Resume

• 2003

  Doctor of Philosophy (Ph.D.)

  Mathematics

  Real Analysis Seminar

  McGill University

  3.87/4

• 2001

  Master of Science (MS)

  Mathematics

  Jordan university for science and technology

  91.7%

• 1997

  Bachelor of Science (BS)

  Deans honor list

  Mathematics

  Yarmouk university

  91.3%

• Algorithms

  University Teaching

  Java

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  Image Processing

  Convexity and matrix means

  In this article we present some mean inequalities for convex functions that lead to some generalized inequalities treating the arithmetic

  geometric and harmonic means for numbers and matrices. Our first main inequality will be\nfor the convex function f

  when and . Moreover

  when

  the inequality will be valid for operator convex functions.\n\nThen by selecting an appropriate convex function

  we obtain certain matrix inequalities. In particular

  we obtain several mixed mean inequalities for operators using real and operator convexity. Our discussion will lead to new multiplicative refinements and reverses of the Heinz and Hölder inequalities for matrices

  new and refined trace and determinant inequalities. The significance of this work is its general treatment

  where convexity is the only needed property.

  Convexity and matrix means

  We prove two probabilistic versions of Hardy's inequality using an argument suggested by Korner in \\cite{korner}.

  We prove that some inequalities

  which are considered to be generalizations of Hardy's inequality on the circle

  \ncan be modified and proved to be true for functions integrable on the real line.\n\nIn fact we would like to show that some constructions which were\nused to prove the Littlewood conjecture can be used similarly to\nproduce real Hardy-type inequalities.\n\nThis discussion will lead to many questions concerning the\nrelationship between Hardy-type inequalities on the circle and\nthose on the real line.

  Hardy Inequalities On The Real Line

  The study of the continuity of the farthest point mapping for uniquely remotal sets has been used extensively in the literature to prove the singletoness of such sets. In this article

  we show that the farthest point mapping is not continuous even if the set is remotal

  rather than being uniquely remotal. Consequently

  we obtain some generalizations of results concerning the singletoness of remotal sets. In particular

  it is proved that a compact set admitting a unique farthest point to its center is a singleton

  generalizing the well known result of Klee. Then Symmetric Remotal Sets are introduced

  and proved to be singletons if uniquely remotal.

  Uniquely remotal sets in Banach spaces

  Roshdi Khalil

  A well known open problem in approximation theory is whether a uniquely\nremotal set in a normed space is necessarily a singleton. In this article

  we introduce the concept\nof isolated remotal points

  and prove that a non singleton closed bounded set with an isolated remotal\npoint

  in any normed space

  cannot be uniquely remotal. In fact

  we prove

  in this article

  that unique remotality with respect to only one point guarantees the singletoness of the underlying set. Stronger results in Hilbert spaces\n will be presented generalizing previous well known results.

  A Study of Uniquely Remotal Sets

  In this article

  we prove that convex functions and log-convex functions obey certain general refinements that lead to several refinements and reverses of well known inequalities for matrices

  including Young’s inequality

  Heinz inequality

  the arithmetic-harmonic and the geometric-harmonic mean inequalities.

  Convex functions and means of matrices

  Mohammad Al Horani

  abdelrahman Yousef

  A New Definition of Fractional Derivative

  In this article

  we present multiple-term refinements of Young's inequality for real numbers and operators. In particular

  given a natural number $N$

  we find $N$ positive terms refining Young's inequality. Detailed properties of this refinement are studied leading

  to a new interesting mean.

  A complete refinement of Young's inequality

  Roshdi Khalil

  In this article we study the connection of remotal points

  extreme points\nand exposed points. Namely

  we prove that a uniquely remotal point is necessarily\nan exposed point but not vice versa. We give examples where some implications are\nnot valid and we propose some questions regarding the problem. Then

  we introduce\na new class of points that play the role of extreme points and prove a Krein-Milman\ntype Theorem.

  Remotal Points and a Krein-Milman Type Theorem

  We prove that some Hardy-type inequalities on the circle can be\nproved to be true on the real line. Namely

  we discuss the idea of getting Hardy inequalities on the\nreal line by the use of corresponding inequalities on the circle. In the last section

  we prove the truth of\na certain open problem under some restrictions.

