Mourad Ben Slimane

King Saud University

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Mourad Ben Slimane has a total of 24 co-authors. Here are the top co-authors by number of shared publications:

Moez Ben Abid
Ecole Supérieure des Sciences et Technologie de Hammam Sousse, Université de Sousse, Sousse, Tunisia
11 shared publications
Ines Ben Omrane
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
10 shared publications
Borhen Halouani
Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia
4 shared publications
Borhen Halouani
Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia
4 shared publications
Hnia Ben Braiek
Université de Tunis El Manar, Faculté des Sciences de Tunis, Laboratoire de Recherche Équations aux Dérivées Partielles et Applications, 2092 Tunis, Tunisia
3 shared publications
Ines Ben Omrane
Department of Mathematics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
3 shared publications

Latest Publications

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- 29 October 2026
- Published by MDPI AG in Fractal and Fractional
An Improved Fractal Analysis of Directional Densities for Borel Probability Measures
by Ines Ben Omrane, Abdulmajeed Alqasem, Mourad Ben Slimane, Hamdi Zorgati
We investigate pointwise rectangular anisotropic and directional densities for Borel probability measures with compact support. Contrary to traditional densities, these new notions have the advantage to weight the side lengths of local boxes differently and take into account long thin boxes. We employ similarities with statistical physics to extract interesting multifractal information using the Hausdorff dimension with respect to the coverings by rectangles with small enough Lebesgue measure (instead of requiring all side lengths to be small). We apply our analysis for invariant measures for some iterated function systems.
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- 29 October 2026
- Published by MDPI AG in Fractal and Fractional
Open Access
An Improved Fractal Analysis of Directional Densities for Borel Probability Measures
by Ines Ben Omrane, Abdulmajeed Alqasem, Mourad Ben Slimane, Hamdi Zorgati
We investigate pointwise rectangular anisotropic and directional densities for Borel probability measures with compact support. Contrary to traditional densities, these new notions have the advantage to weight the side lengths of local boxes differently and take into account long thin boxes. We employ similarities with statistical physics to extract interesting multifractal information using the Hausdorff dimension with respect to the coverings by rectangles with small enough Lebesgue measure (instead of requiring all side lengths to be small). We apply our analysis for invariant measures for some iterated function systems.
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Cited by 3
- 1 January 2026
- Published by World Scientific Pub Co Pte Ltd in Fractals
REMARKS ON THE INVARIANCE THEOREM OF GENERALIZED FRACTAL DIMENSIONS OF GRAPHS OF CONTINUOUS FUNCTIONS
by Binyan Yu, Bilel Selmi, Yongshun Liang, Mourad BEN Slimane
This paper establishes a comprehensive theory for the behavior of the generalized box, Hausdorff, and packing dimensions under composition of continuous functions. We develop two complementary approaches: a measure-theoretic framework based on energy integrals and potential theory for analyzing the generalized Hausdorff and packing dimensions, and a covering-based approach for the generalized box dimensions. Our main results characterize how generalized fractal dimensions of graphs of functions transform under composition with Lipschitz and bi-Lipschitz functions. For outer function variation, it has been shown that composition with Lipschitz functions does not increase dimensions, while bi-Lipschitz functions preserve dimensions exactly. For inner function variation, distinct results for different types of dimensions have been established. For the generalized box dimensions, Lipschitz inner functions yield dimensional variance, while bi-Lipschitz inner functions ensure invariance. For the generalized Hausdorff and packing dimensions, bi-Lipschitz inner functions are required for dimensional invariance. Applications include dimensional invariance for compositions with elementary functions such as power functions, roots, reciprocals, exponentials, and logarithms. Explicit examples of generalized fractal dimensions satisfying our framework have also been provided. Furthermore, we study the effect of the Hölder and counter-Hölder continuity on the generalized lower box dimension of graphs of functions, establishing an exact value [ ] for functions that are simultaneously [ ]-order Hölder and counter-Hölder continuous.
