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Field theory: Classical foundations and multiplicative groups
130
Book Announcements
lems. Algorithms time of oracle proximation polynomial cedures
and Turing
algorithms. and
computation
computations.
(Gaussian
the hermite
machines.
Polynomial
Time and space complexity).
and reduction.
of numbers
elimination.
normal
Encoding.
Transformation
(Encoding
&Y-completeness length
time approximation
Gram-Schmidt
of numbers.
(The running notions).
Polynomial
of real numbers).
orthogonalization.
Oracles
and related
and
Ap-
strongly
Pivoting
and related
pro-
The simplex method.
Computation
of
Chapter 2: Algorithmic Aspects of Convex Sets: Formulation of the Prob-
form).
/ems. Basic algorithmic problems for convex sets. Nondeterministic decision problems for convex sets. Chapter 3: The Ellipsoid Method. Geometric background and an informal description (Properties of ellipsoids.
Description
and polynomiality.
of the basic ellipsoid
Some examples).
method.
Proofs
The central-cut
Chapter 4: Algorithms for Convex Bodies. Summary timization
from
algorithmic
problems
union.
membership.
for convex bodies.
The intersection.
Polars,
Basis Reduction. Continued blems. tion
Basis reduction over
polynomial
of
fractions.
Simultaneous
optimization
algorithms.
Integer
and
Optimization
from
Some
problems
ellipsoid
negative
method.
separation. results.
Op-
Further
on convex bodies (The sum. The convex hull of the
Chapter 5: Diophantine Approximation
antiblockers).
Complexity
Implementation
The shallow-cut
problems.
diophantine
approximation:
Formulation
and
of pro-
Chapter 6: Rational Polyhedra. Optimiza-
More on lattice algorithms.
A preview.
strong
Operations
method.
of results.
of the basic
blockers,
in lattices.
polyhedra:
Equivalence
Equivalence
of some lemmas.
ellipsoid
of rational
polyhedra.
Weak
and
strong
problems.
separation.
Further problems for polyhedra. Strongly in bounded dimension. Chapter 7: Combinatorial Op-
programming
timization: Some Basic Examples. Flows and cuts. Arborescences. Matching. Edge coloring. Matroids. Subset sums. Concluding remarks. Chapter 8: Combinatorial Optimization: A Tour d’Horizon. Blocking hypergraphs and polyhedra. Problems on bipartite branchings, and rooted and directed cuts (Arborescences graphs.
Dicuts and dijoins).
and T-cuts.
Chinese
postmen
Graphs. Odd circuit tiblockers results tions
and
salesmen).
t-perfect
Orthonormal
(Matching.
Multicommodity
graphs.
Clique
representations.
constraints
Coloring
and submodular
intersecting,
and crossing
functions families.
(Packing
b-matching.
and
perfect
perfect
graphs.
More
functions function
graphs
(An-
algorithmic
and polymatroids.
bases of a matroid).
Odd submodular
T-joins
Chapter 9: Stable Sets in
flows.
sets. Chapter 10: Submodular Functions. Submodular
for polymatroids
on lattice,
odd cuts, and generalizations
and traveling
constraints
of hypergraphs).
on stable
Algorithms
Matchings,
graphs. Flows, paths, chains, and cuts. Trees, and rooted cuts. Trees and cuts in undirected
Submodular
minimization
funcand ex-
tensions.
Gregory Karpilovsky, Field Theory: Classical Foundations Groups (Marcel Dekker, New York, 1988) 551 pages Chapter 1: Preliminaries. Notation products.
Module-theoretic
Theory. Algebraic
extensions.
Galois
Finite fields,
extensions.
plications.
Discriminants
Galois theory.
Profinite
and terminology.
prerequisites. Normal
extensions. bases.
Infinite
algebras.
prerequisites.
Separable,
roots of unity and cyclotomic
and integral groups.
Polynomial
Topological
Units in quadratic
Galois theory.
purely
and Multiplicative
Integral
extensions.
Tensor
Chapter 2: Classical Topics in Field inseparable
extensions.
and simple extensions.
Norms,
traces
and their ap-
fields. Units in pure cubic fields. Finite
Witt vectors.
Cyclic extensions.
Kummer
theory.
Radical extensions and related results. Degrees of sums in a separable field extension. Galois cohomology. The Brauer group of a field. An interpretation of Hd(G,E*). A cogalois theory for radical extensions. Abelian p-extensions over fields of characteristic p. Formally real fields. Transcendental extensions. Chapter 3: Valuation Theory. Valuations. Valuation rings and places. Dedekind domains. Completion of a field. Extensions of valuations. Valuations of algebraic number fields. Ramification index and residue degree. Structure of complete discrete valued fields (Notation and terminology. The equal characteristic
case. The unequal
characteristic
case. The inertia
field. Cyclotomic
adic fields). Chapter 4: Multiplicative Groups of Fields. Some general observations. groups. The Dirichlet-ChevalleyHasse Unit Theorem. The torsion subgroup. Global
extensions
of p-
Infinite abelian fields. Algebraic-
Book Announcements ally closed,
real closed
and
the rational
p-adic
fields.
131
Local
fields
characteristic case. The unequal characteristic case). Extensions theorem. Fields with free multiplicative groups modulo torsion. groups.
Multiplicative
groups
under
(Preparatory
results.
The equal
of algebraic number fields. Brandis’ A nonsplitting example. Embedding
field extensions.
Neal Koblitz, A Course in Number Theory and Cryptography
(Springer, New York,
1987) 208 pages Chapter I: Some Topics in Elementary Number Theory. Time estimates for doing arithmetic. Divisibility algorithm. Congruences. Some applications to factoring. Chapter II: Finite Fields and Quadratic Residues. Finite fields. Quadratic residues and reciprocity. Chapter III: Cryptography. Some simple cryptosystems. Enciphering matrices. Chapfer IV: Public Key. The idea of public key cryptography. RSA. Discrete log. Knapsack. Chapter V: Primality and Factoring. Pseudoprimes. The rho method. Fermat factorization and factor bases. The continued fraction method. Chapter VI: Elliptic Curves. Basic facts. Elliptic curve cryptosystems. Elliptic curve factorization. and the Euclidean
Emile Aarts and Jan Korst, Simulated Stochastic Approach to Combinatorial (Wiley, Chichester, 1989) 272 pages
Annealing and Boltzmann Machines: A Optimization and Neural Computing
I: SIMULATED ANNEALING. 1:
Equilibrium Markov
statistics.
theory.
The stationary
tion. Asymptotic ing schedule. algorithm graph
Combinatorial Optimization. Combinatorial optimization problems. Local Annealing. The Metropolis algorithm. The simulated annealing algorithm. Characteristic features. A quantitative analysis. 3: Asymptotic Convergence.
2: Simulated
search.
Empirical
performance
(The travelling
colouring
distribution.
problem.
Inhomogeneous
4: Finite-Time Approximation.
behaviour.
salesman
The placement
chains.
Convergence
schedules.
in distribu-
A polynomial-time
cool-
5: Simulated Annealing in Practice. Implementing
analysis. problem.
Markov Cooling
The max cut problem.
problem).
The independent
A survey of applications
the
set problem.
(Basic problems.
The
Engineer-
ing problems). General performance experiences. 6: Parallel Simulated Annealing Algorithms. Speeding up the simulated annealing algorithm. Parallel-machine models. Designing parallel annealing algorithms. General
algorithms
plementation machine.
and
(The division numerical
Connectionist
Machines. Structural chronous
algorithm.
results).
models.
A historical
description.
parallelism.
Sequential
Asynchronous
binatorial Optimization
The clustering
algorithm.
The error algorithm.
Parallel
II: BOLTZMANN MACHINES. 7: Neural Computing.
parallelism.
and Boltzmann
overview.
Boltzmann
The
Boltzmann
machines.
A parallel
cooling
Machines. General
Parallel
schedule).
strategy.
8: Boltzmann
machine. Boltzmann
The
machines
A taxonomy. max
im-
Man versus
cut
(Syn-
9: Com-
problem.
The
independent set problem. The graph colouring problem. The clique partitioning and clique covering problems. The travelling salesman problem. Numerical results (Graph problems. The travelling salesman problem). 10: Classification and Boltzmann Machines. Classification tural description. Examples (Classification without hidden units. Association. Equilibrium activation
Fault
tolerance).
properties. probabilities).
Learning Learning
11: Learning and Boltzmann without
hidden
with hidden
units (Outline units.
Variants
problems. Extension of the strucClassification with hidden units.
Machines.
Learning
of the learning of the learning
from
algorithm. algorithm.
examples.
Estimation
of
Learning
in
practice (Choosing a desired visible behaviour. Convergence properties. Estimation of the activation probabilities. Termination of the learning algorithm). Robustness aspects (Internal representations. Relearning).
