Women Math Day

King Fahd University of Petroleum and Minerals has recently opened its door to female education, a milestone in the history of the University. “Women Math Day” is the first platform of its kind organized by the Department of Mathematics. It aims to bring together women mathematicians from the Kingdom of Saudi Arabia and beyond to share new ideas and explore new directions with other scientists. One of the expected outcomes of this event is to foster collaboration between female students, faculty and researchers from KFUPM, the Kingdom of Saudi Arabia and the whole world in the field of Mathematics which is vital in science, engineering and economics sectors. This event will include two sessions, each session contains different topics from applied and pure mathematics branches to suite different research interests.

# | Time | Speaker | Presentation |
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Session I - Session Chair: M. Al-Shahrani |
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8:00 - 8:15 | Opening Remarks | ||

I.1 | 8:20 - 8:45 | Zainab Alsheekh Hussain | Rotation sets of dispersing billiards |

I.2 | 8:50 - 9:20 | Manal Alotaibi | Generalized multiscale finite element methods for the reduced model of Darcy flow in fractured porous media |

Session II - Session Chair: K. Furati |
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II.1 | 9:25 - 9:55 | Guanglian Li | Wavelet-based Edge Multiscale Parareal Algorithm for subdiffusion equations with heterogeneous coefficients in a large time domain |

II.2 | 10:00 - 10:25 | Faten Saeed Alamri | A benchmarking suite for geostatistical modeling and Kriging tools |

Session III - Session Chair: A. Laradji |
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III.1 | 10:30 - 10:55 | Eadah Ahmed Al Zahrani | Optimal control for some quasilinear singular elliptic systems |

III.2 | 11:00 - 11:25 | Faiza Shujat | Some differential identities on rings and algebras |

11:25 - 13:00 | Break | ||

Session IV - Session Chair: S.-E. Kabbaj |
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IV.1 | 13:00 - 13:30 | Milica Andelic | Moore-Penrose inverse of the signless Laplacians of bipartite graphs |

IV.2 | 13:35 - 14:00 | Ymnah Salah Alruwaily | Existence results for coupled nonlinear sequential FDEs with coupled Riemann-Stieltjes integro-multipoint boundary conditions |

Session V - Session Chair: B.-Mezerdi |
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V.1 | 14:05 - 14:30 | Mashael AlBaidani | Linear algebra applications in modern world |

14:30 - 14:50 | Break | ||

V.2 | 14:50 - 15:15 | Bana Jawid Al Subaiei | On Rickart-like theorem for non-bijective mappings |

Session VI - Session Chair: B. Smii |
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VI.1 | 15:20 - 15:45 | Weaam Alhejaili | Numerical studies of the Steklov eigenvalue problem via conformal mappings |

VI.2 | 15:50 - 16:15 | Mohra Zayed | Approximation of special monogenic functions by equivalent bases of polynomials in Fréchet modules |

VI.3 | 16:20 - 16:50 | Florentina Tone | Long time stability of the implicit Euler scheme for an incompressible two-phase flow model |

16:55 - 17:00 | Closing Remarks | ||

**Abstracts**

##### - Rotation sets of dispersing billiards

We study the billiard ow in the exterior of a compact subset K of R^{N} (N ≥ 3), where

K = K_{1} ∪ K_{2} ∪ ... ∪ K_{s}

is a union of pairwise disjoint compact and strictly convex bodies with smooth boundaries T_{i} = ∂K_{i}. Here, we do not assume the no-eclipse condition to be satisfied, and the billiard flow is considered to be a special case of dispersing billiards. We associate with it the map related to a starting point of a given billiard trajectory. In this talk, we give brief definitions of what is meant by dispersing billiards and rotation theory. Then, we present our main results and the methodologies we have used to obtain them.

##### - Generalized multiscale finite element methods for the reduced model of Darcy flow in fractured porous media

In this talk, we combine the generalized multiscale finite element method (GMsFEM) with a reduced model based on the discrete fracture model (DFM) to resolve the difficulties of simulating flow in fractured porous media while efficiently and accurately reducing the computational complexity resulted from resolving the fine scale effects of the fractures. The geometrical structure of the fractures is discretely resolved within the model using the DFM. The advantage of using GMsFEM is to represent the fracture effects on a coarse grid via multiscale basis functions constructed using local spectral problems. Solving local problems leads to consideration and usage of small scale information in each coarse grid. On another hand, the multiscale basis functions, generated following GMsFEM framework, are parameter independent and constructed once in what we call offline stage. These basis functions can be re-used for solving the problem for any input parameter when it is needed. Combining GMsFEM and DFM has been introduced in other works assuming continuous pressure across the fractures interface. This continuity is obtained when the fractures are much more permeable than that in the matrix domain. In this talk, we consider a general case for the permeability in both fracture and matrix domain using the reduced model presented in [1]. The proposed reduction technique has significant impact on enabling engineers and scientists to efficiently, accurately and inexpensively solve the large and complex system resulted from modeling ow in fractured porous media.

