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Item Open Access Existence, Uniqueness, and Numerical Analysis of Solutions of a Quasilinear ParabolicProblem(Society for Industrial and Applied Mathematics. SIAM Journal on Numerical Analysis, 1992-04) Wei, DongmingA quasilinear parabolic problem is studied. By using the method of lines, the existence and uniqueness of a solution to the initial boundary value problem with sufficiently smooth initial conditions are shown. Also given are L2 error estimates for the error between the extended fully discrete finite element solutions and the exact solution.Item Open Access A Penalty Method for Approximations of the Stationary Power-Law Stokes Problem(Electronic journal of differential equations, 2001) Lefton, Lew; Wei, DongmingWe study approximations of the steady state Stokes problem governed by the power-law model for viscous incompressible non-Newtonian flow using the penalty formulation. We establish convergence and find error estimates.Item Open Access Decay Estimates of Heat Transfer to Melton Polymer Flow in Pipes with Viscous Dissipation(Electronic journal of differential equations, 2001) Wei, Dongming; Zhang, ZhenbuIn this work, we compare a parabolic equation with an elliptic equation both of which are used in modeling temperature profile of a powerlaw polymer ow in a semi-infinite straight pipe with circular cross section. We show that both models are well-posed and we derive exponential rates of convergence of the two solutions to the same steady state solution away from the entrance. We also show estimates for difference between the two solutions in terms of physical data.Item Open Access Penalty finite element approximations of the stationary power- law Stokes problem(Journal of Numerical Mathematics, 2003) Lefton, Lew; Wei, DongmingFinite element approximations of the stationary power-law Stokes problem using penalty formulation are considered. A priori error estimates under appropriate smoothness assumptions on the solutions are established without assuming a discrete version of the BB condition. Numerical solutions are presented by implementing a nonlinear conjugate gradient methodItem Open Access Join-semidistributive lattices of relatively convex sets(2004) Adaricheva, KiraWe give two sufficient conditions for the lattice Co(Rn,X) of rel- atively convex sets of Rn to be join-semidistributive, where X is a finite union of segments. We also prove that every finite lower bounded lattice can be embedded into Co(Rn,X), for a suitable finite subset X of RnItem Open Access Embedding finite lattices into finite biatomic lattices(2005) Adaricheva, Kira; Wehrung, FreidrichFor a class C of finite lattices, the question arises whether any lattice in C can be embedded into some atomistic, biatomic lattice in C. We provide answers to the question above for C being, respectively, — The class of all finite lattices; — The class of all finite lower bounded lattices (solved by the first author’s earlier work). — The class of all finite join-semidistributive lattices (this problem was, until now, open). We solve the latter problem by finding a quasi-identity valid in all finite, atomistic, biatomic, join-semidistributive lattices but not in all finite join-semidistributive latticesItem Open Access On complex algebras of subalgebras(2006) Adaricheva, Kira; Pilitowska, Agata; Stanovsky, DavidLet V be a variety of algebras. We establish a condition (so called generalized entropic property), equivalent to the fact that for every algebra A 2 V, the set of all subalgebras of A is a subuniverse of the complex algebra of A. We investigate the relationship between the generalized entropic property and the entropic law. Further, provided the generalized entropic property is satisfied in V, we study the identities satisfied by the complex algebras of subalgebras of algebras from VItem Open Access Realization of abstract convex geometries by point configurations. Part I(2007) Adaricheva, Kira; Wild, MarcelThe Edelman-Jamison problem is to characterize those abstract convex geometries that are representable by a set of points in the plane. We show that some natural modification of the Edelman-Jamison problem is equivalent to the well known NP-hard order type problemItem Open Access Positive entropy invariant measures on the space of lattices with escape of mass(2010) Kadyrov, ShiraliOn the space of unimodular lattices, we construct a sequence of invariant probability