tensor products of complete lattices and their application in constructing quantales abstract this paper aims to implement tensor products of complete lattices into fuzzy set theory the most convenient approach for this purpose is to view the tensor product of two complete lattices as the family of all join reversing maps between those lattices we show that some fundamental constructions in fuzzy set theory are tensor products examples of such circumstances include the following to mention only three of them the complete lattice of all lower semicontinuous maps from a topological space into a continuous lattice is the tensor product of the topology of the space and the range lattice a binary operation coming from zadeh s extension principle is the tensor product of the minkowski multiplication with the multiplication of the underlying unital quantale triangle functions on nonnegative left continuous distribution functions are tensor products of the real unit interval and the extended nonnegative half line equipped respectively with a left continuous t norm and the usual addition

an extension of the fuzzy unit interval to a tensor product with completely distributive first factor abstract the original hutton interval i l can algebraically be identified with the tensor product i l of the real unit interval i and a complete lattice l due to this the tensor product m l with m a completely distributive lattice is considered as a generalization of the lattice i l when appropriately endowed with an l topology the tensor product m l becomes also an l topological extension of i l if m is separable it has a countable join base free of supercompact elements many of the l topological features of i l are retained to wit urysohn lemma and tietze urysohn extension theorem for m l valued functions are then proved the relationship of m l to the l fuzzy topological modification of m in the sense of d zhang and y m liu 27 is discussed

representation theorems for l fuzzy quantities abstract in this paper representation theorems are given for l fuzzy quantities which permit a better understanding of fuzziness in particular representation theorems due to negoita and ralescu 17 and to sherwood and taylor 23 are extended to the scope of complete and brouwerian lattices

on r l o we prove there exists a somewhat simpler description of l and its fuzzy topology than the one given by hutton 2 and a considerably simpler description of than the one given by rodabaugh 4 as a result of this simplification we are able to give a direct proof of the continuity of an open problem in 4

of the fuzzy unit interval abstract answering a question raised by gantner steinlage and warren we provide some conditions for the compactness of the fuzzy unit interval

point set lattice theoretic topology this essay attempts to survey in a coherent way certain aspects of point set lattice theoretic or poslat topology by which we mean fuzzy topology grounded in notions of sets functions powersets and powerset operators the latter two being lattice theoretic in nature and examined from a lattice theoretic point of view using methods of category theory connections with other related issues and developments are frequently given

lsrn a parallel iterative solver for strongly over or underdetermined systems we describe a parallel iterative least squares solver named textttlsrn that is based on random normal projection textttlsrn computes the min length solution to min x in mathbb r n a x b 2 where a in mathbb r m times n with m gg n or m ll n and where a may be rank deficient tikhonov regularization may also be included since a is involved only in matrix matrix and matrix vector multiplications it can be a dense or sparse matrix or a linear operator and lsrn automatically speeds up when a is sparse or a fast linear operator the preconditioning phase consists of a random normal projection which is embarrassingly parallel and a singular value decomposition of size lceil gamma min m n rceil times min m n where gamma is moderately larger than 1 e g gamma 2 we prove that the preconditioned system is well conditioned with a strong concentration result on the extreme singular values and hence that the number of iterations is fully predictable

forty years of hutton fuzzy unit interval this note is a very short history of a specific fuzzy topological space the hutton s fuzzy unit interval 0 1 l which occupies one of the central positions in fuzzy topology on that occasion the superiority of l valued topology over 0 1 valued topology is being discussed we then show how the hutton s concept evolved after obvious identifications we can today say that 0 1 l is the tensor product 0 1 l of the lattices 0 1 and l an account of the consequences of this new view point is hoped to appear thereafter

fuzzy points and local properties of fuzzy topological spaces abstract we prove that three theorems of c k wong on local properties of fuzzy topology are wrong thereby we discuss his notions of fuzzy point c 1 separability and local compactness

blendenpik supercharging lapack s least squares solver several innovative random sampling and random mixing techniques for solving problems in linear algebra have been proposed in the last decade but they have not yet made a significant impact on numerical linear algebra we show that by using a high quality implementation of one of these techniques we obtain a solver that performs extremely well in the traditional yardsticks of numerical linear algebra it is significantly faster than high performance implementations of existing state of the art algorithms and it is numerically backward stable more specifically we describe a least squares solver for dense highly overdetermined systems that achieves residuals similar to those of direct qr factorization based solvers lapack outperforms lapack by large factors and scales significantly better than any qr based solver