Book Announcements
lems. Algorithms time of oracle proximation polynomial cedures
and Turing
algorithms. and
computation
computations.
(Gaussian
the hermite
machines.
Polynomial
Time and space complexity).
and reduction.
of numbers
elimination.
normal
Encoding.
Transformation
(Encoding
&Y-completeness length
time approximation
Gram-Schmidt
of numbers.
(The running notions).
Polynomial
of real numbers).
orthogonalization.
Oracles
and related
and
Ap-
strongly
Pivoting
and related
pro-
The simplex method.
Computation
of
Chapter 2: Algorithmic Aspects of Convex Sets: Formulation of the Prob-
form).
/ems. Basic algorithmic problems for convex sets. Nondeterministic decision problems for convex sets. Chapter 3: The Ellipsoid Method. Geometric background and an informal description (Properties of ellipsoids.
Description
and polynomiality.
of the basic ellipsoid
Some examples).
method.
Proofs
The central-cut
Chapter 4: Algorithms for Convex Bodies. Summary timization
from
algorithmic
problems
union.
membership.
for convex bodies.
The intersection.
Polars,
Basis Reduction. Continued blems. tion
Basis reduction over
polynomial
of
fractions.
Simultaneous
optimization
algorithms.
Integer
and
Optimization
from
Some
problems
ellipsoid
negative
method.
separation. results.
Op-
Further
on convex bodies (The sum. The convex hull of the
Chapter 5: Diophantine Approximation
antiblockers).
Complexity
Implementation
The shallow-cut
problems.
diophantine
approximation:
Formulation
and
of pro-
Chapter 6: Rational Polyhedra. Optimiza-
More on lattice algorithms.
A preview.
strong
Operations
method.
of results.
of the basic
blockers,
in lattices.
polyhedra:
Equivalence
Equivalence
of some lemmas.
ellipsoid
of rational
polyhedra.
Weak
and
strong
problems.
separation.
Further problems for polyhedra. Strongly in bounded dimension. Chapter 7: Combinatorial Op-
programming
timization: Some Basic Examples. Flows and cuts. Arborescences. Matching. Edge coloring. Matroids. Subset sums. Concluding remarks. Chapter 8: Combinatorial Optimization: A Tour d’Horizon. Blocking hypergraphs and polyhedra. Problems on bipartite branchings, and rooted and directed cuts (Arborescences graphs.
Dicuts and dijoins).
and T-cuts.
Chinese
postmen
Graphs. Odd circuit tiblockers results tions
and
salesmen).
t-perfect
Orthonormal
(Matching.
Multicommodity
graphs.
Clique
representations.
constraints
Coloring
and submodular
intersecting,
and crossing
functions families.
(Packing
b-matching.
and
perfect
perfect
graphs.
More
functions function
graphs
(An-
algorithmic
and polymatroids.
bases of a matroid).
Odd submodular
T-joins
Chapter 9: Stable Sets in
flows.
sets. Chapter 10: Submodular Functions. Submodular
for polymatroids
on lattice,
odd cuts, and generalizations
and traveling
constraints
of hypergraphs).
on stable
Algorithms
Matchings,
graphs. Flows, paths, chains, and cuts. Trees, and rooted cuts. Trees and cuts in undirected
Submodular
minimization
funcand ex-
tensions.
Gregory Karpilovsky, Field Theory: Classical Foundations Groups (Marcel Dekker, New York, 1988) 551 pages Chapter 1: Preliminaries. Notation products.
Module-theoretic
Theory. Algebraic
extensions.
Galois
Finite fields,
extensions.
plications.
Discriminants
Galois theory.
Profinite
and terminology.
prerequisites. Normal
extensions. bases.
Infinite
algebras.
prerequisites.
Separable,
roots of unity and cyclotomic
and integral groups.
Polynomial
Topological
Units in quadratic
Galois theory.
purely
and Multiplicative
Integral
extensions.
Tensor
Chapter 2: Classical Topics in Field inseparable
extensions.
and simple extensions.
Norms,
traces
and their ap-
fields. Units in pure cubic fields. Finite
Witt vectors.
Cyclic extensions.
Kummer
theory.
