Transcript CA Applications (1)

CA Applications (1)
Lecture 3
General view
 CA has spread into many areas which about 30 years
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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
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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 pointinteractions 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
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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
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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
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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;
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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
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A standard application of computer algebra in GR is the classification of exact
solutions.
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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.
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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.
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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.
Reduce packages were used foe example to search the solutions of the EinsteinMaxwell equations.
Relativity packages are available also in general purpose systems like
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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
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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
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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
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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,
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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.
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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
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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
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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
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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.
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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
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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.
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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:
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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:
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using Singular series of curves of small degree where
build with high singularities.
Automatic Theorem Proving in Geometry
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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.
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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
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The existing methods for geometric proving can be roughly divided into two
classes:
Complex methods:
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The Wu-Ritt method using characteristic sets
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Grobner basis techniques and
complex elimination methods
Real methods:
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decision methods
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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.
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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
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 Reducepackage 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.
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built-in commands in Macaulay can be used (also Singular,
CoCoA and Bergman).
Computing resolutions over various algebras.
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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 noncommutative 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
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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:
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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:
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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
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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.
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Examples follow for the areas of
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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
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signal processing,
wavelets and
algebraic specification.
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,
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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
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hardware design, have become standard in the development of highintegrity 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
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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
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Classical tools for algebraic specification are
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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.
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theorem provers, e.g. Isabelle5, Inka, KIV, or
term rewriting systems, e.g. OBJ, ELAN.
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
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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
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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:
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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
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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
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administering and maintaining network nodes such as switches, crossconnects, 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.