CA Applications (1)

Lecture 3

General view

CA has spread into many areas which about 30 years ago, were only accessible to mathematically skilled experts.

Applications of CA range over the entire spectrum of development, research, production, and education.

CA problems arise in industry, commerce, software engineering, banking and insurance applications.

Classical areas of CA applications are physics and mathematics.

However, more and more methods and systems of CA are also utilized in computer science, in engineering, and in natural sciences.

A very important application area of CA is education.

Physics One of the classical and traditional areas of CA application, most

likely being the reason why physicists always participated at the forefront in the practical development of

Nowadays CA is used in essentially all areas of physics. Some typical applications are:

Elementary Particle Physics: In this field CA tools are indispensable for generation, evaluation, and summing up of Feynman integrals. They are also used in quantum chromodynamics and electroweak interaction.

Gravity: In gravity and general relativity, CA was used right from its beginnings. A classical problem us the N-body problem of Newtonian mechanics with gravitational interaction.

Applications of Differential Geometry: Differential geometry plays a fundamental role in physics such as in mechanics, in field and relativity theory, gauge field theory, string theory etc.

Applications of Differential Equations: In determining Killing tensors and conserved quantities for partial differential equations.

Elementary particle physics Applications of CA in particle physics are mainly concerned with

calculations in the framework of perturbation theory for Quatum Filed Theories (QFT).

The perturbation theory is expressed in terms of Feynman diagrams which are built from vertices, describing point-interactions of various particles, and their connections in terms of particle propagators. Their representation is usually done in momentum space, i.e. the space of four-dimensional momentum vectors.

The order of the perturbation theory can be identified with the number of closed loops in the diagrams, each loop corresponding to a four-dimensional integration in momentum space.

The precision in high energy accelerator experiments is so high that on the side of the theory a corresponding accuracy in precision is needed in order to find new physics in case deviations would show up.

This is the reason why higher order calculations are needed where quite often thousands of diagrams contribute and even one-loop calculations for multiple particle production have to be performed.

Elementary particle physics Technically the calculations are performed in three steps: generation of

the diagrams, simplifications and integration. In the first step for example, if thousand of diagrams are to be

calculated (e.g. for the two-loop anomalous magnetic moment of the moon - 1832 diagrams), one will not even able to investigate each of them separately, one would produce input for CA programs that will provide the momentum representation of each diagram.

The first implementations of computer algebra programs came from particle physics: Schoonschip, Form.

Macsyma, Reduce and Mathematica have partly their rots in particle physics

Several packages have been developed with different areas of applicability: FeynAarts/FyenCalc are Mathematica packages convenient for various aspects of the calculation of radiative corrections;

Mincer is a Form package for evaluating multiloop diagrams; Diana (DIagram ANAlyser) allows automatic Feynman diagram

evaluation; other packages: Geficom, Matad, Ashmedai, Grace, Comphep, Xloops.

Quantized gauge field theories The electroweak and strong interactions of elementary particles

are very successfully described by quantized gauge field theories.

The quantized nature of these theories manifests itself via corrections beyond the lowest order in the perturbative expansion, which is based on Feynman diagrams.

The evaluation of higher-order Feynman diagrams (called loops diagrams) is a very tedious but on the other hand algorithmic procedure.

Some of the first CA programs were in fact developed in order to facilitate this kind of calculations and CA has been applied in this field now for several decades.

In recent years many applications of CA in the theory of electroweak interactions have been based on the collection of Mathematica packages FeynArts, FeynCalc, FormCalc (partially written in Form) and TwoCalc and the Maple package Xloops.

Gravity Einstein’s gravitational theory, general relativity (GR), is the valid theory for

describing gravitational effects In the search for making GR compatible with quantum theory and/or

unifying it with the other interactions of nature (strong, electro-weak etc), different schemes have been developed, like the gauge approach to gravity, including supergravity and metric-affine gravity, string models, Feynman quantization schemes, or noncommutative spacetime geometries.

