麻省理工：量化金融的数学方法 | Mathematical ethods for Quantitative Finance

1. Introduction and Probability Review-

2. More Review; The Bernoulli Process-

3. Law of Large Numbers, Convergence-

4. Poisson (the Perfect Arrival Process)-

5. Poisson Combining and Splitting-

6. From Poisson to Markov-

7. Finite-state Markov Chains; The Matrix Approach-

8. Markov Eigenvalues and Eigenvectors-

9. Markov Rewards and Dynamic Programming-

10. Renewals and the Strong Law of Large Numbers-

11. Renewals Strong Law and Rewards-

12. Renewal Rewards, Stopping Trials, and Wald's Inequality-

13. Little, M G 1, Ensemble Averages-

14. Review-

15. The Last Renewal-

16. Renewals and Countable-state Markov-

17. Countable-state Markov Chains-

18. Countable-state Markov Chains and Processes-

19. Countable-state Markov Processes-

20. Markov Processes and Random Walks-

21. Hypothesis Testing and Random Walks-

22. Random Walks and Thresholds-

23. Martingales (Plain, Sub, and Super)-

24. Martingales Stopping and Converging-

25. Putting It All Together-

01. Tensors For Beginners (-1) Motivation-

02. Tensors for Beginners 0 Tensor Definition-

03. Tensors for Beginners 1 Forward and Backward Transformations (contains err

04. Tensors for Beginners 1.5 Correction on Forward + Backward Transforms-

05. Tensors for Beginners 2 Vector definition-

06. Tensors for Beginners 3 Vector Transformation Rules-

07. Tensors for Beginners 4 What are Covectors-

08. Tensors for Beginners 5 Covector Components

09. Tensors for Beginners 6 Covector Transformation Rules-

10. Tensors for Beginners 7 Linear Maps-

11. Tensors for Beginners 8 Linear Map Transformation Rules-

12. Tensors for Beginners 9 The Metric Tensor-

13. Tensors for Beginners 10 Bilinear Forms-

14. Tensors for Beginners 11 Linear maps are Vector-Covector Pairs-

15. Tensors for Beginners 12 Bilinear Forms are Covector-Covector pairs-

16. Tensors for Beginners 13 Tensor Product vs Kronecker Product-

17. Tensors for Beginners 14 Tensors are general vector covector combinations

18. Tensors for Beginners 15 Tensor Product Spaces-

19. Tensors for Beginners 16 Raising Lowering Indexes (with motivation, sharp

01. Group theory 1 Introduction-

02. Group theory 2 Cayley's theorem-

03. Group theory 3 Homomorphisms-

04. Group theory 4 Lagrange's theorem-

05. Group theory 5 products-

06. Group theory 6 normal subgroups and quotient groups-

07. Group theory 7 Semidirect products-

08. Group theory 8 Extensions-

09. Group theory 9 Quaternions-

10. Group theory 10 Burnside's lemma-

11. Group theory 11 Groups of prime power order-

12. Group theory 12 Cauchy's theorem-

13. Group theory 13 Dihedral groups-

14. Group theory 14 Sylow theorems-

15. Group theory 15Groups of order 12-

16. Group theory 16 Automorphisms of cyclic groups-

17. Group theory 17 Finite abelian groups-

18. Group theory 18 Nilpotent groups-

19. Group theory 19 Wreath products-

20. Group theory 20 Frobenius groups-

21. Group theory 21 Groups of order 24-

22. Group theory 22 Symmetric groups-

23. Group theory 23 Coxeter Todd algorithm-

24. Group theory 24 Extra special groups-

25. Group theory 25 The transfer homomorphism-

26. Group theory 26 Too many p groups-

27. Group theory 27 The icosahedral group-

28. Group theory 28 Groups of order 120, 168-

29. Group theory 30 Outer automorphisms-

30. Group theory 29The Jordan Holder theorem-

31. Group theory 31 Free groups-

32. Group theory 32 Subgroups of free groups-

01. Complex analysis Introduction-

02. Complex analysis Arithmetic-

03. Complex analysis Roots-

04. Complex analysis Exp, log, sin, cos-

05. Complex analysis Holomorphic functions-

06. Complex analysis Harmonic functions-

07. Complex analysis Integration-

08. Complex analysis Cauchy's theorem-

09. Complex analysis Cauchy's integral formula-

10. Complex analysis Analytic continuation-

11. Complex analysis Locally uniform convergence-

12. Complex analysis Residue calculus-

13. Complex analysis Summing series-

14. Complex analysis Zeta function functional equation-

15. Complex analysis Singularities-

16. Complex analysis Gamma function-

17. Complex analysis Maximum modulus principle-

18. Complex analysis Elliptic functions-

19. Complex analysis Weierstrass elliptic functions-

20. Complex analysis Classification of elliptic functions-

1_introduction-solving-data-science-challenges-with-mathematics-

2_motivations-for-linear-algebra-

3_getting-a-handle-on-vectors-

4_operations-with-vectors-

5_summary-

6_introduction-to-module-2-vectors-

7_modulus-inner-product-

8_cosine-dot-product-

9_projection-

10_changing-basis-

11_basis-vector-space-and-linear-independence-

12_applications-of-changing-basis-

13_summary-

14_matrices-vectors-and-solving-simultaneous-equation-problems-

15_how-matrices-transform-space-

16_types-of-matrix-transformation-

17_composition-or-combination-of-matrix-transformations-

18_solving-the-apples-and-bananas-problem-gaussian-elimination-

19_going-from-gaussian-elimination-to-finding-the-inverse-matrix-

20_determinants-and-inverses-

21_summary-

22_introduction-einstein-summation-convention-

23_matrices-changing-basis-

24_doing-a-transformation-in-a-changed-basis-