measures under a singular diagonal element with high entropy and show that the limit measure is 0Item Open Access On the QR decomposition of backslashfancyscript H -matrices(Computing (Vienna/New York), 2010-06-09) Benner, Peter; Mach, ThomasThe hierarchical ( backslashfancyscriptH -) matrix format allows storing a variety of dense matrices from certain applications in a special data-sparse way with linear-polylogarithmic complexity. Many operations from linear algebra like matrix--matrix and matrix--vector products, matrix inversion and LU decomposition can be implemented efficiently using the backslashfancyscriptH -matrix format. Due to its importance in solving many problems in numerical linear algebra like least-squares problems, it is also desirable to have an efficient QR decomposition of backslashfancyscriptH -matrices. In the past, two different approaches for this task have been suggested in Bebendorf (Hierarchical matrices: a means to efficiently solve elliptic boundary value problems. Lecture notes in computational science and engineering (LNCSE), vol 63. Springer, Berlin, 2008) and Lintner (Dissertation, Fakultät für Mathematik, TU München. http://tumb1.biblio.tu-muenchen.de/publ/diss/ma/2002/lintner.pdf , 2002). We will review the resulting methods and suggest a new algorithm to compute the QR decomposition of an backslashfancyscriptH -matrix. Like other backslashfancyscriptH -arithmetic operations, the backslashfancyscriptH QR decomposition is of linear-polylogarithmic complexity. We will compare our new algorithm with the older ones by using two series of test examples and discuss benefits and drawbacks of the new approach.Item Open Access Bernstein-walsh inequalities in higherdimensions over exponential curves(2011) Kadyrov, Shirali; Lawrence, MarkLet x = (x1; : : : ; xd) 2 [1; 1]d be linearly independent over Z, set K = f(ez; ex1z; ex2z : : : ; exdz) : jzj 1g:We prove sharp estimates for the growth of a polynomial of degree n, in terms of En(x) := supfkPk d+1 : P 2 Pn(d + 1); kPkK 1g; where d+1 is the unit polydisk. For all x 2 [1; 1]d with linearly independent entries, we have the lower estimate logEn(x) nd+1 (d 1)!(d + 1) log n O(nd+1); for Diophantine x, we have logEn(x) nd+1 (d 1)!(d + 1) log n + O(nd+1): In particular, this estimate holds for almost all x with respect to Lebesgue measure. The results here generalize those of [6] for d = 1, without relying on estimates for best approximants of rational numbers which do not hold in the vector-valued setting.Item Open Access Escape of mass and entropy for diagonal flows in real rank one situations(2011) Einsiedler, M.; Kadyrov, Shirali; Pohl, A.Let G be a connected semisimple Lie group of real rank 1 with finite center, let be a non-uniform lattice in G and a any diagonalizable element in G. We investigate the relation between the metric entropy of a acting on the homogeneous space \G and escape of mass. Moreover, we provide bounds on the escaping mass and, as an application, we show that the Hausdorff dimension of the set of orbits (under iteration of a) which miss a fixed open set is not full.Item Open Access Representing finite convex geometries by relatively convex sets(2011) Adaricheva, KiraA closure system with the anti-exchange axiom is called a convex geometry. One geometry is called a sub-geometry of the other if its closed sets form a sublattice in the lattice of closed sets of the other. We prove that convex geometries of relatively convex sets in n-dimensional vector space and their nite sub-geometries satisfy the n-Carousel Rule, which is the strengthening of the n-Carath eodory property. We also nd another property, that is similar to the simplex partition property and does not follow from 2-Carusel Rule, which holds in sub-geometries of 2-dimensional geometries of relatively convex sets.Item Open Access Bornological projective limits of inductive limits of normed spaces(Functiones et Approximatio, Commentarii Mathematici, 2011) Bonet, José; Wegner, Sven AkeWe establish a criterion to decide when a countable projective limit of countable inductive limits of normed spaces is bornological. We compare the conditions occurring within our criterion with well-known abstract conditions from the context of homological algebra and with conditions arising within the investigation of weighted PLB-spaces of continuous functions.Item Open Access Algebraic numbers, hyperbolicity, and density modulo one(2011) Kadyrov, Shirali; Gorodnik, A.Item