*[1]V. Martin, J. Jaffré and J. E. Roberts. Modeling fractures and barriers as interfaces for flow in porous media. SIAM Journal on Scientific Computing, 26(5):1667–1691, 2005.*

##### - Wavelet-based Edge Multiscale Parareal Algorithm for subdiffusion equations with heterogeneous coefficients in a large time domain

I will present in this talk the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm recently proposed in [2] to efficiently solve subdiffusion equations with heterogeneous coefficients in long time. This algorithm combines the advantages of multiscale methods that can effectively deal with heterogeneity in the spatial domain, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available. Compared with the previous work for parabolic problem, the main challenge in both the analysis and simulation arise from the nonlocality of the fractional derivative. To conquer this obstacle, an auxiliary problem is constructed on each coarse temporal subdomain to uncouple the temporal variable completely. In this manner, the approximation properties of the correction operator is proved. In addition, a new summation of exponential sums is derived to generate single-step time stepping scheme, with the number of terms of O(|log*τ _{f}*|) independent of final time. Here, O(|log

*τ*|) is the fine-scale time step size. We derive the convergence rate of this algorithm in terms of the mesh size in the spatial domain, the level parameter used in the multiscale method, the coarse-scale time step size and the fine-scale time step size. Several numerical tests are presented to demonstrate the performance of our algorithm, which verify our theoretical results perfectly.

_{f}*[1] Guanglian Li, Wavelet-based Edge Multiscale Parareal Algorithm for subdiffusion equations with heterogeneous coefficients in a large time domain, submitted, 2021.*

[2] Guanglian Li and Jiuhua Hu, Wavelet-based edge multiscale parareal algorithm for parabolic equations with heterogeneous coefficients and rough initial data, J. Comput. Phys, 444, 2021.

[2] Guanglian Li and Jiuhua Hu, Wavelet-based edge multiscale parareal algorithm for parabolic equations with heterogeneous coefficients and rough initial data, J. Comput. Phys, 444, 2021.

##### - A benchmarking suite for geostatistical modeling and Kriging tools

In this talk, we propose a benchmarking suite over ExaGeoStatR, an R package for large-scale geospatial data modeling on manycore systems, to assess existing modeling/kriging tools. The benchmarking suite relies on synthetic datasets generated by the ExaGeoStatR package and predefined assessment metrics. We evaluate five existing R packages in the literature, i.e., Fields, GpGp, GeoR, Gstat and ExaGeoStatR, to explore the internal structure of the proposed benchmarking suite using small-size datasets. Medium-size and large-size modeling/kriging tools are also evaluated to show the capabilities of the proposed benchmarking suite. *This is joint work with Sameh Abdulah, Kesen Wang, Ying Sun, Hatem Ltaief, David E. Keyes, Marc G. Genton, all six from KAUST.*

##### - Optimal control for some quasilinear singular elliptic systems

This talk is concerned with singular systems governed by elliptic operators. We first study the existence and uniqueness of the solution for the scalar case. Secondly, we discuss the (2x2) quasi-linear elliptic systems, then we study the optimal control for these systems.

##### - Some differential identities on rings and algebras

The purpose of this talk is to find an extension of the renowned Chernoff theorem [J. Funct. Anal., 2 (1973)] on standard operator algebra. In fact, we prove the following result: Let *H* be a real (or complex) Banach space and *L(H)* be the algebra of bounded linear operators on *H*. Let *A(H)* ⊂ *L(H)* be a standard operator algebra. Suppose that *D: A(H)* → *L(H)* is a linear mapping satisfying the relation *D(A ^{n}B^{n}) = D(A^{n})B^{n} + A^{n}D(B^{n})* for all

*A, B*∈

*A(H)*. Then

*D*is a linear derivation on

*A(H)*. In particular,

*D*is continuous. Moreover, we discuss similar kind of identities on generalized left derivations on rings and furnish an application in context with Banach algebra.

##### - Moore-Penrose inverse of the signless Laplacians of bipartite graphs

We provide a relation between the Moore-Penrose inverse of the Laplacian and signless Laplacian matrices of a bipartite graph. As a consequence, we present combinatorial formulae for the Moore-Penrose inverse of signless Laplacians of bipartite graphs. We also obtain a combinatorial formula for the Moore-Penrose inverse of an incidence matrix and derive a combinatorial formula for the inverse of signless Laplacians of non-bipartite graphs. These results answer some of the open problems raised in [R. Hessert, S. Mallik, Moore-Penrose inverses of the signless Laplacian and edge-Laplacian of graphs, Discrete Math. 344 (2021) #112451]. *This is joint work with Abdullah Alazemi and Osama Alhalabi, both from Kuwait University.*

##### - Existence results for coupled nonlinear sequential fractional differential equations with coupled Riemann-Stieltjes integro-multipoint boundary conditions

This talk is concerned with the existence of solutions for a fully coupled Riemann-Stieltjes integro-multipoint boundary value problem of Caputo type sequential fractional differential equations. The given system is studied with the aid of Leray-Schauder alternative and contraction mapping principle. A numerical example illustrating the abstract results is also presented.

##### - Linear algebra applications in modern world

This talk demonstrates the use of linear algebra concepts in real-life obligations by presenting linear algebra applications that have waiting to be uncovered or fully explored by mathematicians.

##### - On Rickart-like theorem for non-bijective mappings

Rickart [2] in 1948 proved that every bijective multiplicative mapping of a Boolean ring onto an arbitrary ring is additive. Al Subaiei and Jarboui in [1] have studied the analogous of Rickart for non-bijective mappings. They proved that a multiplicative map φ: *R* → *S*, where *R* is a Boolean ring with identity and *S* is a ring its characteristic is 2, such that *φ(x) + φ(1 _{R} - x) = φ(q_{R})* for any

*x*∈

*R*is additive. The results of this work will be presented in this talk.

*[1] Al Subaiei, B., Jarboui, N. A Rickart-Like Theorem for the Additivity of Multiplicative Maps on Rings. Journal of Mathematics 2022, Article ID 5052308.*

[2] Rickart, C. E. One-to-one mappings of rings and lattices. Bull. Amer. Math. Soc. 1948, 54, 758-764.

[2] Rickart, C. E. One-to-one mappings of rings and lattices. Bull. Amer. Math. Soc. 1948, 54, 758-764.

##### - Numerical studies of the Steklov eigenvalue problem via conformal mappings

A spectral method based on conformal mappings is proposed to solve Steklov eigenvalue problems and their related shape optimization problems in two dimensions. To apply spectral methods, we _rst reformulate the Steklov eigenvalue problem in the complex domain via conformal mappings. The eigenfunctions are expanded in Fourier series so the discretization leads to an eigenvalue problem for coefficients of Fourier series. For shape optimization problem, we use the gradient ascent approach to find the optimal domain which maximizes k-th Steklov eigenvalue with a fixed area for a given k. The coefficients of Fourier series of mapping functions from a unit circle to optimal domains are obtained for several different k. This is joint work with Chiu-Yen Kao from Claremont McKenna College, USA

##### Approximation of special monogenic functions by equivalent bases of polynomials in Fréchet modules

Intensive research efforts have been dedicated to the extension and development of important aspects of, and has resulted in, the theory of one complex variable to higher dimensional spaces. Indeed, Clifford analysis was created several decades ago to provide an elegant and powerful generalization of complex analysis. This talk introduces a new derived base of special monogenic polynomials (SMPs) in Fréchet Cliffordian modules, named the equivalent base. We examine the convergence properties of such a base for several cases according to certain conditions applied on the bases of its related constituent bases. Subsequently, we characterize its effectiveness in various convergence regions such as closed balls, open balls, at the origin, and for all entire special monogenic functions (SMFs). Moreover, the upper and lower bounds of the order of the equivalent base are determined, and proved to be attainable. We eventually discuss the *Τ _{p}*- property of the equivalent base.

##### - Long time stability of the implicit Euler scheme for an incompressible two-phase flow model

In this talk, we present results on the stability for all positive time of the fully implicit Euler scheme for an incompressible two-phase ow model. More precisely, we consider the time discretisation scheme and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prove that the numerical scheme is stable. *This is joint work with T. Tachim Medjo from FIU, USA. *

**Dr. Zainab Alsheekhhussain**

**Title of Presentation:**

Rotation sets of dispersing billiards

**Dr. Manal Alotaibi**

**Title of Presentation:**

Generalized multiscale finite element methods for the reduced model of Darcy flow in fractured porous media

**Dr. Guanglian Li**

**Title of Presentation:**

Wavelet-based Edge Multiscale Parareal Algorithm for subdiffusion equations with heterogeneous coefficients in a large time domain

**Dr. Faten Saeed Alamri**

**Title of Presentation:**

A benchmarking suite for geostatistical modeling and Kriging tools

**Dr. Eada Ahmed Al-zahrani**

**Title of Presentation:**

Optimal control for some quasilinear singular elliptic systems

**Dr. Faiza Shujat**

**Title of Presentation:**

Some differential identities on rings and algebras

**Dr. Milica Andelic**

**Title of Presentation:**

Moore-Penrose inverse of the signless Laplacians of bipartite graphs

**Dr. Ymnah Salah Alruwaily**

**Title of Presentation:**

Existence results for coupled nonlinear sequential fractional differential equations with coupled Riemann-Stieltjes integro-multipoint boundary conditions.

**Dr. Mashael AlBaidani**

**Title of Presentation:**

Linear algebra applications in modern world

**Dr. Bana Jawid Al Subaiei**

**Title of Presentation:**

On Rickart-like theorem for non-bijective mappings

**Dr. Weaam Alhejaili**

**Title of Presentation:**

Numerical studies of the Steklov eigenvalue problem via conformal mappings

**Dr. Mohra Zayed**

**Title of Presentation:**

Approximation of special monogenic functions by equivalent bases of polynomials in Fréchet modules

**Prof. Florentina Tone**

**Title of Presentation:**

Long time stability of the implicit Euler scheme for an incompressible two-phase flow model