25_orthogonal-matrices-

26_the-gram-schmidt-process-

27_example-reflecting-in-a-plane-

28_welcome-to-module-5-

29_what-are-eigenvalues-and-eigenvectors-

30_special-eigen-cases-

31_calculating-eigenvectors-

32_changing-to-the-eigenbasis-

33_eigenbasis-example-

34_introduction-to-pagerank-

35_summary-

36_wrap-up-of-this-linear-algebra-course-

1_welcome-to-multivariate-calculus

2_welcome-to-module-1

3_functions

4_rise-over-run

5_definition-of-a-derivative

6_differentiation-examples-special-cases

7_product-rule

8_chain-rule

9_taming-a-beast

10_see-you-next-module

11_welcome-to-module-2

12_variables-constants-context

13_differentiate-with-respect-to-anything

14_the-jacobian

15_jacobian-applied

16_the-sandpit

17_the-hessian

18_reality-is-hard

19_see-you-next-module

20_welcome-to-module-3

21_multivariate-chain-rule

22_more-multivariate-chain-rule

23_simple-neural-networks

24_more-simple-neural-networks

25_see-you-next-module

26_welcome-to-module-4

27_building-approximate-functions

28_power-series

29_power-series-derivation

30_power-series-details

31_examples

32_linearisation

33_multivariate-taylor

34_see-you-next-module

35_welcome-to-module-5

36_gradient-descent

37_constrained-optimisation

38_see-you-next-module

39_simple-linear-regression

40_general-non-linear-least-squares

41_doing-least-squares-regression-analysis-in-practice

42_wrap-up-of-this-course

1_introduction-to-the-course

2_welcome-to-module-1

3_mean-of-a-dataset

4_variance-of-one-dimensional-datasets

5_variance-of-higher-dimensional-datasets

6_effect-on-the-mean

7_effect-on-the-co-variance

8_see-you-next-module

9_welcome-to-module-2

10_dot-product

11_inner-product-definition

12_inner-product-length-of-vectors

13_inner-product-distances-between-vectors

14_inner-product-angles-and-orthogonality

15_inner-products-of-functions-and-random-variables-optional

16_heading-for-the-next-module

17_welcome-to-module-3

18_projection-onto-1d-subspaces

19_example-projection-onto-1d-subspaces

20_projections-onto-higher-dimensional-subspaces

21_example-projection-onto-a-2d-subspace

22_this-was-module-3

23_welcome-to-module-4

24_problem-setting-and-pca-objective

25_finding-the-coordinates-of-the-projected-data

26_reformulation-of-the-objective

27_finding-the-basis-vectors-that-span-the-principal-subspace

28_steps-of-pca

29_pca-in-high-dimensions

30_other-interpretations-of-pca-optional

31_summary-of-this-module

32_this-was-the-course-on-pca

1_welcome

2_overview-and-motivation

3_distributions

4_factors

5_semantics-factorization

6_reasoning-patterns

7_flow-of-probabilistic-influence

8_conditional-independence

9_independencies-in-bayesian-networks

10_naive-bayes

11_application-medical-diagnosis

12_knowledge-engineering-example-samiam

13_basic-operations

14_moving-data-around

15_computing-on-data

16_plotting-data

17_control-statements-for-while-if-statements

18_vectorization

19_working-on-and-submitting-programming-exercises

20_overview-of-template-models

21_temporal-models-dbns

22_temporal-models-hmms

23_plate-models

24_overview-structured-cpds

25_tree-structured-cpds

26_independence-of-causal-influence

27_continuous-variables

28_pairwise-markov-networks

29_general-gibbs-distribution

30_conditional-random-fields

31_independencies-in-markov-networks

32_i-maps-and-perfect-maps

33_log-linear-models

34_shared-features-in-log-linear-models

35_maximum-expected-utility

36_utility-functions

37_value-of-perfect-information

38_knowledge-engineering

1_overview-conditional-probability-queries

2_overview-map-inference

3_variable-elimination-algorithm

4_complexity-of-variable-elimination

5_graph-based-perspective-on-variable-elimination

6_finding-elimination-orderings

7_belief-propagation-algorithm

8_properties-of-cluster-graphs

9_properties-of-belief-propagation

10_clique-tree-algorithm-correctness

11_clique-tree-algorithm-computation

12_clique-trees-and-independence

13_clique-trees-and-ve

14_bp-in-practice

15_loopy-bp-and-message-decoding

16_max-sum-message-passing

17_finding-a-map-assignment

18_tractable-map-problems

19_dual-decomposition-intuition

20_dual-decomposition-algorithm

21_simple-sampling

22_markov-chain-monte-carlo

23_using-a-markov-chain

24_gibbs-sampling

25_metropolis-hastings-algorithm

26_inference-in-temporal-models

27_inference-summary

1_learning-overview

2_regularization-the-problem-of-overfitting

3_regularization-cost-function

4_evaluating-a-hypothesis

5_model-selection-and-train-validation-test-sets

6_diagnosing-bias-vs-variance

7_regularization-and-bias-variance

8_maximum-likelihood-estimation

9_maximum-likelihood-estimation-for-bayesian-networks

10_bayesian-estimation

11_bayesian-prediction

12_bayesian-estimation-for-bayesian-networks

13_maximum-likelihood-for-log-linear-models

14_maximum-likelihood-for-conditional-random-fields

15_map-estimation-for-mrfs-and-crfs

16_structure-learning-overview

17_likelihood-scores

18_bic-and-asymptotic-consistency

19_bayesian-scores

20_learning-tree-structured-networks

21_learning-general-graphs-heuristic-search

22_learning-general-graphs-search-and-decomposability

23_learning-with-incomplete-data-overview

