Alphabeta MathLandscape Statistics And Selection
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1 · Prerequisites
Abstract Algebra
Differential Geometry
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
Real Analysis
- Areas of Elementary Plane Figures
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
String Theory
2 · Logical flowchart
3 · Items
Definition
Flux-vacuum ensemble and tadpole-bound data
Lemma
Dirac-delta counting formula for regular zeros
Theorem
Parameterized regular-zero counting for a full-rank linear family
Definition
Vacuum-population, measure and selection data
Sources
- Frederik Denef and Michael R. Douglas, Distributions of Flux Vacua, arXiv:hep-th/0404116v2
- Steven B. Giddings, Shamit Kachru, and Joseph Polchinski, Hierarchies from Fluxes in String Compactifications, arXiv:hep-th/0105097v2
- Mariana Graña, Flux Compactifications in String Theory: A Comprehensive Review, arXiv:hep-th/0509003v3
- J. Lebl, Basic Analysis II, Theorem 8.5.1
- Gerald B. Folland, Real Analysis, 2nd ed., Theorem 2.47
- Frederik Denef and Michael R. Douglas, Distributions of Flux Vacua
- Encyclopedia of Mathematics, Sard theorem
- Raphael Bousso, TASI Lectures on the Cosmological Constant, arXiv:0708.4231v2
- Michael R. Douglas, The String Theory Landscape, Universe 2019, 5(7), 176, doi:10.3390/universe5070176
- Michele Cicoli, Joseph P. Conlon, Anshuman Maharana, Susha Parameswaran, Fernando Quevedo, and Ivonne Zavala, String Cosmology: from the Early Universe to Today, arXiv:2303.04819v2
- Bjoern Hassfeld, Arthur Hebecker, Manfred Salmhofer, Jonah Cedric Strauss, and Johannes Walcher, A Local Wheeler–DeWitt Measure for the String Landscape, arXiv:2205.09772v4, Nuclear Physics B 992 (2023) 116230
- Justin Khoury and Sam S. C. Wong, Bayesian Reasoning in Eternal Inflation: A Solution to the Measure Problem, arXiv:2205.11524v2, Physical Review D 108 (2023) 023506