Robert Kragler | Hochschule Ravensburg-Weingarten (original) (raw)
Papers by Robert Kragler
Zeitschrift f�r Physik, 1970
The formalism of 2-nucleon-transfer-reactions developed by Glendenning and Lin is slightly genera... more The formalism of 2-nucleon-transfer-reactions developed by Glendenning and Lin is slightly generalized using a cluster model description for incoming and outgoing particles in a consistent way. The choice of an adequate spin-dependent nucleon-nucleon interaction of Gaussian radial form permits an analytic evaluation of the intrinsic matrix element. A relatives-state motion for the nucleon pair transferred leads to an additional restriction of the reaction model. Further, the polarization of outgoing particles is derived from the general structure of theT-matrix element. If spin-orbit interaction is neglected polarization effects are only expected for the transfer of unlike nucleon pairs.
In the present paper which is an extended version of paper [1] we consider a Mathematica-based pa... more In the present paper which is an extended version of paper [1] we consider a Mathematica-based package for simulation of quantum circuits. It provides a user-friendly graphical interface to specify a quantum circuit, to draw it, and to construct the unitary matrix for quantum computation defined by the circuit. The matrix is computed by means of the linear algebra tools built-in Mathematica. For circuits composed from the Toffoli and Hadamard gates the package can also output the corresponding multivariate polynomial system over F 2 whose number of solutions in F 2 determines the circuit matrix. Thereby the matrix can also be constructed by applying to the polynomial system the Gröbner basis technique based on the corresponding functions built-in Mathematica. We illustrate the package and the method used by a number of examples.
In the present paper which is an extended version of paper [1] we consider a Mathematica-based pa... more In the present paper which is an extended version of paper [1] we consider a Mathematica-based package for simulation of quantum circuits. It provides a user-friendly graphical interface to specify a quantum circuit, to draw it, and to construct the unitary matrix for quantum computation defined by the circuit. The matrix is computed by means of the linear algebra tools built-in Mathematica. For circuits composed from the Toffoli and Hadamard gates the package can also output the corresponding multivariate polynomial system over F2 whose number of solutions in F2 determines the circuit matrix. Thereby the matrix can also be constructed by applying to the polynomial system the Gröbner basis technique based on the corresponding functions built-in Mathematica. We illustrate the package and the method used by a number of examples.
The implementation of the Method of Inverse Differential Operators (MIDO) which is an extension t... more The implementation of the Method of Inverse Differential Operators (MIDO) which is an extension to DSolve in Mathematica is applied to Initial Value Problems (IVP) and Boundary Conditions (BC) of homogeneous and non-homogeneous, linear PDEs of 2nd order such as the Laplace equation, the wave equation and the heat/ diffusion equation with respect to different types of boundary conditions. For selected examples of 2nd order PDEs explicit analytical solutions will be given in order to demonstrate the potential of MIDO.
Lecture Notes in Computer Science, 2009
In this paper we briefly describe a Mathematica package for simulation of quantum circuits and il... more In this paper we briefly describe a Mathematica package for simulation of quantum circuits and illustrate some of its features by simple examples. Unlike other Mathematica-based quantum simulators, our program provides a user-friendly graphical interface for generating quantum circuits and computing the circuit unitary matrices. It can be used for designing and testing different quantum algorithms. As an example we consider a quantum circuit implementing Grover's search algorithm and show that it gives a quadratic speed-up in solving the search problem.
In the present paper which is an extended version of paper [1] we consider a Mathematica-based pa... more In the present paper which is an extended version of paper [1] we consider a Mathematica-based package for simulation of quantum circuits. It provides a user-friendly graphical interface to specify a quantum circuit, to draw it, and to construct the unitary matrix for quantum computation defined by the circuit. The matrix is computed by means of the linear algebra tools built-in Mathematica. For circuits composed from the Toffoli and Hadamard gates the package can also output the corresponding multivariate polynomial system over F 2 whose number of solutions in F 2 determines the circuit matrix. Thereby the matrix can also be constructed by applying to the polynomial system the Gröbner basis technique based on the corresponding functions built-in Mathematica. We illustrate the package and the method used by a number of examples.
Journal of Mathematics and System Science
While the first part was devoted primarily to the main procedures calculateResidues and ContourIn... more While the first part was devoted primarily to the main procedures calculateResidues and ContourIntegration applied to a wide class of complex functions f(z) which are rational polynomials, products of rational and trigonometric/ hyperbolic functions, rational functions consisting of trigonometric/hyperbolic functions. However, the investigations of the second part of this paper are special topics which occur in the context of contour integration and are of interest in itselves. The issues discussed in this paper are : Notation selectPoles_ ,polesRange_ ,onoff_ f_z_ ⟺ ContourIntegral[f_,z_,selectPoles_,polesRange_,onoff_] Replacement Rules and Shortcuts This are substitution rules for {sin(θ), cos(θ) } and {sinh(θ), cosh(θ) } not included in the package ContourIntegration z=.; trigRule:= Sin[θ_] 1 2 z -1 z ,Cos[θ_] 1 2 z + 1 z ,Csc[θ_] 2 z -1 z ,Sec[θ_] 2 z + 1 z , Tan[θ_] - z -1 z z + 1 z ,Cot[θ_] z+ 1 z z-1 z ,θ_ 1 z z ; (* z = θ *) hypRule:= Sinh[θ_] 1 2 z -1 z ,Cosh[θ_] 1 2 z + 1 z ,Csch[θ_] 2 z -1 z ,Sech[θ_] 2 z + 1 z , Tanh[θ_] z -1 z z + 1 z ,Coth[θ_] z + 1 z z -1 z , θ_ 1 z z ; (* z = θ *) ContourIntegration_P2.nb 4
Journal of Mathematics and System Science
The intention of this first part of a sequel of articles is to present an implementation for Cont... more The intention of this first part of a sequel of articles is to present an implementation for Contour Integration which is still missing in Mathematica. There had been some early attempts to establish numerical contour integration with NIntegrate and even line integrals over parametrically defined curves. But no symbolic contour integration procedure is implemented in Mathematica yet although in the Wolfram Functions Site reference is given to many integral representations for special functions in terms of contour integrals. With the package ContourIntegration.m an attempt is made to introduce a rather general procedure ContourIntegration which covers a wide class of functions for the integrand f(z) (rational polynomials, products of rational and trigonometric/hyperbolic functions, rational functions consisting of trigonometric/hyperbolic functions, some special functions). Notation γ_ f_z_ ⟹ NIntegrate[f_,Evaluate[Join[{z_},γ_]]] ,WorkingForm tF Notation θ_a_ θ_b_ f_z_ z_ g_ ⟹ NIntegrateEvaluateSimplify f_ Dt[z_]/.z_g_ Dt[θ_] ,{θ_,a_,b_} ,WorkingForm tF Line integrals ∫ ℒ (t) f (R (t)) · t[R] Notation ℒ_,p_ f_ . t[r_] ⟺ LineIntegral[f_ .Dt[r_],ℒ_,p_,r_] Symbolic contour integrals ∮ selPol,polRange,onoff f (z) z Notation selectPoles_ ,polesRange_ ,onoff_ f_z_ ⟺ ContourIntegral[f_,z_,selectPoles_,polesRange_,onoff_] Replacement Rules This are substitution rules for {sin(θ), cos(θ) } and {sinh(θ), cosh(θ) } not included in the package ContourIntegration ContourIntegration_P1.nb 112 z=.; trigRule:= Sin[θ_] 1 2 z -1 z ,Cos[θ_] 1 2 z + 1 z ,Csc[θ_] 2 z -1 z ,Sec[θ_] 2 z + 1 z ,
ABSTRACT The intention of this paper is to investigate the mathematical algorithms which are invo... more ABSTRACT The intention of this paper is to investigate the mathematical algorithms which are involved in Constructive Solid Geometry (CSG). Besides basic primitives more sophisticated solid are created, such as superquadrics, polyhedra and 3D objects either defined by closed algebraic surfaces or Boolean functions. These 3D objects are subjected to geometric transformations (scaling, rotation, translation) and thus can be deformed, oriented and positioned at any spatial location. By means of various Boolean operators applied to these solids it is possible to combine them to more complex bodies. Operations typical for CSG systems such as extrusion and sweeping are discussed too; they admit creatation of 3D solids from 2D Boolean functions by extrusion. Skewed objects such as prisms etc. will be generated by sweeping techniques. Initialization Introduction CAD systems (see [1], [2a] such as solid modeling systems or volume modeler essentially apply Boolean operations. These mathematical methods used for the construction of 3D solids play an important part in CAD systems for simulation of components too. As regards to the mathematical description of 3D objects several possibilities exist : Boundary Representation (B-rep) : In this case a 3D object is defined by its confining surfaces. B-rep models are preferentially applied for visualization of 3D computer graphics and also in CAD programs because they can be processed algorithmically fast. Furthermore, this method lends itself to the description of volume models too; the object is described by its confining surfaces only where it has to be ensured by an algorithm that the hull is closed. [2b] Constructive Solid Geometry (CSG) :
The implementation of the Method of Inverse Differential Operators (MIDO) which is an extension t... more The implementation of the Method of Inverse Differential Operators (MIDO) which is an extension to DSolve in Mathematica is applied to Initial Value Problems (IVP) and Boundary Conditions (BC) of homogeneous and non-homogeneous, linear PDEs of 2nd order such as the Laplace equation, the wave equation and the heat/ diffusion equation with respect to different types of boundary conditions. For selected examples of 2nd order PDEs explicit analytical solutions will be given in order to demonstrate the potential of MIDO.
In this paper the method of inverse differential operators for solving PDEs as given in [1] is im... more In this paper the method of inverse differential operators for solving PDEs as given in [1] is implemented into Mathematica. A wide class of PDEs for which the differential polynomial can be decomposed into linear factors and for nonhomogenuities which comprise certain combinations of exponential, trigonometric and hyperbolic functions can be solved. In most cases the built-in Mathematica routine DSolve cannot find a solution.
Physics of Particles and Nuclei Letters, 2009
One reason for this is the potential ability of a quantum computer to do certain computational ta... more One reason for this is the potential ability of a quantum computer to do certain computational tasks much more efficiently than they can be done by any classical computer Two the most famous examples of such calculations are Shor's algorithm for efficient factorization of large integers and Grover's algorithm of element search in an unsorted list.
Lecture Notes in Physics, 1979
ABSTRACT Without Abstract
Zeitschrift f�r Physik, 1970
The formalism of 2-nucleon-transfer-reactions developed by Glendenning and Lin is slightly genera... more The formalism of 2-nucleon-transfer-reactions developed by Glendenning and Lin is slightly generalized using a cluster model description for incoming and outgoing particles in a consistent way. The choice of an adequate spin-dependent nucleon-nucleon interaction of Gaussian radial form permits an analytic evaluation of the intrinsic matrix element. A relatives-state motion for the nucleon pair transferred leads to an additional restriction of the reaction model. Further, the polarization of outgoing particles is derived from the general structure of theT-matrix element. If spin-orbit interaction is neglected polarization effects are only expected for the transfer of unlike nucleon pairs.
In the present paper which is an extended version of paper [1] we consider a Mathematica-based pa... more In the present paper which is an extended version of paper [1] we consider a Mathematica-based package for simulation of quantum circuits. It provides a user-friendly graphical interface to specify a quantum circuit, to draw it, and to construct the unitary matrix for quantum computation defined by the circuit. The matrix is computed by means of the linear algebra tools built-in Mathematica. For circuits composed from the Toffoli and Hadamard gates the package can also output the corresponding multivariate polynomial system over F 2 whose number of solutions in F 2 determines the circuit matrix. Thereby the matrix can also be constructed by applying to the polynomial system the Gröbner basis technique based on the corresponding functions built-in Mathematica. We illustrate the package and the method used by a number of examples.
In the present paper which is an extended version of paper [1] we consider a Mathematica-based pa... more In the present paper which is an extended version of paper [1] we consider a Mathematica-based package for simulation of quantum circuits. It provides a user-friendly graphical interface to specify a quantum circuit, to draw it, and to construct the unitary matrix for quantum computation defined by the circuit. The matrix is computed by means of the linear algebra tools built-in Mathematica. For circuits composed from the Toffoli and Hadamard gates the package can also output the corresponding multivariate polynomial system over F2 whose number of solutions in F2 determines the circuit matrix. Thereby the matrix can also be constructed by applying to the polynomial system the Gröbner basis technique based on the corresponding functions built-in Mathematica. We illustrate the package and the method used by a number of examples.
