CiteULike: Tag fractions

Kinetic of crystallization from the melt and chain folding in polyethylene fractions revisited: theory and experiment

John Hoffman — 2008-03-31T16:26:12-00:00

Polymer, Vol. 38, No. 13. (1997), pp. 3151-3212.

The rate of growth of chain-folded lamellar crystals from the subcooled melt of polyethylene fractions is treated in terms of surface nucleation theory with the objective of illuminating the origin of the chain folding phenomenon and associated kinetic effects in molecular terms. An updated version of flux-based nucleation theory in readily usable form is outlined that deals with the nature of polymer chains in more detail than previous treatments. The subjects covered include: (i) the origin of regimes I, II, III, and III-A and the associated crystal growth rates, including the effect of forced steady-state reptation and reptation of `slack' in the subcooled melt; (ii) the variation of the initial lamellar thickness with undercooling; (iii) the origin of the fold surface free energy [sigma]e and the lateral surface free energy [sigma]; (iv) the generation and effect of nonadjacent events (such as tie chains) on the crystallinity and growth rates; and (v) `quantized' chain folding at low molecular weight. The topological limitation on nonadjacent re-entry and the value of the apportionment factor [psi] are discussed. Key experimental data are analysed in terms of the theory and essential parameters determined, including the size of the substrate length L involved in regime I growth. The degree of adjacent and/or `tight' folding that obtains in the kinetically-induced lamellar structures is treated as being a function of molecular weight and undercooling. New evidence based on the quantization effect indicates a high degree of adjacent re-entry in regime I for the lower molecular weight fractions. The quality of the chain folding at higher molecular weights in the various regimes is discussed in terms of kinetic, neutron scattering, i.r., and other evidence. Application of the theory to other polymers is discussed briefly.

A note on the approximation by continued fractions under an extra condition

K Dajani — 2005-11-25T10:52:05-00:00

(1997)

. In this note the distribution of the approximation coefficients Θ n , associated with the regular continued fraction expansion of numbers x 2 [0; 1), is given under extra conditions on the numerators and denominators of the convergents pn=qn . Similar results are also obtained for S-expansions. Further, a Gauss-Kusmin type theorem is derived for the regular continued fraction expansion under these extra conditions. Contents 1. Introduction 69 2. A Natural Extension of a Skew Product by...

A Gauss-Kusmin Theorem for Optimal Continued Fractions

Karma And — 2005-11-25T10:45:52-00:00

this paper is to obtain a Gauss-Kusmin theorem for the OCF. To be more precise, we will show Γ among many other things Γ that for z 2 [Γ ; g] () f 2 [Γ 1 2 zg = ([Γ 1 ; g) with density d (x), given by (x) = ? ? ? ? ? ? ! ? ? ? ? ? ? 2x+1 2x 2 +2x+1 if Γ 2 x ! Γg log G x+1 +2x+2 if Γ g x ! 3 1ΓxΓx (x +2x+2)(2x Γ2x+1) if 2 x ! g; (8) and where T is given by = [ 0; " n+1 b n+1 ; " n+2 b...

Reducibility of rational fractions in several variables

Arnaud Bodin — 2005-11-25T10:08:58-00:00

(20 Oct 2005)

We prove a analogous of Stein theorem for rational fractions in several variables: we bound the number of reducible fibers by a formula depending on the degree of the fraction.

The geometry of continued fractions and the topology of surface singularities

Patrick Popescu-Pampu — 2005-06-22T15:04:30-00:00

(21 June 2005)

We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing structure on the abstract boundaries (also called links) of normal surface singularities. The duality between supplementary cones gives in particular a geometric interpretation of a duality discovered by Hirzebruch between the continued fraction expansions of two numbers l >1 and l/(l - 1).

Toward a More Natural Expression of Quantum Logic with Boolean Fractions

Philip Calabrese — 2005-11-09T21:23:12-00:00

Journal of Philosophical Logic, Vol. 34, No. 4. (August 2005), pp. 363-401.

