich supply of df spaces whose weak duals are quasi suslin but not k analytic the small cardinals mathfrak b and mathfrak d will be used to improve the analysis of kothe s echelon nondistinguished frechet space 1 a

the most obtuse angle row pivot rule for achieving dual feasibility a computational study we recently proposed several new pivot rules for achieving dual feasibility in linear programming which are distinct from existing ones the objective function value will no longer change necessarily monotonically in their solution process in this paper we report our further computational testing with one of them the most obtuse angle rule a two phase dual simplex algorithm in which the rule is used as a row selection rule for phase 1 has been implemented in fortran 77 modules and was tested on a set of standard linear programming problems from netlib we show that if full pricing is applied our code unambiguously will outperform minos 5 3 one of the best implementations of the simplex algorithm at present

a basis deficiency allowing variation of the simplex method for linear programming h one of the most important and fundamental concepts in the simplex methodology basis is restricted to being a square matrix of the order exactly equal to the number of rows of the coefficient matrix such inflexibility might have been the source of too many zero steps taken by the simplex method in solving real world linear programming problems which are usually highly degen erate to dodge this difficulty we first generalize the basis to allow the deficient case characterized as one that has columns fewer than rows of the coefficient matrix variations of the primal and dual simplex procedures are then made and used to form a two phase method based on such a basis the number of whose columns varies dynamically in the solution process generally speaking the more degenerate a problem to be handled is the fewer columns the basis will have so thii renders the possibility of efficiently solving highly degenerate problems we also provide a valuable insight into the distinctive and favorable behavior of the proposed method by reporting our computational experiments 1998 elsevier science ltd all rights reserved

weight of precompact subsets and tightness pfister 1976 and cascales and orihuela 1986 proved that precompact sets in df and lm spaces have countable weight i e are metrizable improvements by valdivia 1982 cascales and orihuela 1987 and kakol and saxon preprint have varying methods of proof for these and other improvements a refined method of upper semi continuous compact valued maps applied to uniform spaces will suffice at the same time this method allows us to dramatically improve kaplansky s theorem that the weak topology of metrizable spaces has countable tightness extending it to include all lm spaces and all quasi barrelled df spaces both in the weak and original topologies one key is showing that for a large class g including all df and lm spaces countable tightness of the weak topology of e in g is equivalent to realcompactness of the weak topology of the dual of e

brief minimal representations of mimo time varying systems and realization of cyclostationary models this paper describes a family of minimally parameterized state space models for completely n step observable multi input multi output time varying systems and their algebraical links with equivalent input output models these results are then used to develop an algorithm for the realization of generic input output sequences generated by a periodic system a complete numerical example is also developed to illustrate all steps of the algorithm described in the paper

representations and structural properties of periodic systems we consider periodic behavioral systems as introduced in kuijper m willems j c 1997 a behavioral framework for periodically time varying systems in proceedings of the 36th conference on decision control vol 3 pp 2013 2016 san diego california usa 10 12 december 1997 and analyze two main issues behavioral representation controllability and autonomy more concretely we study the equivalence and the minimality of kernel representations and introduce latent variable and in particular image representations moreover we relate the controllability of a periodic system with the controllability of an associated time invariant system known as lifted system and derive a controllability test further we prove the existence of an autonomous controllable decomposition similar to the time invariant case finally we introduce a new concept of free variables and inputs which can be regarded as a generalization of the one adopted for time invariant systems but appears to be more adequate for the periodic case

weak barrelledness for c x spaces abstract buchwalter and schmets reconciled cc x and cp x spaces with most of the weak barrelledness conditions of 1973 but could not determine if 0 barrelled l barrelled for cc x the areas grew apart full reconciliation with the fourteen conditions adopted by saxon and sanchez ruiz needs their 1997 characterization of ruess property l which allows us to reduce the cc x problem to its 1973 status and solve it by carefully translating the topology of kunen 1980 and van mill 1982 to find the example that eluded buchwalter and schmets the more tractable cp x readily partitions the conditions into just two equivalence classes the same as for metrizable locally convex spaces instead of the five required for cc x spaces our paper elicits others soon to appear that analytically characterize when the tychonov space x is pseudocompact or warner bounded or when cc x is a df space jarchow s 1981 question

