Arash Rastegar

Sharif University of Technology, Mathematical Sciences, Faculty Member

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Papers by Arash Rastegar

The concept of self-similarity on subsets of algebraic varieties is defined by considering algebr... more The concept of self-similarity on subsets of algebraic varieties is defined by considering algebraic endomorphisms of the variety as 'similarity' maps. Selfsimilar fractals are subsets of algebraic varieties which can be written as a finite and disjoint union of 'similar' copies. Fractals provide a framework in which, one can unite some results and conjectures in Diophantine geometry. We define a well-behaved notion of dimension for self-similar fractals. We also prove a fractal version of Roth's theorem for algebraic points on a variety approximated by elements of a fractal subset. As a consequence, we get a fractal version of Siegel's theorem on finiteness of integral points on hyperbolic curves and a fractal version of Falting's theorem on Diophantine approximation on abelian varieties.

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Particle removal challenges with EUV patterned mask for the sub-22nm HP node

The particle removal efficiency (PRE) of cleaning processes diminishes whenever the minimum defec... more The particle removal efficiency (PRE) of cleaning processes diminishes whenever the minimum defect size for a specific technology node becomes smaller. For the sub-22 nm half-pitch (HP) node, it was demonstrated that exposure to high power megasonic up to 200 W/cm² did not damage 60 nm wide TaBN absorber lines corresponding to the 16 nm HP node on wafer. An

Skills of MATHEMATICAL THINKING in undergraduate level

The goal of this paper is to suggest a package of basic mathematical skills which forms an approp... more The goal of this paper is to suggest a package of basic mathematical skills which forms an appropriate background for an undergraduate student in mathematics. Having such a package available, one could decide about the mathematical content of undergraduate courses designed for math-majors in a way that they provide an appropriate atmosphere for development of the above-mentioned skills. One would be able to afford a reasonable suggestion for this package, only if one has a clear mind about the role of mathematics education in educating students and about the importance of mathematics to human civilization.

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Report on Circular and Spherical Structuring

ABSTRACT

<title>Interaction forces on mask surfaces relevant to EUV lithography</title>

Photomask Technology 2006, 2006

ABSTRACT Due to the increasing impact of smaller particles, mask cleaning continues to become mor... more ABSTRACT Due to the increasing impact of smaller particles, mask cleaning continues to become more and more challenging in EUV lithography. To improve mask cleaning efficiency, advances in the fundamental understanding of the interaction between defect particles and mask surfaces are necessary. For this reason, surface force measurements were performed with an atomic force microscope on various mask surfaces relevant to EUV lithography. Experiments in air were carried out to illustrate particle interaction during mask transport and storage, while measurements in deionized ultrapure water were undertaken to investigate the influence of a basic cleaning chemistry. The effects of particle size were studied using SiNx tips with a nominal radius of 10 nm and spherical SiO2 probes with a radius of 500 nm. Particle interactions with mask surfaces in air were characterized by adhesion. Due to comparable surface roughness and surface chemistry, adhesion forces of a quartz mask substrate and a mask blank were similar. However, for a SiO2 sphere, the absolute values of the measured adhesive forces were greater than for a conventionally fabricated SiNx tip consistent with the probes&#39; relative radii. Using a quartz mask substrate and deionized water as the intervening medium, the probe-substrate interaction observed was no longer characterized by attraction, but dominated by repulsive forces and hence potentially advantageous for cleaning purposes.

Dynamic light scattering study of SmA-SmC*A transition in an antiferroelectric liquid crystal

Proceedings of SPIE - The International Society for Optical Engineering

We have studied the order parameter dynamics close to the SmA-SmC*A phase transition in homeotrop... more We have studied the order parameter dynamics close to the SmA-SmC*A phase transition in homeotropic cells of 4-(1-ethylheptyloxycarbonyl) phenyl-4'-alkylcarbonyloxy biphenyl-4-carboxylate by photon correlation spectroscopy. The order parameter fluctuations in the antiferroelectric SmC*A phase can be decomposed into the fluctuations of the phase and the amplitude of the molecular tilt angle. Considering the unit cell to consist out of two adjacent layers, one can describe these fluctuations with two ferroelectric modes and two antiferroelectric modes. Using photon correlation spectroscopy we measured both ferroelectric phase modes in the backward scattering geometry. This is the first simultaneous observation of these modes. The temperature dependence of the relaxation rates of these modes gives the coupling coefficients of the ferroelectric and antiferroelectric order parameters, whereas the dispersion relation leads to the values of the diffusivity coefficients for antiferroele...