The CA programs applied to GR can be and partially have been extended to these more general framework, but most current programs are applied in the context of GR, gravity-based Feynman integrals or gauge models.

In GR computer algebra was used as soon as it became available. The reason for this is that for solving standard problem it is required to

manipulate a large number of terms and equations. For example a generic problem is gravity is to calculate the Ricci tensor

from a given metric; ExCalc, a Reduce package for exterior calculus, or Mathematica package

MathTensor can solve this problem. Quantities like the Ricci tensor can reach an enormous size; for most

applications (classification, numerics), these objects have to put on the computer.

Gravity A standard application of computer algebra in GR is the classification of exact

solutions. The appropriate algorithms involve an enormous amount of work. Programs for the widely used classification (e.g. Petrov’s classification) are

available for most computer algebra systems. To find out whether two solutions which look different are not just the same

solution in different coordinates, one has to solve the so-called equivalence problem.

This involves differentiation of the curvature tensor up to the seventh order. Computer algebra is also very useful for finding new solutions of the field

equations. Reduce packages were used foe example to search the solutions of the

Einstein-Maxwell equations. Relativity packages are available also in general purpose systems like

Macsyma (CTensor), Maple (tensor, cartan, NPspinor, debever, oframe, GRTensorII, Riemann), Mathematica (Cartan, TTC, MathTensor), Reduce (ExCalc, Redten, GRG, GRGEC, Classim, Crack), Derive.

Newtonian N-Body Problem The Newtonian N-body problem is one of the important issues in

celestial mechanics. It is concerned with describing the motion of N bodies (particles or

point masses) under their mutual gravitational attraction. The differential equations of motion, as given by Newton’s laws, are

not integrable in general. However, there is an important special class of solutions which can

be computer analytically. These particular solutions are called central configurations and

geometrically they describe motions in which the configuration of the bodies remain self-similar in time.

When the center of mass is shifted at the origin, central configurations are described by the fact that the acceleration vector of each body is a common scalar multiple of its position vector.

Central configurations are of fundamental importance in the study of changes in the topology of the integral manifolds of the N-body problem as well as in the analysis of expanding gravitational systems.

5-body problem The resulting eqs. for the case of spatial 5-body problem show the

underlying symmetry of the problem and are tractable, in the simple cases, by many CA methods such as direct Groebner bases computations, triangular sets methods, and a method based on invariant theory of finite groups.

The Groebner bases computations involved can be done with the FGb program, the final system consisting of 11 polynomial equations in 8 variables.

The resulting univariate polynomial is of degree 216 with big integer coefficients

The approximate values for its real roots have been found with RS and Magma

The C.C. program can be used to visualize planar or spatial central configurations.

Some polynomials systems arising in the study of central configurations in the planar 4-body problem and the spatial 5-body problem have been used as benchmarks and included in the FRISCO polynomial test suite and in CABRI.

The search for central configurations in the N-body problem of celestial mechanics offers great computational challenges and CA methods are well suited for attacking them.

CAS for Differential Geometry Modern differential geometry has established itself as a fundamental

mathematical framework for theoretical and mathematical physics. It permits an intrinsic formulation of a wide variety of theories and provides at

the same time an efficient calculus for solving problems in these theories. Substantial fragments of modern differential geometry can be found nowadays

in all major CAS. For Reduce, the package ExCalc (EXterior CALCulus) aims to provides a

syntax that is as close as possible to the notations used in standard textbooks. Applications of exterior calculus in physics are numerous; for example,

computing the component-wise representation of equations modelling the growth of crystals,

computing the variation of super-symmetric Lagrange densities, determining generalized symmetries, higher-dimensional cosmological theories etc.

The CA programs verify lengthy calculations done by hand, or they produce results which could not be obtained by pencil and paper within

a reasonable amount of time. Computing times can vary from seconds up to several days.

CAS for Differential Geometry The Reduce package GRG

has been developed for calculations in theories of gravity, classical field theory and modern differential geometry.