24_expectation-maximization-intro

25_analysis-of-em-algorithm

26_em-in-practice

27_latent-variables

28_summary-learning

29_pgm-course-summary

1_course-overview

2_instructor-introduction

3_module-1-introduction

4_what-are-time-series-and-how-are-they-used

5_getting-started-with-r

6_a-gentle-introduction-to-stationarity-part-1

7_a-gentle-introduction-to-stationarity-part-2

8_a-gentle-introduction-to-stationarity-part-3

9_module-2-introduction

10_weak-and-strong-stationarity-part-1

11_weak-and-strong-stationarity-part-2

12_weak-and-strong-stationarity-part-3

13_weak-and-strong-stationarity-part-4

14_introduction-to-linear-processes-part-1

15_introduction-to-linear-processes-part-2

16_introduction-to-linear-processes-part-3

17_introduction-to-linear-processes-part-4

18_module-3-introduction

19_understanding-arma-p-q-processes-part-1

20_understanding-arma-p-q-processes-part-2

21_understanding-arma-p-q-processes-part-3

22_understanding-arma-p-q-processes-part-4

23_computing-acf-s-of-ar-2-processes-using-difference-equations-part-1

24_computing-acf-s-of-ar-2-processes-using-difference-equations-part-2

25_computing-acf-s-of-ar-2-processes-using-difference-equations-part-3

26_computing-acf-s-of-ar-2-processes-using-difference-equations-part-4

27_computing-acf-s-of-ar-2-processes-using-difference-equations-part-5

28_module-4-introduction

29_acf-s-and-difference-equations-part-1

30_acf-s-and-difference-equations-part-2

31_acf-s-and-difference-equations-part-3

32_acf-s-and-difference-equations-part-3-cont

33_best-linear-predictors-principles-of-forecasting

34_best-linear-predictors-principles-of-forecasting

35_best-linear-predictors-principles-of-forecasting

36_best-linear-predictors-principles-of-forecasting

37_module-5-introduction

38_the-sample-acf-and-sample-pacf-part-1

39_the-sample-acf-and-sample-pacf-part-2

40_preliminary-estimation-and-the-yule-walker-equations-part-1

41_preliminary-estimation-and-the-yule-walker-equations-part-1-cont

42_maximum-likelihood-estimators-for-arma-processes-part-1

43_maximum-likelihood-estimators-for-arma-processes-part-2

44_maximum-likelihood-estimators-for-arma-processes-part-3

45_maximum-likelihood-estimators-for-arma-processes-part-4

46_module-6-introduction

47_model-diagnostics-part-1

48_model-diagnostics-part-2

49_model-diagnostics-part-3

50_order-selection-and-the-aicc-part-1

51_order-selection-and-the-aicc-part-2

52_order-selection-and-the-aicc-part-3

53_module-7-introduction

54_arima-models-part-1

55_arima-models-part-1-cont

56_arima-models-part-2

57_arima-models-part-2-cont

58_arima-models-part-3

59_arima-models-part-4

60_sarima-models-part-1

61_sarima-models-part-2

62_module-8-introduction

63_beyond-one-step-ahead-prediction-part-1

64_beyond-one-step-ahead-prediction-part-1-cont

65_beyond-one-step-ahead-prediction-part-2

66_beyond-one-step-ahead-prediction-part-3

67_beyond-one-step-ahead-prediction-part-3-cont

68_beyond-one-step-ahead-prediction-part-4

69_exponential-smoothing-part-1

70_exponential-smoothing-part-2

1_history-of-complex-numbers

2_algebra-and-geometry-in-the-complex-plane

3_polar-representation-of-complex-numbers

4_roots-of-complex-numbers

5_topology-in-the-plane

6_complex-functions

7_sequences-and-limits-of-complex-numbers

8_iteration-of-quadratic-polynomials-julia-sets

9_how-to-find-julia-sets

10_the-mandelbrot-set

11_the-complex-derivative

12_the-cauchy-riemann-equations

13_the-complex-exponential-function

14_complex-trigonometric-functions

15_first-properties-of-analytic-functions

16_inverse-functions-of-analytic-functions

17_conformal-mappings

18_mobius-transformations-part-1

19_mobius-transformations-part-2

20_the-riemann-mapping-theorem

21_complex-integration

22_complex-integration-examples-and-first-facts

23_the-fundamental-theorem-of-calculus-for-analytic-functions

24_cauchys-theorem-and-integral-formula

25_consequences-of-cauchys-theorem-and-integral-formula

26_infinite-series-of-complex-numbers

27_power-series

28_the-radius-of-convergence-of-a-power-series

29_the-riemann-zeta-function-and-the-riemann-hypothesis

30_the-prime-number-theorem

31_laurent-series

32_isolated-singularities-of-analytic-functions

33_the-residue-theorem

34_finding-residues

35_evaluating-integrals-via-the-residue-theorem

36_bonus-evaluating-an-improper-integral-via-the-residue-theorem

001. A Complete In-Depth Linear Algebra Course with Problems and Exercises

010. Linear Algebra 2ga Subtraction is Often Easier than Addition!

011. Linear Algebra 2h What Else Works like Geometric Vectors

012. Linear Algebra 2i Polynomials Are Vectors, Too!

013. Linear Algebra 2ia Is One Polynomial a Multiple of the Other

014. Linear Algebra 2j Elements of ℝⁿ as Vectors - So Boring, yet so Important!

015. Linear Algebra 2k Linear Combinations - a New Term for an Old Concept

016. Linear Algebra 2k1 Examples of Linear Combinations

017. Linear Algebra 2L The Zero Vector - Does Nothing, Yet so Important!

018. Linear Algebra 2m Span the Man

019. Linear Algebra 2n Spanning Sets

020. Linear Algebra 2o1 Linear Subspaces of the Plane

021. Linear Algebra 2o2 Straight Talk - a Linear Combination Exercise