EPH-classification

A research proposal in mathematical modeling applied to heart and mathematical concepts of mechanisms of heart

Journal of Science and Technology

mathematical intelligence are not without connection with creation in general and with general in... more mathematical intelligence are not without connection with creation in general and with general intelligence. A great proportion of prominent mathematicians have been creators in other fields. One of the greatest, Gauss, carried out important and classical experiments on magnetism; and Newton's fundamental discoveries in optics are well known. And Leibniz influenced by their mathematical abilities on philosophical ones. The research proposals we propose here, are motivated by our background and previous research, and are supposed to indicate my general directions in forthcoming research. Following Poincare, each research subject is formed by an attempt to link two fields in sciences, in order to contribute to at least one of the two. In this proposal, always the mathematical part is the part which takes the benefit. We will try to explain an abstract for each problem and give motivations for importance of the corresponding research direction together with a prediction of expected...

Modeling of Interactions between Physics and Mathematics

There is a spectrum between what mathematicians and physicists do. Moving from mathematics to phy... more There is a spectrum between what mathematicians and physicists do. Moving from mathematics to physics, we find many instances in which, physics influences the way mathematicians do mathematics. In fact, there are instances that pure mathematicians practically do physics. There are mathematicians whose field of research is classical mechanics, relativity, quantum mechanics, string theory or supper-symmetry. Moving from physics to mathematics, there are many instances in which, physicists do mathematics. Physicists do differential equations, infinite dimensional group representations, operator theory, algebraic topology, algebras and operads, deformation of metric, and category theory, while doing physics. It is hard to say if mathematics imitates physics or vice versa. In fact, there is a two way relationship between them and our goal is to model the interaction between the two. Similar developments There are many instances in which, mathematics is developed similar to physical theor...

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رياضيات چيست و چگونه بايد آموزش داده شود؟

دو روش متفاوت برای بررسی تاريخ تحول پاسخهايی که به سؤال رياضيات چيست داده شده است، وجود دارد. روش... more دو روش متفاوت برای بررسی تاريخ تحول پاسخهايی که به سؤال رياضيات چيست داده شده است، وجود دارد. روش سنتی روش فلسفی است و روش جديد روشی انسانشناسانه است. در روش فلسفی به دنبال تعريفی فلسفی از رياضيات چيست میگرديم و در روش انسانشناسانه به ساختار شناختی حاصل از يک انسانشناسی و تعريفی از رياضيات که پيش پا مینهد نظر داريم. در اين مقاله روش سنتی را پيش میگيريم و به سمت اين حرکت میکنيم که مخاطب پله به پله با روش جديد آشنا شود و برای بررسی تاريخ تحول پاسخهايی که به سؤال رياضيات چيست داده
شده است، به زبان انسانشناسی آماده گردد.

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Psychology of Theorization in Mathematics and Physics

There are two kinds of mathematicians or physicists: Problem Solvers are analyzers and move from ... more There are two kinds of mathematicians or physicists: Problem Solvers are analyzers and move from local to global and they are interested in important special cases. Theoreticians are wholists and move from global to local and they are interested in various generalizations. Theorization and problem solving are parallel skills for mathematicians and physicists to further develop mathematics and physics. Skills of problem solving develop in early ages, but theorization skills develop much further in the scientific life of mathematicians and physicists. It rarely happens that someone in the age of Galois is that able in theorization in mathematics or physics. Comparing the research of Abel and Galois clearly indicates the difference between a problem solver and a theorizer.

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What is Mathematical Argument?

There are several possible perspectives towards this question:

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EPH-classifications in Geometry, Algebra, and Arithmetic

Trichotomy of Elliptic-Parabolic-Hyperbolic appears in many different areas of mathematics. All o... more Trichotomy of Elliptic-Parabolic-Hyperbolic appears in many different areas of mathematics. All of these are named after the very first example of trichotomy, which is formed by ellipses, parabolas, and hyperbolas as conic sections. We try to understand if these classifications are justified and if similar
mathematical phenomena is shared among different cases EPH-classification is used.