Toward a More Natural Expression of Quantum Logic with Boolean Fractions Philip G. Calabrese1 Contact Information (1) Data Synthesis, 2919 Luna Avenue, San Diego, CA, 92117 Abstract This paper uses a non-distributive system of Boolean fractions (a|b), where a and b are 2-valued propositions or events, to express uncertain conditional propositions and conditional events. These Boolean fractions, ‘a if b’ or ‘a given b’, ordered pairs of events, which did not exist for the founders of quantum logic, can better represent uncertain conditional information just as integer fractions can better represent partial distances on a number line. Since the indeterminacy of some pairs of quantum events is due to the mutual inconsistency of their experimental conditions, this algebra of conditionals can express indeterminacy. In fact, this system is able to express the crucial quantum concepts of orthogonality, simultaneous verifiability, compatibility, and the superposition of quantum events, all without resorting to Hilbert space. A conditional (a|b) is said to be “inapplicable” (or “undefined”) in those instances or models for which b is false. Otherwise the conditional takes the truth-value of proposition a. Thus the system is technically 3-valued, but the 3rd value has nothing to do with a state of ignorance, nor to some half-truth. People already routinely put statements into three categories: true, false, or inapplicable. As such, this system applies to macroscopic as well as microscopic events. Two conditional propositions turn out to be simultaneously verifiable just in case the truth of one implies the applicability of the other. Furthermore, two conditional propositions (a|b) and (c|d) reside in a common Boolean sub-algebra of the non-distributive system of conditional propositions just in case b=d, their conditions are equivalent. Since all aspects of quantum mechanics can be represented with this near classical logic, there is no need to adopt Hilbert space logic as ordinary logic, just a need perhaps to adopt propositional fractions to do logic, just as we long ago adopted integer fractions to do arithmetic. The algebra of Boolean fractions is a natural, near-Boolean extension of Boolean algebra adequate to express quantum logic. While this paper explains one group of quantum anomalies, it nevertheless leaves no less mysterious the ‘influence-at-a-distance’, quantum entanglement phenomena. A quantum realist must still embrace non-local influences to hold that “hidden variables” are the measured properties of particles. But that seems easier than imaging wave-particle duality and instant collapse, as offered by proponents of the standard interpretation of quantum mechanics. Keywords compatible propositions - conditional events - conditional logic - conditional probability - orthoalgebra - orthogonal - simultaneously measurable - simultaneously observable - simultaneously verifiable - superposition Partial support for this work is gratefully acknowledged from the In-House Independent Research Program and from Code 2737 at the Space & Naval Warfare Systems Center (SSC-SD), San Diego, CA 92152-5001. Presently this work is supported by Data Synthesis, 2919 Luna Avenue, San Diego, CA 92117.

A generalization of the Gauss map and some classical theorems on continued fractions

Simone Cruz — 2004-12-28T16:18:23-00:00

Nonlinearity, Vol. 18, No. 2., 505.

The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Walter Van Assche — 2005-11-29T17:19:47-00:00

(9 Jul 1993)

Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials.

Continued Fractions without Tears

Ian Richards — 2005-11-14T20:40:06-00:00

Mathematics Magazine, Vol. 54, No. 4. (1981), pp. 163-171.

Modified Farey Sequences and Continued Fractions

Maurice Shrader-Frechette — 2005-11-14T20:38:23-00:00

Mathematics Magazine, Vol. 54, No. 2. (1981), pp. 60-63.

Toward a More Natural Expression of Quantum Logic with Boolean Fractions

Philip Calabrese — 2005-11-28T20:11:47-00:00

(1 May 2003)

The fundamental algebraic concepts of quantum mechanics, as expressed by many authors, are reviewed and translated into the framework of the relatively new non-distributive system of Boolean fractions (also called conditional events or conditional propositions). This system of ordered pairs (A|B) of events A, B, can express all of the non-Boolean aspects of quantum logic without having to resort to a more abstract formulation like Hilbert space. Such notions as orthogonality, superposition, simultaneous verifiability, compatibility, orthoalgebras, orthocomplementation, modularity, and the Sasaki projection mapping are translated into this conditional event framework and their forms exhibited. These concepts turn out to be quite adequately expressed in this near-Boolean framework thereby allowing more natural, intuitive interpretations of quantum phenomena. Results include showing that two conditional propositions are simultaneously verifiable just in case the truth of one implies the applicability of the other. Another theorem shows that two conditional propositions (a|b) and (c|d) reside in a common Boolean sub-algebra of the non-distributive system of conditional propositions just in case b=d, that their conditions are equivalent. Some concepts equivalent in standard formulations of quantum logic are distinguishable in the conditional event algebra, indicating the greater richness of expression possible with Boolean fractions. Logical operations and deductions in the linear subspace logic of quantum mechanics are compared with their counterparts in the conditional event realm. Disjunctions and implications in the quantum realm seem to correspond in the domain of Boolean fractions to previously identified implications with respect to various naturally arising deductive relations.