overview.yaml

# The list of topics was originally gathered from
# http://media.devenirenseignant.gouv.fr/file/agreg_externe/59/7/p2020_agreg_ext_maths_1107597.pdf

# 1.
Linear algebra:
  Fundamentals:
    vector space: 'algebra/module.html#vector_space'
    product of vector spaces: 'algebra/pi_instances.html#prod.module'
    vector subspace: 'algebra/module.html#subspace'
    quotient space: 'linear_algebra/basic.html#submodule.quotient'
    sum of subspaces: 'linear_algebra/basic.html#submodule.complete_lattice'
    direct sum: ''
    complementary subspaces: 'linear_algebra/basis.html#submodule.exists_is_compl'
    linear independence: 'linear_algebra/basis.html#linear_independent'
    generating sets: ''
    bases: ''
    existence of bases: 'linear_algebra/basis.html#exists_is_basis'
    linear map: 'algebra/module.html#linear_map'
    range of a linear map: 'linear_algebra/basic.html#linear_map.range'
    kernel of a linear map: 'linear_algebra/basic.html#linear_map.ker'
    algebra of endomorphisms of a vector space: ''
    general linear group: 'linear_algebra/basic.html#linear_map.general_linear_group'
  Duality:
    dual vector space: 'linear_algebra/dual.html#module.dual'
    dual basis: 'linear_algebra/dual.html#is_basis.dual_basis'
    transpose of a linear map: ''
    orthogonality: ''
  Finite-dimensional vector spaces:
    finite-dimensionality : 'linear_algebra/finite_dimensional.html#finite_dimensional'
    isomorphism with $K^n$: 'linear_algebra/basis.html#module_equiv_finsupp'
    rank of a linear map: ''
    rank of a set of vectors: ''
    rank of a system of linear equations: ''
    isomorphism with bidual: 'linear_algebra/dual.html#vector_space.eval_equiv'
  Multilinearity:
    multilinear map: 'linear_algebra/multilinear.html#multilinear_map'
    determinant of vectors: ''
    determinant of endomorphisms: ''
    special linear group: ''
    orientation of a $\R$-valued vector space: ''
  Matrices:
    commutative ring valued matrices: 'data/matrix/basic.html#matrix'
    field-valued matrices: 'data/matrix/basic.html#matrix'
    matrix representation of a linear map: 'linear_algebra/matrix.html#linear_map.to_matrix'
    change of basis: ''
    rank of a matrix: ''
    determinant: 'linear_algebra/determinant.html#matrix.det'
    invertibility: 'linear_algebra/nonsingular_inverse.html#matrix.nonsing_inv'
    elementary row operations: ''
    elementary column operations: ''
    Gauss' pivot: ''
    row-reduced matrices: ''
  Endomorphism polynomials:
    annihilating polynomials: ''
    minimal polynomial: ''
    characteristic polynomial: ''
    Cayley-Hamilton theorem: ''
  Structure theory of endomorphisms:
    eigenvalues: ''
    eigenvectors: ''
    diagonalization: ''
    trigonalization: ''
    endomorphism invariant subspaces: ''
    characteristic subspaces: ''
    kernels lemma: ''
    Dunford decomposition: ''
    Jordan normal form: ''
  Linear representations:
    irreducible representation: ''
    Schur's lemma: ''
    examples: ''
  Exponential:
    endomorphism exponential: ''
    matrix exponential: ''