lf spaces quasi baire spaces and the strongest locally convex topology on e coarser than r n 1 2 if e 0 e and is the finest locally convex n l topology on e which induces on each e a topology coarser than then e is said to be the inductive limit of the sequence e z ithe family e z 1 is a definin9 sequence for the inductive limit space e z note that we are using a narrow definition of the notion of an inductive limit that befits our needs if each e r is a fr6chet banach space e e 1 and z is hausdorff then the inductive limit e z is said to be an lf space lb space and we write

criss cross algorithm based on the most obtuse angle rule and deficient basis to improve computational efficiency a new criss cross algorithm based on the most obtuse angle pivoting index rule and deficient basis algorithm is proposed in this paper to solve linear programming problems we do numerical test values with the 25 standard netlib problems preliminary computational tests demonstrate that the new algorithm overcomes computational difficulties yielding from degeneracy and reduces the number of iterations and the overall computation time in addition the new algorithm has greatly simplified unbound or infeasible conditions to make it more practical so far the numerical experimental results show that the new algorithm is promising and is able to implement under the sparse structure in order to apply to large scale sparse problem

discrete time linear periodic realization in the frequency domain abstract given an appropriate collection of periodic rational matrices we characterize when it has a periodic realization the state space and the input output invariant formulation of discrete time periodic linear systems are involved

an example of cycling in a feasible point algorithm an example shows that if in rosen s 4 algorithm for calculating a point that satisfies linear constraints a natural criterion is used to select constraints to delete from active sets then the algorithm may cycle instead of converging there are no degeneracies in the example

on simplex method with most obtuse angle rule and cosine rule a class of simplex methods for solving linear programming lp problems with cosine pivot rule have been presented in some recent papers in this paper we show that the cosine rule used in this class is equivalent to the most obtuse angle pivot rule proposed by pan 1990 6 the relation between the direct method for lp and the most obtuse angle rule is discussed

phase i cycling under the most obtuse angle pivot rule it has been recently claimed that the most obtuse angle pivot rule is one of the best choices for phase i linear programs based on the simplex method in this short note we give two instances of phase i cycling under such ratio test free rule both when it is used to obtain primal feasibility and when trying to achieve dual feasibility with its unnormalized counterpart a crash procedure that is not objective driven might be the cause and a non simplex active set generalization could be used instead

brief a periodically time varying minimal partial realization algorithm based on twisting the 1975 paper of meyer and burrus gave necessary conditions on the coefficients of a periodically time varying difference equation to generate a prespecified set of impulse responses the main contribution of the present paper consists of an algorithm that constructs the coefficients of such a periodically time varying realization of minimal lag thus the algorithm solves the minimal partial realization problem for the periodically time varying scalar case we use a general technique twisting for associating a time invariant system with a periodically time varying system in such a way that the time step is preserved the technique plays an essential role in the ideas underlying the algorithm

metrizability of precompact subsets in lf spaces in this paper we prove that every precompact subset in any lf space has a metrizable completion as a consequence every lf space is angelic and in this way the answer to a question posed by k floret 3 is given some contributions to the general problem of regularity in inductive limits posed by k floret 3 are also given particularly extensions of well known results of h neuss and m valdivia are provided in the general setting of lf spaces it should also be noted that our results hold for inductive limits of an increasing sequence of metrizable spaces

if cp x is strongly dominated by a second countable space then x is countable abstract we establish that a tychonoff space x is countable if and only if c p x is strongly dominated by a second countable space the same is true for a compact space k such that c p k 0 1 is strongly dominated by a second countable space we also prove that strong domination by a second countable space of the complement of the diagonal of a tychonoff space x implies that x is an 0 space our results solve several published open questions

a new shift invariant representation for periodic linear systems abstract this paper presents a new family of shift invariant representations for periodic linear discrete time systems this representation has the advantages over the standard representation 5 of preserving the time step of the original system and of preserving causality under arbitrary feedback the disadvantage is that the realization is not minimal an application in attitude determination is presented

