mohamed amsaad

University of Benghazi, Mathematics, Faculty Member

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Mohamed Amsaad received his B.Sc. and M.Sc. degrees in Mathematical Sciences in June of 2002 and 2006, respectively, from the University of Benghazi in Libya. He has been appointed as Assistant Lecturer at the Department of Mathematics and Computer Science at Qaminis College of Education in which he also served as Department Chair. In January 2010, Mr. Amsaad has joined the Department of Mathematics at West Virginia University in the USA as a Graduate Student. After earned his second M.Sc. degree in Applied Mathematics, Mr. Amsaad completed all the requirements towards the Doctoral of Philosophy degree in Mathematical Sciences with an emphasis in the Applied Analysis area in May 2016. Mr. Amsaad has been supported by the Department of Mathematics at WVU and granted assistantship on different occasions on both positions of Teaching Assistant and Research Assistant. Throughout his study for the Master and Ph.D. programs, he has researched and covered a wide spectrum of topics in Mathematical Sciences, e.g., Painlevé Analysis and its application, Dynamical Systems, Probability and Mathematical Finance, Stochastic Differential Equations, Partial Differential Equations, Fluid Dynamics, and Optimal Transport Theory taught by distinguished professors in these fields. His interests include Applied Mathematics especially in the area of Applied Analysis such as Differential Equations, Conservation Laws, Oscillations, Numerical Analysis, Fluid Dynamics, Monge-Kantorovich Optimal Mass Transportation Theory and applications to PDE’s, Lagrangian and Eulerian description of Fluid Flow, Mathematical Theory of Finance, and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations.

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On the Lagrangian description of absolutely continuous curves in the Wasserstein space on the line; well-posedness for the Continuity Equation

Indiana University Mathematics Journal, 2015

The Lagrangian description of absolutely continuous curves of probability measures on the real li... more The Lagrangian description of absolutely continuous curves of probability measures on the real line is analyzed. Whereas each such curve admits a Lagrangian description as a well-defined flow of its velocity field, further conditions on the curve and/or its velocity are necessary for uniqueness. We identify two seemingly unrelated such conditions that ensure that the only flow map associated to the curve consists of a time-independent rearrangement of the generalized inverses of the cumulative distribution functions of the measures on the curve. At the same time, our methods of proof yield uniqueness within a certain class for the curve associated to a given velocity, i.e. they provide uniqueness for the solution of the continuity equation within a certain class of curves.

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Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line

Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line Moham... more Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line Mohamed H Amsaad Doctor Of Philosophy in Mathematics throughout my life. Finally, I gratefully acknowledge support from CBIE grant No. 1515 as well as funds from the State of WV and Libyan Ministry of Higher Education. vi Contents Approval Page 2 Acknowledgments v List of Figures ix Notation x Preface It is proved in the book [17] of Ambrosio, Gigli and Savaré that, if 1 < p < +∞, then any such curve admits a unique Borel velocity field v : (0, T) × R d → R d such that (μ, v) satisfies (CE) in the sense of distributions in (0, T) × R d , and v(t, •) ∈ L p (μ(t, •); R d) for a.e. t ∈ (0, T), (0, T) t → v(t, •) L p (μ(t,•);R d) ∈ L q (0, T). This v is called the velocity of minimal norm associated to to the path μ, as it minimizes w(t, •) q L p (μ(t,•);R d) = |μ | q (t) for L 1-a.e. t ∈ (0, T) among all Borel maps w : (0, T) × R d → R d that pair up with μ to satisfy the multidimensional continuity equation above. Moreover, Nguyen and Tudorascu [92] showed that this minimal norm assumption is, in fact, redundant in the case d = 1. They proved that, this "velocity of minimal norm" is unique, in the sense that if v 1 and v 2 satisfy the assumptions above for a.e. t ∈ (0, T), then v 1 (t, •) ≡ v 2 (t, •) μ t-a.e.. They called v a velocity associated to the path μ. Furthermore, within the one-dimensional setting d = 1, the following is true [115], [116]: suppose μ i ∈ P p (R), i = 1, 2 and let M i : I → R be the unique a.e. monotone nondecreasing maps such that M i# χ = μ i , respectively, where χ is the onedimensional Lebesgue measure restricted to I := (0, 1). Then W p (μ 1 , μ 2) = M 1 − M 2 L p (I) , and there is only one Borel velocity v : (0, T) × R → R satisfying (CE), so the minimality of the L p (μ t)-norm as a selection principle is unnecessary here. The most general problem (Flow) discussed here in this thesis assumes only Borel regularity on v and we only study solutions X belonging to some time-Sobolev spaces W 1,q (0, T ; L p (I)) for 1 ≤ q ≤ +∞ and 1 ≤ p < +∞. The reason for this extra-requirement on the object defined in Definition 5.1 will become clear in Chapter 5, where we prove that any map X as in Definition 5.1 which also satisfies X t# χ = μ t for all t ∈ [0, T ] for some μ ∈ AC q (0, T ; P p (R)

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On the Lagrangian description of absolutely continuous curves in the Wasserstein space on the line; well-posedness for the Continuity Equation

Indiana University Mathematics Journal, 2015

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Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line

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