  A Study Of The Real Hardy Inequality

  Invoking the Hermite-Hadamard inequalities for convex functions

  we present different weighted inequalities of the Heinz means

  and any such convex function.

  Integral Inequalities of the Heinz Means as Convex Functions

  We prove a certain type of inequalities concerning the integral of\nthe Fourier transform of a function integrable on the real line.

  Hardy-Type Inequalities On The Real Line

  Khalil

  R

  Proximinality in Operator Space

  Roshdi Khalil

  AbdulRahman Yousef

  In an attempt to solve the Invariant Subspace Problem

  we introduce a certain orthonormal basis of Hilbert spaces

  and prove that a bounded linear operator on a Hilbert space must have an invariant subspace once this basis fulfills certain conditions. Ultimately

  this basis is used to show that every bounded linear operator on a Hilbert space is the sum of a shift and an upper triangular operators

  each of which having an invariant subspace.\n

  On The Invariant Subsace Problem

  In this article we interpolate the well known Young and Heinz inequalities for unitarily invariant norms

  and some of their known refinements. Then we prove new interpolated refinements. In the end

  we use this interpolation idea to prove a hidden monotonicity behavior these inequalities obey.

  Interpolated Inequalities for Unitarily Invariant Norms

  Mustafa Hayajneh

  Saja Hayajneh

  Roshdi Khalil

  Journal of Concrete and Applied Mathematics

  In this article

  we discuss the problem of remotality of exposed points of bounded sets in certain Banach spaces. Indeed

  we present a full characterization of a class of exposed points that are remotal points.

  Remotality of exposed points

  Recent refinements of Young's inequality can be though of certain ratios. In this article

  we present this point of view and prove the relationships between the different ratios induced by the different refinements.

  Inequalities related to the arithmetic

  geometric and harmonic means

  Roshdi Khalil

  Let $X$ \\ be a Banach space and $E$ be a closed bounded subset of $X.$ For $%\nx\\in X$ we set $D(x

  E)=\\sup \\{\\left\\Vert x-e\\right\\Vert :e\\in E\\}.$ The set $%\nE$ \\ is called remotal in $X$ \\ if for any $x\\in X$

  there exists $e\\in E$ \\\nsuch that $D(x

  E)=\\left\\Vert x-e\\right\\Vert .$ It is the object of this\npaper to give new results on remotal sets in $L^{p}(I

  X)

  $ and to simplify\nthe proofs of some results in \\cite{khalil}.

  We discuss a certain generalization of Hardy's inequality concerning the\nFourier coefficients of functions integrable on the circle. More specifically

  \nwe examine a result proved by Ivo Klemes (Klemes

  1993) and treat his construction successively\nin order to get a bounded function with certain equality-properties

  rather than\nhaving inequality-properties. These properties then are used to prove a similar form\nof Klemes' result but allowing gaps in the spectrum of the function. This new form of Klemes'\ninequality happens to be a good generalization of McGehee's inequality (Mcgehee

  1981) which\nis a generalization of the original Hardy's inequality.

  Hardy's Inequality and Bounded Linear Functionals

  Roshdi Khalil

  A set $E$ in a Banach space $X$ is called remotal if for any $x\\in X

  $ there\nexists an $e\\in E$ such that $\\left\\Vert x-e\\right\\Vert =\\sup \\{\\Vert\nx-e\\Vert :e\\in E\\}.$If $e$ is unique

  $E$ is called uniquely remotal. One of\nthe main results of this paper is: a weakly closed bounded set $E$ in a\nreflexive Banach space is uniquely remotal if and only if the closed convex\nhull of $E$ is uniquely remotal.

  Sababheh

  Princess Sumaya University for Technology

  The University of Jordan

  University of Sharjah

  University of Sharjah

  Princess Sumaya University for Technology

  United Arab Emirates

  Associate Professor

  University of Sharjah

  Amman

  Assistant Professor

  The University of Jordan

  University of Sharjah

  Princess Sumaya University for Technology

  Amman

  Associate Professor

  Princess Sumaya University for Technology

  Assistant Professor

  Princess Sumaya University for Technology

  Princess Sumaya University for Technology

  Jordan

  Professor