The implementation of the Method of Inverse Differential Operators (MIDO) which is an extension t... more The implementation of the Method of Inverse Differential Operators (MIDO) which is an extension to DSolve in Mathematica is applied to Initial Value Problems (IVP) and Boundary Conditions (BC) of homogeneous and non-homogeneous, linear PDEs of 2nd order such as the Laplace equation, the wave equation and the heat/ diffusion equation with respect to different types of boundary conditions. For selected examples of 2nd order PDEs explicit analytical solutions will be given in order to demonstrate the potential of MIDO.
Lecture Notes in Computer Science, 2009
In this paper we briefly describe a Mathematica package for simulation of quantum circuits and il... more In this paper we briefly describe a Mathematica package for simulation of quantum circuits and illustrate some of its features by simple examples. Unlike other Mathematica-based quantum simulators, our program provides a user-friendly graphical interface for generating quantum circuits and computing the circuit unitary matrices. It can be used for designing and testing different quantum algorithms. As an example we consider a quantum circuit implementing Grover's search algorithm and show that it gives a quadratic speed-up in solving the search problem.
In the present paper which is an extended version of paper [1] we consider a Mathematica-based pa... more In the present paper which is an extended version of paper [1] we consider a Mathematica-based package for simulation of quantum circuits. It provides a user-friendly graphical interface to specify a quantum circuit, to draw it, and to construct the unitary matrix for quantum computation defined by the circuit. The matrix is computed by means of the linear algebra tools built-in Mathematica. For circuits composed from the Toffoli and Hadamard gates the package can also output the corresponding multivariate polynomial system over F 2 whose number of solutions in F 2 determines the circuit matrix. Thereby the matrix can also be constructed by applying to the polynomial system the Gröbner basis technique based on the corresponding functions built-in Mathematica. We illustrate the package and the method used by a number of examples.
Journal of Mathematics and System Science
While the first part was devoted primarily to the main procedures calculateResidues and ContourIn... more While the first part was devoted primarily to the main procedures calculateResidues and ContourIntegration applied to a wide class of complex functions f(z) which are rational polynomials, products of rational and trigonometric/ hyperbolic functions, rational functions consisting of trigonometric/hyperbolic functions. However, the investigations of the second part of this paper are special topics which occur in the context of contour integration and are of interest in itselves. The issues discussed in this paper are : Notation selectPoles_ ,polesRange_ ,onoff_ f_z_ ⟺ ContourIntegral[f_,z_,selectPoles_,polesRange_,onoff_] Replacement Rules and Shortcuts This are substitution rules for {sin(θ), cos(θ) } and {sinh(θ), cosh(θ) } not included in the package ContourIntegration z=.; trigRule:= Sin[θ_] 1 2 z -1 z ,Cos[θ_] 1 2 z + 1 z ,Csc[θ_] 2 z -1 z ,Sec[θ_] 2 z + 1 z , Tan[θ_] - z -1 z z + 1 z ,Cot[θ_] z+ 1 z z-1 z ,θ_ 1 z z ; (* z = θ *) hypRule:= Sinh[θ_] 1 2 z -1 z ,Cosh[θ_] 1 2 z + 1 z ,Csch[θ_] 2 z -1 z ,Sech[θ_] 2 z + 1 z , Tanh[θ_] z -1 z z + 1 z ,Coth[θ_] z + 1 z z -1 z , θ_ 1 z z ; (* z = θ *) ContourIntegration_P2.nb 4
Journal of Mathematics and System Science
The intention of this first part of a sequel of articles is to present an implementation for Cont... more The intention of this first part of a sequel of articles is to present an implementation for Contour Integration which is still missing in Mathematica. There had been some early attempts to establish numerical contour integration with NIntegrate and even line integrals over parametrically defined curves. But no symbolic contour integration procedure is implemented in Mathematica yet although in the Wolfram Functions Site reference is given to many integral representations for special functions in terms of contour integrals. With the package ContourIntegration.m an attempt is made to introduce a rather general procedure ContourIntegration which covers a wide class of functions for the integrand