022. Linear Algebra 2o3 Equation of a Straight Line - More Important than You'd

023. Linear Algebra 2p A Passionate Appeal - Treat All Objects on Their Own Term

024. Linear Algebra 2q Summary of Terms Encountered so Far

025. Linear Algebra Chapters 1 - 4 Outtakes - as funny as your calculus professo

026. Linear Algebra 3a What is Linear Decomposition

027. Linear Algebra 3b1 Decomposition with Geometric Vectors 1

028. Linear Algebra 3b2 Decomposition with Geometric Vectors 2

029. Linear Algebra 3b3 Decomposition with Geometric Vectors 3

030. Linear Algebra 3b4 Decomposition with Respect to Arbitrary Geometric Vector

031. Linear Algebra 3c1 Decomposition with Polynomials 1

032. Linear Algebra 3c2 Decomposition with Polynomials 2

033. Linear Algebra 3d1 Decomposition in ℝⁿ 1

034. Linear Algebra 3d2 Decomposition in ℝⁿ, 2

035. Linear Algebra 3d3 Decomposition in ℝⁿ, 3

036. Linear Algebra 3e Linear Systems Are a Decomposition Problem

037. Linear Algebra 4a Impossible Decomposition with Geometric Vectors

038. Linear Algebra 4b Impossible Decomposition with Polynomials

039. Linear Algebra 4c Impossible Decomposition in ℝⁿ

040. Linear Algebra 4d What is a Linear Property and Why It's Synonymous with Su

041. Linear Algebra 4d1 A Property Irrelevant in Decomposition Problems

042. Linear Algebra 4e Linear Subspaces in ℝⁿ

043. Linear Algebra 4f Linear Subspaces of Polynomials

044. Linear Algebra 4g A Good Way to Express Linear Subspaces in ℝⁿ

045. Linear Algebra 4h Unions and Intersections of Linear Subspaces

046. Linear Algebra 6a Linear Independence

047. Linear Algebra 6b Alternative Definition of Linear Indpendence

048. Linear Algebra 6c Second Definition of Linear Dependence Fixed!

049. Linear Algebra 6d The Equivalence of the Two Definitions of Linear Dependen

050. Linear Algebra 6da Linear Independence Implies Uniqueness

051. Linear Algebra 6e Linear Dependence Example 1 - Geometric Vectors

052. Linear Algebra 6f Linear Dependence Example 2 - Geometric Vectors

053. Linear Algebra 6g Linear Dependence Example 3 - Geometric Vectors

054. Linear Algebra 6h Linear Dependence Example 4 - Polynomials

055. Linear Algebra 6i Linear Dependence Example 5 - Vectors in ℝⁿ

056. Linear Algebra 6j Linear Systems for the Impatient

057. Linear Algebra 6k The Null Space

058. Linear Algebra 6z Outtakes from Chapters 5 and 6

059. Linear Algebra 7a Three Linearly Independent Vectors in the Plane

060. Linear Algebra 7b Four Linearly Independent Vectors in the Space

061. Linear Algebra 7c Concepts of Basis and Dimension

062. Linear Algebra 7d Relationship among Four Quadratic Polynomials

063. Linear Algebra 7d1 Linear Dependence by Simple Counting

064. Linear Algebra 7e Counting Solutions of a Linear System

065. Linear Algebra 7f1 Null Space of a Matrix

066. Linear Algebra 7f2 21 Easy Null Space Exercises

067. Linear Algebra 7g The Nullspace of a 3x3 Matrix with a Specific Pattern

068. Linear Algebra 8a Linear Systems Terminology

069. Linear Algebra 8b Our First Linear System - A Thorough Discussion

070. Linear Algebra 8c Linear Systems as Collections of Equations - Not Such a G

071. Linear Algebra 8d Another 2x4 Linear System

072. Linear Algebra Comment Reusing the Same Letter in Different Expressions

073. Linear Algebra 8e A 3x3 Linear System

074. Linear Algebra 8f A Tall Linear System

075. Linear Algebra 8g A Wide Linear System

076. Linear Algebra 8h A Linear System with Zero Columns

077. Linear Algebra 8p The Relationship Between the Column Space and the Null Sp

078. Linear Algebra 8i An Exceedingly Simple Linear System

079. Linear Algebra 8j The Simplest Possible System

080. Linear Algebra 8k You and I Have a Problem - We Need Some Distance!

081. Linear Algebra 9a Introduction to Gaussian Elimination

082. Linear Algebra 9b The Operations of Gaussian Elimination

083. Linear Algebra 9c- Review - Column Space Versus Null Space

084. Linear Algebra 9c Why Gaussian Elimination Works

085. Linear Algebra 9c1 Gaussian Elimination Cannot Be Used to Determine the Col

086. Linear Algebra 9d First Gaussian Elimination Example

087. Linear Algebra 9e Gaussian Elimination and Systems Without Solutions

088. Linear Algebra 9f Row Switching in Gaussian Elimination

089. Linear Algebra 9g Which Columns Are Pivot Columns in Gaussian Elimination

090. Linear Algebra 9i Gaussian Elimination Example 1

091. Linear Algebra 9j Gaussian Elimination Example 2

092. Linear Algebra 9k Gaussian Elimination Example 3

093. Linear Algebra 9L Gaussian Elimination Example 4

094. Linear Algebra 9m Gaussian Elimination Example 5

095. Linear Algebra 9n Gaussian Elimination Example 6

096. Linear Algebra 9o Gaussian Elimination Example 7

097. Linear Algebra 9p Pragmatic Gaussian Elimination

098. Linear Algebra 9q Gaussian Thinning

099. Linear Algebra 10a Introduction to the Row Reduced Echelon Form (RREF) of a

100. Linear Algebra 10b Calculating the Null Space from the Row Reduced Echelon

101. Linear Algebra 10c A Row Reduced Echelon Form (RREF) Challenge

102. Linear Algebra 10d The Rank of a Matrix - A Spectacular Application of the

103. Linear Algebra 10e An Application of the Matrix Rank

