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Mortaza Abtahi

Associate Professor of Pure Mathematics

Selected Publications

Abtahi, M., Farhangi, S. Vector-valued spectra of Banach algebra valued continuous functions (2018) Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas, 112 (1), pp. 103-115.

DOI: 10.1007/s13398-016-0367-2

Let E be a commutative unital Banach algebra, X a compact Hausdorff space, and A⊂ C(X, E) a Banach E-valued function algebra. An E-valued spectrum E-SP(f) of every f∈ A is introduced and investigated, and it is shown that E-SP(f) can be determined by E-valued characters of A. For the so-called natural E-valued function algebras, such as C(X, E) and Lip (X, E) , we get E-SP(f)=f(X). When E= C, E-valued characters reduce to characters and E-valued spectra reduce to classical spectra. © 2016, Springer-Verlag Italia.

AUTHOR KEYWORDS: Algebras of continuous vector-valued functions; Commutative Banach algebras; Vector-valued characters; Vector-valued spectra
PUBLISHER: Springer-Verlag Italia s.r.l.

Abtahi, M., Kadelburg, Z., Radenović, S. Fixed points and coupled fixed points in partially ordered ν-generalized metric spaces (2018) Applied General Topology, 19 (2), pp. 189-201.

DOI: 10.4995/agt.2018.7409

New fixed point and coupled fixed point theorems in partially ordered ν-generalized metric spaces are presented. Since the product of two ν-generalized metric spaces is not in general a ν-generalized metric space, a different approach is needed than in the case of standard metric spaces. © AGT, UPV, 2018.

AUTHOR KEYWORDS: Coupled fixed point theorems; Meir-Keeler contractions; Proinov-type contractions; Ćirić-matkowski contractions; ν-generalized metric space
PUBLISHER: Universitat Politecnica de Valencia

Abtahi, M., Farhangi, S. A Characterization of Polynomially Convex Sets in Banach Spaces (2017) Results in Mathematics, 72 (4), pp. 2013-2021.

DOI: 10.1007/s00025-017-0695-3

Let E be a Banach space and ▵∗ be the closed unit ball of the dual space E∗. For a compact set K in E, we prove that K is polynomially convex in E if and only if there exist a unital commutative Banach algebra A and a continuous function f:A such that (i) A is generated by f(▵∗), (ii) the character space of A is homeomorphic to K, and (iii) K=SP→(f) the joint spectrum of f. In case E= C(X) , where X is a compact Hausdorff space, we will see that ▵∗ can be replaced by X. © 2017, Springer International Publishing.

AUTHOR KEYWORDS: commutative banach algebras; joint spectrum; polynomially convexity; Polynomials on Banach spaces
PUBLISHER: Birkhauser Verlag AG

Abtahi, M. On the character space of Banach vector-valued function algebras (2017) Bulletin of the Iranian Mathematical Society, 43 (5), pp. 1195-1207.

Given a compact space X and a commutative Banach algebra A, the character spaces of A-valued function algebras on X are investigated. The class of natural A-valued function algebras, those whose characters can be described by means of characters of A and point evaluation homomorphisms, is introduced and studied. For an admissible Banach A-valued function algebra A on X, conditions under which the character space M(A) is homeomorphic to M(A) × M(A) are presented, where A = C(X) ∩ A is the subalgebra of A consisting of scalar-valued functions. An illustration of the results is given by some examples. © 2017 Iranian Mathematical Society.

AUTHOR KEYWORDS: Banach function algebras; Commutative banach algebras; Vector-valued characters; Vector-valued function algebras
PUBLISHER: Iranian Mathematical Society

Abtahi, M. Common fixed point theorems of Meir-Keeler type in metric spaces (2017) Fixed Point Theory, 18 (1), pp. 17-18.

DOI: 10.24193/fpt-ro.2017.1.02

This paper is a continuation of the recent work [M. Abtahi, Fixed point theorems for Meir-Keeler type contractions in metric spaces, Fixed Point Theory, 17(2016), No. 2, 225-236]. After establishing a criterion for sequences in metric spaces to be Cauchy, unified simple proofs for several known results on the existence of a common fixed point for compatible pairs of mappings of complete metric spaces satisfying a contractive condition of Meir-Keeler type are obtained. A very general common fixed point theorem, corresponding to the fixed point theorem of Proinov [P.D. Proinov, Fixed point theorems in metric spaces, Nonlinear Anal. 64 (2006) 546-557], is presented. Examples are given to support the results. © 2017, House of the Book of Science. All rights reserved.

