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Nicolae Cotfas

University of Bucharest, Physics, Faculty Member

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Born April 23, 1954.
1961-1973 student at Subcetate High School, Harghita County.
1974-1978 student at the Faculty of Mathematics, University of Bucharest.
1978-1990 professor of mathematics at the School 131 Bucharest.
1990-2019 teacher at the Faculty of Physics, University of Bucharest.
1995-1996, 1998 PhD student at the National Polytechnic Institute of Grenoble.
1996 Doctorate in Mathematics (University of Bucharest).
1998 Doctorate in Physics (Institut National Polytechnique de Grenoble).
2007-2011 six research internships at Universite Paris 7 Denis Diderot.

For more details, see https://unibuc.ro/user/nicolae.cotfas/
Address: https://unibuc.ro/user/nicolae.cotfas/

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Papers by Nicolae Cotfas

Quantum states defined by using the finite frame quantization

arXiv (Cornell University), Jun 19, 2022

Finite frame quantization is a discrete version of the coherent state quantization. In the case o... more Finite frame quantization is a discrete version of the coherent state quantization. In the case of a quantum system with finite-dimensional Hilbert space, the finite frame quantization allows us to associate a linear operator to each function defined on the discrete phase space of the system. We investigate the properties of the density operators which can be defined by using this method.

Model sets with predefined local structure

Model sets play a fundamental role in structure analysis of quasicrystals. The diffraction diagra... more Model sets play a fundamental role in structure analysis of quasicrystals. The diffraction diagram of a quasicrystal admits as symmetry group a finite group G, and there is a G-cluster C (union of orbits of G) such that the quasicrystal can be regarded as a quasiperiodic packing of interpenetrating copies of C. We present an algorithm which leads from any G-cluster C directly to a multi-component model set Q such that the arithmetic neighbours of any point x ∈ Q are distributed on the sites of the translated copy x + C of C. Our mathematical algorithm may be useful in quasicrystal physics.

$Research paper thumbnail of New definition of the discrete fractional Fourier transform$

New definition of the discrete fractional Fourier transform

The Hamiltonian of the harmonic oscillator is usually defined as a differential operator, but an ... more The Hamiltonian of the harmonic oscillator is usually defined as a differential operator, but an integral representation can be obtained by using the coherent state quantization. The finite frame quantization is a finite counterpart of the coherent state quantization and it allows us to define a finite oscillator by starting from the integral representation of the harmonic oscillator. Our purpose is to investigate the oscillator obtained in this way, and to present a possible application to the discrete fractional Fourier transform.

Quantum states defined by using the finite frame quantization

arXiv (Cornell University), Jun 19, 2022

Finite frame quantization is a discrete version of the coherent state quantization. In the case o... more Finite frame quantization is a discrete version of the coherent state quantization. In the case of a quantum system with finite-dimensional Hilbert space, the finite frame quantization allows us to associate a linear operator to each function defined on the discrete phase space of the system. We investigate the properties of the density operators which can be defined by using this method.

Finite tight frames and some applications

Journal of Physics A, Apr 20, 2010

A finite-dimensional Hilbert space is usually described in terms of an orthonormal basis, but in ... more A finite-dimensional Hilbert space is usually described in terms of an orthonormal basis, but in certain approaches or applications a description in terms of a finite overcomplete system of vectors, called a finite tight frame, may offer some advantages. The use of a finite tight frame may lead to a simpler description of the symmetry transformations, to a simpler and more symmetric form of invariants or to the possibility to define new mathematical objects with physical meaning, particularly in regard with the notion of a quantization of a finite set. We present some results concerning the use of integer coefficients and frame quantization, several examples and suggest some possible applications.

Linear representations of SU(2) described by using Kravchuk polynomials

arXiv (Cornell University), Jan 24, 2016

We show that a new unitary transform with characteristics almost similar to those of the finite F... more We show that a new unitary transform with characteristics almost similar to those of the finite Fourier transform can be defined in any finite-dimensional Hilbert space. It is defined by using the Kravchuk polynomials, and we call it Kravchuk transform. Some of its properties are investigated and used in order to obtain a simple alternative description for the irreducible representations of the Lie algebra su(2) and group SU (2). Our approach offers a deeper insight into the structure of the linear representations of SU (2) and new possibilities of computation, very useful in applications in quantum mechanics, quantum information, signal and image processing.

