Space Of Real Numbers Research Articles

Homeomorphism Problems of Fuzzy Real Number Space and The Space of Bounded Functions with Same Monotonicity on [-1,1

<p>In this paper, based on the fuzzy structured element, we prove that there is a bijection function between the fuzzy number space ε1 and the space B[−1, 1], which defined as a set of standard monotonic bounded functions with monotonicity on interval [−1, 1]. Furthermore, a new approach based upon the monotonic bounded functions has been proposed to create fuzzy numbers and represent them by suing fuzzy structured element. In order to make two different metrics based space in B[−1, 1], Hausdorff metric and Lp metric, which both are classical functional metrics, are adopted and their topological properties are discussed. In addition, by the means of introducing fuzzy functional to space B[−1, 1], we present two new fuzzy number’s metrics. Finally, according to the proof of homeomorphism between fuzzy number space ε1 and the space B[−1, 1], it’s argued that not only does it give a new way to study the fuzzy analysis theory, but also makes the study of fuzzy number space easier.</p>

Fiber bundle description of number scaling in gauge theory and geometry

This work uses fiber bundles as a framework to describe some effects of number scaling on gauge theory and some geometric quantities. A description of number scaling and fiber bundles over a flat space time manifold, M, is followed by a description of gauge theory. A fiber at point x of M contains a pair of scaled complex number and vector space structures, $C^{c}_{x}\times V^{c}_{x} $ for each c in GL(1,C). A space time dependent scalar field, g, determines, for each x, the scaling value of the vector space structures that contain the values of a vector valued matter field at x. The vertical components of connections between neighboring fibers are taken to be the gradient field A(x)+iB(x), of g. Abelian gauge theory for these fields gives the result that B is massless and no mass restrictions for A. Addition of an electromagnetic field dies not change these results. In the Mexican hat Higgs mechanism B combines with a Goldstone boson to create massive vector bosons, the photon field, and the Higgs field. For geometric quantities the fiber bundle is a tangent bundle with a fiber at point x containing scaled pairs, $R^{r}_{x}\times T^{r}_{x}$ of real number and tangent space structures for each x and and nonnegative real r. B is zero everywhere. The A field affects path lengths and the proper times of clocks along paths. It also appears in the geodesic equation. The lack of physical evidence for the gradient field, A(x)+iB(x) means that it either couples very weakly to matter fields or that it is close to zero for all x in a local region of cosmological space and time. It says nothing about the values outside the local region.

Holographic spectroscopy: analysis of convergent phase development

A real-number representation of wave amplitudes and phases inside the photosensitive medium with holographic elementary grating is deduced from coupled-wave equations by choosing a separation ansatz. Using numerical simulations, we show that so-called in-Bragg and off-Bragg cases are closely tied to the phase relationship between the reference and signal waves. Since the formulation is, as a whole, performed in the real number space, the results allow a strong intuitive interpretation. Asymptotic convergence to constant boundary values of the phase difference (in-Bragg) and to linear functions with slope (off-Bragg case) are discussed and unambiguously attributed to the type of recorded grating (pure index, mixed, or pure absorption grating). Finally, an analytical examination of the real amplitudes dependent on the waves’ phase parameters is derived. All results are discussed in the framework of energy transfer between two waves coupled at a recorded holographic grating.

A Computational Method of Real-valued Discrete Fourier Series Transform (real-DFST) and Application

In this paper, we propose a real-valued discrete Fourier series transform (real-DFST) and a computational method for the circular cross correlation. For images, the real-DFST presents the sub elements of frequency components in the DFT. And the space of the real-DFST is a real number space, therefore the real-DFST can be represented with a memory space for image-size real numbers. The real-DFST is a memory saving model for real-valued data and is faster than the DFT in computation. These advantages are important to implement hardware and to compute with large sequences/images. In implementation of system, proposed computational method could be used to enhance their systems through the better MCU computation-time by internal register memory or cache memory. According to increase the size of images, memory saving model is required necessarily. Experiments are suggested for the circular cross correlation processing in the real-DFST and experiments show the computational result for images.

Riemann Integral of Functions from ℝ into Real Banach Space

Summary In this article we deal with the Riemann integral of functions from R into a real Banach space. The last theorem establishes the integrability of continuous functions on the closed interval of reals. To prove the integrability we defined uniform continuity for functions from R into a real normed space, and proved related theorems. We also stated some properties of finite sequences of elements of a real normed space and finite sequences of real numbers. In addition we proved some theorems about the convergence of sequences. We applied definitions introduced in the previous article [21] to the proof of integrability.

A green pump scheduling algorithm for minimizing power consumption and land depletion

In a water distribution system, water and electricity are two limited resources. Previous research has shown that pump scheduling is a useful technique for dealing with the resource leveling problem. This study aims to develop a genetic algorithm for achieving near-optimal resource leveling. Three contributions are made in this study. First, the problem is modeled in the real-number space, that is, it is closer to the real-world case. Second, the proposed algorithm converges faster and achieves higher solution quality than traditional ones. Finally, the proposed algorithm generates cost-economic and eco-friendly schedules.