Open Access Entropy and escape of mass for SL3(Z)n SL3(R)(2011) Einsiedler, Manfred; Kadyrov, ShiraliWe study the relation between measure theoretic entropy and escape of mass for the case of a singular diagonal flow on the moduli space of three-dimensional unimodular latticesItem Open Access Stasheff polytope as a sublattice of permutohedron(2011) Adaricheva, KiraAn assosiahedron Kn, known also as Stasheff polytope, is a multifaceted combinatorial object, which, in particular, can be realized as a convex hull of certain points in Rn, forming (n − 1)-dimensional polytope1. A permutahedron Pn is a polytope of dimension (n−1) in Rn with vertices forming various permutations of n-element set. There exists well-known orderings of vertices of Pn and Kn that make these objects into lattices: the first known as permutation lattices, and the latter as Tamari lattices. We provide a new proof to the statement that the vertices of Kn can be naturally associated with particular vertices of Pn in such a way that the corresponding lattice operations are preserved. In lattices terms, Tamari lattices are sublattices of permutation lattices. The fact was established in 1997 in paper by Bjorner and Wachs, but escaped the attention of lattice theorists. Our approach to the proof is based on presentation of points of an associahedron Kn via so-called bracketing functions. The new fact that we establish is that the embedding preserves the height of elementsItem Open Access Positive undecidable numberings in the Ershov hierarchy(2012) Mustafa, M.; Sorbi, AndreaWe give a su cient condition for an in nite computable family of 1 a sets, to have computable positive but undecidable numberings, where a is a notation for a nonzero computable ordinal. This extends a theorem proved by Talasbaeva for the nite levels of the Ershov hierarchy. In par- ticular the family of all 1 a sets has positive undecidable numberings: this veri es for all levels of the Ershov hierarchy a conjecture due to Badaev and Goncharov. We point out also that for every ordinal notation a of a nonzero ordinal, there are families of 1 a sets having positive numberings, but no Friedberg numberings: this answers for all levels (whether nite or in nite) of the Ershov hierarchy, a question originally raised, only for the nite levels over level 1, by Badaev and Goncharov.Item Open Access Ordered direct implication basis of a finite closure system(2012) Adaricheva, Kira; Nation, J.B.; Rand, R.Closure system on a nite set is a unifying concept in logic programming, relational data bases and knowledge systems. It can also be presented in the terms of nite lattices, and the tools of economic description of a nite lattice have long existed in lattice theory. We present this approach by describing the so-called D-basis and introducing the concept of ordered direct basis of an implicational system. A direct basis of a closure operator, or an implicational system, is a set of implications that allows one to compute the closure of an arbitrary set by a single iteration. This property is preserved by the D-basis at the cost of following a prescribed order in which implications will be attended. In particular, using an ordered direct basis allows to optimize the forward chaining procedure in logic programming that uses the Horn fragment of propositional logic. One can extract the D-basis from any direct unit basis in time polynomial in the size s( ), and it takes only linear time of the cardinality of the D-basis to put it into a proper order. We produce examples of closure systems on a 6-element set, for which the canonical basis of Duquenne and Guigues is not ordered directItem Open Access On the Global Solvability of a Class of Fourth- Order Nonlinear Boundary Value Problems(Mathematics Faculty Publications, 2012) Elgindi, Mohamed B. M.; Wei, DongmingIn this paper we prove the global solvability of a class of fourth-order nonlinear boundary value problems that govern the deformation of a Hollomon’s power-law plastic beam subject to an axial compression and nonlinear lateral constrains. For certain ranges of the acting axial compression force, the solvability of the equations follows from the monotonicity of the fourth order nonlinear differential operator. Beyond these ranges the monotonicity of the operator is lost. It is shown that, in this case, the global solvability may be generated by the lower order nonlinear terms of the equations for a certain type of constrains.