Model sets with predefined local structure

arXiv (Cornell University), May 10, 2004

Model sets play a fundamental role in structure analysis of quasicrystals. The diffraction diagra... more Model sets play a fundamental role in structure analysis of quasicrystals. The diffraction diagram of a quasicrystal admits as symmetry group a finite group G, and there is a G-cluster C (union of orbits of G) such that the quasicrystal can be regarded as a quasiperiodic packing of interpenetrating copies of C. We present an algorithm which leads from any G-cluster C directly to a multi-component model set Q such that the arithmetic neighbours of any point x belonging to Q are distributed on the sites of the translated copy x+C of C. Our mathematical algorithm may be useful in quasicrystal physics.

Hypergeometric type functions, raising and lowering operators

arXiv (Cornell University), Jun 2, 2003

The classical orthogonal polynomials are usually defined as particular solutions of some hypergeo... more The classical orthogonal polynomials are usually defined as particular solutions of some hypergeometric type equations. It is known that the class of these equations is larger and they are related to some Schrodinger type equations. We show that some of them define finite systems of orthogonal polynomials and analyse the corresponding associated special functions and raising/lowering operators. These functions correspond to the square integrable solutions in the case of Morse, Scarf hyperbolic and generalized Poschl-Teller type potentials.

On the Asymptotic Behaviour of Some Elliptic Problems in Perforated Domains

The asymptotic behavior of the solution of a class of elliptic problems modeling diffusion in som... more The asymptotic behavior of the solution of a class of elliptic problems modeling diffusion in some periodic perforated media is analyzed. We consider, at the microscale, an elliptic equation with various nonlinear conditions prescribed on the boundary of the perforations and we prove that the effective behavior of the solution of such a problem is governed by another elliptic equation, which, depending on the type of the conditions imposed on the surface of the cavities, can contain extra zeroorder terms.

Gaussian type functions defined on a finite set and finite oscillators

arXiv (Cornell University), Dec 8, 2015

The mathematical description of the quantum harmonic oscillator is essentially based on the Gauss... more The mathematical description of the quantum harmonic oscillator is essentially based on the Gaussian function. In the case of a quantum oscillator with finite-dimensional Hilbert space, the position space consists in a finite number of points, and the ground state can be regarded as a finite version of the Gaussian function. Conversely, each finite Gaussian can be used in order to define some finite oscillators. We investigate certain finite Gaussians as concern their Fourier transform, Wigner function and associated oscillators.

Associated hypergeometric-type functions and coherent states

International Conference on Applied Mathematics, Dec 16, 2005

We present in a unified and explicit way the systems of orthogonal polynomials defined by hyperge... more

On the self-similarities of the Penrose tiling

arXiv (Cornell University), Mar 31, 2004

We show that the well known two-dimensional Penrose tiling admits an infinite number of independe... more

Probabilistic aspects of finite frame quantization

arXiv (Cornell University), Mar 2, 2008

A finite-dimensional Hilbert space is usually described in terms of an orthonormal basis, but in ... more A finite-dimensional Hilbert space is usually described in terms of an orthonormal basis, but in certain approaches or applications a description in terms of a finite overcomplete system of vectors, called a finite tight frame, may offer some advantages. The use of a finite tight frame may lead to a simpler description of the symmetry transformations, to a simpler and more symmetric form of invariants or to the possibility to define new mathematical objects with physical meaning, particularly in regard with the notion of a quantization of a finite set. We present some results concerning the use of integer coefficients and frame quantization, several examples and suggest some possible applications.

Quasiperiodic packings of two-shell decagonal clusters

arXiv (Cornell University), Apr 11, 2005

Special functions, raising and lowering operators

arXiv (Cornell University), Jun 2, 2003

The Schrödinger equations which are exactly solvable in terms of associated special functions are... more The Schrödinger equations which are exactly solvable in terms of associated special functions are directly related to some self-adjoint operators defined in the theory of hypergeometric type equations. The fundamental formulae occurring in a supersymmetric approach to these Hamiltonians are consequences of some formulae concerning the general theory of associated special functions. We use this connection in order to obtain a general theory of Schrödinger equations exactly solvable in terms of associated special functions, and to extend certain results known in the case of some particular potentials.