non baire hyperplanes in non separable baire spaces abstract given an infinite dimensional banach space e one may ask does e have 1 a properly separable quotient 2 a dense non baire hyperplane every closed hyperplane in a baire space is baire the famous separable quotient problem 1 remains unsolved and question 2 is question 13 1 1 of p perez carreras and j bonet north holland math stud vol 131 north holland amsterdam 1987 in 1966 wilansky klee conjectured the answer to 2 is always no j arias de reyna denied the conjecture math ann 249 1980 111 114 proving the answer to 2 is yes whenever the answer to 1 is yes under assumption of c additivity c a a condition weaker than martin s axiom m valdivia extended the result to e a baire 2nd category in itself topological vector space collect math 34 1983 287 296 in general 2 is non trivial only in case e is baire with e e an extension of valdivia s extension yields the complete answer for 2 in the general locally convex setting affirming question 13 1 1 of perez carreras and bonet referenced above in particular a not necessarily hausdorff locally convex space e has a dense non baire hyperplane if and only if dim e and e e even for e non locally convex dim e and e e suffices but is not necessary whereas e e is obviously always necessary in order that e have a dense non baire hyperplane whether e e suffices in the hausdorff non locally convex case and whether the assumption of c a can be omitted we do not know contrastingly any infinite dimensional e is the union of a sequence of hyperplanes and if e is baire one of the hyperplanes must be dense and baire and every hyperplane of e must be unordered baire like if e is

periodic systems a sequence algebra approach to realisation parameterisation and topology a characterisation of discrete time linear periodic systems is presented based on a general framework using cyclic projection operators on sequences two known liftings of periodic systems to a time invariant one the monodromy and the cyclic representation are readily derived in this framework this approach also leads to the definition of the operational transfer function for such systems and more generally the operational transfer inclusion it is shown that the cyclic realisation allows the multiplexing of individual realisations of a periodic system these descriptions are useful in the realisation problem and the search for canonical forms parameterisations for the class of reachable systems are recalled and their geometric and topological features are illustrated

a basis deficiency allowing primal phase i algorithm using the most obtuse angle column rule the dual phase 1 algorithm using the most obtuse angle row pivot rule is very efficient for providing a dual feasible basis in either the classical or the basis deficiency allowing context in this paper we establish a basis deficiency allowing phase i algorithm using the so called most obtuse angle column pivot rule to produce a primal deficient or full basis our computational experiments with the smallest test problems from the standard netlib set show that a dense projected gradient implementation largely outperforms that of the variation of the primal simplex method from the commercial code matlab linprog v1 17 and that a sparse projected gradient implementation of a normalized revised version of the proposed algorithm runs 34 faster than the sparse implementation of the primal simplex method included in the commercial code tomlab lpsolve v3 0

bounded tightness for locally convex spaces and spaces c x abstract we show that metrizability and bounded tightness are actually equivalent for a large class g of locally convex spaces including lf spaces df spaces the space of distributions d etc a consequence of this fact is that for x g the bounded tightness for the weak topology of x is equivalent to the following one x is linearly homeomorphic to a subspace of r n this nicely supplements very recent results of cascales and raja moreover we show that a metric space x is separable if the space cp x has bounded tightness

a canonical decomposition of linear periodic discrete time systems in this paper a linear periodic discrete time system is studied on the basis of simple properties of the subspaces of controllable states and of unreconstructible ones a canonical structure theorem is derived this generalizes 1o such a system the classical kalman decomposition while preserving the constant dimensionality of the four subsystems which arise when a periodic continuous time system is decomposed the dynamic matrices of the non controllable and or non reconstructible subsystems are shown to be non singular at each time instant as those for a time invariant discrete time system are

a phase 1 approach for the generalized simplex algorithm abstract a new simplex variant allowing basis deficiency has recently been proposed to attack the degeneracy 1 as a generalization of the simplex algorithm it uses a phase 1 procedure solving an auxiliary problem with piecewise linear sums of infeasibilities as its objective in this paper we develop another phase 1 approach that only introduces a single artificial variable unlike the former which needs a crash procedure to supply an initial basis the proposed phase 1 is able to get itself started from scratch with an artificial basis having a single column computational results with a set of standard test problems from netlib are also reported

on countable bounded tightness for spaces cp x abstract it is well known that the space cp 0 1 has countable tightness but it is not frechet urysohn let x be a cech complete topological space we prove that the space cp x of continuous real valued functions on x endowed with the pointwise topology is frechet urysohn if and only if cp x has countable bounded tightness i e for every subset a of cp x and every x in the closure of a in cp x there exists a countable and bounding subset of a whose closure contains x we study also the problem when the weak topology of a locally convex space has countable bounded tightness additional results in this direction are provided

lf spaces with more than separable quotients abstract the separable quotient problem remains open for banach spaces the recent solution for barrelled spaces counters saxon narayanaswami s solution for lf spaces w j robertson proposed properly separable quotients we fully answer her quarter century old question proving lf spaces admit properly separable quotients with rare exception corollary every lf space except admits properly separable quotients if and only if banach s classic separable quotient problem has a positive solution