It performs analytic calculations with all kinds of geometrical objects: spinors, vectors, tensors, exterior differential forms, connection and related structures defined on a smooth manifold of arbitrary dimension.

Answers can be obtained to standard computational problems in gravitational theory such derivation of curvature, finding the field equations, verifying symmetry properties, calculating covariant and Lie derivatives, etc.

Another Reduce package GRGEC is also intended for applications to the theory of gravity and related

problems of geometry and classical field theory. It is aware of the majority of basic characteristics of the geometry of a

curved space-time utilized in Einstein’s gravitational theory and operates with major characteristics of many classical fields (electromagnetic field, massless spinor field, massive vector field, pressure-free dust matter, etc.).

The package maintains the input language which is maximally close to the one used for the representation of the relevant notions an the relationships taking place in the application field itself.

Differential Equations in Physics A special computer algebra package Crack

has been used to solve problems related to the computation of conservation laws of geodesic motion in curved space or of arbitrary systems of partial differential equations (PDEs).

Another program, ConLaw can be used to find conservation laws that are

nonpolynomial and have explicit variable dependence. MathLie

is a package written in Mathematica supporting the calculation of classical, nonclassical, potential, approximative, and generalized symmetries.

It is able to solve ordinary differential equations of orders greater than two by quadratures.

The theory behind MathLie is the symmetry analysis of Lie which is useful in solving any kind of differential equations in an algorithmic way.

Mathematics the development of algorithms for computer algebra, computer

algebra systems and applications often are intertwined. A short list of few selected mathematical subjects which require

the use of sufficiently powerful computer algebra systems and algorithms is the following: classification of finite groups, and their presentations systematic investigations of algebraic number and function

fields, and determination of their invariants study of systems of nonlinear algebraic equations with regard

to problems from commutative and non-commutative algebra, and algebraic geometry

experimental and theoretical investigation of special classes of diophantine equations

phenomenological and structural investigation of dynamic systems

coupling of symbolic and numerical methods for solving numerical problems effectively, while reducing round-off errors at the same time.

Group theory Computational group theory was much more systematically and

constantly developed than computational methods in other parts of mathematics.

The most significant problem is the classification of finite simple groups. A typical task is to construct a new sporadic simple group. Meat-Axe for example was used to prove the existence of the group J4,

the largest of Janko’s sporadic groups. A sporadic simple group of order more then 4 · · · 1033, named Baby

Monster, was build as permutation group on more than 13 milliards points.

Meat-Axe was also used to explicit construction of the largest sporadic group, the Monster, as group of (196882×196882)-matrices over the field with two elements.

Many explicit matrix representations for specific finite groups are available in the Atlas of Finite Group Representations (http://www.math.bham.ac.uk/atlas).

Group theory another important task is the systematic investigation of the quasi-simple

groups. The character tables and information about the subgroup lattices of all

sporadic groups as well as the first few members of the infinite series can be obtained through Gap, as well as generic character tables of groups of Lie type through Chevie.

Such generic character tables are applied, for example, to prove the existence of rigid generating systems proving Galois realizations for the groups in question.

Another typical area of applications is the analysis of finitely presented groups. One of the problems is to decide whether such a group is finite of infinite. A further objective of finite group theory is the classification of p-groups, or,

more generally, of solvable groups. To compute with such groups, special presentations, such as power commutator presentations and special algorithms such as p-quotient method and collection have been developed.

The approaches led to new results ad the classification and construction of all groups of order 256 (more than 56 thousands) and the enumeration of the groups of order 512 (more than 10 millions).

Classification of groups of small order, primitive and transitive permutation groups, perfect groups, matrix groups, crystallographic groups, cohomology groups.

Theory of singularities

There are several interesting conjectures and problems in local algebraic geometry that were decided with the help of CAS, like Singular: related to the isolated hypersurface singularities, using a tangent cone algorithm examples in Singular,

can be derived examples for which the Poincare complex is exact by they are not quasi-homogeneous;

structure of the moduli space of curve singularities. Another way to use computer algebra is to construct

interesting explicit examples: using Singular series of curves of small degree where

build with high singularities.