Radical extensions and related results. Degrees of sums in a separable field extension. Galois cohomology. The Brauer group of a field. An interpretation of Hd(G,E*). A cogalois theory for radical extensions. Abelian p-extensions over fields of characteristic p. Formally real fields. Transcendental extensions. Chapter 3: Valuation Theory. Valuations. Valuation rings and places. Dedekind domains. Completion of a field. Extensions of valuations. Valuations of algebraic number fields. Ramification index and residue degree. Structure of complete discrete valued fields (Notation and terminology. The equal characteristic
case. The unequal
characteristic
case. The inertia
field. Cyclotomic
adic fields). Chapter 4: Multiplicative Groups of Fields. Some general observations. groups. The Dirichlet-ChevalleyHasse Unit Theorem. The torsion subgroup. Global
extensions
of p-
Infinite abelian fields. Algebraic-
Book Announcements ally closed,
real closed
and
the rational
p-adic
fields.
131
Local
fields
characteristic case. The unequal characteristic case). Extensions theorem. Fields with free multiplicative groups modulo torsion. groups.
Multiplicative
groups
under
(Preparatory
results.
The equal
of algebraic number fields. Brandis’ A nonsplitting example. Embedding
field extensions.
Neal Koblitz, A Course in Number Theory and Cryptography
(Springer, New York,
1987) 208 pages Chapter I: Some Topics in Elementary Number Theory. Time estimates for doing arithmetic. Divisibility algorithm. Congruences. Some applications to factoring. Chapter II: Finite Fields and Quadratic Residues. Finite fields. Quadratic residues and reciprocity. Chapter III: Cryptography. Some simple cryptosystems. Enciphering matrices. Chapfer IV: Public Key. The idea of public key cryptography. RSA. Discrete log. Knapsack. Chapter V: Primality and Factoring. Pseudoprimes. The rho method. Fermat factorization and factor bases. The continued fraction method. Chapter VI: Elliptic Curves. Basic facts. Elliptic curve cryptosystems. Elliptic curve factorization. and the Euclidean
Emile Aarts and Jan Korst, Simulated Stochastic Approach to Combinatorial (Wiley, Chichester, 1989) 272 pages
Annealing and Boltzmann Machines: A Optimization and Neural Computing
I: SIMULATED ANNEALING. 1:
Equilibrium Markov
statistics.
theory.
The stationary
tion. Asymptotic ing schedule. algorithm graph
Combinatorial Optimization. Combinatorial optimization problems. Local Annealing. The Metropolis algorithm. The simulated annealing algorithm. Characteristic features. A quantitative analysis. 3: Asymptotic Convergence.
2: Simulated
search.
Empirical
performance
(The travelling
colouring
distribution.
problem.
Inhomogeneous
4: Finite-Time Approximation.
behaviour.
salesman
The placement
chains.
Convergence
schedules.
in distribu-
A polynomial-time
cool-
5: Simulated Annealing in Practice. Implementing
analysis. problem.
Markov Cooling
The max cut problem.
problem).
The independent
A survey of applications
the
set problem.
(Basic problems.
The
Engineer-
ing problems). General performance experiences. 6: Parallel Simulated Annealing Algorithms. Speeding up the simulated annealing algorithm. Parallel-machine models. Designing parallel annealing algorithms. General
algorithms
plementation machine.
and
(The division numerical
Connectionist
Machines. Structural chronous
algorithm.
results).
models.
A historical
description.
parallelism.
Sequential
Asynchronous
binatorial Optimization
The clustering
algorithm.
The error algorithm.
Parallel
II: BOLTZMANN MACHINES. 7: Neural Computing.
parallelism.
and Boltzmann
overview.
Boltzmann
The
Boltzmann
machines.
A parallel
cooling
Machines. General
Parallel
schedule).
strategy.
8: Boltzmann
machine. Boltzmann
The
machines
A taxonomy. max
im-
Man versus
cut
(Syn-
9: Com-
problem.
The
independent set problem. The graph colouring problem. The clique partitioning and clique covering problems. The travelling salesman problem. Numerical results (Graph problems. The travelling salesman problem). 10: Classification and Boltzmann Machines. Classification tural description. Examples (Classification without hidden units. Association. Equilibrium activation
Fault
tolerance).
properties. probabilities).
Learning Learning
11: Learning and Boltzmann without
hidden
with hidden
units (Outline units.
Variants
problems. Extension of the strucClassification with hidden units.
Machines.
Learning
of the learning of the learning
from
algorithm. algorithm.
examples.
Estimation
of
Learning
in
practice (Choosing a desired visible behaviour. Convergence properties. Estimation of the activation probabilities. Termination of the learning algorithm). Robustness aspects (Internal representations. Relearning).