104. Linear Algebra 10f The RREF of a Matrix Whose the Null Space is Known Befor

105. Linear Algebra 11a Introduction to Matrix Multiplication and Matrix Algebra

106. Linear Algebra 11b A Look Ahead - An Illustration of Matrix Algebra

107. Linear Algebra 11c Definition of Matrix Multiplication for Single Columns o

108. Linear Algebra 11d Matrix Products as Actions

109. Linear Algebra 11g Complete Definition of Matrix Multiplication for Arbitra

110. Linear Algebra 11f An Example of an Incompatible Matrix Product

111. Linear Algebra 11h A Few Matrix Multiplication Examples

112. Linear Algebra 11e A Few Matrix Product Examples with Crooked Matrices

113. Linear Algebra 11j The Associative Property (a.k.a. Associativity) of Matri

114. Linear Algebra 11k The Commutative Property of Matrix Multiplication - Or L

115. Linear Algebra 11L The Three Equivalent Perspectives on Matrix Multiplicati

116. Linear Algebra 11L1 A Matrix Algebraic Expression for the Null Space

117. Linear Algebra 11L2 The Null Space of the Product AB

118. Linear Algebra 11L3 The Null Space of the Product AB, Proof by Matrix Algbr

119. Linear Algebra 11L4 The Relationship Between the Old and the New Expres

120. Linear Algebra 11L5 The Column Space of AB

121. Linear Algebra 11m The Identity Matrix - The Number One of Matrix Algebra

122. Linear Algebra 11n The Inspired Idea of the Matrix Inverse

123. Linear Algebra 11o There Is Only One Inverse Matrix

124. Linear Algebra 11p Some Matrices Are Not Invertible - I.e. They Don't Have

125. Linear Algebra 11q Algorithm for Calculating the Inverse Matrix

126. Linear Algebra 11r First Explanation for the Inversion Algorithm

127. Linear Algebra 11s An Even Better Explanation for the Inversion Algorithm B

138. Linear Algabra 11zb The Combination xᵀAy

139. Linear Algebra 11zc The Combination x ᵀAy for a Symmetric matrix A

140. Linear Algebra 12a Applications Series - Polynomial Interpolation

141. Linear Algebra 12b Applications Series - Non-Polynomial Interpolation

142. Linear Algebra 12c Applications Series - Polynomial Interpolation According

143. Linear Algebra 13a Introduction to Elementary Matrices

144. Linear Algebra 13b The Effect of an Elementary Matrix on the Other Matrix i

145. Linear Algebra 13c The Inverse of an Elementary Matrix

146. Linear Algebra 13d Elementary Matrix Exercises

147. Linear Algebra 13e The LU Decomposition

148. Linear Algebra 13f An LU Decomposition Example

149. Linear Algebra 13g Third Explanation of the Matrix Inversion Algorithm

01方程组的几何解释

02矩阵消元

03乘法和逆矩阵

04矩阵A的LU分解

05转置 置换 向量空间

06列空间和零空间

07求解Ax=0：主变量 特解

08可解性和解的结构

09线性相关性 基 维数

10四个基本子空间

11矩阵空间 秩1矩阵 小世界图

12图和网络

13复习一

14正交向量与子空间

15子空间投影

16投影矩阵 最小二乘

17正交矩阵 Schmidt正交化

18行列式及其性质

19行列式公式 代数余子式

20克拉默法则 逆矩阵 体积

21特征值 特征向量

22对角化 A的幂

23微分方程 exp At

24马尔可夫矩阵 傅立叶级数

25复习二

26对称矩阵及正定性

27复数矩阵 快速傅里叶变换

28正定矩阵 最小值

29相似矩阵 若尔当形

30奇异值分解

31线性变换及对应矩阵

32基变换 图像压缩

33单元检测3复习

34左右逆 伪逆

35期末复习

1-Probability and Measure Lecture 1 What is a Measure

2-Probability and Measure, Lecture 2 Carathéodory's Extension Theorem

3-Probability and Measure, Lecture 3 Uniqueness and Dykin's Pi-Lambda Theorem

4-Probability and Measure, Lecture 4 Lebesgue Measure

5-Probability and Measure, Lecture 5 Measurable Functions

6-Probability and Measure, Lecture 6 Integration and Convergence Theorems

7-Probability and Measure, Lecture 7 Lebesgue-Stieltjes Measure and Monotone

8-Probability and Measure, Lecture 8 Product Measure and Fubini-Tonelli

9-Probability and Measure, Lecture 9 Lp spaces and Inequalities

10-Probability and Measure, Lecture 10 Convergence of Measures

11-Probability and Measure, Lecture 11 The Strong Law of Large Numbers

12-Probability and Measure, Lecture 12 The Central Limit Theorem

13-Probability and Measure, Lecture 13 The Ergodic Theorem

1. History of Algebraic Topology; Homotopy Equivalence - Pierre Albin

2. Path Homotopy; the Fundamental Group - Pierre Albin

3. Fundamental Group of the Circle - Pierre Albin

4-4. Applications of π₁(S¹); Induced Homomorphisms - Pierre Albin-480P 标清-AVC

5. Induced Homomorphisms; van Kampen Theorem - Pierre Albin

6. Applications and Proof of the van Kampen Theorem - Pierre Albin

7. Covering Spaces; Lifting Criterion - Pierre Albin

8. Universal Covers; Application to Free Groups - Pierre Albin

9. Classification of Covering Spaces; Deck Transformations - Pierre Albin

10. Covering Space Actions; the idea of Homology - Pierre Albin

11. ∆-Complexes; Simplicial Homology - Pierre Albin

12. Singular Homology; Chain Homotopy - Pierre Albin

13. Homotopy Invariance of Homology; Exact Sequences - Pierre Albin

14. Long Exact Sequence of Pairs Triples; Excision - Pierre Albin

15. Proof of the Excision Theorem - Pierre Albin

16. Equivalence of Simplicial and Singular Homology - Pierre Albin

17. Manifolds and Local Homology; Degree of a map - Pierre Albin

18. Degree of maps between oriented manifolds - Pierre Albin

19. Homology of Cell Complexes - Pierre Albin

20. Euler Characteristic; Mayer-Vietoris Sequence - Pierre Albin

21. Homology with Coefficients; Orientation & Covering Spaces - Pierre Albin

22. Orientability with Coefficients - Pierre Albin

23. Orientation; Categories - Pierre Albin