Automatic Theorem Proving in Geometry

Particularly successful application area of CA within mathematics. Extensive and detailed studies have produced automatic proofs for a large

number of classical and recent geometric theorems. They have even supported the discovery of new such theorems. The general approach follows Descartes’ idea of algebraization of geometry: by

introducing a Cartesian coordinate system the considered geometric configurations are described via polynomial equations.

Many geometric theorems in the plane are closure theorems. Their conclusions is of type ”three points lie on a straight line or on a circle” or

”three straight lines meet in one point”. Such a conclusion translates algebraically via the chosen coordinate system into

a polynomial equation. An obvious approach is to try and prove the geometric statement by verifying its

algebraic translation for all values of the variables. Non-degeneracy conditions should be considered in the hypothesis. Such

conditions would, e.g., guarantee that certain points do not coincide or that a certain triangle does not collapse to a line.

Non-degeneracy conditions are reflected in the algebraic description as additional polynomial disequations, i.e. negated equations.

Adding non-degeneracy conditions by hand is extremely tedious. Such conditions can be generated automatically.

Automatic Theorem Proving in Geometry

The existing methods for geometric proving can be roughly divided into two classes:

Complex methods: The Wu-Ritt method using characteristic sets

extends a given system of polynomial equations to an equivalent disjunction of systems of polynomial equations and disequations by iterated pseudo-division with remainder; there resulting systems are called characteristic sets

Grobner basis techniques and complex elimination methods

Real methods: decision methods

answers the question of universal validity of a formula in real numbers with either yes or no, while a quantifier elimination method

Quantifier elimination methods. assigns to an arbitrary formula with quantified variables an equivalent formula

without quantified variables An example for an implemented read decision and quantifier elimination method

is the CAD method implemented in the QEPCAD3 package. An alternative real decision and quantifier elimination method based on the

virtual substitution of parametric test points was implemented in the Reduce-package REDLOG.

Homological Algebra deals with derived functors and related concepts. The cohomology of groups and Lie algebras, which play a role in

theoretical physics, are examples of derived functors. Homological algebra also plays a role in the theory of differential

equations and is used in algebraic geometry and topology. There are two main areas in which computer algebra has been

employed in investigating derived functors. One is in computing the ranks of the objects an the other is in

using symbolic manipulation to actually computing resolutions. built-in commands in Macaulay can be used (also Singular,

CoCoA and Bergman). Computing resolutions over various algebras.

Macaulay and Axiom can be used. Gap and Magma can be used to computer the ranks of the first

and second (co)homology of finite groups given a suitable presentation of the group.

Study of Differential Structures on Quantum Groups Non-commutative differential calculi are important tools in

studying non-commutative differential geometry on quantum spaces.

There are specific methods in the study of differential structures on quantum groups that involve Groebner basis computations in very general noncommutative situations and the size of the problems to be solved is remarkable.

Through several CAS contain implementations of non-commutative Groebner bases the noncommutative algebras and modules appearing in these specific studies are still not supported in most systems.

Felix was and can be applied to handle the calculations.

Symmetric Bifurcation Theory

Theoretical investigation of bifurcation problems with symmetry has been a very active area in the last decade.

The investigation of symmetric bifurcation problems typical start with a general vector field having the symmetry of a certain group action being relevant in applications.

The algorithmic determination of such a generic equivariant vector field involves symbolic computations, especially Groebner basis computations.

The Hilbert basis of an invariant ring and the generators of the module of equivariants are computed.

The packages InVar and Symmetry provide software for these tasks; the first package concentrating on invariants, while the second one includes computations for equivariants and complete description.

Symbolic-Numeric Treatment of Equivariant Systems of Equations In certain applications, one encounters a class of nonlinear systems of

equations which depend on an additional parameter, and which feature symmetries originating from geometric properties. Examples: discrete versions of reaction-diffusion equations, problems in structural engineering, and neural nets.

Symcon is a Reduce package which exploits symmetries and finally perform the actual computation of solutions in a numeric part written in C; the symbolic part takes care of many derivations, e.g. it determines the bifurcation groups and isotropy groups, computes the systems of equations resulting from the reductions by symmetry and the block structure of the Jacobian.

The combination of symbolics and numerics has considerable advantages: the equivalence condition is checked, therefore application instances are

implemented reliably and error-free; Symbolic differentiation used to compute bifurcation points is by far superior

compared to numeric differentiation; special structure of the Jacobian matrices is exploited by generating the

functions which are evaluated numerically.

Computer Science A connection between CA and CS is not only given by CAS being computer

programs, but also by formal and algebraic methods used in CA. Examples follow for the areas of

signal processing, wavelets and algebraic specification.

Another link between CA and CS is the use of of inference on mathematical databases for knowledge based systems in mathematics.

Also algebra provides the foundations for coding theory and cryptography. CA yields a systematic approach to efficient algorithms. Algebraic techniques are also used for the design of hardware architectures and

VLSI design. Further examples for applications are

decision problem solving within algebraic structures by term rewrite and reduction systems

and automatic theorem proving for which Groebner bases and characteristic set methods are important tools.

Algorithms from CA are successfully applied to the theory of lattices and ordered structures, with applications to data analysis and knowledge representation.

Furthermore the correspondence of the theory of monoids and automata theory allows the application of CA in theoretical CS.

Signal processing Signal transforms as the Fourier transform, the Hartley transform,

or the cosine transform often exhibit a symmetric structure. One problem is to factor the signal matrix into a product of sparse

matrices; this yields a fast algorithm for the matrix multiplication with the signal matrix.

IAKS CAS can be used to derive these sparse base transform matrices with methods of representation theory.

The discrete wavelet transform is a recent technique in computational harmonic analysis.

There is a close connection between the fast algorithms for these transforms and perfect reconstruction filter banks, which are studied in signal processing.

A typical application of fast wavelet transforms is image compression, where the transform is used to reduce the correlation between adjacent image pixels.

CAS like IAKS are used to reduce the computational complexity of fast wavelet transform algorithms.

Algebraic specification Formal methods, i.e. the systematic use of mathematics in software or

hardware design, have become standard in the development of high-integrity systems for safety-critical applications.

For software systems, algebraic specification allows for a formal design process in terms of abstract datatypes, starting from loose requirement specifications and ending up with executable specifications close to program code.

A specification language CASL4 (Common Algebraic Specification Language) was designed with a formal semantics.

For high-integrity system design, correctness is of course the crucial point. In algebraic specification, correctness is achieved by

proving the consistency of specifications (i.e. there exists a model which has the properties described in the specification),

validating requirement specifications (i.e. the specification describes the class of models which one has in mind), and

proving that a specification refines another one (i.e. that a development step towards a computer program is correct).

Tool support is needed to deal with these items within a large system’s design

Algebraic specification Classical tools for algebraic specification are

theorem provers, e.g. Isabelle5, Inka, KIV, or term rewriting systems, e.g. OBJ, ELAN.

Usually the development of a complex system requires a great variety of specialized tools as no single tool is able to deal with all its aspects.

At this point, CAS come into scope as a welcome supplement to the established tools.

The reason is that CAS are able to deal effectively and efficiently with certain datatypes.

From an algebraic specification point of view, datatypes of special interest are not only the classical algebraic structures like groups, rings, and fields, but also the more practical datatypes, for example numbers (naturals, integers, rationals, reals) or structured datatypes like lists and bags, which all exhibit a large amount of algebraic structure.

The CASL library of standard datatypes, for example, includes specifications of groups, rings, and fields, explicitly states the algebraic properties of datatypes, and makes even use of algebraic properties to specify standard datatypes.

An integration of CAS into the algebraic specification development process lacks a semantically sound basis. First solution to this problem have been stated and partially realized by the OpenMath initiative.

Decomposable Structures, Generating Functions and Average-Case of Algorithms

Although the use of generating functions has a long tradition in enumerative combinatorics, a systematic investigation and exploitation of this tool with its mechanization in mind has been made only quire recently. It has become evident that a considerable portion of enumeration problems can be dealt with in a routine and highly efficient way.

Using a setup which is not accidentally reminiscent of context-free grammars and languages, one may specify many interesting classes of combinatorial structures from atomic building blocks by using a few standard constructors, such as union, product, sequence, set, multi-set, cycle.

As a simple specimen, the class FD of functional digraphs may e specified as FD=set(CFD), CFD=cycle(RT), RT=product(Z,set(RT)) which expresses the fact that a functional digraph is a set of connected

components (CFD), each of which is a cycle of rooted trees (RT), where rooted trees of nodes Z are defined recursively in an obvious way.

Such a specification for decomposable structures can be compiled into a system of equations for the corresponding generating functions, which in a universe with distinguishable atoms

fd(z) = exp(cfd(z)), cfd(z) = −log(1 − rt(z)), rt(z) = z · exp(rt(z)) where the exponential generating function fd(z) = sum(fd_nz^n/n!, n>=0) the

coefficient fd_n denotes the number of functional digraphs on n points.

Decomposable Structures, Generating Functions and Average-Case of Algorithms

In fortunate case such a system of equations can be solved explicitly, but even if this is not possible, lots of useful information can be obtained:

initial segments of the counting sequence (such as (fd_n)n>=0 in the example) can be computed,

efficient algorithms for random and exhaustive generation of the structures under consideration can be constructed automatically,

and detailed information about the asymptotic behavior of the counting sequence can be extracted using methods such as singularity analysis or saddle point methods.

Using the technique of tagging and bivariate generating functions, parameters recording the number of occurrences of substructures, such as leaves in a tree or the number of connected components of a graph, can be analyzed within the same setup, giving precise and/or asymptotic information about the average, the variance etc. of the corresponding distribution over structures of fixed size.

For many algorithms the task of analyzing the quantitative behavior can be reduced to combinatorial counting problems, hence generating functions and recurrence relations play an important role in the analysis of average-case complexity.

Viewing the definitions of decomposable structures as specifications of datatypes, one may analyze algorithms which systematically traverse these structures and operate relative to the substructures encountered. Typical examples are tree searching methods, rewriting algorithms, unification, pattern matching etc.

Decomposable Structures, Generating Functions and Average-Case of Algorithms The Maple package combstruct represents much of the current status

of the implementation of the above ideas. The package gdev provides a function equivalent responsible for

asymptotic analysis. The package gfun

deals with holomonic generating functions, i.e. generating functions defined by linear differential equations with polynomial coefficients, or equivalently linear recurrent sequences with polynomial coefficients, quite often encountered in combinatorial situations and otherwise;

implements the closure properties of this class of functions and has strong guessing capabilities which lets one finds plausible

candidates for a differential equation (recurrence relation) satisfied by a generating function (its sequence of coefficients) once one knows a sufficiently long initial segment of the sequence.

Finding such equations (recurrences) can serve for various purposes: providing identities, fast computation of coefficients, search for closed form solutions, and asymptotics.

Telecommunication Management Networks

A telecommunication management network (TMN) is a data network for administering and maintaining network nodes such as switches, cross-connects, and large telecommunication networks, like the synchronous digital hierarchy, from a central operations center.

For a commercial telecommunication operator, real-time response and performance are of prime importance.

Malfunctioning network nodes and lines have to be detected in a timely manner in order to keep down-time at a minimum.

Massive data transfer within short time intervals is required, e.g. to collect billing information, or to update hundreds of network nodes with new software.

The protocols used in TMNs have been standardized. For manufacturers and operators of networks, there still remains the task

to determine a large number of design parameters, to achieve smooth operation and performance.

CAS are used to improve performances of nodes and networks. Commercial general purpose CA packages with their libraries and their

flexible programming language allowed to generate quickly simulations which included generation of statical input data, Fourier transform, filtering and discrete mathematics.