BookofProofs is a growing library of open books with a mathematical context.However, it is not always possible to write a complete online book from the very beginning to the end. It might happen that some theorems remain unproven, some sections remain without text or exercises remain without solutions.

This is where you might help this project! If you like our site, please give it a little support and check out our todo list below.You are welcome to contribute any of the entries below which are work in progress.


Missing Proofs (1)
A proof for this theorem is still missing.

Isomorphism of Rings [171]

Unchained Proofs (0)
Everything is fine!

Missing Text (278)
This node has only a title, but no text was provided.

Basic Concepts of Logic [26]Topological Vector Spaces [69]Analytic Number Theory [72]Probability Theory and Statistics [73]Homotopy [83]Theory of Algorithms [84]Elliptic Geometry [86]Optimization Methods [88]Paradigms [89]Prime Number Theorem [91]Randomness and Probability Calculus [93]Languages [94]Correctness [95]First and Second Axiom of Countability [98]Linear Integral Equations [99]Hyperbolic Geometry [102]Inclusion and Exclusion [104]Automata Theory [107]Projective Geometry [110]Proof Theory [113]Galois Theory [114]Functional Analysis [116]Differential Geometry [124]Vector Analysis [128]Continuity [129]Complex Infinite Series [130]Classification of Differential Equations [131]CW Complexes [132]Computational Complexity Theory [133]Tables [135]Genetic Programming [136]Real Infinite Series [142]Differentiability [144]Calculus of Variations [145]Geometric Aspects [146]Methods to Solve Ordinary DE [149]Computability Theory [151]Dynamic Programming [152]Linked Lists [153]Trees [154]Fuzzy Logic [157]Decidability [158]Paths, Cycles and Connectivity [159]Methods to Solve Partial DE [160]Topological Aspects [161]Moebius Transformations [162]Integrability [163]Implicit Functions [164]Types of Real Functions [166]Algebraic Geometry [168]Eigenvalues and Eigenvectors [169]Symmetry Groups [172]Hashing [176]Numerical Algorithms [177]Planar Graphs [178]Prime Number Theorem for Arithmetic Progressions [180]Constructions with Ruler and Compass [193]Criteria for Convergence of Real Series [196]Properties of Real Functions [200]Types of Complex Functions [201]Fredholm Integral Equations [202]Entropy [204]Neural Networks [205]Sets [206]Higher-Order Logics [207]Peano Arithmetic [208]Universal Algebras [210]Principal Axis Transformation [213]Subspaces [214]Taylor Series [220]Banach Spaces and Banach Algebra [221]Fixed Point Theory [223]Properties of Complex Functions [224]Vertex Colourings and Decompositions [226]Greedy Algorithms [228]Sorting [229]Positional Notation Systems [230]Linear Programming [231]Laws of Large Numbers [236]Transitive Hull and Irreducible Kernels [238]Bipartite Graphs [239]Infinite Series Approximations [241]Roman Numbers [242]String Matching [243]Randomized Rounding [245]Volterra Integral Equation [246]Linear 1. Order DE [248]Linear 1. Order DE [249]Laurant Series [250]Paths [253]Accumulation Points [255]Path Integrals [257]Differential Operations on Vector Fields [258]Hilbert Spaces [259]Power Series [260]Jordan Normal Form [262]Categories [264]Continuum Hypothesis [268]Gödel's Incompleteness Theorems [270]Solving Strategies and Sample Solutions to Problems in Set Theory [271]Rotations and Basic Transformations [278]Fourier Series [283]Linear Operators and Linear Functionals [284]Integral Operations on Vector Fields [285]Criteria for Absolute Convergence of Real Series [286]Multiple Riemann Integrals [287]Riemann's Plane and Sphere [289]Homology and Winding Numbers [290]Complex Rational Functions [291]Differentiable Complex Functions [292]Stirling Numbers [293]Touring's Maschine [297]Determinants [298]Solving Strategies and Sample Solutions to Problems in Logic [300]Logarithms [301]Non-linear Operator [303]Extremal Values with Side Conditions [304]Multiple Lebesque Integrals [305]Criteria for Convergent Sequences [307]Inversion [309]Isogonality [310]Homotopy [311]Periodic Complex Functions [313]Isolated Singularities [314]Linear vs. Non-Linear DE [315]Linear 2. Order DE [316]Linear 2. Order DE [317]Liouville-Neumann Series [318]Tychonoff's Theorem [319]Worst Case [320]Simulated Annealing [321]Stacks [322]External Sorting [323]Discrete Distributions [324]Continuous Distributions [325]Chebyshev's Lemma [326]Multidimensional Random Variables [327]Graph Transformations [328]Zeta Functions [329]Analytic Continuation [331]Jordan Measurability [332]Brouwer's Theorem [333]Complex Logarithms [334]Invertible Complex Functions [335]Meromorphic Functions [336]Queues [337]Mixing [338]Matrices [343]Dual Space [344]Solving Strategies and Sample Solutions Related to Arithmetics [345]Trigonometric Functions [346]Lebesque Measure and Lebesque Integral [348]Schauder's Theorem [349]Fixed Points [350]Complex Trigonometric Functions [351]Residue Theorem [353]Order of DE [354]Linear n-th Order DE [355]Abel Integral Equations [356]Average Case [357]Strings [358]Graph Traversing [363]Dirichlet Convolution [365]Sieve Methods [366]Random Sample Models [367]Weak Law of Large Numbers [369]Markov Chains [370]Subalgebras [371]Vectors [372]Some Important Constants [373]Cyclometric Functions [375]Kakutani's Theorem [378]Cross-Ratio [379]Complex Cyclometric Functions [380]Integrable Complex Functions [381]Discrete Fourier Transform [382]Systems of DE [383]Linear Systems of 1. Order DE [384]Uniform Complexity Measure and Bit Complexity [385]Files [386]Data Compression [387]Pseudo-Random Numbers [388]Interpolation [389]Shortest Paths [392]Uniform Distribution [396]Normal Distribution [397]Strong Law of Large Numbers [398]Discrete Time Markov Chains [399]Estimation Theory [400]Cross Product [402]Hyporbolic Functions [406]Complex Hyporbolic Functions [409]Holomorphic Functions [410]Polynomial Time [411]Kernel and Image [413]Inverse Hyperbolic Functions [415]Simple Connectivity [417]Complex Inverse Hyperbolic Functions [418]Indefinite Integral [419]Non-Linear DE [420]NP-Completeness [421]Branch and Bound [422]Multiplication [423]Extrapolation [424]Flows [425]Reduction [426]Logarithmic Normal Distribution [430]Continuous Time Markov Chains [431]Matrices of Linear Maps [433]Mixed Functions [435]Integrable Functions [436]The $C(X)$ Algebra [437]Complex Mixed Functions [438]Path Integrals [439]Criteria for Holomorphic Functions [440]Boundary Value Problems [441]Probabilistic Complexity [442]And / Or Trees [443]Differentiation [445]Matchings [446]Similarity of Graphs [447]Goldbach Conjecture [448]Student Distribution [451]Affine Maps [452]Properties [453]Min-Max and Alpha-Beta [454]Normal Forms [455]Indefinite Integral [456]Gamma Function [457]Harmonic Functions [458]Satisfaction Problem Solving [459]Integration [461]Contraction and Minors [462]Network Design and Routing [463]Hypergeometric Distribution [465]$\chi^2$ Distribution [466]Fisher Distribution [467]Poisson Distribution [468]Infinite Graphs [470]Graph Representations [471]Linear Equation Systems [472]Quadrics [475]Improper Integral [476]Cryptography [477]Elimination [478]Non-Linear Equation Systems [479]Extremal Problems in Graph Theory [480]Riemann-Stieltjes Integral [482]Calendar [483]Combinatorial Algorithms [484]Iterative Methods [485]Orthogonalisation [486]Graph-Theoretic Algorithms [488]Iterative Methods [489]Eigenvalues and Eigenvectors [490]Network Paths [491]Integration of Starting Value Problems (Ordinary DE) [492]Network Flows [493]Error Analysis [494]One-Step Methods [495]Multi-Step Methods [496]Fibonacci Numbers [498]Quantum Computation [622]General Relativity [643] [1016] [1018] [2427] [2788]600 BC [2792]569 BC [3248]547 BC [3249]Graphs [6217]Algebraic Topology [6285]Quantum Field Theory [6302]Classical Physics [6303]Fractional Arithmetic [6630]Closed Formulas for Infinite Real Series [6655]Complex Numbers [6788]