24. Axioms for Homology; Bordism Homology - Pierre Albin

25. Relating π₁ and H₁; Jordan Curve Theorem - Pierre Albin

26. Jordan Curve; Invariance of Domain; Lefschetz Fixed Point - Pierre Albin

27. Beyond Homology Classifying Lens Spaces - Pierre Albin

01 -Topology & Geometry - LECTURE 01 Part 01_02 - by Dr Tadashi Tokieda

02 - Topology & Geometry - LECTURE 01 Part 02_02 - by Dr Tadashi Tokieda

03 - Topology & Geometry - LECTURE 02 Part 01_03 - by Dr Tadashi Tokieda

04 - Topology & Geometry - LECTURE 02 Part 02_03 - by Dr Tadashi Tokieda

05 - Topology & Geometry - LECTURE 02 Part 03_03 - by Dr Tadashi Tokieda

06 - [06]Topology & Geometry - LECTURE 03 Part 01_03 - by Dr Tadashi Tokieda

07 -Topology & Geometry - LECTURE 03 Part 02_03 - by Dr Tadashi Tokieda

08 - Topology & Geometry - LECTURE 03 Part 03_03 - by Dr Tadashi Tokieda

09 -Topology & Geometry - LECTURE 04 Part 01_02 - by Dr Tadashi Tokieda

10 - Topology & Geometry - LECTURE 04 Part 02_02 - by Dr Tadashi Tokieda

11 - Topology & Geometry - LECTURE 05 Part 01_02 - by Dr Tadashi Tokieda

12 - Topology & Geometry - LECTURE 05 Part 02_02 - by Dr Tadashi Tokieda

13 - Topology & Geometry - LECTURE 06 Part 01_02 - by Dr Tadashi Tokieda

14 - Topology & Geometry - LECTURE 06 Part 02_02 - by Dr Tadashi Tokieda

15 - Topology & Geometry - LECTURE 07 Part 01_02 - by Dr Tadashi Tokieda

16 - Topology & Geometry - LECTURE 07 Part 02_02 - by Dr Tadashi Tokieda

17 - Topology & Geometry - LECTURE 08 Part 01_02 - by Dr Tadashi Tokieda

18 - Topology & Geometry - LECTURE 08 Part 02_02 - by Dr Tadashi Tokieda

19 - Topology & Geometry - LECTURE 09 Part 01_03 - by Dr Tadashi Tokieda

20 -Topology & Geometry - LECTURE 09 Part 02_03 - by Dr Tadashi Tokieda

21 -Topology & Geometry - LECTURE 09 Part 03_03 - by Dr Tadashi Tokieda

22 - Topology & Geometry - LECTURE 10 Part 01_02 - by Dr Tadashi Tokieda

23 - Topology & Geometry - LECTURE 10 Part 02_02 - by Dr Tadashi Tokieda

24 - Topology & Geometry - LECTURE 11 Part 01_03 - by Dr Tadashi Tokieda

25 - Topology & Geometry - LECTURE 11 Part 02_03 - by Dr Tadashi Tokieda

26 - Topology & Geometry - LECTURE 11 Part 03_03 - by Dr Tadashi Tokieda

27 - Topology & Geometry - LECTURE 12 Part 01_02 - by Dr Tadashi Tokieda

28 -Topology & Geometry - LECTURE 12 Part 02_02 - by Dr Tadashi Tokieda

29 - Topology & Geometry - LECTURE 13 Part 01_02 - by Dr Tadashi Tokieda

30 - Topology & Geometry - LECTURE 13 Part 02_02 - by Dr Tadashi Tokieda

31 - Topology & Geometry - LECTURE 14 Part 01_02 - by Dr Tadashi Tokieda

32 - Topology & Geometry - LECTURE 14 Part 02_02 - by Dr Tadashi Tokieda

33 - Topology & Geometry - LECTURE 15 Part 01_03 - by Dr Tadashi Tokieda

34 - Topology & Geometry - LECTURE 15 Part 02_03 - by Dr Tadashi Tokieda

35 - Topology & Geometry - LECTURE 15 Part 03_03 - by Dr Tadashi Tokieda

1 变化率

2 极限

3 导数

4 链式法则

5 隐函数求导

6 指数与对数

7 第一次考试复习

9 线性和二次近似

10 曲线描绘

11 最大值与最小值

12 相关变化率

13 牛顿法

14 均值定理

15 原函数

16 微分方程

18 定积分

19 第一基本定理

20 第二基本定理

21 对数的应用

22 体积

23 功与概率

24 数值分析

25 第三次考试复习

27 三角积分

28 反代换

29 部分分式

30 分部积分

31 参数方程

32 极坐标

33 第四次考试复习

35 未定型

36 广义积分

37 无穷级数

38 泰勒系列

39 期末复习

01-点积

02-行列式，叉积

03-矩阵，逆矩阵

04-矩阵方程，平面方程

05-曲线和直线的参数方程

06-速度，加速度和开普勒第二定律

07-复习

08-等值面，偏导数，切平面逼近

09-极大极小问题，最小二乘法

10-二阶导检验：边界与无穷

11-微分，链式法则

12-梯度，方向导数，切平面

13-拉格朗日乘数法

14-非独立变量

15-偏微分方程，复习

16-二重积分

17-极坐标的二重积分，应用

18-变量的变化

19-平面向量场的线积分

20-路径独立和保守场

21-梯度场和势函数

22-格林公式

23-通量，格林公式的标准形式

24-单连通区域，复习

25-直角坐标和柱坐标下的三重积分

26-球面坐标，表面积

27-三维向量场，面积分和通量

28-散度定理

29-散度定理（续）,.应用和证明

30-空间线积分，旋度，恰当微分和势

31-斯托克斯定理

32-斯托克斯定理（续）

33-拓扑思考——麦克斯韦方程组

34-期末复习

35-期末复习续

1.1 多元函数简介

1.2 空间中的点和向量

1.3 三维函数图

1.4 切片法理解三维函数图

1.5 等高线图

1.6 参数曲线

1.7 参数曲面

1.8 向量场

1.9 向量场和流体流动

1.10 三维向量场

1.11 三维复杂向量场

1.12 变换，变换介绍

1.13 二维映二维

1.14 参数曲面

2.1 偏导介绍

2.2 偏导和函数图

2.3 偏导的正式定义

2.4 二阶偏导及二阶混导的对称性

3.1 梯度的计算

3.2 梯度和函数图

3.3 方向导数介绍

3.4 方向导数正式定义

3.5 方向导数和斜率

3.6 为何梯度指向最速上升

3.7 梯度垂直等高线或等势面

4.1 多元链式法则简介

4.2 多元链式法则的图形直觉

4.3 多元链式法则向量形式

4.4 多元链式法则和方向导数的关系

4.5 多元链式法则的正式证明

5.1 曲率

5.2 曲率公式1

5.3 曲率公式2

5.4 曲率公式3

5.5 曲率公式4

5.6 曲率公式5

5.7 曲率实例：螺线曲率1

5.8 曲率实例：螺线曲率2

5.9 曲率实例：摆线曲率

6.1 向量偏导：如何计算向量函数偏导

6.2 向量偏导：参数曲面的偏导1

6.3 向量偏导：参数曲面的偏导2

6.4 向量偏导：向量场偏导1

6.5 向量偏导：向量场偏导2

7.1 散度：散度的直观感受

7.2 散度，+ 0 -，散度的感受

7.3 散度：散度公式1

7.4 散度：散度公式2

7.5 散度：散度例子

7.6 散度：散度符号

8.1 二维旋度

8.2 2D旋度公式

8.3 2D旋度例子

8.4 2D旋度情形补充

9.1 三维旋度

9.2 3D旋度直觉1

9.3 3D旋度直觉2

9.4 3D旋度公式1

9.5 3D旋度公式2

9.6 3D旋度实例

10.1 拉普拉斯算子介绍

10.2 拉普拉斯算子的计算

10.3 直接化的显式拉普拉斯算子公式

10.4 调和函数简介

11.1 雅可比矩阵前置知识：矩阵和线性变

11.2 多元函数的局部线性属性

11.3 雅可比矩阵

11.4 计算雅可比矩阵

11.5 雅可比行列式

01. Probability Models and Axioms

02. Conditioning and Bayes' Rule

03. Independence

04. Counting

05. Discrete Random Variables I

06 Discrete Random Variables II

07. Discrete Random Variables III

08. Continuous Random Variables

09. Multiple Continuous Random Variable

10. Continuous Bayes' Rule; Derived D

11. Derived Distributions (ctd.)

12. Iterated Expectations

13. Bernoulli Process

14. Poisson Process I

15. Poisson Process II

16. Markov Chains I

17. Markov Chains II

18. Markov Chains III

19. Weak Law of Large Numbers

20. Central Limit Theorem

21. Bayesian Statistical Inference I

22. Bayesian Statistical Inference II