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- 1 January 2026
- Published by World Scientific Pub Co Pte Ltd in Fractals
REMARKS ON THE INVARIANCE THEOREM OF GENERALIZED FRACTAL DIMENSIONS OF GRAPHS OF CONTINUOUS FUNCTIONS
by Binyan Yu, Bilel Selmi, Yongshun Liang, Mourad BEN Slimane
This paper establishes a comprehensive theory for the behavior of the generalized box, Hausdorff, and packing dimensions under composition of continuous functions. We develop two complementary approaches: a measure-theoretic framework based on energy integrals and potential theory for analyzing the generalized Hausdorff and packing dimensions, and a covering-based approach for the generalized box dimensions. Our main results characterize how generalized fractal dimensions of graphs of functions transform under composition with Lipschitz and bi-Lipschitz functions. For outer function variation, it has been shown that composition with Lipschitz functions does not increase dimensions, while bi-Lipschitz functions preserve dimensions exactly. For inner function variation, distinct results for different types of dimensions have been established. For the generalized box dimensions, Lipschitz inner functions yield dimensional variance, while bi-Lipschitz inner functions ensure invariance. For the generalized Hausdorff and packing dimensions, bi-Lipschitz inner functions are required for dimensional invariance. Applications include dimensional invariance for compositions with elementary functions such as power functions, roots, reciprocals, exponentials, and logarithms. Explicit examples of generalized fractal dimensions satisfying our framework have also been provided. Furthermore, we study the effect of the Hölder and counter-Hölder continuity on the generalized lower box dimension of graphs of functions, establishing an exact value [ ] for functions that are simultaneously [ ]-order Hölder and counter-Hölder continuous.
Cited by 3
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Cited by 1
- 1 January 2026
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems - S
On joint profile multifractal formalisms
by Mourad Ben Slimane, Ines Ben Omrane, Moez Ben Abid, Imtithal Alzughaibi, Obaid Algahtani, Hamdi Zorgati
In this paper, we formulate joint profile formalisms based on statistics of histograms of joint multiscale quantities. These formalisms allow the simultaneous multifractal analysis of several pointwise exponents derived from collections of measures, functions, signals, or images. We first show that joint profiles improve the upper bounds for joint spectra of regularities that can be derived from joint Legendre spectra, and have the advantage to recover non-concave joint spectra. We illustrate these facts for some bivariate scale-lacunary wavelet series. These series are very important in applications since they mimic the typical behavior of sequences of independent random variables better suited to real-world data modeling. We also show the importance of synchronicity. Then, we focus on the setting of joint $ p_i $-wavelet leaders, $ 1 \leq i \leq I $, $ I \in \mathbb{N} $. We prove that the upper bounds are optimal since they become equalities for some multivariate functions. Finally, we prove generic related results in the sense of Baire's categories.
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- 1 January 2026
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems - S
On joint profile multifractal formalisms
by Mourad Ben Slimane, Ines Ben Omrane, Moez Ben Abid, Imtithal Alzughaibi, Obaid Algahtani, Hamdi Zorgati
In this paper, we formulate joint profile formalisms based on statistics of histograms of joint multiscale quantities. These formalisms allow the simultaneous multifractal analysis of several pointwise exponents derived from collections of measures, functions, signals, or images. We first show that joint profiles improve the upper bounds for joint spectra of regularities that can be derived from joint Legendre spectra, and have the advantage to recover non-concave joint spectra. We illustrate these facts for some bivariate scale-lacunary wavelet series. These series are very important in applications since they mimic the typical behavior of sequences of independent random variables better suited to real-world data modeling. We also show the importance of synchronicity. Then, we focus on the setting of joint $ p_i $-wavelet leaders, $ 1 \leq i \leq I $, $ I \in \mathbb{N} $. We prove that the upper bounds are optimal since they become equalities for some multivariate functions. Finally, we prove generic related results in the sense of Baire's categories.
Cited by 1
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Cited by 7
- 31 July 2024
- Published by World Scientific Pub Co Pte Ltd in Analysis and Applications
A new regularity criterion for the 3D nematic liquid crystal flows
by Ines Ben Omrane, Mourad Ben Slimane, Sadek Gala, Maria Alessandra Ragusa
The aim of this study was to establish a new regularity criterion for local-in-time smooth solutions for the 3D nematic liquid crystal flows in terms of the horizontal gradient of two horizontal velocity components in the framework of the homogeneous Besov [ ] space and one directional derivative of molecular orientations in the framework of the homogeneous Morrey–Campanato [ ] space. More precisely, we show that the smooth solution [ ] can be extended beyond [ ], provided that [ ]
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- 31 July 2024
- Published by World Scientific Pub Co Pte Ltd in Analysis and Applications
A new regularity criterion for the 3D nematic liquid crystal flows
by Ines Ben Omrane, Mourad Ben Slimane, Sadek Gala, Maria Alessandra Ragusa
The aim of this study was to establish a new regularity criterion for local-in-time smooth solutions for the 3D nematic liquid crystal flows in terms of the horizontal gradient of two horizontal velocity components in the framework of the homogeneous Besov [ ] space and one directional derivative of molecular orientations in the framework of the homogeneous Morrey–Campanato [ ] space. More precisely, we show that the smooth solution [ ] can be extended beyond [ ], provided that [ ]
Cited by 7
Open Access
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Cited by 1
- 13 November 2023
- Published by MDPI AG in Fractal and Fractional
On Mixed Fractional Lifting Oscillation Spaces
by Imtithal Alzughaibi, Mourad Ben Slimane, Obaid Algahtani
We introduce hyperbolic oscillation spaces and mixed fractional lifting oscillation spaces expressed in terms of hyperbolic wavelet leaders of multivariate signals on $R^{d}$ , with $d \geq 2$ . Contrary to Besov spaces and fractional Sobolev spaces with dominating mixed smoothness, the new spaces take into account the geometric disposition of the hyperbolic wavelet coefficients at each scale $(j_{1}, \dots, j_{d})$ , and are therefore suitable for a multifractal analysis of rectangular regularity. We prove that hyperbolic oscillation spaces are closely related to hyperbolic variation spaces, and consequently do not almost depend on the chosen hyperbolic wavelet basis. Therefore, the so-called rectangular multifractal analysis, related to hyperbolic oscillation spaces, is somehow ‘robust’, i.e., does not change if the analyzing wavelets were changed. We also study optimal relationships between hyperbolic and mixed fractional lifting oscillation spaces and Besov spaces with dominating mixed smoothness. In particular, we show that, for some indices, hyperbolic and mixed fractional lifting oscillation spaces are not always sharply imbedded between Besov spaces or fractional Sobolev spaces with dominating mixed smoothness, and thus are new spaces of a really different nature.
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- 13 November 2023
- Published by MDPI AG in Fractal and Fractional
Open Access
On Mixed Fractional Lifting Oscillation Spaces
by Imtithal Alzughaibi, Mourad Ben Slimane, Obaid Algahtani
We introduce hyperbolic oscillation spaces and mixed fractional lifting oscillation spaces expressed in terms of hyperbolic wavelet leaders of multivariate signals on $R^{d}$ , with $d \geq 2$ . Contrary to Besov spaces and fractional Sobolev spaces with dominating mixed smoothness, the new spaces take into account the geometric disposition of the hyperbolic wavelet coefficients at each scale $(j_{1}, \dots, j_{d})$ , and are therefore suitable for a multifractal analysis of rectangular regularity. We prove that hyperbolic oscillation spaces are closely related to hyperbolic variation spaces, and consequently do not almost depend on the chosen hyperbolic wavelet basis. Therefore, the so-called rectangular multifractal analysis, related to hyperbolic oscillation spaces, is somehow ‘robust’, i.e., does not change if the analyzing wavelets were changed. We also study optimal relationships between hyperbolic and mixed fractional lifting oscillation spaces and Besov spaces with dominating mixed smoothness. In particular, we show that, for some indices, hyperbolic and mixed fractional lifting oscillation spaces are not always sharply imbedded between Besov spaces or fractional Sobolev spaces with dominating mixed smoothness, and thus are new spaces of a really different nature.
