

2018:

He has been invited and elected member of the ESF College of Expert Reviewers. The
European Science Foundation—ESF
is an independent and nonprofit organization that aims at facilitating research management processes across Europe.
ESF is committed to promoting the highest quality science in Europe to drive progress in research and innovation.
To be able to provide the best possible scientific assessments, ESF has setup a
College of Expert Reviewers.

2017:

He was honored with the N.K. Artemiadis Award of the
Academy of Athens
for the best scientific publication in the last three years
in the area of Mathematical Analysis authored by Greek researchers working in Greece or abroad.
The award, which is endowed with €3,000, was presented to him at the official ceremony Panegyric Meeting on December 21, 2017, in the presence of Academy members and Government and Parliament representatives.
The scientific publication for which the award was received (for the time period of January 1, 2014 through December 31, 2016) is:
○
Vrahatis M.N., Generalization of the Bolzano theorem for simplices [Topology and its Applications 202 (2016) 40–46],
according to which the excerpt from the report read at the official ceremony of the Academy concludes as follows:
With this study, the award winner continues a consistent scientific progress .
In this publication, a generalization of the Bolzano theorem for simplices is given and its proof is based on the Knaster–Kuratowski–Mazurkiewicz covering principle.
The first proofs of the very important and pioneering Bolzano's theorem also called intermediate value theorem ,
given independently by Bernard Bolzano in 1817 and
Augustin–Louis Cauchy in 1821, were crucial in the procedure of Arithmetization of Analysis
(which was a research program in the foundations of mathematics during the second half of the 19th century).
A straightforward generalization of Bolzano's theorem to continuous mappings of an n–dimensional cube (parallelotope) into ℝ^{n}
was proposed without proof by Jules Henri Poincaré in 1883 and 1884 in his work on the Three Body Problem .
For this reason, some authors call this theorem Poincaré's intermediate value theorem .
The Poincaré theorem was soon forgotten and it has come to be known as Miranda's theorem which was independently proposed and proved by Carlo Miranda in 1940
that partly explains the nomenclature Poincaré–Miranda theorem or Bolzano–Poincaré–Miranda theorem .
This theorem is closely related to important theorems in mathematical analysis and topology as well as it is an invaluable tool for verified solutions of numerical problems by means of interval arithmetic.
Also, it has been shown by Carlo Miranda in 1940 that this theorem is equivalent to the Brouwer fixed point theorem (1912) due to Luitzen Egbertus Jan Brouwer and
it is closely related to the existence theorems of Karol Borsuk (1933), Leonid Kantorovich (1948)
[Nobel Laureate in Economic Sciences in 1975] and Stephen Smale (1986) [Fields Medalist in 1966].
In the awarded paper, the obtained proof is based on the
covering principle of Bronisław Knaster, Kazimierz Kuratowski and Stefan Mazurkiewicz (1929) known as
Knaster–Kuratowski–Mazurkiewicz lemma or KKM lemma for short.
This lemma constitutes the basis for the proof of many theorems including, among others, the famous Brouwer's fixed point theorem (1912) due to Luitzen Egbertus Jan Brouwer.
It is worth noting that three pioneering classical results, namely,
(a) the Brouwer fixed point theorem (1912), (b) the Sperner lemma (1928) due to Emanuel Sperner, and (c) the KKM lemma (1929)
are mutually equivalent in the sense that each one can be deduced from another.
The KKM lemma has numerous applications in various fields of pure and applied mathematics.
In particular, among others, in the field of mathematical economics, the very important and pioneering extension of the KKM lemma
due to Lloyd Stowell Shapley [Nobel Laureate in Economic Sciences in 2012] customarily called
the Knaster–Kuratowski–Mazurkiewicz–Shapley theorem (1973) constitutes the basis for
the proof of many theorems on the existence of solutions in Game Theory and in the General Equilibrium Theory of economic analysis.
The generalizations of Bolzano's theorem are particular useful for tackling various problems including, among others,
the existence of solutions of systems of nonlinear algebraic and/or transcendental equations,
the existence of fixed points of functions of several variables, the localization of extrema of objective functions,
the localization of periodic orbits of area preserving mappings and the localization of periodic orbits of conservative dynamical systems
by using the Poincaré mapping on a surface of section.
These generalized theorems require only the algebraic sign of the function values that is the smallest amount of information (one bit of information) necessary for the purpose needed,
and not any additional information. Thus, these theorems are of major importance for tackling problems with imprecise (not exactly known) information.
This kind of problems occurs in various fields in science and technology.
This is so, because, in a large variety of applications, precise function values are either impossible or time consuming and computationally expensive to obtain.

