On A Sequence of Cantor Fractals

In this paper we discuss some topological and geometrical properties of terms in a sequence of Cantor fractals and the limit of the sequence in order to obtain an exact relation between positive real numbers and Hausdorff dimensions of fractals of Euclidean spaces

On Operator-valued Semicircular Random Variables

In this paper, we discuss some special properties of operator- valued semicircular random variables including representation of the Cauchy transform of a compactly supported probability measure in terms of their operator-valued Cauchy transforms and existence of nonzero discrete part of their associated distributions.Comment: 14 page

A Classification of Elements of Function Space F(R,R)F(\mathbb{R},\mathbb{R})

In this paper, we classify the function space of all real-valued functions on R denoted as F(R, R) into 28 distinct blocks. Each block contains elements that share common features in terms of the cardinality of their sets of continuity and differentiability. Alongside this classification, we introduce the concept of the Connection, which reveals a special relationship structure between four wellknown real-valued functions in real analysis: the Cantor function, Dirichlet function, the Thomae function, and the Weierstrass function. Despite the significance of this field, several perspectives remain unexplore

A Multilevel Analysis of the Contribution of Individual, Socioeconomic and Geographical Factors on Kindergarten Children’s Developmental Health: A Saskatchewan Province-Wide Study

In current literature of child public health, a growing number of studies has been dedicated to early childhood development with a focus on child developmental health measured via the teacher completed Early Development Instrument (EDI). Using multilevel modeling as the optimal statistical method to analyze hierarchical EDI data, this study determines the strength of the effect and significance of predictors of children’s 5 EDI outcomes, vulnerability, and the multiple vulnerability by taking into account the hierarchy present in its design. In addition, this study conducts an extensive epidemiological review of the risk factors associated with a child’s developmental health at each level of the hierarchy, at cross-levels of the hierarchy and their variations across different levels of the hierarchy. This cross-sectional study considered 9045 Saskatchewan children who were ages 4-8 years in the 2008-2009 school years. Individual child characteristics, EDI domains, and vulnerability data were collected by the Ministry of Education teachers in the provincial 2008 EDI project; neighborhood contextual Census data were compiled by SPHERU staff at the University of Saskatchewan. Multilevel linear and logistic models were used to analyze the data. According to the results, individual characteristics, such as being Aboriginal, an ESL learner, male, and being absent from school; neighborhood characteristics such as income inequality; and geographical characteristics such as living in a large city have negative effects on EDI scores and exacerbating the odds of vulnerability. Compounding effects of Aboriginal-special skills, large city-Aboriginal, and large city-neighborhood median income were positive on the above outcomes with considerable either significance or strength, while those of neighborhood income inequality-Aboriginal, and large city-neighborhood income inequality were negative with notable significance and strength. Furthermore, neighborhood contextual variables contribute to a considerable proportion of health outcome variations and the results associated with neighborhood income inequality give further evidence of the income inequality hypothesis. The findings of this study recommend provincial child public health policy makers’ extended attention to Aboriginal children, children with ESL status, those children living in neighborhoods with high income inequality and children from Regina

Some Results on the Distributions of Operator Valued Semicircular Random Variables

The operator-valued free central limit theorem and operator-valued semi-circular random variables were first introduced by D. Voiculescu in 1995 as operator-valued free analogues of the classical central limit theorem and normal random variables, respectively. In 2007, R. Speicher and others showed that the operator-valued Cauchy transform of the semicircular distribution satisfies a functional equation involving the variance of the semicircular distribution. In this thesis, we consider a non - commutative probability space (A;E_B;B) where in which A is a unital C -algebra, B is a C -subalgebra of A containing its unit and E_B A B is a conditional expectation. For a given B−valued self-adjoint semicircular random variable s > A with variance ; it is still an open question under what conditions the distribution of s has an atomic part. We provide a partial answer in terms of properties of when B is the algebra of n × n complex matrices. In addition, we show that for a given compactly supported probability measure its associated Cauchy transform can be represented in terms of the operator-valued Cauchy transforms of a sequence of finite dimensional matrix-valued semicircular random variables in two ways. Finally, we give another representation of its Cauchy transform in terms of operator-valued Cauchy transform of an in finite dimensional matrix-valued semicircular random variable

Cayley's hyperdeterminant: a combinatorial approach via representation theory

Cayley's hyperdeterminant is a homogeneous polynomial of degree 4 in the 8 entries of a 2 x 2 x 2 array. It is the simplest (nonconstant) polynomial which is invariant under changes of basis in three directions. We use elementary facts about representations of the 3-dimensional simple Lie algebra sl_2(C) to reduce the problem of finding the invariant polynomials for a 2 x 2 x 2 array to a combinatorial problem on the enumeration of 2 x 2 x 2 arrays with non-negative integer entries. We then apply results from linear algebra to obtain a new proof that Cayley's hyperdeterminant generates all the invariants. In the last section we show how this approach can be applied to general multidimensional arrays.Comment: 20 page

