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AuthorCassels, J. W. S. author
TitleAn Introduction to the Geometry of Numbers [electronic resource] / by J. W. S. Cassels
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1997
Connect tohttp://dx.doi.org/10.1007/978-3-642-62035-5
Descript VIII, 345 p. online resource

SUMMARY

Reihentext + Geometry of Numbers From the reviews: "The work is carefully written. It is well motivated, and interesting to read, even if it is not always easy... historical material is included... the author has written an excellent account of an interesting subject." (Mathematical Gazette) "A well-written, very thorough account ... Among the topics are lattices, reduction, Minkowski's Theorem, distance functions, packings, and automorphs; some applications to number theory; excellent bibliographical references." (The American Mathematical Monthly)


CONTENT

Prologue -- I. Lattices -- 1. Introduction -- 2. Bases and sublattices -- 3. Lattices under linear transformation -- 4. Forms and lattices -- 5. The polar lattice -- II. Reduction -- 1. Introduction -- 2. The basic process -- 3. Definite quadratic forms -- 4. Indefinite quadratic forms -- 5. Binary cubic forms -- 6. Other forms -- III. Theorems of BLICHFELDT and MINKOWSKI -- 1. Introduction -- 2. BLICHFELDTโ{128}{153}S and MINKOWSKIโ{128}{153}S theorems -- 3. Generalisations to non-negative functions -- 4. Characterisation of lattices -- 5. Lattice constants -- 6. A method of MORDELL -- 7. Representation of integers by quadratic forms -- IV. Distance functions -- 1. Introduction -- 2. General distance-functions -- 3. Convex sets -- 4. Distance functions and lattices -- V. MAHLERโ{128}{153}S compactness theorem -- 1. Introduction -- 2. Linear transformations -- 3. Convergence of lattices -- 4. Compactness for lattices -- 5. Critical lattices -- 6. Bounded star-bodies -- 7. Reducibility -- 8. Convex bodies -- 9. Spheres -- 10. Applications to diophantine approximation -- VI. The theorem of MINKOWSKI-HLAWKA -- 1. Introduction -- 2. Sublattices of prime index -- 3. The Minkowski-Hlawka theorem -- 4. SCHMIDTโ{128}{153}S theorems -- 5. A conjecture of ROGERS W -- 6. Unbounded star-bodies -- VII. The quotient space -- 1. Introduction -- 2. General properties -- 3. The sum theorem -- VIII. Successive minima -- 1. Introduction -- 2. Spheres -- 3. General distance-functions -- 4. Convex sets -- 5. Polar convex bodies -- IX. Packings -- 1. Introduction -- 2. Sets with V(L) = 2n?(L) -- 3. VORONOIโ{128}{153}S results -- 4. Preparatory lemmas -- 5. FEJES Tร{147}Thโ{128}{153}S theorem -- 6. Cylinders -- 7. Packing of spheres -- 8. The product of n linear forms -- X. Automorphs -- 1. Introduction -- 2. Special forms -- 3. A method of MORDELL -- 4. Existence of automorphs -- 5. Isolation theorems -- 6. Applications of isolation -- 7. An infinity of solutions -- 8. Local methods -- XI. Inhomogeneous problems -- 1. Introduction -- 2. Convex sets -- 3. Transference theorems for convex sets -- 4. The product of n linear forms -- References


Mathematics Geometry Number theory Mathematics Number Theory Geometry



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