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Coherent upper conditional expectations are introduced through fractal outer measures to consider conditioning events that have zero probability with respect to the initial probability, as fractal sets. The model can be applied when the probability is concentrated on sets with zero Lebesgue measure and the density does not exist. It can be applied to provide a probabilistic representation of quantum states in cases where the density probability is absent. However, when the density probability exists, the two probabilistic representations align. This new approach can have intriguing data analytics applications, ranging from addressing probabilistic challenges in quantum state representation in quantum physics and managing extreme events in financial markets to capturing rare occurrences with significant implications in biomedical research. [ABSTRACT FROM AUTHOR]

Let ν be a Borel probability measure on ℝ d {\mathbb{R}^{d}} and q , t ∈ ℝ {q,t\in\mathbb{R}} . This study takes a broad approach to the multifractal and fractal analysis problem and proposes an intrinsic definition of the general Hausdorff and packing measures by taking into account sums of the type ∑ i h - 1 ⁢ ( q ⁢ h ⁢ ( ν ⁢ ( B ⁢ ( x i , r i ) ) ) + t ⁢ g ⁢ ( r i ) ) \sum_{i}h^{-1}(qh(\nu(B(x_{i},r_{i})))+tg(r_{i})) for some prescribed functions h and g. The aim of this paper is to study the descriptive set-theoretic complexity and measurability of these measures and related dimension maps. [ABSTRACT FROM AUTHOR]

Recently, many number of people various countries contracted the COVID-19 (C-19) disease. The identification of this harmful disease and other pneumonia diseases is an important process in the medical world. CT-Scan is mainly prescribed to predict the severity of lung diseases affected by C-19 and other pneumonia diseases. Generally, a CT-Scan image is represented as a grayscale image (GI). Biomedical-oriented GIs are more complex and they are too difficult to recognize the status of lung diseases directly from the experimental images. To address this type of medical problem, a multifractal approach can be applied to analyze and illustrate the GIs in detail. Therefore, the multifractal dimensional analysis is used to diagnose and explore the vehemence of the contamination levels in the lungs. In this study, the complexity of CT-Scan images of C-19 patients is analyzed in order to perform a comparison in terms of age and noise levels. It was also discovered that the intricate of CT-Scan images is considerably varied for patients aged 50 and above when compared to younger subjects. This comparative study indicates that the deadly virus affects elderly persons compared to youngsters. The proposed age-based classification is supported by image processing techniques, qualitative measures, and statistical tools. The obtained results are demonstrated graphically by Generalized Fractal Dimensions (GFD) spectrum, 3D visualization, ANOVA table, and box plots to expose the rate of discrimination of complexity in CT-Scan lung images. [ABSTRACT FROM AUTHOR]

This study provides a general multifractal formalism that overcomes the limitations of the traditional one. The generic Hewitt-Stromberg measures are used to introduce and study a multifractal formalism. The generic Hewitt-Stromberg dimensions' upper and lower bounds are estimated, producing results even at places q where the upper and lower multifractal Hewitt-Stromberg dimension functions diverge. [ABSTRACT FROM AUTHOR]

There are only two kinds of measures in which the mixed multifractal formalism applies, which are self-similar and self-conformal measures. This paper studies the validity and non-validity of the mixed multifractal formalism of other kinds of measures, called irregular/homogeneous Moran measures. [ABSTRACT FROM AUTHOR]

Sets of infinite multifractal measures are awkward to work with, and reducing them to sets of positive finite multifractal measures is a very useful simplification. The aim of this paper is to show that the multifractal Hausdorff measures satisfy the "subset of positive and finite measure" property. We apply our main result to prove that the multifractal function dimension is defined as the supremum over the multifractal dimension of all Borel probability measures. [ABSTRACT FROM AUTHOR]

In this work, we focus on the centered Hausdorff measure, the packing measure, and the Hewitt–Stromberg measure that determines the modified lower box dimension Moran fractal sets. The equivalence of these measures for a class of Moran is shown by having a strong separation condition. We give a sufficient condition for the equality of the Hewitt–Stromberg dimension, Hausdorff dimension, and packing dimensions. As an application, we obtain some relevant conclusions about the Hewitt–Stromberg measures and dimensions of the image measure of a τ -invariant ergodic Borel probability measures. Moreover, we give some statistical interpretation to dimensions and corresponding geometrical measures. [ABSTRACT FROM AUTHOR]