Semiotic Analysis of Religious Identity in Iranian Animations (Case Study: Semiotics of the Prophets' Animation)

Today, media representation has become a fundamental concept in cultural and media studies and plays an important and central role in analyzing media texts. On the other hand, religious identity as a form of various forms of social identity is influenced by the growing influence of modern media and cultural goods and products, including animations. The main purpose of this research is to study how religious identity is represented in Iranian animations. This research has been conducted in the cultural studies approach, with emphasis on Stuart Hall's theory of representation and using theories related to religion, in particular the Glock and Stark's theory. The statistical society includes Iranian animations that have been produced and broadcasted over the past years by the Islamic Republic of Iran, with a focus on religion and religious issues. Sampling in this research was carried out through non-random purposeful sampling and 3 parts of the animation of "Prophets"(Noah's ship, Cold Fire, Patient) were selected as the sample. To analyze the semiotics of animations, Barthes's narrative model and also Fisk and Hartley's triple analysis, have been used. The results of the research showed that the representation of religious identity in Prophets animation is associated with orientation, media frameworks and ideological functions. In these animations, various aspects of religious identity, such as ideological, ritualistic, experimental and consequential, are represented

The AL-Gaussian Distribution as the Descriptive Model for the Internal Proactive Inhibition in the Standard Stop Signal Task

Measurements of response inhibition components of reactive inhibition and proactive inhibition within the stop signal paradigm have been of special interest for researchers since the 1980s. While frequentist nonparametric and Bayesian parametric methods have been proposed to precisely estimate the entire distribution of reactive inhibition, quantified by stop signal reaction times(SSRT), there is no method yet in the stop-signal task literature to precisely estimate the entire distribution of proactive inhibition. We introduce an Asymmetric Laplace Gaussian (ALG) model to describe the distribution of proactive inhibition. The proposed method is based on two assumptions of independent trial type(go/stop) reaction times, and Ex-Gaussian (ExG) models for them. Results indicated that the four parametric, ALG model uniquely describes the proactive inhibition distribution and its key shape features; and, its hazard function is monotonically increasing as are its three parametric ExG components. In conclusion, both response inhibition components can be uniquely modeled via variations of the four parametric ALG model described with their associated similar distributional features.Comment: KEYWORDS Proactive Inhibition, Reaction Times, Ex-Gaussian, Asymmetric Laplace Gaussian, Bayesian Parametric Approach, Hazard functio

Stop Signal Reaction Times: New Estimations with Longitudinal, Bayesian and Time Series based Methods

The stop signal reaction times (SSRT), a measure of the unobserved latency of the stop signal process in Stop Signal Task (SST) has been a focal point of the estimation. It was theoretically formulated as constant and random variable indices. The first index has been formulated using the independent horse race model of go and stop trials by Gordon Logan in 1994. The second index was estimated by Eric Jan Wagenmaker and colleagues in 2012 using the Bayesian Parametric Approach (BPA) and Ex-Gaussian assumption for the underlying go reaction times (GORT), signal respond reaction times (SRRT) and SSRT. Both of the mentioned estimation methods of SSRT assume equal impact of the preceding trial type (go/stop) on the current stop trial for its measurement. In case of violation of this assumption, the appropriate estimations of SSRT are required to address the measurement error. In this dissertation, we estimate SSRT under violation of the assumption in three frequentist longitudinal, mixture Bayesian and time series based methods. The frequentist longitudinal estimation method considers two clusters of SST trials and introduces Mixture SSRT and Weighted SSRT as two new distinct indices of SSRT being asymptotically equivalent under special conditions. The Bayesian estimation method considers degenerate mixture Bayesian estimation of SSRT using the cluster type SSRT estimation with underlying Ex-Gaussian assumption for GORT, SRRT and SSRT. In our proposed method we use the Two Stage Bayesian Parametric Approach (TSBPA) with uninformative priors. We compare the new Bayesian Mixture estimation of SSRT with the current single estimation in stochastic order and discuss the results for various underlying assumptions in terms of types of distributions, priors and weights. Finally, the time series based estimation method assumes lognormal distributions for GORT and SRRT; and, applies state-space missing data EM algorithm on each subjects SST data encompassing the order of (go/stop) trials yielding the order SST data. Then, using the Logan 1994 formula for the ordered SST data, the state-space index of SSRT is calculated. In all three methods, our examples of empirical SST data and simulated data are investigated to compare the new index to the established ones.Ph.D

A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions