Bernd Bank - Academia.edu (original) (raw)

Papers by Bernd Bank

Research paper thumbnail of Intrinsic complexity estimates in polynomial optimization

Journal of Complexity, 2014

It is known that point searching in basic semialgebraic sets and the search for globally minimal ... more It is known that point searching in basic semialgebraic sets and the search for globally minimal points in polynomial optimization tasks can be carried out using (s d) O(n) arithmetic operations, where n and s are the numbers of variables and constraints and d is the maximal degree of the polynomials involved. Subject to certain conditions, we associate to each of these problems an intrinsic system degree which becomes in worst case of order (n d) O(n) and which measures the intrinsic complexity of the task under consideration. We design non-uniform deterministic or uniform probabilistic algorithms of intrinsic, quasi-polynomial complexity which solve these problems.

Research paper thumbnail of Constraint Databases, Geometric Elimination and Geographic Information Systems, 20.05. - 25.05.2007

Research paper thumbnail of 07212 Abstracts Collection -- Constraint Databases, Geometric Elimination ang Geographic Information Systems

Dagstuhl Seminars, 2007

From 20.05. to 25.05., the Dagstuhl Seminar 07212 Constraint Databases, Geometric Elimination and... more From 20.05. to 25.05., the Dagstuhl Seminar 07212 Constraint Databases, Geometric Elimination and Geographic Information Systems was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The rst section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.

Research paper thumbnail of OpTERA - Optimization and Real Equation Solving

German-Argentinian Workshop on Information Technology, 1999

Research paper thumbnail of 07212 Manifesto -- Constraint Databases, Geometric Elimination ang Geographic Information Systems

Dagstuhl Seminars, 2007

Dagstuhl Seminar 07212 1 Description and goals of the seminar During the past 15 years the topic ... more Dagstuhl Seminar 07212 1 Description and goals of the seminar During the past 15 years the topic of constraint databases (CDB) [2, 3] has evolved into a mature area of computer science with sound mathematical foundations and with a profound theoretical understanding of the expressive power of a variety of query languages. Constraint databases are especially suited for applications in which possibly infinite sets of continuous data, which have a geometric interpretation, have to be stored in a computer. Today, the most important application domains of constraint databases are geographic information systems (GIS), spatial databases and spatio-temporal databases . In these applications infinite geometrical sets of continuous data are finitely represented by means of finite combinations of polynomial equality and inequality constraints that describe these data sets (in mathematical terms these geometrical data sets are known as semi-algebraic sets and they have been extensively studied in real algebraic geometry). On the other hand, constraint databases provide us with a new view of classic (linear and nonlinear) optimization theory.

Research paper thumbnail of Combining Digital Waveguide and Functional Transformation Methods for Physical Modeling of Musical I

Digital sound synthesis based on physical models is realized in real-time applications mostly wit... more Digital sound synthesis based on physical models is realized in real-time applications mostly with the well known digital waveguide method (DWG). It approximates the underlying physical behavior of a vibrating structure in a computationally efficient way. Due to these computational efficient approximations, the waveguide method looses the direct connection to the parameters of the underlying physical model. The recently introduced functional transformation method (FTM) on the other hand solves the underlying physical model ...

Research paper thumbnail of Variétés bipolaires et résolution d'une équation polynomiale réelle

In previous work we designed an ecient procedure that finds an algebraic sample point for each co... more In previous work we designed an ecient procedure that finds an algebraic sample point for each connected component of a smooth real complete intersection variety. This procedure exploits geometric properties of generic polar varieties and its complexity is intrinsic with respect to the problem. In the present paper we introduce a natural construction that allows to tackle the case of

Research paper thumbnail of The light is polar

Research paper thumbnail of Polar varieties and efficient real elimination

Let S 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f... more Let S 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f 1 , . . . , f p . This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S 0 . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of S 0 . This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f 1 , . . . , f p and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system f 1 , . . . , f p .

Research paper thumbnail of Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case

Journal of Complexity, 1996

The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais... more The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo \cite{gihemorpar} can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in \cite{gihemorpar} yields a method for

Research paper thumbnail of Stability of mixed-integer quadratic programming problems

Mathematical Programming Studies, 1984

The purpose of this paper is to present a stability analysis for mixed-integer quadratic programs... more The purpose of this paper is to present a stability analysis for mixed-integer quadratic programs under perturbations of the linear part of the quadratic objective function and the right-hand sides of the linear inequality constraints in the absence of boundedness on the feasible region and convexity on the objective function. The only hypothesis needed is the rationality of the matrices

Research paper thumbnail of A geometrical bound for integer programming with polynomial constraints

Lecture Notes in Computer Science, 1991

... Bernd Bank*, Teresa Krick**, Reinhard Mandel* & Pablo Solern6** ... Let X1,'--,Xn be... more ... Bernd Bank*, Teresa Krick**, Reinhard Mandel* & Pablo Solern6** ... Let X1,'--,Xn be indeterminates over R. One calls quasiconvex a polynomial FER[X1,-.-,X~] if for all AER the lower level set {x CR; F(x) < A } is a convex subset of R. It is clear that the convex polynomials form a ...

Research paper thumbnail of Computability and Complexity of Polynomial Optimization Problems

Lecture Notes in Economics and Mathematical Systems, 1992

It is known that point searching in basic semialgebraic sets and the search for globally minimal ... more It is known that point searching in basic semialgebraic sets and the search for globally minimal points in polynomial optimization tasks can be carried out using (s d) O(n) arithmetic operations, where n and s are the numbers of variables and constraints and d is the maximal degree of the polynomials involved. Subject to certain conditions, we associate to each of these problems an intrinsic system degree which becomes in worst case of order (n d) O(n) and which measures the intrinsic complexity of the task under consideration. We design non-uniform deterministic or uniform probabilistic algorithms of intrinsic, quasi-polynomial complexity which solve these problems.

Research paper thumbnail of On the geometry of polar varieties

We have developed in the past several algorithms with intrinsic complexity bounds for the problem... more We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these algorithms. Our results form also the geometrical main ingredients for the computational treatment of singular hypersurfaces.

Research paper thumbnail of Bipolar varieties and real solving of a singular polynomial equation

In this paper we introduce the concept of a bipolar variety of a real algebraic hypersurface. Thi... more In this paper we introduce the concept of a bipolar variety of a real algebraic hypersurface. This notion is then used for the design and complexity estimations of a novel type of algorithms that finds algebraic sample point for the connected component of a singular real hypersurface. The complexity of these algorithms is polynomial in the maximal geometric degree of

Research paper thumbnail of Polar varieties and efficient real elimination

Mathematische Zeitschrift, 2001

Let S 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f... more Let S 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f 1 , . . . , f p . This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S 0 . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of S 0 . This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f 1 , . . . , f p and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system f 1 , . . . , f p .

Research paper thumbnail of Polar Varieties, Real Equation Solving, and Data Structures: The Hypersurface Case

Journal of Complexity, 1997

In this paper we apply for the first time a new method for multivariate equationsolving which was... more In this paper we apply for the first time a new method for multivariate equationsolving which was developed in [18], [19], [20] for complex root determination to thereal case. Our main result concerns the problem of finding at least one representativepoint for each connected component of a real compact and smooth hypersurface.The basic algorithm of [18], [19], [20] yields a new method for symbolically solvingzero-dimensional polynomial equation systems over the complex numbers. One...

Research paper thumbnail of Analysis of handling noises on wound strings

The Journal of the Acoustical Society of America, 2007

This study analyzes the handling noises that occur when a finger is slid along a wound string. Th... more This study analyzes the handling noises that occur when a finger is slid along a wound string. The resulting noise has a harmonic structure due to the periodic texture of the wound string. The frequency of the harmonics and the root-mean-square amplitude of the noise were found to be linearly proportional to the sliding speed. In addition, the sliding excites the longitudinal modes of the string, thus resulting in a set of static harmonics in the noise spectrum. The sliding excites different longitudinal modes depending on the sliding location.

Research paper thumbnail of Generalized polar varieties: geometry and algorithms

Let W be a closed algebraic subvariety of the n -dimensional projective space over the complex or... more Let W be a closed algebraic subvariety of the n -dimensional projective space over the complex or real numbers and suppose that W is non-empty and equidimensional. In this paper we generalize the classic notion of polar variety of W associated with a given linear subvariety of the ambient space of W . As particular instances of this new notion of generalized polar variety we reobtain the classic ones and two new types of polar varieties, called dual and (in case that W is affine) conic. We show that for a generic choice of their parameters the generalized polar varieties of W are empty or equidimensional and smooth in any regular point of W . In the case that the variety W is affine and smooth and has a complete intersection ideal of definition, we are able, for a generic parameter choice, to describe locally the generalized polar varieties of W by explicit equations. Finally, we use this description in order to design a new, highly efficient elimination procedure for the following algorithmic task: In case, that the variety W is Q -definable and affine, having a complete intersection ideal of definition, and that the real trace of W is non-empty and smooth, find for each connected component of the real trace of W a representative point.

Research paper thumbnail of On the intrinsic complexity of point finding in real singular hypersurfaces

In previous work we designed an efficient procedure that finds an algebraic sample point for each... more In previous work we designed an efficient procedure that finds an algebraic sample point for each connected component of a smooth real complete intersection variety. This procedure exploits geometric properties of generic polar varieties and its complexity is intrinsic with respect to the problem. In the present paper we introduce a natural construction that allows to tackle the case of a non-smooth real hypersurface by means of a reduction to a smooth complete intersection.

Research paper thumbnail of Intrinsic complexity estimates in polynomial optimization

Journal of Complexity, 2014

It is known that point searching in basic semialgebraic sets and the search for globally minimal ... more It is known that point searching in basic semialgebraic sets and the search for globally minimal points in polynomial optimization tasks can be carried out using (s d) O(n) arithmetic operations, where n and s are the numbers of variables and constraints and d is the maximal degree of the polynomials involved. Subject to certain conditions, we associate to each of these problems an intrinsic system degree which becomes in worst case of order (n d) O(n) and which measures the intrinsic complexity of the task under consideration. We design non-uniform deterministic or uniform probabilistic algorithms of intrinsic, quasi-polynomial complexity which solve these problems.

Research paper thumbnail of Constraint Databases, Geometric Elimination and Geographic Information Systems, 20.05. - 25.05.2007

Research paper thumbnail of 07212 Abstracts Collection -- Constraint Databases, Geometric Elimination ang Geographic Information Systems

Dagstuhl Seminars, 2007

From 20.05. to 25.05., the Dagstuhl Seminar 07212 Constraint Databases, Geometric Elimination and... more From 20.05. to 25.05., the Dagstuhl Seminar 07212 Constraint Databases, Geometric Elimination and Geographic Information Systems was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The rst section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.

Research paper thumbnail of OpTERA - Optimization and Real Equation Solving

German-Argentinian Workshop on Information Technology, 1999

Research paper thumbnail of 07212 Manifesto -- Constraint Databases, Geometric Elimination ang Geographic Information Systems

Dagstuhl Seminars, 2007

Dagstuhl Seminar 07212 1 Description and goals of the seminar During the past 15 years the topic ... more Dagstuhl Seminar 07212 1 Description and goals of the seminar During the past 15 years the topic of constraint databases (CDB) [2, 3] has evolved into a mature area of computer science with sound mathematical foundations and with a profound theoretical understanding of the expressive power of a variety of query languages. Constraint databases are especially suited for applications in which possibly infinite sets of continuous data, which have a geometric interpretation, have to be stored in a computer. Today, the most important application domains of constraint databases are geographic information systems (GIS), spatial databases and spatio-temporal databases . In these applications infinite geometrical sets of continuous data are finitely represented by means of finite combinations of polynomial equality and inequality constraints that describe these data sets (in mathematical terms these geometrical data sets are known as semi-algebraic sets and they have been extensively studied in real algebraic geometry). On the other hand, constraint databases provide us with a new view of classic (linear and nonlinear) optimization theory.

Research paper thumbnail of Combining Digital Waveguide and Functional Transformation Methods for Physical Modeling of Musical I

Digital sound synthesis based on physical models is realized in real-time applications mostly wit... more Digital sound synthesis based on physical models is realized in real-time applications mostly with the well known digital waveguide method (DWG). It approximates the underlying physical behavior of a vibrating structure in a computationally efficient way. Due to these computational efficient approximations, the waveguide method looses the direct connection to the parameters of the underlying physical model. The recently introduced functional transformation method (FTM) on the other hand solves the underlying physical model ...

Research paper thumbnail of Variétés bipolaires et résolution d'une équation polynomiale réelle

In previous work we designed an ecient procedure that finds an algebraic sample point for each co... more In previous work we designed an ecient procedure that finds an algebraic sample point for each connected component of a smooth real complete intersection variety. This procedure exploits geometric properties of generic polar varieties and its complexity is intrinsic with respect to the problem. In the present paper we introduce a natural construction that allows to tackle the case of

Research paper thumbnail of The light is polar

Research paper thumbnail of Polar varieties and efficient real elimination

Let S 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f... more Let S 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f 1 , . . . , f p . This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S 0 . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of S 0 . This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f 1 , . . . , f p and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system f 1 , . . . , f p .

Research paper thumbnail of Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case

Journal of Complexity, 1996

The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais... more The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo \cite{gihemorpar} can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in \cite{gihemorpar} yields a method for

Research paper thumbnail of Stability of mixed-integer quadratic programming problems

Mathematical Programming Studies, 1984

The purpose of this paper is to present a stability analysis for mixed-integer quadratic programs... more The purpose of this paper is to present a stability analysis for mixed-integer quadratic programs under perturbations of the linear part of the quadratic objective function and the right-hand sides of the linear inequality constraints in the absence of boundedness on the feasible region and convexity on the objective function. The only hypothesis needed is the rationality of the matrices

Research paper thumbnail of A geometrical bound for integer programming with polynomial constraints

Lecture Notes in Computer Science, 1991

... Bernd Bank*, Teresa Krick**, Reinhard Mandel* & Pablo Solern6** ... Let X1,'--,Xn be... more ... Bernd Bank*, Teresa Krick**, Reinhard Mandel* & Pablo Solern6** ... Let X1,'--,Xn be indeterminates over R. One calls quasiconvex a polynomial FER[X1,-.-,X~] if for all AER the lower level set {x CR; F(x) < A } is a convex subset of R. It is clear that the convex polynomials form a ...

Research paper thumbnail of Computability and Complexity of Polynomial Optimization Problems

Lecture Notes in Economics and Mathematical Systems, 1992

It is known that point searching in basic semialgebraic sets and the search for globally minimal ... more It is known that point searching in basic semialgebraic sets and the search for globally minimal points in polynomial optimization tasks can be carried out using (s d) O(n) arithmetic operations, where n and s are the numbers of variables and constraints and d is the maximal degree of the polynomials involved. Subject to certain conditions, we associate to each of these problems an intrinsic system degree which becomes in worst case of order (n d) O(n) and which measures the intrinsic complexity of the task under consideration. We design non-uniform deterministic or uniform probabilistic algorithms of intrinsic, quasi-polynomial complexity which solve these problems.

Research paper thumbnail of On the geometry of polar varieties

We have developed in the past several algorithms with intrinsic complexity bounds for the problem... more We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these algorithms. Our results form also the geometrical main ingredients for the computational treatment of singular hypersurfaces.

Research paper thumbnail of Bipolar varieties and real solving of a singular polynomial equation

In this paper we introduce the concept of a bipolar variety of a real algebraic hypersurface. Thi... more In this paper we introduce the concept of a bipolar variety of a real algebraic hypersurface. This notion is then used for the design and complexity estimations of a novel type of algorithms that finds algebraic sample point for the connected component of a singular real hypersurface. The complexity of these algorithms is polynomial in the maximal geometric degree of

Research paper thumbnail of Polar varieties and efficient real elimination

Mathematische Zeitschrift, 2001

Let S 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f... more Let S 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f 1 , . . . , f p . This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S 0 . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of S 0 . This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f 1 , . . . , f p and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system f 1 , . . . , f p .

Research paper thumbnail of Polar Varieties, Real Equation Solving, and Data Structures: The Hypersurface Case

Journal of Complexity, 1997

In this paper we apply for the first time a new method for multivariate equationsolving which was... more In this paper we apply for the first time a new method for multivariate equationsolving which was developed in [18], [19], [20] for complex root determination to thereal case. Our main result concerns the problem of finding at least one representativepoint for each connected component of a real compact and smooth hypersurface.The basic algorithm of [18], [19], [20] yields a new method for symbolically solvingzero-dimensional polynomial equation systems over the complex numbers. One...

Research paper thumbnail of Analysis of handling noises on wound strings

The Journal of the Acoustical Society of America, 2007

This study analyzes the handling noises that occur when a finger is slid along a wound string. Th... more This study analyzes the handling noises that occur when a finger is slid along a wound string. The resulting noise has a harmonic structure due to the periodic texture of the wound string. The frequency of the harmonics and the root-mean-square amplitude of the noise were found to be linearly proportional to the sliding speed. In addition, the sliding excites the longitudinal modes of the string, thus resulting in a set of static harmonics in the noise spectrum. The sliding excites different longitudinal modes depending on the sliding location.

Research paper thumbnail of Generalized polar varieties: geometry and algorithms

Let W be a closed algebraic subvariety of the n -dimensional projective space over the complex or... more Let W be a closed algebraic subvariety of the n -dimensional projective space over the complex or real numbers and suppose that W is non-empty and equidimensional. In this paper we generalize the classic notion of polar variety of W associated with a given linear subvariety of the ambient space of W . As particular instances of this new notion of generalized polar variety we reobtain the classic ones and two new types of polar varieties, called dual and (in case that W is affine) conic. We show that for a generic choice of their parameters the generalized polar varieties of W are empty or equidimensional and smooth in any regular point of W . In the case that the variety W is affine and smooth and has a complete intersection ideal of definition, we are able, for a generic parameter choice, to describe locally the generalized polar varieties of W by explicit equations. Finally, we use this description in order to design a new, highly efficient elimination procedure for the following algorithmic task: In case, that the variety W is Q -definable and affine, having a complete intersection ideal of definition, and that the real trace of W is non-empty and smooth, find for each connected component of the real trace of W a representative point.

Research paper thumbnail of On the intrinsic complexity of point finding in real singular hypersurfaces

In previous work we designed an efficient procedure that finds an algebraic sample point for each... more In previous work we designed an efficient procedure that finds an algebraic sample point for each connected component of a smooth real complete intersection variety. This procedure exploits geometric properties of generic polar varieties and its complexity is intrinsic with respect to the problem. In the present paper we introduce a natural construction that allows to tackle the case of a non-smooth real hypersurface by means of a reduction to a smooth complete intersection.