The multifractal formalism for vector-valued measures holds whenever the existence of corresponding Gibbs-like measures, supported on the singularities sets holds. We tried through this article to improve a result developed by Menceur et al. in [29] and to suggest a new sufficient condition for a valid mixed multifractal formalism for vectorvalued measures. We describe a necessary condition of validity for the formalism which is very close to the sufficient one. [ABSTRACT FROM AUTHOR]

We are interested in the behavior of relative multifractal box-dimensions, density results and multifractal dimensions under projections onto a lower-dimensional linear subspace. The results in this paper establish the connections with various multifractal and fractal dimensions and their projections, and generalize many known results about Hausdorff and packing measures and dimensions of a subset of ℝn. [ABSTRACT FROM AUTHOR]

Consider (Y , ρ) as a complete metric space and S as the space of probability Borel measures on Y. Let dim ‾ B Ψ , Φ (E) be the general upper box dimension of the set E ⊂ Y. We begin by proving that the general packing dimension of the typical compact set, in the sense of the Baire category, is at least inf ⁡ { dim ‾ B Ψ , Φ (B (x , r)) | x ∈ Y , r > 0 } where B (x , r) is the closed ball in Y with center at x and radii r > 0. Next, we obtain some estimates of the general upper and lower box dimensions of typical measures in the sense of the Baire category. Finally, we demonstrate that if S is equipped with the weak topology and under some assumptions then the set of measures possessing the general upper and lower correlation dimension zero are residual. Furthermore, the general upper correlation dimension of typical measures (in the sense of the Baire category) is approximated through the general local lower and upper entropy dimensions of Y. [ABSTRACT FROM AUTHOR]

This paper studies the behavior of the lower and upper multifractal Hewitt–Stromberg functions under slices onto (n - m) {(n-m)} -dimensional subspaces. More precisely, we discuss the relationship between the multifractal Hewitt–Stromberg functions of a compactly supported Borel probability measure and those of slices or sections of the measure. In addition, we prove that if μ has a finite m-energy and q lies in a certain somewhat restricted interval, then these functions satisfy the expected adding of co-dimensions formula. [ABSTRACT FROM AUTHOR]

In this paper. we intend to estimate the multifractal Hausdorff and the packing dimensions of product measures. The results in this paper establish the connections with various multifractal dimensions of measures on Rn. and generalize many known results about Hausdorff and packing dimensions of measures. [ABSTRACT FROM AUTHOR]

The concept of dimension is an important task in geometry. It permits a description of the growth process of objects. It may be seen as an invariant measure characterizing the object. Fractal dimensions are a kind of invariants permitting essentially to describe the irregularity hidden in irregular objects, by providing a suitable growth law. Among fractal geometrical objects, Moran's types play an important role in explaining many situations, in pure mathematics as the general context of Cantor's, and in applied physics as a suitable context for studying scaling laws. In the present paper, some non-regular homogeneous Moran measures are investigated, by establishing some new sufficient conditions permitting an explicit computation of the relative multifractal dimensions of the level sets for which the classical formulation does not hold. Besides, the mutual singularity of the relative multifractal measures for the homogeneous Moran case with different multifractal dimensions is investigated. This is very important, as in quasi-all existing situations, the validity of the multifractal formalism passes through the equality of the multifractal Hausdorff dimension with the packing one. [ABSTRACT FROM AUTHOR]

In studying physical systems, it is usually convenient to consider their dimensions. In the classical sense, this turns around the dimension of the Euclidean space where the variables live. Next, with the discovery of non-Euclidean geometry, hidden structures, and with the technological developments, the concept of dimension have been extended to fractal cases such as Billingsley and topological ones and which are also kinds of invariants permitting to describe the irregularity hidden in irregular objects via growth laws. In the present paper, the main purpose was to extend the concept of fractal dimension by introducing a variant of the Billingsley dimension called the ϕ -topological Billingsley dimension, relative to a non-negative function ϕ defined on a collection of subsets of a metric space. Some connections with the topological and Hausdorff dimensions have been also discussed on the basis of the well-known self-similar Sierpiński carpet. Besides, a class of functions has been provided, for which the computation of the new dimension is possible, and where the equality holds for the upper and lower bounds of the ϕ -topological Billingsley dimension. [ABSTRACT FROM AUTHOR]

The aim of this article is to study the behavior of the multifractal packing function under slices in Euclidean space. We discuss the relationship between the multifractal packing and pre-packing functions of a compactly supported Borel probability measure and those of slices or sections of the measure. More specifically, we prove that if μ satisfies a certain technical condition and q lies in a certain somewhat restricted interval, then Olsen's multifractal dimensions satisfy the expected adding of co-dimensions formula. [ABSTRACT FROM AUTHOR]

We are interested in the behavior of Billingsley dimensions under projections onto a lower dimensional linear subspace. The results in this paper establish the connections with various dimensions of subsets E of ℝ n and their projections, and generalize many known results about Hausdorff and packing dimensions of projections of E. In particular, we improve, through these results, one of the main theorems of Selmi et al. in [Multifractal variation for projections of measures, Chaos Solitons Fractals  91 (2016) 414–420] and treat an unsolved case in their work. [ABSTRACT FROM AUTHOR]

In the present paper, we suggest new proofs of many known results about the relative multifractal formalism. We provide results even at points q for which the relative multifractal Hausdorff and packing functions differ. We also give some examples of two measures where the multifractal functions are different and the Hausdorff dimension of the level sets of the v-local Holder exponent is given by the Legendre transform of the multifractal Hausdorff function and their packing dimension by the Legendre transform of the multifractal packing function. [ABSTRACT FROM AUTHOR]

In the present work, we are concerned with some multifractal dimensions estimations of vector-valued measures in the framework of the so-called mixed multifractal analysis. We precisely consider some Borel probability measures that are no longer Gibbs and introduce some mixed multifractal generalizations of Hausdorff and packing dimensions of measures in a framework of relative mixed multifractal analysis. As an application, we are interested in the '-unidimensionality of those measures and to the calculus of its mixed multifractal Hausdorff and packing dimensions. In particular, we give a necessary and sufficient condition for the existence of the '-mixed multifractal Hausdorff and packing dimensions of a Borel probability measure. Finally, concrete examples satisfying the above property are developed. [ABSTRACT FROM AUTHOR]

Conditional aggregation operators are defined by the Choquet integral and the Sugeno integral with respect to a monotone set function that assesses positive measure of the conditioning set. General Hausdorff and packing measures are introduced and examples of infinite s-sets with positive and finite generalized Hausdorff and packing measures are constructed and their fractal dimensions are compared. Coherent upper conditional previsions on the linear space of all Choquet integrable random variables are defined by the Choquet integral with respect to the general Hausdorff and packing measures when the conditioning event has positive and finite generalized Hausdorff and packing measures in its respective fractal dimensions. Conditional aggregation operators are defined by the Sugeno integral with respect to general Hausdorff and packing measures on the class of all Sugeno integrable random variables. Actually, the general Hausdorff and packing dimensions are proven to be the Sugeno integral with respect to the Lebesgue measure of the general Hausdorff and packing measures respectively. [ABSTRACT FROM AUTHOR]

Consider a separable metric space (X , d) , and let (K (X) , d ˜) denote the space of non-empty compact subsets of X equipped with the Hausdorff metric. This paper aims to introduce and investigate the concepts of two general fractal dimensions and general dimensions within the framework of (K (X) , d ˜). In particular, we explore a relationship between the general fractal dimensions of a set Z of a self-similar sequence space and their counterparts in the space of compact subsets K (Z). [ABSTRACT FROM AUTHOR]

By introducing a novel expression for the Hausdorff and packing measures with respect to gauge functions, we establish a connection between different approaches to defining the exact Hausdorff and packing dimensions of measures. Furthermore, we define the upper and lower Hausdorff and packing dimensions of a Borel probability measure and demonstrate that these dimensions can be represented using an entropy formula. This paper also addresses an intriguing question raised multiple times regarding the structural representation of singular probability measures. We provide an affirmative response to this question by utilizing characteristic measures. [ABSTRACT FROM AUTHOR]