Weak theories of linear algebra

We investigate the theories of linear algebra, which were originally defined to study the question of whether commutativity of matrix inverses has polysize Frege proofs. We give sentences separating quantified versions of these theories, and define a fragment in which we can interpret a weak theory V¹ of bounded arithmetic and carry out polynomial time reasoning about matrices - for example, we can formalize the Gaussian elimination algorithm. We show that, even if we restrict our language, proves the commutativity of inverses.This work was done while a postdoctoral research fellow at the Department of Computer Science, University of Toronto, Canada.     

A symplectic framework of linear field theories

Summary A framework for linear field theories based on the geometry of Lagrangian subspaces of symplectic spaces is formulated. Fields defined on finite compact domains and infinitesimal limits are treated in terms of de Rham currents. Field sources are included in the formulation. No distinction between elliptic and hyperbolic field equations is made.     

Deformation of short composite beam using refined theories

High-order flexural theories for short laminated composite beams subjected to mechanical and thermal loading are presented. The formulation allows for warping of the cross-section of the beam and eliminates the need for using arbitrary shear correction coefficients as in other theories. Based on higher-order shear deformation theories, the governing equations are obtained using the principle of virtual work (PVW). The justification for use of higher-order shear deformation theories is established for short and composite beams where cross-sectional warping is predominant.     

Prime models of theories of computable linear orderings

We answer a long-standing question of Rosenstein by exhibiting a complete theory of linear orderings with both a computable model and a prime model, but no computable prime model. The proof uses the relativized version of the concept of limitwise monotonic function.

     

Integrating learning theories and application-based modules in teaching linear algebra

The research team of The Linear Algebra Project developed and implemented a curriculum and a pedagogy for parallel courses in (a) linear algebra and (b) learning theory as applied to the study of mathematics with an emphasis on linear algebra. The purpose of the ongoing research, partially funded by the National Science Foundation, is to investigate how the parallel study of learning theories and advanced mathematics influences the development of thinking of individuals in both domains. The researchers found that the particular synergy afforded by the parallel study of math and learning theory promoted, in some students, a rich understanding of both domains and that had a mutually reinforcing effect. Furthermore, there is evidence that the deeper insights will contribute to more effective instruction by those who become high school math teachers and, consequently, better learning by their students. The courses developed were appropriate for mathematics majors, pre-service secondary mathematics teachers, and practicing mathematics teachers. The learning seminar focused most heavily on constructivist theories, although it also examined socio-cultural and historical perspectives. A particular theory, Action-Process-Object-Schema (APOS) [10], was emphasized and examined through the lens of studying linear algebra. APOS has been used in a variety of studies focusing on student understanding of undergraduate mathematics. The linear algebra courses include the standard set of undergraduate topics. This paper reports the results of the learning theory seminar and its effects on students who were simultaneously enrolled in linear algebra and students who had previously completed linear algebra and outlines how prior research has influenced the future direction of the project.     

Comparison of linear differential operators

In this paper, we study the question of the existence of the inequality $$\left\| {Q(D)f} \right\|_{L_q } \leqslant \gamma _0 \left\| {P(D)f} \right\|_{L_p } $$ , where P and Q are algebraic polynomials, D = d/dx, and γ ₀ is independent of the function f. We obtain criteria (necessary and simultaneously sufficient conditions) for the existence of such inequalities for functions on the circle, on the whole line, and on the semiaxis. Besides, for the semiaxis, we obtain an inequality for q = ∞ and any p ≥ 1 with the smallest constant γ ₀.     

Asymptotic error estimation in linear elastic beam models

In this work, we establish that the error in norm H¹ between the solution of the three-dimensional linear elasticity system and that of the classical Bernoulli-Navier model, for a clamped rod with transversal section having a diameter of order s. is ()(ɛ^1/2).     

Derivation of linear beam equations using nonlinear continuum mechanics

Familiar linear elastic and viscoelastic beam equations (Euler-Bernoulli, Rayleigh, Kelvin-Voigt, Timoshenko, and Shear Diffusion) and boundary conditions are derived from a nonlinear theory of large motions rather than the usual variational techniques. Also included is a fairly detailed derivation of the nonlinear theory and a careful discussion of the hypotheses.This work has been partially supported by the Office of Naval Research under grant number N00014-88-K0417 and by the National Science Foundation under grant number DMS-8801412.     

A rapprochement of the theories of radiative transfer and linear stochastic estimation

Parallels between linear stochastic estimation theory and radioactive transfer in the atmosphere are investigated. It is shown that the two theories are virtually equivalent and that basic functions in one may be meaningfully interpreted in the context of the other.     

A justification of the one-dimensional linear model of elastic beam

In this paper we prove that the one-dimensional model of elastic beam is an approximation to the three-dimensional linear theory of elasticity.     

The complexity of isomorphism for complete theories of linear orders with unary predicates

Suppose A is a linear order, possibly with countably many unary predicates added. We classify the isomorphism relation for countable models of \(\text {Th}(A)\) up to Borel bi-reducibility, showing there are exactly five possibilities and characterizing exactly when each can occur in simple model-theoretic terms. We show that if the language is finite (in particular, if there are no unary predicates), then the theory is \(\aleph _0\)-categorical or Borel complete; this generalizes a theorem due to Schirmann (Theories des ordres totaux et relations dequivalence. Master’s thesis, Universite de Paris VII, 1997).     