Algorithm for generating new explicitly solvable Schrodinger type equations

arXiv (Cornell University), Feb 13, 2011

In this note we present an algorithm to generate new Schrödinger type equations explicitly solvab... more

Algorithm for Generating Quasiperiodic Packings of Multi-Shell Clusters

arXiv (Cornell University), Apr 27, 2005

Many of the mathematical models used in quasicrystal physics are based on tilings of the plane or... more Many of the mathematical models used in quasicrystal physics are based on tilings of the plane or space obtained by using strip projection method in a superspace of dimension four, five or six. We present some mathematical results which allow one to use this very elegant method in spaces of dimension much higher and to generate directly quasiperiodic packings of multi-shell clusters. We show that in the case of a two-dimensional (resp. three-dimensional) cluster we have to compute only determinants of order three (resp. four), independently of the dimension of the superspace we use. The computer program based on our mathematical results is very efficient. For example, we can easily generate quasiperiodic packings of three-shell icosahedral clusters (icosahedron + dodecahedron + icosidodecahedron) by using strip projection method in a 31-dimensional space (hundreds of points are obtained in a few minutes on a personal computer).

Quantum harmonic oscillator in option pricing

arXiv (Cornell University), Oct 15, 2013

The Black-Scholes model anticipates rather well the observed prices for options in the case of a ... more The Black-Scholes model anticipates rather well the observed prices for options in the case of a strike price that is not too far from the current price of the underlying asset. Some useful extensions can be obtained by an adequate modification of the coefficients in the Black-Scholes equation. We investigate from a mathematical point of view an extension directly related to the quantum harmonic oscillator. In the considered case, the solution is the sum of a series involving the Hermite-Gauss functions. A finite-dimensional version is obtained by using a finite oscillator and the Harper functions. This simplified model keeps the essential characteristics of the continuous one and uses finite sums instead of series and integrals.

Associated special functions and coherent states

arXiv (Cornell University), Oct 30, 2005

A hypergeometric type equation satisfying certain conditions defines either a finite or an infini... more A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated special functions and some systems of coherent states. This general formalism allows us to extend some results known only in particular cases.

Modified strip projection method

arXiv (Cornell University), May 14, 2006

The diffraction image of a quasicrystal admits a finite group G as a symmetry group, and the quas... more The diffraction image of a quasicrystal admits a finite group G as a symmetry group, and the quasicrystal can be regarded as a quasiperiodic packing of copies of a G-cluster C, joined by glue atoms. The physical space E containing C can be embedded into a higher-dimensional space R k such that, up to an inflation factor, C is the orthogonal projection of the set {(

Quantum states defined by using the finite frame quantization

arXiv (Cornell University), Jun 19, 2022

Finite frame quantization is a discrete version of the coherent state quantization. In the case o... more Finite frame quantization is a discrete version of the coherent state quantization. In the case of a quantum system with finite-dimensional Hilbert space, the finite frame quantization allows us to associate a linear operator to each function defined on the discrete phase space of the system. We investigate the properties of the density operators which can be defined by using this method.

Model sets with predefined local structure

Model sets play a fundamental role in structure analysis of quasicrystals. The diffraction diagra... more Model sets play a fundamental role in structure analysis of quasicrystals. The diffraction diagram of a quasicrystal admits as symmetry group a finite group G, and there is a G-cluster C (union of orbits of G) such that the quasicrystal can be regarded as a quasiperiodic packing of interpenetrating copies of C. We present an algorithm which leads from any G-cluster C directly to a multi-component model set Q such that the arithmetic neighbours of any point x ∈ Q are distributed on the sites of the translated copy x + C of C. Our mathematical algorithm may be useful in quasicrystal physics.

$Research paper thumbnail of New definition of the discrete fractional Fourier transform$

New definition of the discrete fractional Fourier transform

The Hamiltonian of the harmonic oscillator is usually defined as a differential operator, but an ... more The Hamiltonian of the harmonic oscillator is usually defined as a differential operator, but an integral representation can be obtained by using the coherent state quantization. The finite frame quantization is a finite counterpart of the coherent state quantization and it allows us to define a finite oscillator by starting from the integral representation of the harmonic oscillator. Our purpose is to investigate the oscillator obtained in this way, and to present a possible application to the discrete fractional Fourier transform.

Quantum states defined by using the finite frame quantization

arXiv (Cornell University), Jun 19, 2022

Finite frame quantization is a discrete version of the coherent state quantization. In the case o... more Finite frame quantization is a discrete version of the coherent state quantization. In the case of a quantum system with finite-dimensional Hilbert space, the finite frame quantization allows us to associate a linear operator to each function defined on the discrete phase space of the system. We investigate the properties of the density operators which can be defined by using this method.