AUTHOR KEYWORDS: Common fixed points; Compatible mappings; Complete metric spaces; Meir-Keeler contractions
PUBLISHER: House of the Book of Science

Abtahi, M., Kadelburg, Z., Radenović, S. Fixed points of Ćirić-Matkowski-type contractions in ν -generalized metric spaces (2017) Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas, 111 (1), pp. 57-64.

DOI: 10.1007/s13398-016-0275-5

In this paper, fixed point theorems for Ćirić-Matkowski-type contractions in ν-generalized metric spaces are presented. Then, by replacing the distance function d(x, y) with function of the form m(x, y) = d(x, y) + γ(d(x, Tx) + d(y, Ty)) , where γ> 0 , results analogue to those due to Proinov (Nonlinear Anal 64:546–557, 2006) are obtained. An example is provided to demonstrate a possible usage of these results. © 2016, Springer-Verlag Italia.

AUTHOR KEYWORDS: Cauchy sequence; Fixed point; Ćirić-Matkowski-type contraction; ν-Generalized metric space
PUBLISHER: Springer-Verlag Italia s.r.l.

Abtahi, M. Vector-valued characters on vector-valued function algebras (2016) Banach Journal of Mathematical Analysis, 10 (3), pp. 608-620.

DOI: 10.1215/17358787-3607486

Let A be a commutative unital Banach algebra and let X be a compact space. We study the class of A-valued function algebras on X as subalgebras of C(X;A) with certain properties. We introduce the notion of A-characters of an A-valued function algebra A(Script) as homomorphisms from A(Script) into A that basically have the same properties as evaluation homomorphisms εx: f 7(mapping) f(x), with x ∈ X. We show that, under certain conditions, there is a one-to-one correspondence between the set of A-characters of A(Script) and the set of characters of the function algebra A = A(Script) ∩ C(X) of all scalar-valued functions in A(Script) . For the so-called natural A-valued function algebras, such as C(X,A) and Lip(X,A), we show that εx (x ∈ X) are the only A-characters. Vector-valued characters are utilized to identify vector-valued spectra. © 2016 by the Tusi Mathematical Research Group.

AUTHOR KEYWORDS: Algebras of continuous vector-valued functions; Banach function algebras; Characters; Maximal ideals; Vector- valued function algebras
PUBLISHER: Duke University Press

Abtahi, M. Fixed point theorems for Meir-Keeler type contractions in metric spaces (2016) Fixed Point Theory, 17 (2), pp. 225-236.

We establish a simple and powerful lemma that provides a criterion for sequences in metric spaces to be Cauchy. Using the lemma, it is then easily verified that the Picard iterates {Tnx}, where T is a contraction or asymptotic contraction of Meir-Keeler type, are Cauchy sequences. As an application, new and simple proofs for several known results on the existence of a fixed point for continuous and asymptotically regular self-maps of complete metric spaces satisfying a contractive condition of Meir-Keeler type are derived. These results include the remarkable fixed point theorem of Proinov in [Petko D. Proinov, Fixed point theorems in metric spaces, Nonlinear Anal. 46 (2006) 546–557], the fixed point theorem of Suzuki for asymptotic contractions in [Tomonari Suzuki, A definitive result on asymptotic contractions, J. Math. Anal. Appl. 335 (2007) 707–715], and others. We also prove some new fixed point theorems. © 2016, House of the Book of Science. All rights reserved.

AUTHOR KEYWORDS: Asymptotic contractions; Complete metric spaces; Fixed point theorems; Meir-Keeler contractions
PUBLISHER: House of the Book of Science

Abtahi, M. Suzuki-type fixed point theorems for generalized contractive mappings that characterize metric completeness (2015) Bulletin of the Iranian Mathematical Society, 41 (4), pp. 931-943.

