Intersection of the Sierpinski carpet with its rational translate

Motivated by Mandelbrot’s idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, Nekka and Li studied the properties of these translation sets and showed how they can be used for a classification purpose. In this paper, we pursue this study on the Sierpinski carpet with its rational translate. We also get the fractal structure of intersection I(x, y) of the Sierpinski carpet with its translate. We find that the packing measure of these sets forms a discrete spectrum whose non-zero values come only from shifting numbers with a finite triadic expansion. Concretely, when x and y have a finite triadic expansion, a very brief calculation formula of the measure is given.     

Biquaternion (Complexified Quaternion) Roots of −1

The roots of −1 in the set of biquaternions (quaternions with complex components, or complex numbers with quaternion real and imaginary parts) are derived. There are trivial solutions (the complex operator, and any unit pure real quaternion), and non-trivial solutions consisting of complex numbers with perpendicular pure quaternion real and imaginary parts. The moduli of the two perpendicular pure quaternions are expressible by a single parameter by using a hyperbolic trigonometric identity.     

Intersection of triadic Cantor sets with their translates— I. Fundamental properties

Motivated by Mandelbrot's [The Fractal Geometry of Nature, Freeman, San Francisco, 1983] idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, we study the properties of these translation sets and show how they can be used for a classification purpose. This first paper of a series of two will be devoted to set up the fundamental properties of Hausdorff measures of those intersection sets. Using the triadic expansion of the shifting number, we determine the fractal structure of intersection of triadic Cantor sets with their translates. We found that the Hausdorff measure of these sets forms a discrete spectrum whose non-zero values come only from those shifting numbers with a finite triadic expansion. We characterize this set of shifting numbers by giving a partition expression of it and the steps of its construction from a fundamental root set. Finally, we prove that intersection of Cantor sets with their translates verify a measure-conservation law with scales. The second paper will take advantage of the properties exposed here in order to utilize them in a classification context. Mainly, it will deal with the use of the discrete spectrum of measures to distinguish two Cantor-like sets of the same fractal dimension.     

Wandering sets for a class of borel isomorphisms of [0,1)

A wandering set for a map ϕ is a set containing precisely one element from each orbit of ϕ. We study the existence of Borel wandering sets for piecewise linear isomorphisms. Such sets need not exist even when the parameters involved are rational, but they do exist if in addition all the slopes are powers of 2. For ϕ having at most one discontinuity, the existence of a Borel wandering set is equivalent to rationality of the Poincaré rotation number. We compute the rotation numbers for a special class of such functions. The main result provides a concrete method of connecting certain pairs of wavelet sets.     

Computation of class numbers of quadratic number fields

We explain how one can dispense with the numerical computation of approximations to the transcendental integral functions involved when computing class numbers of quadratic number fields. We therefore end up with a simpler and faster method for computing class numbers of quadratic number fields. We also explain how to end up with a simpler and faster method for computing relative class numbers of imaginary abelian number fields.

     

<Emphasis Type="Italic">p</Emphasis>-adic interpolation of Taylor coefficients of modular forms

In this paper we show that the Taylor coefficients of a Hecke eigenform at a CM-point, suitably modified, form a sequence of algebraic numbers that satisfy the Kubota–Leopoldt generalization of the Kummer congruences for primes that split in the imaginary quadratic field associated with a CM-point. More generally, we show that these numbers are moments of a certain p-adic measure. In addition, we write down explicitly the “Euler factor” at p in terms of the p th Hecke eigenvalue of the modular form in question and certain data of the CM-point. P. Guerzhoy is supported by NSF grant DMS-0700933.     

Rotation intervals for chaotic sets

Chaotic invariant sets for planar maps typically contain periodic orbits whose stable and unstable manifolds cross in grid-like fashion. Consider the rotation of orbits around a central fixed point. The intersections of the invariant manifolds of two periodic points with distinct rotation numbers can imply complicated rotational behavior. We show, in particular, that when the unstable manifold of one of these periodic points crosses the stable manifold of the other, and, similarly, the unstable manifold of the second crosses the stable manifold of the first, so that the segments of these invariant manifolds form a topological rectangle, then all rotation numbers between those of the two given orbits are represented. The result follows from a horseshoe-like construction.