f(z) (rational polynomials, products of rational and trigonometric/hyperbolic functions, rational functions consisting of trigonometric/hyperbolic functions, some special functions). Notation γ_ f_z_ ⟹ NIntegrate[f_,Evaluate[Join[{z_},γ_]]] ,WorkingForm tF Notation θ_a_ θ_b_ f_z_ z_ g_ ⟹ NIntegrateEvaluateSimplify f_ Dt[z_]/.z_g_ Dt[θ_] ,{θ_,a_,b_} ,WorkingForm tF Line integrals ∫ ℒ (t) f (R (t)) · t[R] Notation ℒ_,p_ f_ . t[r_] ⟺ LineIntegral[f_ .Dt[r_],ℒ_,p_,r_] Symbolic contour integrals ∮ selPol,polRange,onoff f (z) z Notation selectPoles_ ,polesRange_ ,onoff_ f_z_ ⟺ ContourIntegral[f_,z_,selectPoles_,polesRange_,onoff_] Replacement Rules This are substitution rules for {sin(θ), cos(θ) } and {sinh(θ), cosh(θ) } not included in the package ContourIntegration ContourIntegration_P1.nb 112 z=.; trigRule:= Sin[θ_] 1 2 z -1 z ,Cos[θ_] 1 2 z + 1 z ,Csc[θ_] 2 z -1 z ,Sec[θ_] 2 z + 1 z ,
ABSTRACT The intention of this paper is to investigate the mathematical algorithms which are invo... more ABSTRACT The intention of this paper is to investigate the mathematical algorithms which are involved in Constructive Solid Geometry (CSG). Besides basic primitives more sophisticated solid are created, such as superquadrics, polyhedra and 3D objects either defined by closed algebraic surfaces or Boolean functions. These 3D objects are subjected to geometric transformations (scaling, rotation, translation) and thus can be deformed, oriented and positioned at any spatial location. By means of various Boolean operators applied to these solids it is possible to combine them to more complex bodies. Operations typical for CSG systems such as extrusion and sweeping are discussed too; they admit creatation of 3D solids from 2D Boolean functions by extrusion. Skewed objects such as prisms etc. will be generated by sweeping techniques. Initialization Introduction CAD systems (see [1], [2a] such as solid modeling systems or volume modeler essentially apply Boolean operations. These mathematical methods used for the construction of 3D solids play an important part in CAD systems for simulation of components too. As regards to the mathematical description of 3D objects several possibilities exist : Boundary Representation (B-rep) : In this case a 3D object is defined by its confining surfaces. B-rep models are preferentially applied for visualization of 3D computer graphics and also in CAD programs because they can be processed algorithmically fast. Furthermore, this method lends itself to the description of volume models too; the object is described by its confining surfaces only where it has to be ensured by an algorithm that the hull is closed. [2b] Constructive Solid Geometry (CSG) :
The implementation of the Method of Inverse Differential Operators (MIDO) which is an extension t... more The implementation of the Method of Inverse Differential Operators (MIDO) which is an extension to DSolve in Mathematica is applied to Initial Value Problems (IVP) and Boundary Conditions (BC) of homogeneous and non-homogeneous, linear PDEs of 2nd order such as the Laplace equation, the wave equation and the heat/ diffusion equation with respect to different types of boundary conditions. For selected examples of 2nd order PDEs explicit analytical solutions will be given in order to demonstrate the potential of MIDO.
In this paper the method of inverse differential operators for solving PDEs as given in [1] is im... more In this paper the method of inverse differential operators for solving PDEs as given in [1] is implemented into Mathematica. A wide class of PDEs for which the differential polynomial can be decomposed into linear factors and for nonhomogenuities which comprise certain combinations of exponential, trigonometric and hyperbolic functions can be solved. In most cases the built-in Mathematica routine DSolve cannot find a solution.
Physics of Particles and Nuclei Letters, 2009
One reason for this is the potential ability of a quantum computer to do certain computational ta... more One reason for this is the potential ability of a quantum computer to do certain computational tasks much more efficiently than they can be done by any classical computer Two the most famous examples of such calculations are Shor's algorithm for efficient factorization of large integers and Grover's algorithm of element search in an unsorted list.
Lecture Notes in Physics, 1979
ABSTRACT Without Abstract