# 2.
Group Theory:
  Basic definitions:
    group: 'core/init/algebra/group.html#group'
    group morphism: 'algebra/group/hom.html#monoid_hom'
    direct product of groups: 'algebra/group/prod.html#prod.group'
    subgroup: 'group_theory/bundled_subgroup.html#subgroup'
    subgroup generated by a subset: 'group_theory/bundled_subgroup.html#subgroup.closure'
    order of an element: 'group_theory/order_of_element.html'
    normal subgroup: 'group_theory/subgroup.html#normal_subgroup'
    quotient group: 'group_theory/quotient_group.html#quotient_group.group'
    group action: 'group_theory/group_action.html#mul_action'
    stabilizer of a point: 'group_theory/group_action.html#mul_action.stabilizer'
    orbit: 'group_theory/group_action.html#mul_action.orbit'
    quotient space: 'group_theory/group_action.html#mul_action.orbit_equiv_quotient_stabilizer'
    class formula: ''
    conjugacy classes: ''
  Abelian group:
    cyclic group: 'group_theory/order_of_element.html#is_cyclic'
    finite type abelian groups: ''
    complex roots of unity: ''
    primitive complex roots of unity: ''
  Permutation group:
    permutation group of a type: 'data/equiv/basic.html#equiv.perm'
    decomposition into transpositions: ''
    decomposition into cycles with disjoint support: ''
    signature: 'group_theory/perm/sign.html#equiv.perm.sign'
    alternating group: ''
  Classical automorphism groups:
    general linear group: 'linear_algebra/basic.html#linear_map.general_linear_group'
    special linear group: 'linear_algebra/special_linear_group.html#matrix.special_linear_group'
    orthogonal group: ''
    special orthogonal group: ''
    unitary group: ''
    special unitary group: ''
  Representation theory of finite groups:
    representations of abelian groups: ''
    dual groups: ''
    Maschke theorem: ''
    orthogonality of irreducible characters: ''
    Fourier transform for finite abelian groups: ''
    convolution: ''
    class function over a group: ''
    characters of a finite dimensional representation: ''
    orthonormal basis of irreducible characters: ''
    examples of groups with small cardinality: ''

# 3.
Ring Theory:
  Fundamentals:
    ring: 'algebra/ring.html#ring'
    subrings: 'ring_theory/subring.html#top'
    ring morphisms: 'algebra/ring.html#ring_hom'
    Ring structure $\Z$: 'init_/data/int/basic.html#int.comm_ring'
    Product of rings: 'algebra/pi_instances.html#pi.ring'
  Ideals and Quotients:
    Ideal of a commutative ring: 'algebra/module.html#ideal'
    Quotient rings: 'ring_theory/ideals.html#ideal.quotient'
    Prime ideals: 'ring_theory/ideals.html#ideal.is_prime'
    Maximal ideals: 'ring_theory/ideals.html#ideal.is_maximal'
    Chinese remainder theorem: 'ring_theory/ideal_operations.html#ideal.quotient_inf_ring_equiv_pi_quotient'
  Algebra:
    algebra over a commutative ring: 'ring_theory/algebra.html#algebra'
    associative algebra over a commutative ring:
  Divisibility in integral domains:
    irreducible elements: 'algebra/associated.html#irreducible'