Finite tight frames and some applications

Journal of Physics A, Apr 20, 2010

A finite-dimensional Hilbert space is usually described in terms of an orthonormal basis, but in ... more A finite-dimensional Hilbert space is usually described in terms of an orthonormal basis, but in certain approaches or applications a description in terms of a finite overcomplete system of vectors, called a finite tight frame, may offer some advantages. The use of a finite tight frame may lead to a simpler description of the symmetry transformations, to a simpler and more symmetric form of invariants or to the possibility to define new mathematical objects with physical meaning, particularly in regard with the notion of a quantization of a finite set. We present some results concerning the use of integer coefficients and frame quantization, several examples and suggest some possible applications.

Linear representations of SU(2) described by using Kravchuk polynomials

arXiv (Cornell University), Jan 24, 2016

We show that a new unitary transform with characteristics almost similar to those of the finite F... more We show that a new unitary transform with characteristics almost similar to those of the finite Fourier transform can be defined in any finite-dimensional Hilbert space. It is defined by using the Kravchuk polynomials, and we call it Kravchuk transform. Some of its properties are investigated and used in order to obtain a simple alternative description for the irreducible representations of the Lie algebra su(2) and group SU (2). Our approach offers a deeper insight into the structure of the linear representations of SU (2) and new possibilities of computation, very useful in applications in quantum mechanics, quantum information, signal and image processing.

Model sets with predefined local structure

arXiv (Cornell University), May 10, 2004

Model sets play a fundamental role in structure analysis of quasicrystals. The diffraction diagra... more Model sets play a fundamental role in structure analysis of quasicrystals. The diffraction diagram of a quasicrystal admits as symmetry group a finite group G, and there is a G-cluster C (union of orbits of G) such that the quasicrystal can be regarded as a quasiperiodic packing of interpenetrating copies of C. We present an algorithm which leads from any G-cluster C directly to a multi-component model set Q such that the arithmetic neighbours of any point x belonging to Q are distributed on the sites of the translated copy x+C of C. Our mathematical algorithm may be useful in quasicrystal physics.

Hypergeometric type functions, raising and lowering operators

arXiv (Cornell University), Jun 2, 2003

The classical orthogonal polynomials are usually defined as particular solutions of some hypergeo... more The classical orthogonal polynomials are usually defined as particular solutions of some hypergeometric type equations. It is known that the class of these equations is larger and they are related to some Schrodinger type equations. We show that some of them define finite systems of orthogonal polynomials and analyse the corresponding associated special functions and raising/lowering operators. These functions correspond to the square integrable solutions in the case of Morse, Scarf hyperbolic and generalized Poschl-Teller type potentials.

On the Asymptotic Behaviour of Some Elliptic Problems in Perforated Domains

The asymptotic behavior of the solution of a class of elliptic problems modeling diffusion in som... more The asymptotic behavior of the solution of a class of elliptic problems modeling diffusion in some periodic perforated media is analyzed. We consider, at the microscale, an elliptic equation with various nonlinear conditions prescribed on the boundary of the perforations and we prove that the effective behavior of the solution of such a problem is governed by another elliptic equation, which, depending on the type of the conditions imposed on the surface of the cavities, can contain extra zeroorder terms.

Gaussian type functions defined on a finite set and finite oscillators

arXiv (Cornell University), Dec 8, 2015

The mathematical description of the quantum harmonic oscillator is essentially based on the Gauss... more The mathematical description of the quantum harmonic oscillator is essentially based on the Gaussian function. In the case of a quantum oscillator with finite-dimensional Hilbert space, the position space consists in a finite number of points, and the ground state can be regarded as a finite version of the Gaussian function. Conversely, each finite Gaussian can be used in order to define some finite oscillators. We investigate certain finite Gaussians as concern their Fourier transform, Wigner function and associated oscillators.

Associated hypergeometric-type functions and coherent states

International Conference on Applied Mathematics, Dec 16, 2005

We present in a unified and explicit way the systems of orthogonal polynomials defined by hyperge... more

On the self-similarities of the Penrose tiling

arXiv (Cornell University), Mar 31, 2004

We show that the well known two-dimensional Penrose tiling admits an infinite number of independe... more

Probabilistic aspects of finite frame quantization

arXiv (Cornell University), Mar 2, 2008

A finite-dimensional Hilbert space is usually described in terms of an orthonormal basis, but in ... more A finite-dimensional Hilbert space is usually described in terms of an orthonormal basis, but in certain approaches or applications a description in terms of a finite overcomplete system of vectors, called a finite tight frame, may offer some advantages. The use of a finite tight frame may lead to a simpler description of the symmetry transformations, to a simpler and more symmetric form of invariants or to the possibility to define new mathematical objects with physical meaning, particularly in regard with the notion of a quantization of a finite set. We present some results concerning the use of integer coefficients and frame quantization, several examples and suggest some possible applications.