Cited by 1
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Cited by 5
- 8 November 2021
- Published by Springer Nature in Journal of Fourier Analysis and Applications
Multifractal Analysis of Rectangular Pointwise Regularity with Hyperbolic Wavelet Bases
by Moez Ben Abid, Mourad Ben Slimane, Ines Ben Omrane, Maamoun Turkawi
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- 8 November 2021
- Published by Springer Nature in Journal of Fourier Analysis and Applications
Multifractal Analysis of Rectangular Pointwise Regularity with Hyperbolic Wavelet Bases
by Moez Ben Abid, Mourad Ben Slimane, Ines Ben Omrane, Maamoun Turkawi
Cited by 5
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Cited by 2
- 3 September 2021
- Published by Springer Nature in Acta Mathematica Hungarica
Dimension prints for continuous functions on the unit square
by M. Ben Abid, I. Ben Omrane, M. Ben Slimane, M. Turkawi
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- 3 September 2021
- Published by Springer Nature in Acta Mathematica Hungarica
Dimension prints for continuous functions on the unit square
by M. Ben Abid, I. Ben Omrane, M. Ben Slimane, M. Turkawi
Cited by 2
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Cited by 2
- 5 April 2021
- Published by Taylor & Francis in Applicable Analysis
Multivariate multifractal formalism for simultaneous pointwise (Tuipi)i regularities
by Mourad Ben Slimane, Moez Ben Abid, Ines Ben Omrane, Borhen Halouani
Recently, a multivariate multifractal analysis for pointwise regularities based on hierarchical multiresolution quantities was developed. General bounds between the Hausdorff dimension of the intersection of single fractal sets and that of the original sets were derived. Equalities were checked for some synthetic signals that include multiplicative cascades. In this paper, we focus on the setting supplied by simultaneous pointwise $(T_{u_{i}}^{p_{i}})_{i = 1, \dots, L}$ regularities. The $T_{u}^{p}$ regularity for $1 \leq p < \infty$ was first introduced in order to better study elliptic partial differential equations where the natural function space setting is $L^{p}$ or a Sobolev space which includes unbounded functions. We will prove that both corresponding multivariate multifractal formalism and equalities above hold Baire generically in a given product of Besov spaces $\prod_{i = 1, \dots, L} B_{t_{i}}^{s_{i}, r_{i}} (R^{d})$ , $s_{i}, t_{i}, r_{i} > 0, 1 \leq p_{i} \leq \infty$ such that $s_{i} - \frac{d}{t_{i}} > - \frac{d}{p_{i}}$ . We therefore extend previous results where only cases ( $p_{i} = \infty$ and $s_{i} - \frac{d}{t_{i}} > 0$ for all i) and ( $p_{i} = \infty$ , $s_{i} = \frac{d}{t_{i}}$ and $r_{i} \leq 1$ for all i) for simultaneous pointwise Hölder regularities have been proved.
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- 5 April 2021
- Published by Taylor & Francis in Applicable Analysis
Multivariate multifractal formalism for simultaneous pointwise (Tuipi)i regularities
by Mourad Ben Slimane, Moez Ben Abid, Ines Ben Omrane, Borhen Halouani
Recently, a multivariate multifractal analysis for pointwise regularities based on hierarchical multiresolution quantities was developed. General bounds between the Hausdorff dimension of the intersection of single fractal sets and that of the original sets were derived. Equalities were checked for some synthetic signals that include multiplicative cascades. In this paper, we focus on the setting supplied by simultaneous pointwise $(T_{u_{i}}^{p_{i}})_{i = 1, \dots, L}$ regularities. The $T_{u}^{p}$ regularity for $1 \leq p < \infty$ was first introduced in order to better study elliptic partial differential equations where the natural function space setting is $L^{p}$ or a Sobolev space which includes unbounded functions. We will prove that both corresponding multivariate multifractal formalism and equalities above hold Baire generically in a given product of Besov spaces $\prod_{i = 1, \dots, L} B_{t_{i}}^{s_{i}, r_{i}} (R^{d})$ , $s_{i}, t_{i}, r_{i} > 0, 1 \leq p_{i} \leq \infty$ such that $s_{i} - \frac{d}{t_{i}} > - \frac{d}{p_{i}}$ . We therefore extend previous results where only cases ( $p_{i} = \infty$ and $s_{i} - \frac{d}{t_{i}} > 0$ for all i) and ( $p_{i} = \infty$ , $s_{i} = \frac{d}{t_{i}}$ and $r_{i} \leq 1$ for all i) for simultaneous pointwise Hölder regularities have been proved.