23. Classical Statistical Inference I

24. Classical Inference II

25. Classical Inference III

1-认识你的老师

2-陶哲轩的旅程

3-神秘的数学

4-数学在你的日常生活

5-选择要解决的问题

6-通过故事解决问题

7-转换的问题

8-游戏、作弊和谜题

9-数学与失败

10-困惑

11-从数字中寻找力量

12-勇往直前

1.第一讲 课程介绍 证明

2.第二讲 反证法 数学归纳法

3.第三讲 不变式 第二数学归纳法

4.第四讲 状态机 最大公约数 欧几里得算法

5.第五讲 同余式 欧拉定理 RSA加密算法

6.第六讲 图着色问题 NP完备性

7.第七讲 匹配 二分图 稳定婚姻算法

8.第八讲 最小权重生成树

9-片尾

[第1集] 课程简介

[第2集] 多重回归

[第3集] 信任区间的修正练习

[第4集] 多重回归第二部分

[第5集] 使用SPSS软件研究实例

[第6集] 一个真实项目的统计学分析介绍

[第7集] 主成分因素分析

[第1集] 第一课

[第2集] 第二课

[第3集] 第三课

[第4集] 第四课

[第5集] 第五课

[第6集] 第六课

[第7集] 第七课

[第8集] 第八课

[第9集] 第九课

[第10集] 第十课

1-M7Q4BLENR

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24-M7Q4BLENR

25-M7Q4BLENR

26-M7Q4BLENR

27 - M7Q4BLENR

28 - M7Q4BLENR

29 - M7Q4BLENR

01 - [01]math e222 L01 20030915.zh_en

02 - [02]math e222 L02 20030917.zh_en_2025-08-24_20-29-40

03 - [03]math e222 L03 20030919.zh_en

04 - [04]math e222 L04 20030922.zh_en

05 - [05]math e222 L05 20030924.zh_en

06 - [06]math e222 L06 20030926.zh_en

07 - [07]math e222 L07 20030929.zh_en

08 - [08]math e222 L08 20031001.zh_en_2025-08-25_06-13-35

09 - [09]math e222 L09 20031003.zh_en

10 - [10]math e222 L10 20031006.zh_en_2025-08-25_12-14-33

11 - [11]math e222 L11 20031008.zh_en_2025-08-25_14-03-07

12 - [12]math e222 L12 20031010.zh_en

13 - [13]math e222 L13 20031017.zh_en

14 - [14]math e222 L14 20031020.zh_en

15 - [15]math e222 L15 20031022.zh_en

16 - [16]math e222 L16 20031024.zh_en_2025-08-25_15-13-07

17 - [17]math e222 L17 20031027.zh_en

18 - [18]math e222 L18 20031029.zh_en

19 - [19]math e222 L19 20031031.zh_en

20 - [20]math e222 L20 20031103.zh_en

21 - [21]math e222 L21 20031105.zh_en

22 - [22]math e222 L22 20031107.zh_en

23 - [23]math e222 L23 20031110.zh_en_2025-08-25_16-35-20

24 - [24]math e222 L24 20031112.zh_en

25 - [25]math e222 L25 20031114.zh_en_2025-08-25_18-55-13

26 - [26]math e222 L26 20031117.zh_en

27 - [27]math e222 L27 20031119.zh_en

28 - [28]math e222 L28 20031121.zh_en_2025-08-25_20-51-57

29 - [29]math e222 L29 20031126.zh_en

30 - [30]math e222 L30 20031201.zh_en_2025-08-25_21-47-00

31 - [31]math e222 L31 20031203.zh_en

32 - [32]math e222 L32 20031205.zh_en_2025-08-25_22-19-41

33 - [33]math e222 L33 20031208.zh_en

34 - [34]math e222 L34 20031210.zh_en_2025-08-26_00-03-59

35 - [35]math e222 L35 20031212.zh_en

36 - [36]math e222 L36 20031215.zh_en

37 - [37]math e222 Review1 20040114.zh_en

38 - [38]math e222 Review2 20040116.zh_en_2025-08-25_11-12-49

01 Gilbert and Cleve Introduction

02 Overview of Differential Equations

03 The Calculus You Need

04 Response to Exponential Input

05 Response to Oscillating Input

06 Solution for Any Input

07 Step Function and Delta Function

08 Response to Complex Exponential

09 Integrating Factor for Constant Rate

10 Integrating Factor for a Varying Rate

11 The Logistic Equation

12 The Stability and Instability of Steady States

13 Separable Equations

14 Second Order Equations

15 Forced Harmonic Motion

16 Unforced Damped Motion

17 Impulse Response and Step Response

18 Exponential Response – Possible Resonance

19 Second Order Equations with Damping

20 Electrical Networks - Voltages and Currents

21 Method of Undetermined Coefficients

22 An Example of Undetermined Coefficients

23 Variation of Parameters

24 Laplace Transform - First Order Equation

25 Laplace Transform - Second Order Equation

26 Laplace Transforms and Convolution

27 Pictures of Solutions

28 Phase Plane Pictures - Source, Sink, Saddle

29 Phase Plane Pictures - Spirals and Centers

30 Two First Order Equations - Stability

31 Linearization at Critical Points

32 Linearization of two nonlinear equations

33 Eigenvalues and Stability - 2 by 2 Matrix, A

34 The Tumbling Box in 3-D

35 The Column Space of a Matrix

36 Independence, Basis, and Dimension

37 The Big Picture of Linear Algebra

38 Graphs

39 Incidence Matrices of Graphs

40 Eigenvalues and Eigenvectors

41 Diagonalizing a Matrix

42 Powers of Matrices and Markov Matrices

43 Solving Linear Systems

44 The Matrix Exponential

45 Similar Matrices

46 Symmetric Matrices, Real Eigenvalues, Orthogonal Eigenvectors

47 Second Order Systems

48 Positive Definite Matrices

49 Singular Value Decomposition (the SVD)

50 Boundary Conditions Replace Initial Conditions

51 Laplace Equation

52 Fourier Series

53 Examples of Fourier Series

54 Fourier Series Solution of Laplaces Equation

55 Heat Equation

56 Wave Equation

57 Euler, ODE1

58 Midpoint Method, ODE2

59 Classical Runge-Kutta, ODE4

60 Order, Naming Conventions

61 Estimating Error, ODE23

62 ODE45

63 Stiffness, ODE23s, ODE15s

64 Systems of Equations~1

65 The MATLAB ODE Suite

66 Tumbling Box

67 Predator-Prey Equations

68 Lorenz Attractor and Chaos

01_第一讲-实数导论

02_第二讲-实数导论（续）

03_第三讲-如何写证明_阿基米德性质

04_第四讲-序列_收敛

05_第五讲-单调收敛定理

06_第六讲-柯西收敛定理

07_第七讲-波尔扎诺-魏尔斯特拉斯定理_柯西序列_级数

08_第八讲-级数收敛判别法_幂级数

09_第九讲-上极限与下极限_幂级数_连续函数_指数函数

10_第十讲-连续函数_指数函数（续）

11_第十一讲-极值定理与介值定理_度量空间

12_18.100B 实分析期中复习

13_第十二讲-度量空间中的收敛_集合运算

14_第十三讲-开集与闭集_覆盖_紧致性

15_第十四讲-序列紧致性_度量空间中的波尔扎诺-魏尔斯特拉斯定理

16_第十五讲-导数_微分法则

17_第十六讲-罗尔定理_中值定理_洛必达法则_泰勒展开

18_第十七讲-泰勒多项式_余项_黎曼积分

19_第十八讲-可积函数

20_第十九讲-微积分基本定理

21_第二十讲-逐点收敛_一致收敛

22_第二十一讲-一致收敛下的积分与导数

23_第二十二讲-幂级数的微分与积分_常微分方程(ODE)

