Conjecture

About: Conjecture is a research topic. Over the lifetime, 24353 publications have been published within this topic receiving 366042 citations. The topic is also known as: Expectation, forecast, conjecture & mathematical conjecture.

A bound on chaos

Abstract: We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent λ L ≤ 2πk B T/ℏ. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.

Introduction to Cyclotomic Fields

Abstract: 1 Fermat's Last Theorem.- 2 Basic Results.- 3 Dirichlet Characters.- 4 Dirichlet L-series and Class Number Formulas.- 5 p-adic L-functions and Bernoulli Numbers.- 5.1. p-adic functions.- 5.2. p-adic L-functions.- 5.3. Congruences.- 5.4. The value at s = 1.- 5.5. The p-adic regulator.- 5.6. Applications of the class number formula.- 6 Stickelberger's Theorem.- 6.1. Gauss sums.- 6.2. Stickelberger's theorem.- 6.3. Herbrand's theorem.- 6.4. The index of the Stickelberger ideal.- 6.5. Fermat's Last Theorem.- 7 Iwasawa's Construction of p-adic L-functions.- 7.1. Group rings and power series.- 7.2. p-adic L-functions.- 7.3. Applications.- 7.4. Function fields.- 7.5. = 0.- 8 Cyclotomic Units.- 8.1. Cyclotomic units.- 8.2. Proof of the p-adic class number formula.- 8.3. Units of $$ \mathbb{Q}\left( {{\zeta _p}} \right)$$ and Vandiver's conjecture.- 8.4. p-adic expansions.- 9 The Second Case of Fermat's Last Theorem.- 9.1. The basic argument.- 9.2. The theorems.- 10 Galois Groups Acting on Ideal Class Groups.- 10.1. Some theorems on class groups.- 10.2. Reflection theorems.- 10.3. Consequences of Vandiver's conjecture.- 11 Cyclotomic Fields of Class Number One.- 11.1. The estimate for even characters.- 11.2. The estimate for all characters.- 11.3. The estimate for hm-.- 11.4. Odlyzko's bounds on discriminants.- 11.5. Calculation of hm+.- 12 Measures and Distributions.- 12.1. Distributions.- 12.2. Measures.- 12.3. Universal distributions.- 13 Iwasawa's Theory of $$ {\mathbb{Z}_p} -$$ extensions.- 13.1. Basic facts.- 13.2. The structure of A-modules.- 13.3. Iwasawa's theorem.- 13.4. Consequences.- 13.5. The maximal abelian p-extension unramified outside p.- 13.6. The main conjecture.- 13.7. Logarithmic derivatives.- 13.8. Local units modulo cyclotomic units.- 14 The Kronecker-Weber Theorem.- 15 The Main Conjecture and Annihilation of Class Groups.- 15.1. Stickelberger's theorem.- 15.2. Thaine's theorem.- 15.3. The converse of Herbrand's theorem.- 15.4. The Main Conjecture.- 15.5. Adjoints.- 15.6. Technical results from Iwasawa theory.- 15.7. Proof of the Main Conjecture.- 16 Miscellany.- 16.1. Primality testing using Jacobi sums.- 16.2. Sinnott's proof that = 0.- 16.3. The non-p-part of the class number in a $$ {\mathbb{Z}_p} -$$ extension.- 1. Inverse limits.- 2. Infinite Galois theory and ramification theory.- 3. Class field theory.- Tables.- 1. Bernoulli numbers.- 2. Irregular primes.- 3. Relative class numbers.- 4. Real class numbers.- List of Symbols.

Liouville Correlation Functions from Four-dimensional Gauge Theories

Abstract: We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of \({\mathcal{N}=2}\) SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0, 1.

Sphere packings I

Abstract: We describe a program to prove the Kepler conjecture on sphere packings. We then carry out the first step of this program. Each packing determines a decomposition of space into Delaunay simplices, which are grouped together into finite configurations called Delaunay stars. A score, which is related to the density of packings, is assigned to each Delaunay star. We conjecture that the score of every Delaunay star is at most the score of the stars in the face-centered cubic and hexagonal close packings. This conjecture implies the Kepler conjecture. To complete the first step of the program, we show that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.

An Introduction to the Geometry of Numbers

Abstract: Notation Prologue Chapter I. Lattices 1. Introduction 2. Bases and sublattices 3. Lattices under linear transformation 4. Forms and lattices 5. The polar lattice Chapter II. Reduction 1. Introduction 2. The basic process 3. Definite quadratic forms 4. Indefinite quadratic forms 5. Binary cubic forms 6. Other forms Chapter III. Theorems of Blichfeldt and Minkowski 1. Introduction 2. Blichfeldt's and Mnowski'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 Chapter IV. Distance functions 1. Introduction 2. General distance-functions 3. Convex sets 4. Distance functions and lattices Chapter V. Mahler'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. Speres 10. Applications to diophantine approximation Chapter VI. The theorem of Minkowski-Hlawka 1. Introduction 2. Sublattices of prime index 3. The Minkowski-Hlawka theorem 4. Schmidt's theorems 5. A conjecture of Rogers 6. Unbounded star-bodies Chapter VII. The quotient space 1. Introduction 2. General properties 3. The sum theorem Chapter VIII. Successive minima 1. Introduction 2. Spheres 3. General distance-functions Chapter IX. Packings 1. Introduction 2. Sets with V(/varphi) =n^2/Delta(/varphi) 3. Voronoi's results 4. Preparatory lemmas 5. Fejes Toth's theorem 6. Cylinders 7. Packing of spheres 8. The proudctio of n linear forms Chapter 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 Chapter XI. Ihomogeneous problems 1. Introduction 2. Convex sets 3. Transference theorems for convex sets 4. The producti of n linear forms Appendix References Index quotient space. successive minima. Packings. Automorphs. Inhomogeneous problems.