suitability in fuzzy topological spaces abstract an l fuzzy topological space is said to be suitable if it possesses a nontrivial crisp closed subset basic properties of and sufficient conditions for suitable spaces are derived characterizations of the suitable subspaces of the fuzzy unit interval the fuzzy open unit interval and the fuzzy real line are obtained suitability is l fuzzy productive nondegenerate 1 hausdorff spaces are suitable the fuzzy unit interval the fuzzy open unit interval and the fuzzy real line are not suitable and no suitable subspace of the fuzzy unit interval the fuzzy open unit interval or the fuzzy real line is a fuzzy retract of the fuzzy unit interval the fuzzy open unit interval or the fuzzy real line respectively without restrictions there cannot be a fuzzy extension theorem

on l tychonoff spaces topics of this paper include 1 basic terminology and auxiliary observations 2 pointfree characterizations of completely l regular spaces 3 embedding spaces of prescribed weight applications to l tychonoff and l to spaces 4 remarks on liu s characterizations of complete l regularity involving points 5 extending continuous l real functions from h compact subspaces 6 conditional completeness of c x r l 7 remarks on the compactification of liu and luo 8 l creg l creg and l tych l tych 9 lower order axioms for the l line a new coherent system of separation axioms for l topological spaces 10 open questions

generalizing the l fuzzy unit interval abstract given an arbitrary connected topological space an l fuzzy space is constructed when the original space is the real line unit interval the constructed space is the l fuzzy real line l fuzzy unit interval for some spaces including r n for all n the original space is embedded as a subspace of the constructed space lastly the construction yields a non trivial fuzzy topology on certain classically important spaces of monotone mappings

some questions in fuzzy topology the purpose of this paper is to discuss some basic questions related mainly to hutton s l fuzzy unit interval some of those questions are stated here for the first time while some of them are long standing related results are surveyed in some cases

separation axioms and the fuzzy real lines we show that for each lattice l which is completely distributive and equipped with an order reversing involution the canonical topology of the fuzzy real line l is uniformizable in the sense of hutton furthermore this uniformity induces a pseudometric which induces the canonical topology consequently l satisfies the higher order separation axioms of hutton and sarkar more generally we answer most of the open questions concerning l and its subspaces with respect to each of the separation axiom schemes of hutton and sarkar answer some of these same questions for the stratified fuzzy real line c l and its subspaces and analyze c l and its subspaces with respect to the separation axioms of rodabaugh

l fuzzy normal spaces and tietze extension theorem abstract in this paper we prove a characterization theorem for normal l fuzzy topological spaces l is an infinitely distributive lattice with an order reversing involution in the particular case l 0 1 this theorem reduces to a known result of kat tov fund math 38 1951 85 91 and tong duke math j 19 1952 289 292 as an important application we obtain a fuzzy version of tietze extension theorem this yields an affirmative answer to a question raised in a series of papers by rodabaugh fuzzy sets and systems 11 1983 163 183

a statistical perspective on randomized sketching for ordinary least squares we consider statistical as well as algorithmic aspects of solving large scale least squares ls problems using randomized sketching algorithms for a ls problem with input data x y rn p rn sketching algorithms use a sketching matrix s rr n where r n then rather than solving the ls problem using the full data x y sketching algorithms solve the ls problem using only the sketched data sx sy prior work has typically adopted an algorithmic perspective in that it has made no statistical assumptions on the input x and y and instead it has been assumed that the data x y are fixed and worst case wc prior results show that when using sketching matrices such as random projections and leverage score sampling algorithms with p r n the wc error is the same as solving the original problem up to a small constant from a statistical perspective we typically consider the mean squared error performance of randomized sketching algorithms when data x y are generated according to a statistical linear model y x e where e is a noise process in this paper we provide a rigorous comparison of both perspectives leading to insights on how they differ to do this we first develop a framework for assessing in a unified manner algorithmic and statistical aspects of randomized sketching methods we then consider the statistical prediction efficiency pe and the statistical residual efficiency re of the sketched ls estimator and we use our framework to provide upper bounds for several types of random projection and random sampling sketching algorithms among other results we show that the re can be upper bounded when p r n while the pe typically requires the sample size r to be substantially larger lower bounds developed in subsequent results show that our upper bounds on pe can not be improved

