Information, Representation, and Structure

This paper investigates the consequences of the information-theoretic result that representations of numbers in base-<i>e</i> are most efficient. Since theories on complex system behavior in both natural and physical systems assume that Nature is optimal, as is done, for example, in the principle of least action, natural representations must be to the base <i>e</i>. Another way to interpret this fact is to take <i>e</i> as the information dimension of the data space. Some implications of this noninteger dimensionality are investigated. The approximate equivalent to such a space is the Menger sponge in which the recursion is taken to be random.

Information, Representation, and Structure

This paper investigates the consequences of the information-theoretic result that representations of numbers in base-<i>e</i> are most efficient. Since theories on complex system behavior in both natural and physical systems assume that Nature is optimal, as is done, for example, in the principle of least action, natural representations must be to the base <i>e</i>. Another way to interpret this fact is to take <i>e</i> as the information dimension of the data space. Some implications of this noninteger dimensionality are investigated. The approximate equivalent to such a space is the Menger sponge in which the recursion is taken to be random.

On absolutely Baire nonmeasurable functions

We shall show that under Martin’s axiom, there exist absolutely Baire nonmeasurable additive functions. This provides a Baire category counterpart of an analogous measure-theoretic result of A. B. Kharazishvili.

On commutativity of rings and Banach algebras with generalized derivations

In the present paper, it is shown that if a prime ring R admits a generalized derivation f associated with a nonzero derivation d such that either f([x^{m},y^{n}])+[x^{m},y^{n}]\in Z(R)\quad\text{for all }x,y\in R or f([x^{m},y^{n}])-[x^{m},y^{n}]\in Z(R)\quad\text{for all }x,y\in R, then R is commutative. We apply this purely ring theoretic result to obtain commutativity of Banach algebras and prove that if A is a prime Banach algebra which admits a continuous linear generalized derivation f associated with a nonzero continuous linear derivation d such that either {f([x^{m},y^{n}])-[x^{m},y^{n}]\in Z(A)} or {f([x^{m},y^{n}])+[x^{m},y^{n}]\in Z(A)} for an integer {m=m(x,y)>1} and sufficiently many {x,y} in A, then A is commutative. A similar result is obtained for a unital prime Banach algebra A which admits a nonzero continuous linear generalized derivation f associated with a continuous linear derivation d such that {d(Z(A))\neq 0} satisfying either {f((xy)^{m})-x^{m}y^{m}\in Z(A)} or {f((xy)^{m})-y^{m}x^{m}\in Z(A)} for each {x\in G_{1}} and {y\in G_{2}} , where {G_{1},G_{2}} are open sets in A and {m=m(x,y)>1} is an integer: then A is commutative.

Spatiality of countably presentable locales (proved with the Baire category theorem)

The first part of the paper presents a generalization of the well-known Baire category theorem. The generalization consists in replacing the dense open sets of the original formulation by dense UCO sets, where UCO means union of closed and open. This topological theorem is exactly what is needed to prove in the second part of the paper the locale-theoretic result that locales whose frame of opens has a countable presentation (countably many generators and countably many relations) are spatial. This spatiality theorem does not require choice.

Analysis on Static Character of Piezoelectric Tubular Composite Actuator

The static character of piezoelectric composite actuator with four tubes is studied. Taking into account the non-uniform electric field distribution in piezoelectric tube, the bending deflection, axial strain, output axial force and bending moment are discussed. The results calculated by proposed method are compared to that calculated by traditional thin wall theory and the proposed method is more accurate and reasonable. A prototype actuator was fabricated and tested to validate the theory results. The theoretic result and tested results agree well with each other. This research provides theoretical guidance and basis for the design of piezoelectric tubular composites actuators.

LOGARITHMIC STACKS AND MINIMALITY

Given a category fibered in groupoids over schemes with a log structure, one produces a category fibered in groupoids over log schemes. We classify the groupoid fibrations over log schemes that arise in this manner in terms of a categorical notion of "minimal" objects. The classification is actually a purely category-theoretic result about groupoid fibrations over fibered categories, though most of the known applications occur in the setting of log geometry, where our categorical framework encompasses many notions of "minimality" previously extant in the literature.

SUBFACTORS OF INDEX LESS THAN 5, PART 4: VINES

We eliminate 39 infinite families of possible principal graphs as part of the classification of subfactors up to index 5. A number-theoretic result of Calegari–Morrison–Snyder, generalizing Asaeda–Yasuda, reduces each infinite family to a finite number of cases. We provide algorithms for computing the effective constants that are required for this result, and we obtain 28 possible principal graphs. The Ostrik d-number test and an algebraic integer test reduce this list to seven graphs in the index range (4,5) which actually occur as principal graphs.