24_第二十三讲-ODE的存在性与唯一性-皮卡-林德洛夫定理

25_18.100B 实分析期末复习

01_凸包算法

02_平面扫描算法用于查找线段交点

03_用于平面细分的双向链表

04_多边形三角剖分_美术馆问题

05_用于点定位的垂直分解_第1部分-数据结构

06_用于点定位的垂直分解_第2部分-随机增量构建

07_用于点定位的垂直分解_第3部分-随机增量构建的分析

08_Voronoi图（Delaunay三角剖分和Voronoi图_第1部分）

09_Delaunay三角剖分（Delaunay三角剖分和Voronoi图_第2部分）

10_几何对偶

11_线排列

12_机器人运动规划-引言_路线图_和配置空间（3部分之1）

13_机器人运动规划-计算（凸）多边形的闵可夫斯基和（3部分之2）

14_机器人运动规划-旋转机器人_可见性图（3部分之3）

15_良好分离对分解和生成树

16_范围搜索-简短示例

17_范围搜索

18_窗口查询1_3-区间树

19_窗口查询2_3-优先搜索树

20_窗口查询3_3-线段树

21_四叉树_平衡四叉树_和网格化

22_几何算法-超越理论

讲座 1：简介

讲座 2：命题逻辑语言的形式定义

讲座 3：命题公式的语法性质

讲座 4：命题逻辑的语义

讲座 5：公式集合的语义概念

讲座 6：公式集合的语义概念（续）

讲座 7：紧致性定理

讲座 8：命题逻辑证明的语法

讲座 9：紧致性定理的推论

讲座 10：证明系统、公理与蕴含规则（Modus Ponens）

讲座 11：正确性定理、一致性定理、极大一致性定理、单调性

讲座 12：一致性（续）

讲座 13：一致性与其他语义到语法的联系

讲座 14：谓词演算简介

讲座 15：结构与语法

讲座 16：语义

讲座 17：一阶逻辑的表达能力——特定结构内的可定义性

无限集合简介

讲座 18：紧致性定理的应用

讲座 19：紧致性定理的延续 & 一阶逻辑的证明系统

讲座 20：一阶逻辑的证明系统

讲座 21：证明系统、全称推广 & 逻辑在计算机科学中的应用（柯尔莫哥洛夫复杂性）

讲座 22：柯尔莫哥洛夫复杂性、停机问题 & 哥德尔不完备性定理

第0讲-欢迎课

第1讲-导论材料

第2讲-逻辑组合子

作业1辅导

作业2辅导

习题集1辅导

第3讲-蕴涵

第4讲-等价关系

补充-课程评分标准使用指南

作业3辅导

作业4辅导

习题集2辅导

第5讲-量词

补充-如何阅读数学公式

作业5辅导

习题集3辅导

第6a讲-量词运用1

第6b讲-量词运用2

作业6辅导

习题集4辅导

第7a讲-证明方法1

第7b讲-证明方法2

作业7辅导

习题集5辅导

第8a讲-含量词的证明1

第8b讲-含量词的证明2

作业8辅导

习题集6辅导

第9a讲-数论基础1

第9b讲-数论基础2

作业9辅导

习题集7辅导

第10a讲-实分析基础1

第10b讲-实分析基础2

第10c讲-实分析基础3

作业10辅导

习题集8辅导

测试飞行-导言

测试飞行辅导1

测试飞行辅导2

测试飞行辅导3

数学思维阅读-《声障》

01_课程欢迎_01_02_01

02_认识-根瑟-瓦尔特_01_02_02

03_介绍_01_03_01

04_饼图-条形图-和直方图_01_03_02

05_箱须图-和散点图_01_03_03

06_提供背景信息是统计分析的关键_01_03_04

07_可视化信息时的陷阱_01_03_05

08_均值和中位数_01_04_01

09_百分位数-五数概括-和标准差_01_04_02

10_额外的行业见解-安德鲁-拉丁的介绍_01_05_01

11_介绍_02_01_01

12_简单随机抽样和分层随机抽样_02_02_01

13_偏差和偶然误差_02_02_02

14_观察与实验-混淆和安慰剂效应_02_03_01

15_随机对照实验的逻辑_02_03_02

16_额外的行业见解-为TwoXar申请专利_02_04_01

17_概率的解释_03_01_01

18_补集-等可能结果-加法和乘法_03_02_01

19_四个规则示例-如何处理至少一个_03_02_02

20_通过完全枚举解决问题_03_03_01

21_贝叶斯法则_03_03_02

22_贝叶斯分析_03_04_01

23_沃纳的随机响应模型_03_04_02

24_额外的行业见解-TwoXar的药物发现_03_05_01

25_正态曲线_04_01_01

26_经验法则_04_02_01

27_标准化数据和标准正态曲线_04_02_02

28_正态近似_04_02_03

29_用正态近似计算百分位数_04_02_04

30_二项式设置和二项式系数_04_03_01

31_二项式公式_04_03_02

32_随机变量和概率直方图_04_03_03

33_二项式正态近似-无放回抽样_04_03_04

34_额外的行业见解-生命科学领域的机遇_04_04_01

35_参数和统计量_05_01_01

36_期望值和标准误差_05_02_01

37_总和-百分比以及模拟时的EV和SE_05_02_02

38_平方根定律_05_02_03

39_抽样分布_05_02_04

40_三个直方图_05_02_05

41_大数定律_05_03_01

42_中心极限定理_05_03_02

43_中心极限定理何时适用_05_03_03

44_预测是统计的关键任务_06_01_01

45_相关系数_06_02_01

46_相关性衡量线性关联_06_02_02

47_回归线和最小二乘法_06_03_01

48_回归到均值-回归谬误_06_03_02

49_从x预测y和从y预测x_06_03_03

50_给定x的正态近似_06_03_04

51_残差图-异方差性和变换_06_04_01

52_异常值和有影响力的点_06_04_02

53_额外的行业见解-在医学中使用数据科学的挑战_06_05_01

54_置信区间的解释_07_01_01

55_使用中心极限定理找到置信区间_07_02_01

56_用bootstrap原则估计标准误差_07_02_02

57_更多关于置信区间_07_02_03

58_假设检验背后的思想_08_01_01

59_建立检验统计量_08_02_01

60_p值作为证据的度量_08_02_02

61_通过口味区分可口可乐和百事可乐_08_03_01

62_t检验_08_03_02

63_统计显著性与重要性_08_03_03

64_双样本z检验_08_04_01

65_配对_08_04_02

66_额外的行业见解-招聘数据科学人才_08_05_01

67_使用计算机模拟代替计算_09_01_01

68_使用大数定律来近似感兴趣的量_09_02_01

69_插入原则_09_03_01

70_参数bootstrap和bootstrap置信区间_09_04_01

71_回归中的Bootstrapping_09_04_02

72_两个类别变量之间的关系_10_01_01

73_M&M豆的颜色比例_10_02_01

74_同质性和独立性的卡方检验_10_03_01

75_比较几个均值_11_01_01

76_方差分析的思想_11_02_01

77_使用F分布评估方差分析_11_02_02

78_更多关于方差分析_11_02_03

79_额外的行业见解-开始你的数据科学职业生涯_11_03_01

80_数据窥探和多重检验谬误-可重现性和可复验性_12_01_01

81_Bonferroni校正-错误发现率和数据拆分_12_02_01

82_总结_12_03_01

01 -Single Neuron Models 单神经元模型

02 - Neural Networks (Hopfield, Perceptron, Machine learning) 神经网络（Ho

03 -Wilson Cowan Model Wilson Cowan 模型

04 - Neural fields 神经场

05 - Oscillations and synchrony 振荡与同步

06 - Large Scale Simulations 大规模模拟

07 - Probabilistic Brain 概率大脑

08 - Effect of Noise in Neural Systems 神经系统中噪声的影响

09 - Anaesthesia & Plasticity 麻醉与可塑性

10 - Presentations & Conclusions 展示与总结

01_介绍

02_多变量微积分回顾 (已更新并修正)

03_笛卡尔-极坐标与基向量

04_雅可比矩阵

05_导数是向量

06_导数变换规则（逆变性）

07_导数算符是向量的讨论

08_微分形式是余向量

09_余向量场分量

10_余向量场变换规则（协变性）

11_用微分形式积分

12_用微分形式积分的例子

13_度量张量与弧长（平坦空间）

14_弯曲空间中用于测量弧长的度量张量

15_梯度与d算符（外微分-微分）

16_梯度解释+例子

17_测地线与克里斯托费尔符号（外在几何）

18_平面和球面上的测地线例子

19_协变导数（平坦空间）

20_协变导数（分量定义）-可选

21_协变导数（外在）和平行输运

22_协变导数（内在）和测地线

23_抽象协变导数（Levi-Civita联络）

24_李括号__流__挠率张量（包含错误__见置顶评论）

25_黎曼曲率张量几何意义（完整性+测地线偏差）

26_黎曼曲率张量分量和对称性

27_里奇张量几何意义（截面曲率）

28_几何意义-里奇张量-标量（体积形式）

29_里奇张量-标量性质

01-表征理论导论

02-表示

03-模

04-intertwining算子

05-单纯表示

06-不可分解表示

07-若尔当-赫尔德定理

08-半单性

09-舒尔引理

10-特征标

11-特征标表

12-特征标表示例

13-舒尔正交关系

14-正则表示

15-对偶群

16-有限傅里叶分析

17-拉格朗日表示定理

18-伯恩赛德定理

19-诱导与限制

20-弗罗贝尼乌斯互反律

21-杨氏图

22-斯佩赫特模

23-弗罗贝尼乌斯特征标公式