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We compute a lower bound of Billingsley–Hausdorff dimension, defined by Gibbs measure, of the level set related to Birkhoff average in the countable symbolic space $\mathbb{N}^{\mathbb{N}}$.

This appendix gives a lower bound of the Billingsley-Hausdorff dimension of a level set related to Birkhoff average in the “non-compact” symbolic space $\mathbb{N}^{\mathbb{N}}$, defined by Gibbs measure.

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]

M. Das proved that the relative multifractal measures are mutually singular for the self-similar measures satisfying the significantly weaker open set condition. The aim of this paper is to show that these measures are mutually singular in a more general framework. As examples, we apply our main results to quasi-Bernoulli measures.

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]

This paper proves some results concerning the lower Hewitt-Stromberg dimension of projections. We give non-trivial lower bounds on the lower Hewitt-Stromberg dimension of the projections onto almost all $m$-dimensional subspaces. In addition, we estimate the dimension of the exceptional sets of $m$-dimensional subspaces where the dimensions of projection are smaller than the typical value. The condition where the quasi-Assouad dimension is no greater than $m$ shows that the lower Hewitt-Stromberg dimension is preserved under orthogonal projections for almost all $m$-dimensional subspaces. [ABSTRACT FROM AUTHOR]

The properties of various fractal and multifractal measures and dimensions have been under extensive study in the real-line and higher-dimensional Euclidean spaces. In non-Euclidean spaces, it is often impossible to construct non-trivial self-similar or self-conformal sets, etc. We consider in the present paper the proper way to phrase the definitions for use in general metric spaces. We investigate the relative Hausdorff measures and the relative packing measures defined in a separable metric space. We give some product inequalities which are a consequence of a new version of density theorems for these measures. Moreover, we prove that and can be expressed as Henstock-Thomson variation measures. The question of the weak-Vitali property arises in this context. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study average Hewitt-Stromberg and box dimensions of typical compact metric spaces belonging to the Gromov-Hausdorff space equipped with the Gromov-Hausdorff metric. [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]

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]