first and second order energy stable methods for the modified phase field crystal equation abstract the phase field crystal pfc model was extended to the modified phase field crystal mpfc model which is a sixth order nonlinear damped wave equation to include not only diffusive dynamics but also elastic interactions in this paper we present temporally first and second order accurate methods for the mpfc equation which are based on an appropriate splitting of the energy for the pfc equation and we use the fourier spectral method for the spatial discretization the first and second order methods are shown analytically to be unconditionally stable with respect to the energy and pseudoenergy of the mpfc equation respectively numerical experiments are presented demonstrating the accuracy and energy stability of the proposed methods

faster least squares approximation least squares approximation is a technique to find an approximate solution to a system of linear equations that has no exact solution in a typical setting one lets n be the number of constraints and d be the number of variables with n gg d then existing exact methods find a solution vector in o nd 2 time we present two randomized algorithms that provide accurate relative error approximations to the optimal value and the solution vector of a least squares approximation problem more rapidly than existing exact algorithms both of our algorithms preprocess the data with the randomized hadamard transform one then uniformly randomly samples constraints and solves the smaller problem on those constraints and the other performs a sparse random projection and solves the smaller problem on those projected coordinates in both cases solving the smaller problem provides relative error approximations and if n is sufficiently larger than d the approximate solution can be computed in o nd ln d time

on l tychonoff spaces ii this paper is a sequel to the 1995 paper on l tychonoff spaces the embedding theorem for l topological spaces is shown to hold true for l an arbitrary complete lattice without imposing any order reversing involution on l some results on completely l regular spaces and on l tychonoff spaces which have previously been known to hold true for l a frame are exhibited as ones holding for l a meet continuous lattice for such a lattice an insertion theorem for completely l regular spaces is given some weak forms of separating families of maps are discussed we also clarify the dependence between the sub t0 separation axiom of liu and the l t0 separation axiom of rodabaugh

fast monte carlo algorithms for matrices i approximating matrix multiplication motivated by applications in which the data may be formulated as a matrix we consider algorithms for several common linear algebra problems these algorithms make more efficient use of computational resources such as the computation time random access memory ram and the number of passes over the data than do previously known algorithms for these problems in this paper we devise two algorithms for the matrix multiplication problem suppose a and b which are m times n and n times p respectively are the two input matrices in our main algorithm we perform c independent trials where in each trial we randomly sample an element of 1 2 ldots n with an appropriate probability distribution cal p on 1 2 ldots n we form an m times c matrix c consisting of the sampled columns of a each scaled appropriately and we form a c times n matrix r using the corresponding rows of b again scaled appropriately the choice of cal p and the column and row scaling are crucial features of the algorithm when these are chosen judiciously we show that cr is a good approximation to ab more precisely we show that left ab cr right f o left a right f left b right f sqrt c where cdot f denotes the frobenius norm i e a 2 f sum i j a ij 2 this algorithm can be implemented without storing the matrices a and b in ram provided it can make two passes over the matrices stored in external memory and use o c m n p additional ram to construct c and r we then present a second matrix multiplication algorithm which is similar in spirit to our main algorithm in addition we present a model the pass efficient model in which the efficiency of these and other approximate matrix algorithms may be studied and which we argue is well suited to many applications involving massive data sets in this model the scarce computational resources are the number of passes over the data and the additional space and time required by the algorithm the input matrices may be presented in any order of the entries and not just row or column order as is the case in many applications where e g the data has been written in by multiple agents in addition the input matrices may be presented in a sparse representation where only the nonzero entries are written

fuzzy addition in the l fuzzy real line abstract under the hypothesis l is a chain we construct a binary operation on the l fuzzy real line r l which reduces to the usual addition on r if is restricted to the embedded image of r in r l which yields a partially ordered abelian cancellation semigroup with identity and which is jointly fuzzy continuous on r l we show is unique i e it is the only extension of addition to r l which is consistent we study the relationship between and other fuzzy continuous extensions of the usual addition we also show that fuzzy translation is a weak fuzzy homeomorphism and under certain conditions a fuzzy homeomorphism