Cited by 2
Open Access
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Cited by 4
- 17 July 2020
- Published by MDPI AG in Mathematics
Pointwise Rectangular Lipschitz Regularities for Fractional Brownian Sheets and Some Sierpinski Selfsimilar Functions
by Mourad Ben Slimane, Moez Ben Abid, Ines Ben Omrane, Mohamad Maamoun Turkawi
We consider pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities for continuous functions f on the unit cube $I^{2}$ in $R^{2}$ . Firstly, we provide characterizations by simple estimates on the decay rate of the coefficients (resp. leaders) of the expansion of f in the rectangular Schauder system, near the point considered. We deduce that pointwise rectangular Lipschitz regularity yields pointwise level coordinate axes Lipschitz regularities. As an application, we refine earlier results in Ayache et al. (Drap brownien fractionnaire. Potential Anal. 2002, 17, 31–43) and Kamont (On the fractional anisotropic Wiener field. Probab. Math. Statist. 1996, 16, 85–98), where uniform rectangular Lipschitz regularity of the trajectories of the fractional Brownian sheet over the total $I^{2}$ (or any cube) was considered. Actually, we prove that fractional Brownian sheets are pointwise rectangular and level coordinate axes monofractal. On the opposite, we construct a class of Sierpinski selfsimilar functions that are pointwise rectangular and level coordinate axes multifractal.
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- 17 July 2020
- Published by MDPI AG in Mathematics
Open Access
Pointwise Rectangular Lipschitz Regularities for Fractional Brownian Sheets and Some Sierpinski Selfsimilar Functions
by Mourad Ben Slimane, Moez Ben Abid, Ines Ben Omrane, Mohamad Maamoun Turkawi
We consider pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities for continuous functions f on the unit cube $I^{2}$ in $R^{2}$ . Firstly, we provide characterizations by simple estimates on the decay rate of the coefficients (resp. leaders) of the expansion of f in the rectangular Schauder system, near the point considered. We deduce that pointwise rectangular Lipschitz regularity yields pointwise level coordinate axes Lipschitz regularities. As an application, we refine earlier results in Ayache et al. (Drap brownien fractionnaire. Potential Anal. 2002, 17, 31–43) and Kamont (On the fractional anisotropic Wiener field. Probab. Math. Statist. 1996, 16, 85–98), where uniform rectangular Lipschitz regularity of the trajectories of the fractional Brownian sheet over the total $I^{2}$ (or any cube) was considered. Actually, we prove that fractional Brownian sheets are pointwise rectangular and level coordinate axes monofractal. On the opposite, we construct a class of Sierpinski selfsimilar functions that are pointwise rectangular and level coordinate axes multifractal.
Cited by 4
Open Access
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- 29 August 2019
- Published by Wiley in Journal of Function Spaces
On the Baire Generic Validity of the $t$ -Multifractal Formalism in Besov and Sobolev Spaces
by Moez Ben Abid, Mourad Ben Slimane, Ines Ben Omrane, Borhen Halouani
The $t -$ multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce the $t -$ spectrum of a function $f$ from the knowledge of the $(p, t) -$ oscillation exponent of $f$ . The $t -$ spectrum is the Hausdorff dimension of the set of points where $f$ has a given value of pointwise $L^{t}$ regularity. The $(p, t) -$ oscillation exponent is measured by determining to which oscillation spaces $O_{p, t}^{s}$ (defined in terms of wavelet coefficients) $f$ belongs. In this paper, we first prove embeddings between oscillation and Besov-Sobolev spaces. We deduce a general lower bound for the $(p, t) -$ oscillation exponent. We then show that this lower bound is actually equality generically, in the sense of Baire’s categories, in a given Sobolev or Besov space. We finally investigate the Baire generic validity of the $t -$ multifractal formalism.
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- 29 August 2019
- Published by Wiley in Journal of Function Spaces
Open Access
On the Baire Generic Validity of the $t$ -Multifractal Formalism in Besov and Sobolev Spaces
by Moez Ben Abid, Mourad Ben Slimane, Ines Ben Omrane, Borhen Halouani
The $t -$ multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce the $t -$ spectrum of a function $f$ from the knowledge of the $(p, t) -$ oscillation exponent of $f$ . The $t -$ spectrum is the Hausdorff dimension of the set of points where $f$ has a given value of pointwise $L^{t}$ regularity. The $(p, t) -$ oscillation exponent is measured by determining to which oscillation spaces $O_{p, t}^{s}$ (defined in terms of wavelet coefficients) $f$ belongs. In this paper, we first prove embeddings between oscillation and Besov-Sobolev spaces. We deduce a general lower bound for the $(p, t) -$ oscillation exponent. We then show that this lower bound is actually equality generically, in the sense of Baire’s categories, in a given Sobolev or Besov space. We finally investigate the Baire generic validity of the $t -$ multifractal formalism.

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