Inspired by the work of Suzuki in [T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc., 136 (2008), 1861-1869], we prove a fixed point theorem for contractive mappings that generalizes a theorem of Geraghty in [M.A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604-608] and characterizes metric completeness. We introduce the family A of all nonnegative functions ϕ with the property that, given a metric space (X; d) and a mapping T: X ! X, the condition x; y 2 X; x ̸= y; d(x; Tx) _ d(x, y) =) d(Tx; Ty) < ϕ(d(x; y)); implies that the iterations xn = Tnx, for any choice of initial point x 2 X, form a Cauchy sequence in X. We show that the family of L-functions, introduced by Lim in [T.C. Lim, On characterizations of Meir-Keeler contractive maps, Nonlinear Anal., 46 (2001), 113-120], and the family of test functions, introduced by Geraghty, belong to A. We also prove a Suzuki-type fixed point theorem for nonlinear contractions. © 2015 Iranian Mathematical Society.


AUTHOR KEYWORDS: Banach contraction principle; Contractive mappings; Fixed points; Metric completeness; Suzuki-type fixed point theorem
PUBLISHER: Iranian Mathematical Society

Pourgholi, R., Abtahi, M., Tabasi, S.H. A numerical approach for solving an inverse parabolic problem with unknown control function (2015) International Journal of Computational Science and Engineering, 10 (4), pp. 395-401.

DOI: 10.1504/IJCSE.2015.070994

In this paper, we will first study the existence and uniqueness of the solution for a one dimensional inverse heat conduction problem (IHCP) via an auxiliary problem. Then the present work is motivated by desire to obtain numerical approach for solving this IHCP. Our method begins with the utilisation of some transformations. These transformations allow us to eliminate an unknown term from parabolic equation to obtain an inverse parabolic problem with two unknown boundary conditions. To solve this inverse problem, we use the fundamental solution method. The effectiveness of the algorithm is illustrated by numerical example. Copyright 2015 Inderscience Enterprises Ltd.

AUTHOR KEYWORDS: Fundamental solution method; Inverse heat conduction problem; Stability; The L-curve method; Tikhonov regularisation method
INDEX KEYWORDS: Convergence of numerical methods; Heat conduction; Problem solving, Existence and uniqueness; Fundamental solution method; Inverse heat conduction problem; L-curve methods; Numerical approaches; Parabolic Equations; Parabolic problems; Regularisation, Inverse problems
PUBLISHER: Inderscience Enterprises Ltd.

Pourgholi, R., Esfahani, A., Abtahi, M. A numerical solution of a two-dimensional IHCP (2013) Journal of Applied Mathematics and Computing, 41 (1-2), pp. 61-79.

DOI: 10.1007/s12190-012-0592-6

In this paper, we will first study the existence and uniqueness of the solution of a two-dimensional inverse heat conduction problem (IHCP) which is severely ill-posed, i.e.; the solution does not depend continuously on the data. We propose a stable numerical approach based on the finite-difference method and the least-squares scheme to solve this problem in the presence of noisy data. We prove the convergence of the numerical solution, then to regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization 0th, 1st and 2nd method to obtain the stable numerical approximation to the solution. The stability and accuracy of the scheme presented is evaluated by comparison with the Singular Value Decomposition (SVD) method. © 2012 Korean Society for Computational and Applied Mathematics.

AUTHOR KEYWORDS: Consistency; Convergence; Existence; Finite difference method; Inverse heat conduction problem; Least-square method; Stability; Tikhonov regularization method; Uniqueness
INDEX KEYWORDS: Consistency; Convergence; Existence; Inverse heat conduction problem; Least square methods; Tikhonov regularization method; Uniqueness, Convergence of numerical methods; Finite difference method; Linear systems; Singular value decomposition; Two dimensional, Least squares approximations

Abel, M., Abtahi, M. Description of Closed Maximal Ideals in Topological Algebras of Continuous Vector-Valued Functions (2013) Mediterranean Journal of Mathematics, pp. 1185-1193.

DOI: 10.1007/s00009-013-0366-x

Let X be a completely regular Hausdorff space, A be a unital locally convex algebra with jointly continuous multiplication and C(X,A) be the algebra of all continuous A-valued functions on X equipped with the topology of K(X) -convergence. Moreover, let Mℓ(A) and M(A) denote the set of all closed maximal left and two-sided ideals in A, respectively. In this note, we describe all closed maximal left and two-sided ideals in C(X,A) and show that there exist bijections from Mℓ(C(X,A)) onto X×Mℓ(A) and M(C(X,A)) onto X×M(A). We also present new characterizations of closed maximal ideals in C(X, A) when A is a unital commutative locally convex Gelfand–Mazur algebra with jointly continuous multiplication. © 2013, Springer Basel.