Choban operators in generalized ordered spaces

In this paper we investigate generalized ordered (GO) spaces that have a flexible diagonal in the sense of Arhangelʼskii (2009) [2]. Spaces with a flexible diagonal are generalizations of topological groups and include spaces that are continuously homogeneous, Choban spaces, and rotoid spaces. We prove some paracompactness and metrization theorems for such spaces and construct examples of generalized ordered spaces that clarify how the types of spaces with a flexible diagonal are interrelated. We show, for example, that any GO Choban space is hereditarily paracompact, that any continuously homogeneous, first-countable GO-space is metrizable, that the space of real numbers is the only non-degenerate connected LOTS that is a Choban space, and that the Sorgenfrey line and the Michael line are Choban spaces. We extend some results of Arhangelʼskii and pose a family of questions.

Generalized Groups that Distribute Over Stars

In the paper “Groups that Distribute over Mathematical Structures”, [5], we stated that a group (S, ·) on a set S left (or right) distributes over an arbitrary mathematical structure (S, ∗) on the same set S if and only if respectively for all fixed t ∈ S the permutation Lt (x) = t · x (or Rt (x) = x · t) is a similarity mapping on (S, ∗). A similarity mapping f on (S, ∗) is a permutation on S that preserves the structure of (S, ∗) such as a homeomorphism on a topological space, an automorphism on a binary operator or a similarity mapping on a binary relation. Also, Lt (x) and Rt (x) are called the left and right translations by t. For example, the group (R, ◦,+) both left and right distributes over the space of real numbers (R, T ) with the usual topology. In other words, for all subsets U of R, and for all x ∈ R, U + x = x+U is an open subset of R if and only if U is an open subset of R. See [5] for the details. In this paper we define and give a reasonably complete solution for a naturally occurring example that involves what we call an n-star which is structurally the same as n lines in the plane intersecting in ( n 2 ) district points. However, an equally important purpose of this paper is to show that if a structure (S, ∗) is given, then a fundamental idea is to see if a group (S, ·) exists such that (S, ·) left-distributes or right-distributes over (S, ∗) . This paper takes the reader on a long journey, but we hope that it is a reasonably fast and easy journey.

A continuous ant colony system framework for fuzzy data mining

The goal of data mining is to find out interesting and meaningful patterns from large databases. In some real applications, many data are quantitative and linguistic. Fuzzy data mining was thus proposed to discover fuzzy knowledge from this kind of data. In the past, two mining algorithms based on the ant colony systems were proposed to find suitable membership functions for fuzzy association rules. They transformed the problem into a multi-stage graph, with each route representing a possible set of membership functions, and then, used the any colony system to solve it. They, however, searched for solutions in a discrete solution space in which the end points of membership functions could be adjusted only in a discrete way. The paper, thus, extends the original approaches to continuous search space, and a fuzzy mining algorithm based on the continuous ant approach is proposed. The end points of the membership functions may be moved in the continuous real-number space. The encoding representation and the operators are also designed for being suitable in the continuous space, such that the actual global optimal solution is contained in the search space. Besides, the proposed approach does not have fixed edges and nodes in the search process. It can dynamically produce search edges according to the distribution functions of pheromones in the solution space. Thus, it can get a better nearly global optimal solution than the previous two ant-based fuzzy mining approaches. The experimental results show the good performance of the proposed approach as well.

Inequalities of Hlawka type for fuzzy real numbers

In this paper, we give and prove inequalities of Hlawka type and related results in the fuzzy real number space. Mathematics subject classification (2010): 26D15.

Effects of Disparate Bases and Norms on Minimizing a Least-Squares Reconstruction Error

Nomenclature b = least-squares coefficient of gappy proper orthogonal decomposition c = generalized least-squares coefficient d = dimension of an observed variable diag = diagonals of matrix N = Gaussian probability distribution q = dimension of a latent variable, i.e., model selection R = n-dimensional real number space r = estimation error V = eigenvector matrix, i.e., a proper orthogonal decomposition basis Ve = guessed Vq obtained from ~ Y 0 W = factor loading matrix x = latent variable Y = collection of observed variables y = observed variable ML = maximum likelihood estimate = general norm = basis _ = mean-centered data h i = expectation k kL2 = L norm k kn = gappy norm ~ = estimation

Methods of critical value reduction for type-2 fuzzy variables and their applications

A type-2 fuzzy variable is a map from a fuzzy possibility space to the real number space; it is an appropriate tool for describing type-2 fuzziness. This paper first presents three kinds of critical values (CVs) for a regular fuzzy variable (RFV), and proposes three novel methods of reduction for a type-2 fuzzy variable. Secondly, this paper applies the reduction methods to data envelopment analysis (DEA) models with type-2 fuzzy inputs and outputs, and develops a new class of generalized credibility DEA models. According to the properties of generalized credibility, when the inputs and outputs are mutually independent type-2 triangular fuzzy variables, we can turn the proposed fuzzy DEA model into its equivalent parametric programming problem, in which the parameters can be used to characterize the degree of uncertainty about type-2 fuzziness. For any given parameters, the parametric programming model becomes a linear programming one that can be solved using standard optimization solvers. Finally, one numerical example is provided to illustrate the modeling idea and the efficiency of the proposed DEA model.