     

Comparison of topological and uniform structures for fuzzy numbers and the fixed point problem

The goal of this paper is to compare certain topological and uniform structures on sets of fuzzy numbers. Since fuzzy sets can be viewed as a kind of subsets they can be endowed with uniform structures in ways similar to those used for classic subsets. The Hausdorff construction is one such approach. The structures we study are the myope structure, the uniform convergence of the cuts in the sense of Hausdorff, and a structure induced by a metric defined by Goetschel and Voxman. We show how these structures are related to fixed point problems for some classes of fuzzy numbers.     

Number Types

<正>In mathematics,number is a mathematical object used to count and measure.There are so many numbers like negative numbers,zero,positive numbers,real numbers and they are said to be the main source of mathematics.All numbers are classified into sets and every number represented a unique representation.Today we will talk about"number types".     

Betti and Tachibana numbers of compact Riemannian manifolds

We present definitions and properties of conformal Killing forms on a Riemannian manifold and determine Tachibana numbers as analogs of the well known Betti numbers of a compact Riemannian manifold. We show some sets of conditions which characterize these numbers. Finally, we prove some results which establish relationships between Betti and Tachibana numbers.     

Discrete symmetries and chirality invariance in quantum field dynamics

The aim of the present paper is to discuss systematically the discrete symmetry operations on a quantized field in interaction; and to base the introduction of the new quantum number “chirality” for spinor fields on these symmetry properties. In the course of this investigation, several general results on the group of symmetry operations are proved and relation between certain sets of discrete symmetry operations and the spinor representation of the rotation group in 3 and 4 dimensions is established. An attempt has been made to present clearly the connection between additive and multiplicative quantum numbers, gauge transformations, unitary transformations and invariance laws. The chirality invariance of spinor fields in interaction is discussed in some detail. The emphasis throughout is on the systematic development rather than on details of application. The paper is divided into two parts, the first dealing with the general theory of discrete symmetry operations and the second concerned with chirality invariance for spinor fields.     

Minimizing the error of linear separators on linearly inseparable data

Given linearly inseparable sets $R$ of red points and $B$ of blue points, we consider several measures of how far they are from being separable. Intuitively, given a potential separator (“classifier”), we measure its quality (“error”) according to how much work it would take to move the misclassified points across the classifier to yield separated sets. We consider several measures of work and provide algorithms to find linear classifiers that minimize the error under these different measures.     

Some properties for the intersection of Moran sets with their translates

Motivated by Mandelbrot’s idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, Nekka and Li studied the properties of these translation sets and showed how they can be used for the classification purpose. In this paper, we pursue this study on a class of Moran sets with their rational translates. We also get the fractal structure of intersection I(x, y) of a class of Moran sets with their rational translates, and the formula of the box-counting dimension. We find that the Hausdorff measures of these sets form a discrete spectrum whose non-zero values come only from shifting vector with the expansion in fraction of (x, y). Concretely, when (x, y) has a finite expansion in fraction, a very brief calculation formula of the measure is given.     

A generalization of the notion of microscopic sets

We investigate families of subsets of the real line defined by nonincreasing sequences of positive real numbers. One of these families coincides with the σ-ideal of microscopic sets. We prove that the union of our families is equal to the σ-ideal of Lebesgue measure zero sets and the intersection of all such families is the σ-ideal of sets of strong measure zero. We also study other properties concerning homeomorphisms between sets of the first category and sets from our families.     