AUTHOR KEYWORDS: 46J20; Primary 46H10; Secondary 46J10
PUBLISHER: Birkhauser Verlag AG

Abtahi, M. Normed algebras of infinitely differentiable Lipschitz functions (2012) Quaestiones Mathematicae, 35 (2), pp. 195-202.

DOI: 10.2989/16073606.2012.696855

Let X be a perfect compact plane set, ω ={w n} be a weight sequence of positive numbers, and 0 < α ≤ 1. Then Lip(X, α, ω) denotes the algebra of infinitely differentiable functions f on X such that f (n), for n = 0, 1,..., satisfies the Lipschitz condition of order α, and such that, where {pipe}{pipe}·{pipe}{pipe} α is the Lipschitz norm on X. Using some formulae from combinatorial analysis, we show that, under certain conditions on ω, if f ∈ Lip(X, α, ω) and f (z)≠0 for every z ∈ X, then 1/f ∈ Lip(X, α, ω). We then conclude that if, moreover, the maximal ideal space of Lip̄(X, α, ω), the uniform closure of Lip(X, α, ω), equals X then every non-zero continuous complex homomorphism on Lip(X, α, ω) is an evaluation character at some point of X. © 2012 Copyright NISC Pty Ltd.


AUTHOR KEYWORDS: complex homomorphisms; differentiable functions on perfect compact plane sets; evaluation characters; Lipschitz functions; Normed function algebras

Abtahi, M., Pourgholi, R., Shidfar, A. Existence and uniqueness of a solution for a two dimensional nonlinear inverse diffusion problem (2011) Nonlinear Analysis, Theory, Methods and Applications, 74 (7), pp. 2462-2467.

DOI: 10.1016/j.na.2010.12.001

The problem of identifying the coefficient in a square porous medium is considered. It is shown that under certain conditions of data f,g, and for a properly specified class A of admissible coefficients, there exists at least one a∈A such that (a,u) is a solution of the corresponding inverse problem. © 2011 Elsevier Ltd. All rights reserved.


AUTHOR KEYWORDS: Inverse problem; Nonlinear diffusion problem; Square porous medium
INDEX KEYWORDS: Class A; Existence and uniqueness; Nonlinear diffusion problems; Nonlinear inverse diffusion; Porous medium; Square porous medium, Diffusion; Inverse problems, Porous materials

Abtahi, M., Zhang, Y. A new proof of the amenability of C(X) (2010) Bulletin of the Australian Mathematical Society, 81 (3), pp. 414-417.

DOI: 10.1017/S0004972709001063

In this paper, we present a constructive proof of the amenability of C(X), where X is a compact space. © 2010 Australian Mathematical Publishing Association Inc.

AUTHOR KEYWORDS: amenability; approximate diagonal; Banach algebra of continuous functions

Abtahi, M., Honary, T.G. On the maximal ideal space of Dales-Davie algebras of infinitely differentiable functions (2007) Bulletin of the London Mathematical Society, 39 (6), pp. 940-948.

DOI: 10.1112/blms/bdm084

Let X be a perfect, compact plane set, and let M=(Mn) be a sequence of positive numbers such that M0 =1 and Mn/M kMn-k≥(kn) for all n ε ℕ and k=0, 1, 2, ..., n. We consider a remarkable class of Banach function algebras of infinitely differentiable functions f on X such that ∑n=0∞||f(n)||x/Mn ≤ ∞. These algebras, called ales-Davie algebras, are denoted by D(X, M), and they are complete under certain conditions on X. The main aim of this work is to find conditions on the sequence M=(Mn) to guarantee that D(X, M) is natural; that is, its maximal ideal space is identified with X. We present a general result on the naturality of D(X, M) using some formulas from combinatorial analysis. In particular, it is shown that if X is uniformly regular, and if the sequence (Pn)=(Mn/n!) satisfies any one of the following conditions, then D(X, M) is natural: (i) sup P iPj/Pi+j-1:i, j∈ℕ}<∞; (ii); Pn2 ≤ Pn-1Pn+1 for all n ∈ ℕ and (iii) Bn=max PkPn-k/Pn: 1≤k≤n-1→0 as n→∞. © 2007 London Mathematical Society.


PUBLISHER: John Wiley and Sons Ltd

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