arXiv subjects

Mohsen Soltanifar

Publications and source records attributed to Mohsen Soltanifar.

12 recordsLinked to original sources

arXiv2026

Descriptions of Cantor Sets: A Set-Theoretic Survey and Open Problems

This survey synthesizes the principal descriptive set-theoretic perspectives on deterministic Cantor sets on the real line and charts directions for future study. After recounting their historical genesis and compiling an up-to-date taxonomy, we review the Borel hierarchy and four hierarchically ordered representations-general, nested, iterated-function-system (IFS), and q-ary expansion-presented from the most general to the most specific set-theoretic description of deterministic Cantor sets. We then present explicit and recursive descriptions for two thin families of measure-zero Cantor sets and an augmented "tick" family of positive measure, respectively, showing that the classical middle-third set lies in the intersection of all three families of after-mentioned Cantor sets. The survey closes by isolating several open problems in four directions, aiming to provide mathematicians with a coherent platform for further descriptive set-theoretic investigations into Cantor-type sets on the real line.

math.CA↗

arXiv2025

A Classification of Elements of Sequence Space $Seq(\mathbb{R})$

The sequence space of all real-valued sequences, denoted $Seq(\mathbb{R})$, is typically investigated through the lens of infinite-dimensional vector spaces, utilizing Banach space norms or Schauder bases. This work proposes a complementary, constructive classification based instead on the asymptotic limit profile encoded by the pair $(\liminf a_n, \limsup a_n)$. We demonstrate that this perspective naturally partitions $Seq(\mathbb{R})$ into seven mutually disjoint macroscale blocks, covering behaviors from finite convergence to bounded and unbounded oscillation. For each block, we provide explicit closed-form representative sequences and establish that every constituent class possesses the cardinality of the continuum. Furthermore, we investigate the structural relationships between these blocks at two distinct levels of granularity. At the macroscale, we employ injective mappings to define an idealized connectivity graph, while at the microscale, we introduce a connection relation governed by the Hadamard (pointwise) product. This dual analysis reveals a rich directed graph structure where the block of finite convergent sequences functions both as the only block subspace and as a global attractor with no outgoing connections. Statistical comparisons between the idealized and realized adjacency matrices indicate that the pointwise product structure realizes approximately two-thirds of the theoretically possible macroscale relations. Ultimately, this partition-based framework endows the seemingly chaotic space $Seq(\mathbb{R})$ with a transparent, geometrically interpretable internal structure.

math.GM↗

arXiv2025

CMHSU: An R Statistical Software Package to Detect Mental Health Status, Substance Use Status, and their Concurrent Status in the North American Healthcare Administrative Databases

The concept of concurrent mental health and substance use (MHSU) and its detection in patients has garnered growing interest among psychiatrists and healthcare policymakers over the past four decades. Researchers have proposed various diagnostic methods, including the Data-Driven Diagnostic Method (DDDM), for the identification of MHSU. However, the absence of a standalone statistical software package to facilitate DDDM for large healthcare administrative databases has remained a significant gap. This paper introduces the R statistical software package CMHSU, available on the Comprehensive R Archive Network (CRAN), for the diagnosis of mental health (MH), substance use (SU), and their concurrent status (MHSU). The package implements DDDM using hospital and medical service physician visit counts along with maximum time span parameters for MH, SU, and MHSU diagnoses. A working example using a simulated real-world dataset is presented to examine various analytical aspects, including three key dimensions of MHSU detection based on the DDDM framework, as well as temporal analysis to demonstrate the package's application for healthcare policymakers. Additionally, the limitations of the CMHSU package and potential directions for its future extension are discussed.

stat.ME↗

arXiv2024

GenTwoArmsTrialSize: An R Statistical Software Package to estimate Generalized Two Arms Randomized Clinical Trial Sample Size

The precise calculation of sample sizes is a crucial aspect in the design of clinical trials particularly for pharmaceutical statisticians. While various R statistical software packages have been developed by researchers to estimate required sample sizes under different assumptions, there has been a notable absence of a standalone R statistical software package that allows researchers to comprehensively estimate sample sizes under generalized scenarios. This paper introduces the R statistical software package "GenTwoArmsTrialSize" available on the Comprehensive R Archive Network (CRAN), designed for estimating the required sample size in two-arm clinical trials. The package incorporates four endpoint types, two trial treatment designs, four types of hypothesis tests, as well as considerations for noncompliance and loss of follow-up, providing researchers with the capability to estimate sample sizes across 24 scenarios. To facilitate understanding of the estimation process and illuminate the impact of noncompliance and loss of follow-up on the size and variability of estimations, the paper includes four hypothetical examples and one applied example. The discussion encompasses the package's limitations and outlines directions for future extensions and improvements.

stat.ME↗

arXiv2023

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

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 unexplored

math.GM↗

arXiv2022

Some Results on the Additivity and Multiplication Order Preserving Properties of Stochastic Orders

In this paper, we consider the problem of order preservation under addition and multiplication operators over the vector space of univariate real-valued random variables. Consistent with the case of usual order over the real numbers-as constant random variables, we prove that the usual stochastic order is both additive and multiplicative. We additionally discuss the situation for nine extra univariate stochastic orders presenting counterexamples in some cases. The problem remains unsolved for other univariate stochastic orders.

math.PR↗

arXiv2021

The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals

In this paper, we present a second partial solution for the problem of cardinality calculation of the set of fractals for its subcategory of the random virtual ones. Consistent with the deterministic case, we show that for the given quantities of Hausdorff dimension and Lebesgue measure, there are aleph-two virtual random fractals with almost surely Hausdorff dimension of a bivariate function of them and the expected Lebesgue measure equal to the later one. The associated results for three other fractal dimensions are similar to the case given for the Hausdorff dimension. The problem remains unsolved for the case of non-Euclidean abstract fractal spaces.

math.PR↗

arXiv2021

A Generalization of the Hausdorff Dimension Theorem for Fractals

How many fractals exist in nature or the virtual world In this work, we partially answer the second question using Mandelbrots fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of beth-two of virtual fractals with a Hausdorff dimension of a bivariate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

math.DS↗

arXiv2021

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.

stat.ME↗

arXiv2021

A Bayesian Mixture Modelling of Stop Signal Reaction Time Distributions

The distribution of single Stop Signal Reaction Times (SSRT) in the stop signal task (SST) as a measurement of the latency of the unobservable stopping process has been modeled with a nonparametric method by Hans Colonius (1990) and with a Bayesian parametric method by Eric-Jan Wagenmakers and colleagues (2012). These methods assume equal impact of the preceding trial type (go/stop) in the SST trials on the SSRT distributional estimation without addressing the case of the violated assumption. This study presents the required model by considering two-state mixture model for the SSRT distribution. It then compares the Bayesian parametric single SSRT and mixture SSRT distributions in the usual stochastic order at the individual and the population level under the ex-Gaussian distributional format. It shows that compared to a single SSRT distribution, the mixture SSRT distribution is more diverse, more positively skewed, more leptokurtic, and larger in stochastic order. The size of the disparities in the results also depends on the choice of weights in the mixture SSRT distribution. This study confirms that mixture SSRT indices as a constant or distribution are significantly larger than their single SSRT counterparts in the related order. This offers a vital improvement in the SSRT estimations.

stat.ME↗

arXiv2011

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.

math.FA↗

arXiv2011

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.

math.RT↗