2016:

One of his theorems was named in the title of the paper by:
●
Prof. Dr., Emeritus, Gerhard Heindl: Generalizations of theorems of Rohn and Vrahatis [Reliable Computing 21 (2016) 109–116].
The cited theorem is the Theorem 2 in the paper:
○
Vrahatis M.N., A short proof and a generalization of Miranda's existence theorem [Proceedings of the American Mathematical Society 107(3) (1989) 701–703],
where a short proof of the Miranda existence theorem , (also known as Poincaré's Intermediate Value Theorem ,
Poincaré–Miranda theorem or Bolzano–Poincaré–Miranda theorem ),
is given. The proof is based on the homotopy invariance of the topological degree theory
and the Kronecker existence theorem (1869) due to Leopold Kronecker. Following this proof, a generalization (stated as Theorem 2)
of the Miranda existence theorem is proved. This generalization is defined with respect to an arbitrary basis of ℝ^{n} and
eliminates the dependence of the classical Poincaré's intermediate value theorem on the standard basis of ℝ^{n}.

2014:

He has been invited and elected panel member of the ERC Starting Grant 2014 peer review process for the
European Research Council—ERC,
in the domain
Physical Sciences and Engineering (PE)—Computer Science and Informatics (PE6) .
The main goal of the ERC is to encourage high quality research in Europe through competitive funding.

2006:

The
Empeirikion Foundation in Athens,
at the award ceremony on February 25, 2006,
awarded him financial support in memory of Miltiadis Empeirikos for the organization of a laboratory for research purposes.
The competitive applications were submitted in response to a call issued by the Empeirikion Foundation on July 16, 2004
and his application, entitled:
Financial support of the
Computational Intelligence Laboratory
of the Department of Mathematics of the University of Patras
was approved on February 10, 2006 for financial support of €13,300.

2001:

His research activities were supported by the
Deutsche Forschungsgemeinschaft—DFG/German National Research Foundation
for research collaboration and lectures presentation at the
Collaborative Research Center/Sonderforschungsbereich—SFB 531:
Design and management of complex technical process and systems by means of computational intelligence methods ,
Technische Universität Dortmund, Germany.

2000:

He was supported by the
American Mathematical Society—AMS,
and the
US National Science Foundation—NSF,
in order to present an invited lecture at the
AMSIMSSIAM Joint Summer Research Conference in the Mathematical Sciences,
Algorithms and their Complexity for Nonlinear Problems , July 1620, 2000, Mount Holyoke College, South Hadley, MA, USA.

1999:

One of his methods was named in the title of the thesis by:
●
Dr. Angelika Christina Hailer:
Bisektion im ℝ^{n}—Betrachtung eines Verfahrens von M.N. Vrahatis [Diplomarbeit, Abteilung Mathematik VI (Numerik), Universität Ulm, Ulm (Donau), Deutschland, Juni 1999].
The cited method is a generalized method of bisection for solving systems of nonlinear algebraic and/or transcendental equations in ℝ^{n}.
This method requires only the algebraic sign of the function values and can be applied to problems with imprecise (not exactly known) information.
Details of this method and its implementation, as well as details of an illustrative application for the computation of all
the periodic orbits (stable and unstable) of any period and accuracy which occur in the study of beam dynamics
in circular particle accelerators like the Large Hadron Collider—LHC machine at the
European Organization for Nuclear Research—CERN,
are provided in:
○
Vrahatis M.N., Solving systems of nonlinear equations using the nonzero value of the topological degree [ACM Transactions on Mathematical Software 14(4) (1988) 312–329],
○
Vrahatis M.N., CHABIS: A mathematical software package for locating and evaluating roots of systems of nonlinear equations [ACM Transactions on Mathematical Software 14(4) (1988) 330–336] and
○
Vrahatis M.N., An efficient method for locating and computing periodic orbits of nonlinear mappings [Journal of Computational Physics 119, (1995) 105–119].

1997:

The Hellenic Club of Writers in Athens, on March 28, 1997, awarded him an honorary diploma,
stating that: Honoring the virtue and the literary interest, we award this honorary diploma to reward his excellent efforts and achievements .

1992:

His research activities were supported by the
Ministero Italiano degli Affari Esteri e della Cooperazione Internazionale/Italian Ministry of Foreign Affairs and International Cooperation,
for research collaboration and lectures presentation at the
Istituto Nazionale di Fisica Nucleare—INFN/Dipartimento di Fisica, Università degli studi di Bologna, Bologna, Italy
and at the
European Organization for Nuclear Research—CERN/Organisation Européenne pour la Recherche Nucléaire—CERN,
Geneva, Switzerland.

———

Till now, his scientific research work has been cited in 250 international patents.
More specifically, these citations have been appeared in (the alphabetically ordered list):
(i) 7 Canadian patents,
(ii) 47 Chinese patents,
(iii) 23 European patents,
(iv) 5 French patents,
(v) 7 Japanese patents,
(vi) 8 Korean patents,
(vii) 121 United States patents and
(viii) 32 World Intellectual Property Organization—WIPO patents.