Missing Informal Examples, Explanations, etc. (24)
This node includes an important theoretical result, but no informal examples, applications or explanations have yet been provided.

Nested Closed Subset Theorem [127]Law of Total Probability [449]Bayes' Theorem [464]Every Bounded Real Sequence has a Convergent Subsequence [1152]Simulating LOOP Programs Using WHILE Programs [1199]Intermediate Value Theorem [1259]Completeness Principle For Complex Numbers [1709]Supremum Property, Infimum Property [1756]Indefinite Integral, Antiderivative [1768]Mean Value Theorem For Riemann Integrals [1772]Multinomial Theorem [1822]Theorem of Large Numbers for Relative Frequencies [1838]Characterization of Eulerian Graphs [6381]Heine-Borel Theorem [6596]Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains [6604]Theorem of Bolzano-Weierstrass [6608]Continuous Functions on Compact Domains are Uniformly Continuous [6614]Continuous Real Functions on Closed Intervals are Uniformly Continuous [6616]Reverse Triangle Inequalities [6637]Intermediate Root Value Theorem [6692]Rolle's Theorem [6778]Darboux's Theorem [6779]Fundamental Theorem of Calculus [6807]Integration by Substitution [6813]

Incomplete Biographies of Mathematicians (0)
Everything is fine!

Missing Solution (5)
A solution to this problem/exercise is still missing.

Are there infinitely many primorial primes? [511]TSP - The Traveling Salesman Problem [1780]Even Perfect Numbers Problem [8132]Odd Perfect Numbers Problem [8133]Twin Prime Problem [8203]

Hidden (0)
Everything is fine!