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What is the Meaning of Deep for a Mathematician?

A mathematician could be thought of, both as a problem solver and also as a theoretician; as a sc... more A mathematician could be thought of, both as a problem solver and also as a theoretician; as a scientist and also as an artist; as an arguer and also as a conjecturer; as a coach and also as a teacher; As an educator and also as an evaluator; As a philosopher and also as a physician. So, there are several roles a mathematician is playing in the mathematical society and each of these roles has its own definition and relevant skills and each has a different perspective towards what is called a deep for a mathematician.

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Cognition types for mathematicians

Some people have wrong believes about cognition types: They believe verbals become scientists; pi... more Some people have wrong believes about cognition types: They believe verbals become scientists; pictorials become artists; kinetics become sportsmen; wholists become philosophers; and analyzers become Scientist. To us, cognition abilities of verbals are associated to speech and hearing; and those of pictorials are associated to vision and image processing.

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Engineering the Correlation of Scientific and Educational Systems

We construct mathematical models for research in science and evolution of educational systems and... more We construct mathematical models for research in science and evolution of educational systems and then try to understand the correlation between science and educational systems using these models. We have chosen the language of “Atlas of Concepts”. This notion has been used before as a mathematical model for mathematical communication. It provides an appropriate tool for understanding of science and educational systems in a common framework. The main question is the following: How is it possible, to guide scientific research via control of educational systems and vice versa, by engineering their correlation? We are particularly interested in the correlation of research in mathematics and evolution of mathematical educational systems.

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On the Psychology of Teaching Mathematics

It is evident to any able mathematics teacher that the process of mathematical education of stude... more It is evident to any able mathematics teacher that the process of mathematical education of students should be organized according to their particular capacities, interests, abilities, culture and the particular nature of their mind and creativity. In this paper, we shall focus on the personality of the educators in making educational decisions for students and their scientific perspectives in mathematical education. Psychology of mathematics teachers is an important component in teachers’ education and also in curriculum planning and the process of implementing the curriculum. We try to introduce mathematical models for different personalities of mathematics teachers and discuss how these models can be used in classroom and in curriculum planning.

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Who is a good problem solver?

It is more than one decade now, that Iran is participating in “International Mathematical Olympia... more It is more than one decade now, that Iran is participating in “International Mathematical Olympiad”. Many of Iranian Olympiad contestants have pursued their studies in mathematics up to the PhD level. Students in each generation become the coaches of the next generation. This way, every generation of Olympiad students develops a new culture of problem solving. As a member of this club, I have been through all the different layers of scientific and administrative works involved in this movement, and at the same time, trying to record people’s experiences in problem solving, I have been communicating and collaborating with members of different generations of Olympiad club, some of whom are well-experienced mathematicians now. This paper is a gist of what I have been learning from my colleagues and students about the good problem solvers; their personality, habits and preferences. I believe recording of this group-experience, sincerely serves the educational purposes in mathematics.

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On Assessment of Problem Solving

We discuss a few different approaches to the question of how we manage to solve problems and intr... more We discuss a few different approaches to the question of how we manage to solve problems and introduce a mathematical model for the process of problem solving. Then we use our model in order to formulate an assessment strategy which has positive improving effect on development of students’ skills and their ability of solving problems.

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Psychology of Communication in Mathematics

In this paper we try to characterize a successful mathematical communication. In this regard, we ... more In this paper we try to characterize a successful mathematical communication. In this regard, we utilize the notions of “Parent”, “Child”, “Adult” introduced by T. Harris to recognize different internal mathematical personifications in students and teachers [1]. We introduce “Emotion” and “Reason” as two external personifications, which represent social mathematical interactions. We analyze these educational personifications in two educational systems. The first educational system is a system which is oriented towards problem solving. Then we try to give a model for mathematical creativity and implement the above personifications in that model. As a result a few educational perspectives are introduced. Then we try to analyze these personifications in an educational system which is oriented towards developing mathematical maturity rather than emphasizing on problem solving. We give another model for mathematical creativity with this new perspective and end up with a few different educational perspectives. Along the path, we try to give a model for group thinking and introduce the notion of “atlas of history of concepts”.

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Arash Rastegar

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