Quasiperiodic packings of two-shell decagonal clusters

arXiv (Cornell University), Apr 11, 2005

Special functions, raising and lowering operators

arXiv (Cornell University), Jun 2, 2003

The Schrödinger equations which are exactly solvable in terms of associated special functions are... more The Schrödinger equations which are exactly solvable in terms of associated special functions are directly related to some self-adjoint operators defined in the theory of hypergeometric type equations. The fundamental formulae occurring in a supersymmetric approach to these Hamiltonians are consequences of some formulae concerning the general theory of associated special functions. We use this connection in order to obtain a general theory of Schrödinger equations exactly solvable in terms of associated special functions, and to extend certain results known in the case of some particular potentials.

Algorithm for generating new explicitly solvable Schrodinger type equations

arXiv (Cornell University), Feb 13, 2011

In this note we present an algorithm to generate new Schrödinger type equations explicitly solvab... more

Algorithm for Generating Quasiperiodic Packings of Multi-Shell Clusters

arXiv (Cornell University), Apr 27, 2005

Many of the mathematical models used in quasicrystal physics are based on tilings of the plane or... more Many of the mathematical models used in quasicrystal physics are based on tilings of the plane or space obtained by using strip projection method in a superspace of dimension four, five or six. We present some mathematical results which allow one to use this very elegant method in spaces of dimension much higher and to generate directly quasiperiodic packings of multi-shell clusters. We show that in the case of a two-dimensional (resp. three-dimensional) cluster we have to compute only determinants of order three (resp. four), independently of the dimension of the superspace we use. The computer program based on our mathematical results is very efficient. For example, we can easily generate quasiperiodic packings of three-shell icosahedral clusters (icosahedron + dodecahedron + icosidodecahedron) by using strip projection method in a 31-dimensional space (hundreds of points are obtained in a few minutes on a personal computer).

Quantum harmonic oscillator in option pricing

arXiv (Cornell University), Oct 15, 2013

The Black-Scholes model anticipates rather well the observed prices for options in the case of a ... more The Black-Scholes model anticipates rather well the observed prices for options in the case of a strike price that is not too far from the current price of the underlying asset. Some useful extensions can be obtained by an adequate modification of the coefficients in the Black-Scholes equation. We investigate from a mathematical point of view an extension directly related to the quantum harmonic oscillator. In the considered case, the solution is the sum of a series involving the Hermite-Gauss functions. A finite-dimensional version is obtained by using a finite oscillator and the Harper functions. This simplified model keeps the essential characteristics of the continuous one and uses finite sums instead of series and integrals.

Associated special functions and coherent states

arXiv (Cornell University), Oct 30, 2005

A hypergeometric type equation satisfying certain conditions defines either a finite or an infini... more A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated special functions and some systems of coherent states. This general formalism allows us to extend some results known only in particular cases.

Modified strip projection method

arXiv (Cornell University), May 14, 2006

The diffraction image of a quasicrystal admits a finite group G as a symmetry group, and the quas... more The diffraction image of a quasicrystal admits a finite group G as a symmetry group, and the quasicrystal can be regarded as a quasiperiodic packing of copies of a G-cluster C, joined by glue atoms. The physical space E containing C can be embedded into a higher-dimensional space R k such that, up to an inflation factor, C is the orthogonal projection of the set {(

DIFFERENTIAL EQUATIONS -Solved Exercises 1

The paper presents several solved differential equations

COMPLEX ANALYSIS -Solved Problems 1

The paper presents several solved problems in Complex Analysis

LINEAR ALGEBRA -Solved Exercises 1

The paper presents several solved exercises in Linear Algebra

LINEAR ALGEBRA -Solved Exercises 2

The paper contains several solved exercises in Linear Algebra

LINEAR ALGEBRA -Solved Exercises 3

This paper presents some solved exercises in Linear Algebra

LINEAR ALGEBRA -Solved Exercises 4

The paper presents several solved exercises in Linear Algebra

LINEAR ALGEBRA -Solved Exercises 5

The paper contains several solved exercises in Linear Algebra

ELEMENTS OF LINEAR ALGEBRA AND SOME APPLICATIONS

This is a tutorial on Linear Algebra and some of its applications to the description of quantum s... more