Wavelet Codes for Algorithm-Based Fault Tolerance Applications

Algorithm-based fault tolerance (ABFT) methods, which use real number parity values computed in two separate comparable ways to detect computer-induced errors in numerical processing operations, can employ wavelet codes for establishing the necessary redundancy. Wavelet codes, one form of real number convolutional codes, determine the required parity values in a continuous fashion and can be intertwined naturally with normal data processing. Such codes are the transform coefficients associated with an analysis uniform filter bank which employs downsampling, while parity-checking operations are performed by a syndrome synthesis filter bank that includes upsampling. The data processing operations are merged effectively with the parity generating function to provide one set of parity values. Good wavelet codes can be designed starting from standard convolutional codes over finite fields by relating the field elements with the integers in the real number space. ABFT techniques are most efficient when employing a systematic form and methods for developing systematic codes are detailed. Bounds on the ABFT overhead computations are given and ABFT protection methods for processing that contains feedback are outlined. Analyzing syndromes' variances guide the selection of thresholds for syndrome comparisons. Simulations demonstrate the detection and miss probabilities for some high-rate wavelet codes.

A Comparison Study of Transcription Factor – DNA Binding Models

The comparison study is drawn between two widely used motif representations i.e. Positional Weight Matrices (PWM) and Consensus Sequences. In the case of motif finding, where the binding sites are not known a priori but the algorithm must search a large space of possible binding sites, the PWM model may be difficult to learn as the search space is very large even for the PWM of short length (RN for a PWM of length N, where R is the space of real numbers between 0 to 1). Optimization methods used to search for the best PWM may converge to a local minimum. On the other hand the consensus sequence has a smaller search space (15N for a motif of length N) which is easier to search for the global optimum.

MANDELBROT-LIKE BIFURCATION STRUCTURES IN CHAOS BAND SCENARIO OF SWITCHED CONVERTER WITH DELAYED-PWM CONTROL

In this paper, we report Mandelbrot-like bifurcation structures in a one-dimensional parameter space of real numbers corresponding to a dc-dc power converter modeled as a piecewise-smooth system with three zones. These fractal patterns have been studied in two-dimensional parameter space for smooth systems, but for nonsmooth systems has not been reported yet. The Mandelbrot-like sets we found are created in transition from the torus band to chaos band scenarios exhibited by a dc-dc buck power converter controlled by Delayed Pulse-Width Modulator (PWM) based on Zero Average Dynamics (or ZAD strategy), which corresponds to a piecewise-smooth system (PWS). The real parameter is provided by the PWM control strategy, namely ZAD strategy, and it can be varied in a large range, ideally (-∞, +∞). At -∞ and +∞ the dynamical behavior is the same, and thus we will describe the synamics in an ring-like parameter space. Mandelbrot-like borders are built by four chaotic bands, therefore these structures can be thought as instability islands where the state variables cannot be located. Using the Poincaré map approach we characterize the bifurcation structures and we describe recurrent patterns in different scales.

Bounded variation double sequence space of fuzzy real numbers

In this paper we have introduced the notion of fuzzy real-valued bounded variation double sequence space b 2 v F . We have studied some of its properties like convergence free, solidness, symmetricity, monotonicity, etc. We have proved some inclusion results too.

A note on the sendograph metric of fuzzy numbers

In this paper, by studying the attainable properties of sendograph metric in fuzzy number space, we generalize Monotone convergence theorem and Nested theorem of intervals from the space of real numbers to the space of fuzzy numbers.

When does [formula omitted] hold?

A map f : X → Y between topological spaces is called scatteredly continuous if for each non-empty subspace A ⊂ X the restriction f | A has a point of continuity. By S C ( X ) we denote the set of all scatteredly continuous maps from X to the space of real numbers R . We consider the following problem: What conditions must satisfy space X so that S C ( X ) = R X ?

Simulation of electron mirrors by the differential algebraic method

A differential algebraic (DA) method has been developed for the aberration analysis of electron mirrors, and a software package MIRROR_DA has been developed. Time is used instead of axial position as our independent variable, and a reference ray is also employed. A single DA ray is traced in the fields of the mirror system, performed on a non-standard extension of real number space called nDv, and all aberrations coefficients can be obtained simultaneously, in principle up to arbitrary order and with extremely high accuracy. To demonstrate our software, an example of tetrode mirror is presented. From the results by MIRROR_DA, it is shown that when adjusting the voltages of two middle electrodes, Cs and Cc change signs from positive to negative, and their values vary in a wide range: therefore this tetrode mirror can be used to cancel Cs and Cc of round lenses.

The limit theorems for random walk with state space ℝ in a space-time random environment

We consider a discrete time random walk on real number space in a space-time random environment. We state that when the random environment is i.i.d., under the marginal annealed law, the law of large numbers, iterated law and CLT of the process are correct. Using a martingale approach, we also state an a.s. invariance principle for random walks in general random environment whose hypothesis requires a subdiffusive bound on the variance of the quenched mean, under an ergodic invariant measure for the environment chain.