Stabilization of nonlinear systems in compound critical cases

In this paper, we study the stabilization of nonlinear systems in critical cases by using the center manifold reduction technique. Three degenerate cases are considered, wherein the linearized model of the system has two zero eigenvalues, one zero eigenvalue and a pair of nonzero pure imaginary eigenvalues, or two distinct pairs of nonzero pure imaginary eigenvalues; while the remaining eigenvalues are stable. Using a local nonlinear mapping (normal form reduction) and Liapunov stability criteria, one can obtain the stability conditions for the degenerate reduced models in terms of the original system dynamics. The stabilizing control laws, in linear and/or nonlinear feedback forms, are then designed for both linearly controllable and linearly uncontrollable cases. The normal form transformations obtained in this paper have been verified by using code MACSYMA.     

On the Geometric Densities of Random Closed Sets

Abstract

In many applications it is of great importance to handle evolution equations about random closed sets of different (even though integer) Hausdorff dimensions, including local information about initial conditions and growth parameters. Following a standard approach in geometric measure theory such sets may be described in terms of suitable measures. For a random closed set of lower dimension with respect to the environment space, the relevant measures induced by its realizations are singular with respect to the Lebesgue measure, and so their usual Radon–Nikodym derivatives are zero almost everywhere. In this paper we suggest to cope with these difficulties by introducing random generalized densities (distributions) á la Dirac–Schwarz, for both the deterministic case and the stochastic case. In this last one we analyze mean generalized densities, and relate them to densities of the expected values of the relevant measures. Many models of interest in material science and in biomedicine are based on time dependent random closed sets, as the ones describing the evolution of (possibly space and time inhomogeneous) growth processes; in such a situation, the Delta formalism provides a natural framework for deriving evolution equations for mean densities at all (integer) Hausdorff dimensions, in terms of the local relevant kinetic parameters of birth and growth. In this context connections with the concepts of hazard function, and spherical contact distribution function are offered.     

Standard simplices and pluralities are not the most noise stable

The Standard Simplex Conjecture and the Plurality is Stablest Conjecture are two conjectures stating that certain partitions are optimal with respect to Gaussian and discrete noise stability respectively. These two conjectures are natural generalizations of the Gaussian noise stability result by Borell (1985) and the Majority is Stablest Theorem (2004). Here we show that the standard simplex is not the most stable partition in Gaussian space and that Plurality is not the most stable low influence partition in discrete space for every number of parts k ≥ 3, for every value ρ ≠ 0 of the noise and for every prescribed measure for the different parts as long as they are not all equal to 1/k. Our results do not contradict the original statements of the Plurality is Stablest and Standard Simplex Conjectures in their original statements concerning partitions to sets of equal measure. However, they indicate that if these conjectures are true, their veracity and their proofs will crucially rely on assuming that the sets are of equal measures, in stark contrast to Borell’s result, the Majority is Stablest Theorem and many other results in isoperimetric theory. Given our results it is natural to ask for (conjectured) partitions achieving the optimum noise stability.     

A novel similarity measure between intuitionistic fuzzy sets based on the mid points of transformed triangular fuzzy numbers with applications to pattern recognition and medical diagnosis

Similarity measure is an essential tool to compare and determine the degree of similarity between intuitionistic fuzzy sets(IFSs). In this paper, a new similarity measure between intuitionistic fuzzy sets based on the mid points of transformed triangular fuzzy numbers is proposed. The proposed similarity measure provides reasonable results not only for the sets available in the literature but also gives very reasonable results, especially for fuzzy sets as well as for most intuitionistic fuzzy sets. To provide supportive evidence, the proposed similarity measure is tested on certain sets available in literature and is also applied to pattern recognition and medical diagnosis problems. It is observed that the proposed similarity measure provides a very intuitive quantification.     

Rotation number for non-autonomous linear Hamiltonian systems II: The Floquet coefficient

This paper is the second of a two-part series in which we review the properties of the rotation number for a random family of linear non-autonomous Hamiltonian systems. In Part I, we defined the rotation number for such a family and discussed its basic properties. Here we define and study a complex quantity - the Floquet coefficient w - for such a family. The rotation number is the imaginary part of w. We derive a basic trace formula satisfied by w, and give applications to Atkinson-type spectral problems. In particular we use w to discuss the convergence properties of the Weyl M-functions, the Kotani theory, and the gap-labelling phenomenon for these problems.