    invertible elements: 'algebra/invertible.html#invertible'
    coprime elements: 'ring_theory/ideals.html#ideal.is_coprime'
    unique factorisation domain (UFD): 'ring_theory/unique_factorization_domain.html'
    greatest common divisor: 'algebra/gcd_domain.html#gcd_domain.gcd'
    least common multiple: 'algebra/gcd_domain.html#gcd_domain.lcm'
    $A[X]$ is a UFD when $A$ is a UFD: ''
    principal ideal domain: 'ring_theory/principal_ideal_domain.html#submodule.is_principal'
    Euclidean rings: 'algebra/euclidean_domain.html#euclidean_domain'
    Euclid's' algorithm: 'data/int/gcd.html#nat.xgcd'
    $\Z$ is a euclidean ring: 'algebra/euclidean_domain.html#int.euclidean_domain'
    congruence in $\Z$: 'data/int/modeq.html#int.modeq'
    Prime numbers: 'algebra/associated.html#associates.prime'
    Bezout's identity: 'data/int/gcd.html#nat.gcd_eq_gcd_ab'
    $\Z/n\Z$ and its invertible elements: 'data/zmod/basic.html#zmod.unit_of_coprime'
    Euler's totient function ($\varphi$): 'data/nat/totient.html#nat.totient'
  Polynomial rings:
    $K[X]$ is a euclidean ring when $K$ is a field: 'data/polynomial.html#polynomial.euclidean_domain'
    irreducible polynomial: 'algebra/associated.html#irreducible'
    cyclotomic polynomials in $\Q[X]$: ''
    Eisenstein's criterion: ''
    polynomial algebra in one or several indeterminates over a commutative ring: 'data/mv_polynomial.html#mv_polynomial'
    roots of a polynomial: 'data/polynomial.html#polynomial.roots'
    multiplicity: 'data/polynomial.html#polynomial.root_multiplicity'
    relationship between the coefficients and the roots of a split polynomial:
    Newton's sums:
    polynomial derivative: 'data/polynomial.html#polynomial.derivative'
    decomposition into sums of homogeneous polynomials:
    symmetric polynomials:
  Field Theory:
    fields: 'algebra/field.html#field'
    characteristic of a ring: 'algebra/char_p.html#ring_char'
    characteristic zero: 'algebra/char_zero.html#char_zero'
    characteristic p: 'algebra/char_p.html#char_p'
    Subfields: 'field_theory/subfield.html'
    Frobenius morphisms: 'algebra/char_p.html#frobenius'
    field $\Q$ of rational numbers: 'data/rat/basic.html#rat.division_ring'
    field $\R$ of real numbers: 'data/real/basic.html#real.division_ring'
    field $\C$ of complex numbers: 'data/complex/basic.html#complex.field'
    fundamental theorem of algebra: 'analysis/complex/polynomial.html#complex.exists_root'
    field of fractions of an integral domain: 'ring_theory/localization.html#fraction_map'
    algebraic elements: 'ring_theory/algebraic.html#is_algebraic'
    transcendental elements:
    algebraic extensions: 'ring_theory/algebraic.html#algebra.is_algebraic'
    algebraically closed fields:
    rupture fields: 'ring_theory/adjoin_root.html#adjoin_root'
    splitting fields:
    finite fields: 'field_theory/finite.html'
    rational fraction fields with one indeterminate over the field:
  Partial fraction decomposition:
    General:
    $\R(X)$-partial fraction decomposition:
    $\C(X)$-partial fraction decomposition:

# 4.
Bilinear and Quadratic Forms Over a Vector Space:
  Bilinear forms:
    bilinear forms: 'linear_algebra/bilinear_form.html#bilin_form'
    alternating bilinear forms: 'linear_algebra/bilinear_form.html#aly_bilin_form.is_alt'
    symmetric bilinear forms: 'linear_algebra/bilinear_form.html#sym_bilin_form.is_sym'
  Quadratic forms:
    quadratic form: 'linear_algebra/quadratic_form.html#quadratic_form'
    polar form of a quadratic: 'linear_algebra/quadratic_form.html#quadratic_form.polar'
  Orthogonal elements: 'linear_algebra/bilinear_form.html#bilin_form.is_ortho'
  Nondegenerate forms:
  Adjoint endomorphism:
  Matrix representation: 'linear_algebra/bilinear_form.html#bilin_form.to_matrix'
  Change of coordinates: 'linear_algebra/bilinear_form.html#bilin_form.to_matrix_comp'
  Rank of a bilinear form:
  Orthogonality:
    inertia law of Sylvester:
    real classification:
    complex classification:
    Schmidt orthogonalisation:
  Vector spaces:
    Euclidean vector spaces: 'analysis/normed_space/real_inner_product.html#inner_product_space'
    Hermitian vector spaces:
    dual isomorphism in the euclidean case:
    orthogonal supplementary:
    Cauchy-Schwarz inequality: 'analysis/normed_space/real_inner_product.html#inner_mul_inner_self_le'
    norm: 'analysis/normed_space/real_inner_product.html#inner_product_space_has_norm'
    orthonormal bases:
  Orthogonal group:
  Unitary group:
  Special orthogonal group:
  Special unitary group:
  Symmetrical endomorphism: 'linear_algebra/bilinear_form.html#bilin_form.is_self_adjoint'
  Normal endomorphism:
  Diagonalization of a symmetrical endomorphism:
  Diagonalization of normal endomorphisms:
  Simultaneous reduction of two real quadratic forms with a definite positive one:
  Polar decompositions in $\mathrm{GL}(n, \R)$:
  Polar decompositions in $\mathrm{GL}(n, \C)$:
  Triple product:
  Vector product:
  Examples:
    decomposition of an orthogonal automorphism in product of reflections:
    euclidean vector spaces of dimension 2:
    euclidean vector spaces of dimension 3:
    classification of $\mathrm{O}(2, \R)$:
    classification of $\mathrm{O}(3, \R)$:

# 5.
Affine and Euclidian Geometry (finite dimensional only):
  Affine spaces and associated vector spaces:
  Affine functions and linear associated functions:
  Affine subspaces:
  Barycenters:
  Affine spans:
  Equations of affine subspaces:
  Affine groups:
  Affine property:
  Homothetic transformation groups:
  Affinity:
  Convex subsets:
  Convex hull of a subset of an affine real space:
  Extreme point:
  Isometries of a Euclidian affine subspace:
  Euclidian affine space isometry group:
  Euclidean affine space isometries:
  Isometries that do and do not preserve orientation:
  Direct and indirect similarities of the plane:
  Isometric classification in two and three dimensions:
  Angles of vectors:
  Angles formed by planes:
  Inscribed angle theorem:
  Cocyclicity:
  Group of isometries stabilizing subset of the plane or space:
  Regular polygons:
  Metric relations in the triangle:
  Using complex numbers in plane geometry:
  Application of quadratic forms to study proper conic sections of the affine euclidean plane:
    foyer:
    eccentricity:
    quadratics on 3 dimensional euclidean affine spaces:

# 6.
Single Variable Real Analysis:
  Real Numbers:
    definition of $\R$: 'data/real/basic.html#real'
    field structure: 'data/real/basic.html#real.division_ring'
    order: 'data/real/basic.html#real.linear_order'
  Sequences of real numbers:
    convergence: 'order/filter/basic.html#filter.tendsto'
    limit point:
    recurrent sequences: 'core/init/core.html#nat'
    Limit infimum and supremum: 'order/liminf_limsup.html'
    Cauchy sequences: 'topology/uniform_space/cauchy.html#cauchy_seq'
  Topology of R:
    metric structure: 'topology/metric_space/basic.html#real.metric_space'
    Completeness of R: 'topology/instances/real.html#real.complete_space'
    Bolzano-Weierstrass theorem:
    Compact subsets of $\R$: 'topology/metric_space/basic.html#metric.compact_iff_closed_bounded'
    Connected subsets of $\R$:
    Additive subgroups of $\R$:
  Numerical Series:
    Convergence of real valued-series:
    Geometric series: 'analysis/specific_limits.html#has_sum_geometric_of_abs_lt_1'
    Riemann series:
    Positive valued series:
    Summation of comparison relations:
    Comparison of a series and an integral:
    Error estimation:
    Absolute convergence:
    Products of series:
    Alternating series:
  Real-valued functions defined on a subset of $\R$:
    Continuity: 'topology/basic.html#continuous'
    Limits: 'order/filter/basic.html#filter.tendsto'
    Intermediate value theorem: 'topology/algebra/ordered.html#intermediate_value_Icc'
    Image of a segment:
    Continuity of monotonic functions:
    Continuity of reciprocal functions:
  Differentiability:
    Derivative at a point: 'analysis/calculus/deriv.html#has_deriv_at'
    Differentiable functions: 'analysis/calculus/deriv.html#has_deriv_at'
    Derivative of a composite function: 'analysis/calculus/deriv.html#deriv.comp'
    Derivative of a reciprocal function: 'analysis/calculus/deriv.html#has_strict_deriv_at.of_local_left_inverse'
    Rolle's theorem: 'analysis/calculus/local_extr.html#exists_deriv_eq_zero'
    Mean value theorem: 'analysis/calculus/mean_value.html#exists_ratio_deriv_eq_ratio_slope'
    Higher order derivatives of functions: 'analysis/calculus/iterated_deriv.html#iterated_deriv'
    $C^k$ functions: 'analysis/calculus/times_cont_diff.html#times_cont_diff'
    piecewise $C^k$ functions:
    Leibniz formula: 'analysis/calculus/deriv.html#deriv_mul'
  Taylor-like theorems:
    Taylor with rough error estimation:
    Taylor with integral error estimation:
    Taylor-Lagrange:
    Series expansions:
  Usual functions (trigonometric, rational, $\exp$, $\log$, etc):
    Polynomial functions: 'data/polynomial.html#polynomial.eval'
    Rational functions:
    Logarithms: 'analysis/special_functions/exp_log.html#real.log'
    Exponential: 'data/complex/exponential.html#real.exp'
    Power functions: 'algebra/group_power.html#monoid.pow'
    Circular trigonometric functions: 'data/complex/exponential.html#real.sin'
    Hyperbolic trigonometric functions: 'data/complex/exponential.html#real.sinh'
    Reciprocal circular trigonometric functions: 'analysis/special_functions/trigonometric.html#real.arcsin'
    Reciprocal hyperbolic trigonometric functions:
  Integration:
    Integral over a segment of piecewise continuous functions:
    Antiderivatives:
    Riemann sums:
    Antiderivative of a continuous function:
    Usual methods of  calculating integrals:
    Change of variable:
    Integration by parts:
    Generalized integrals:
    Absolutely convergent integrals:
    Integration of asymptotic comparison relationships:
    Semi-convergent integrals:
  Sequences and series of functions:
    Pointwise convergence:
    Uniform convergence: 'topology/uniform_space/uniform_convergence.html#tendsto_uniformly'
    Normal convergence:
    Continuity of the limit: 'topology/uniform_space/uniform_convergence.html#continuous_of_uniform_approx_of_continuous'
    Differentiability of the limit:
    Weierstrass polynomial approximation theorem:
    Weierstrass trigonometric approximation theorem:
  Convexity:
    Convex functions of a real variable: 'analysis/convex/basic.html#convex_on'
    Continuity and differentiability of convex functions:
    Characterizations of convexity: 'analysis/calculus/mean_value.html#convex_on_of_deriv2_nonneg'
    Convexity inequalities: 'analysis/mean_inequalities.html'