Comparison between the universal theories of partially commutative metabelian groups

We find necessary and sufficient conditions for the coincidence of the universal theories of partially commutative groups of metabelian varieties defined by acyclic graphs.     

KAM tori of Hamiltonian perturbations of 1D linear beam equations

In this paper, one-dimensional (1D) nonlinear beam equations

     

New linear spring stiffness definition for displacement analysis of cracked beam elements

The paper describes the reasons for the derivation of a new definition of a rotational spring that can be utilised in the simplified computational model for the computation of transverse displacements of cracked beam structures due to transverse load. This definition plays an extremely important role in the inverse identification of cracks. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparison of duality models in fractional linear programming

Summary This paper deals with the duality models in fractional linear programming presented in the last years bySwarup, Kaka, Sharma andSwarup and other authors.

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Zusammenfassung Der Aufsatz befaßt sich mit Dualitätsmodellen für Linear Fractional Programming, die in den letzten Jahren vonSwarup, Kaka, Sharma undSwarup sowie von anderen Autoren angegeben wurden.

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This work was sponsored by the Grant No. A 7329 from the National Research Council of Canada.     

Comparison of strategies when building linear prediction models

In statistical and biometric sciences, one often uses predictive linear models. The initial form of such models is usually obtained by fitting the coefficients of the model to a set of observed data according to the classical least squares method. Newborn models that are obtained in this way will be referred to as raw models. Such raw models are often subject of efforts to improve them as to their predictive performance on external datasets. Several methods can be followed to fine‐tune raw models, thus leading to a variety of model building strategies. In this paper, the idea of so‐called victory rates is introduced to compare the performance of building strategies mutually.Copyright © 2013 John Wiley & Sons, Ltd.     

Dynamics of a linear beam with an attached local nonlinear energy sink

We provide numerical evidence of passive and broadband targeted energy transfer from a linear flexible beam under shock excitation to a local essentially nonlinear lightweight attachment that acts, in essence, as nonlinear energy sink—NES. It is shown that the NES absorbs shock energy in a one-way, irreversible fashion and dissipates this energy locally, without ‘spreading’ it back to the linear beam. Moreover, we show numerically that an appropriately designed and placed NES can passively absorb and locally dissipate a major portion of the shock energy of the beam, up to an optimal value of 87%. The implementation of the NES concept to the shock isolation of practical engineering structures and to other applications is discussed.     

Comparison results for solving preconditioned linear systems

In this paper we present some comparison theorems between two different modified Gauss–Seidel (MGS) methods. The second preconditioning based on the first preconditioning is also discussed in this paper.     

On elementary theories of linear elastic beams,plates and shells (review paper)

Summary This paper presents a review of the elementary theories on the bending of straight and curved beams, on plates and shells, using asymptotic approximations of the basic linearized equations of elasticity in three dimensions. The maximun norm has been chosen to specify the orders of magnitude of the quantities involved. The expansions are given as usual in terms of the small geometrical parameter characterizing the thinness of the structure. Most of the ideas and results are well known. Nevertheless, in the cases where more than one small parameter may be involved, such as small curvatures (shallow structures) or the small loading parameter used to linearize the equations of elasticity, the discussion on the limits of validity of the different theories lead to some interesting newer aspects. Moreover, the main ideas presented in this paper concerning multiple parameter expansions may be applied to discuss the behaviour of the structures and to obtain valuable analytical results in more complicated situations such as moderate and strong anisotropy, dynamic problems, stability etc.

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Zusammenfassung Die vorliegende Arbeit enthält einen Ueberblick über die elementaren Theorien der Biegung von geraden und krummen Stäben, von Platten und Schalen unter Verwendung von asymptotischen Approximationen der dreidimensionalen linearisierten Grundgleichungen der Elastizität, Zur Festlegung der Grössenordnung der massgebenden Funktionen wurde die Maximum-Norm verwendet. Die asymptotischen Entwicklungen sind wie üblich in Abhängigkeit des kleinen geometrischen Parameters gegeben, der die dünne Struktur charakterisiert. Die meisten Ideen und Resultate sind gut bekannt. Dennoch in den mehrparametrigen Fällen, so etwa bei kleinen Krümmungen (flache Strukturen) oder bei der expliziten Berücksichtigung des kleinen Belastungparameters, der zur Linearisierung der elastizitätsgleichungen verwendet wurde, führt die Diskussion über die Gültigkeitsgrenzen der verschiedenen Theorien zu einigen neueren und interessanten Aspekten. Ausserdem können die in der vorliegenden Arbeit dargelegten Hauptideen über mehrparametrige Entwicklungen zur Diskussion und zur Herleitung wertvoller Resultate über das Verhalten dünner Tragwerke in komplizierten Fällen wie mässige und starke Anisotropie, dynamische Probleme, Stabilität usw. übertragen werden.