# 7.
Single Variable Complex Analysis:
  Complex Valued series:
    Radius of convergence:
    Properties of sums of complex valued series on their disks of convergence:
    Continuity:
    differentiability with respect to the complex variable:
    Antiderivative:
    Complex exponentials:
    Extension of circular functions to the complex plane:
    Power series expansion of usual functions:
  Functions on one complex variable:
    Holomorphic functions:
    Cauchy-Riemann conditions:
    Line integrals of continuous functions in $\C$:
    Antiderivatives of a holomorphic function:
    Representations of the $\log$ function on $\C$:
    Theorem of holomorphic functions under integral domains:
    Winding number of a closed curve in $\C$ with respect to a point:
    Cauchy formulas:
    Analyticity of a holomorphic function:
    Principle of isolated zeros:
    Cauchy formulas:
    Analyticity of a holomorphic function:
    Principle of analytic continuation:
    Maximum principle:
    Isolated singularities:
    Laurent series:
    Meromorphic functions:
    Residue theorem:
    Sequences and series of holomorphic functions:
    Holomorphic stability by uniform convergence:

# 8.
Topology:
  Topology and Metric Spaces:
    Topology of a metric space: 'topology/metric_space/basic.html#metric.is_open_iff'
    Induced topology: 'topology/order.html#topological_space.induced'
    Finite product of metric spaces: 'topology/metric_space/basic.html#metric_space_pi'
    Limits of sequences: 'topology/metric_space/basic.html#metric.tendsto_at_top'
    cluster points:
    Continuous functions: 'topology/basic.html#continuous'
    Homeomorphisms: 'topology/homeomorph.html#homeomorph'
    Compactness: 'topology/subset_properties.html#compact_space'
    Equivalence of definitions in terms of cluster points (Bolzano-Weierstrass) or open covers (Borel-Lebesgue):
    Connectedness: 'topology/subset_properties.html#connected_space'
    Connected components: 'topology/subset_properties.html#connected_component'
    Arc connectedness:
    Lipschitz functions: 'topology/metric_space/lipschitz.html#lipschitz_with'
    uniformly continuous functions: 'topology/metric_space/basic.html#metric.uniform_continuous_iff'
    Heine-Cantor theorem:
    Complete metric spaces: 'topology/metric_space/basic.html#metric.complete_of_cauchy_seq_tendsto'
    Fixed point theorem for contraction mapping: 'topology/metric_space/contracting.html#contracting_with.exists_fixed_point'
  Normed vector spaces on $\R$ and $\C$:
    Topology on a normed vector space: 'analysis/normed_space/basic.html#normed_space.topological_vector_space'
    Equivalent norms:
    Continuity of linear maps in finite dimension: 'normed_space/finite_dimension.html#linear_map.continuous_of_finite_dimensional'
    Normes $\lVert\cdot\rVert_p$ on $\R^n$ and $\C^n$:
    Continuous linear functions: 'topology/algebra/module.html#continuous_linear_map'
    Norm of a continuous linear function: 'analysis/normed_space/operator_norm.html#linear_map.mk_continuous'
    Absolutely convergent series on Banach spaces: 'analysis/normed_space/basic.html#summable_of_summable_norm'
    Banach open mapping theorem:  'analysis/normed_space/banach.html#open_mapping'
    Uniform convergence norm (sup-norm): 'topology/metric_space/emetric_space.html#emetric.tendsto_uniformly_on_iff'
    Complete space of continuous bounded complete space valued functions: 'topology/bounded_continuous_function.html#bounded_continuous_function.complete_space'
    Closed and bounded subsets are compact in finite-dimension: 'analysis/normed_space/finite_dimension.html#finite_dimensional.proper'
    Riesz' characterization of finite dimension:
    Ascoli's Theorem: 'topology/bounded_continuous_function.html#bounded_continuous_function.arzela_ascoli'
  Hilbert Spaces:
    Hilbert projection theorem:
    Orthogonal projection onto closed vector subspaces:
    Dual space:
    Riesz representation theorem:
    $l^2$ and $L^2$ cases:
    Hilbert bases (in the separable case):
    Basis of trigonometric polynomials:
    Basis of orthogonal polynomials:
    Lax-Milgram theorem:
    $H^1_0([0,1])$ and its application to the Dirichlet problem in one dimension:

# 9.
Differential Calculus:
  Differential Calculus:
    Differentiable functions on an open subset of $\R^n$:
    Differentials (linear tangent functions):
    Derivatives with respect to a vector:
    Partial derivatives:
    Jacobian matrix:
    gradient vector:
    Hessian matrix:
    Composition of differentiable functions:
    Mean value theorem:
    Differentiable functions:
    Functions that can be differentiated k times:
    Kth partial derivative:
    Inversion of differentiation order:
    Taylor-style theorems:
      Taylor theorem with rough error estimation:
      Taylor theorem with integral error estimation:
    Local study of real valued functions:
    Series representations:
    Local extrema:
    Convexity of functions on an open convex subset of $\R^n$:
    Diffeomorphisms:
    Inverse function theorem:
    Implicit function theorem:
  Differential equations:
    Differential equations of the form X' = f(t, X):
    Cauchy-Lipschitz Theorem:
    Maximal solutions:
    Grönwall lemma:
    Exit theorem of a compact subspace:
    Autonomous differential equations:
    Phase portraits:
    qualitative behavior:
    Stability of equilibrium points (linearisation theorem):
    Linear differential systems:
    Method of constant variation (Duhamel’s formula):
    Constant coefficient case:
    Solving systems of differential equations of order > 1:
  Differential Geometry:
    Generalizations to R^n:
    Equivalent definitions:
      local graphs:
      local parameterization:
      local equation:
    tangent space:
    gradient:
    $\R^3$ case:
      Position with respect to the plane of the tangent:
    construction of curves/planes represented by a parametric equation:
    Metric study of curves:
    line integral:
    curve length:
      $\mathcal{C}^1$ case:
    Lagrange multipliers:

# 10.
Integral Calculus:
  Measure theory:
    measurable spaces:
    sigma-algebras:
    product of sigma-algebras:
    examples of sigma-algebras:
      borel sigma-algebras:
    positive measure:
    examples of measures:
      counting measure:
      Lebesgue measure:
      probability measures:
    product measure:
    measurable functions:
    approximation by step functions:
Integration:
  Integral of positive measurable functions:
  Monotone convergence theorem:
  Fatou's lemma:
  integrable functions:
  dominated convergence theorem:
  finite dimensional vector-valued integrable functions:
  Integral with parameters:
    Continuity theorem:
    Differentiability under the integral sign theorem:
  $\mathrm{L}^p$ spaces where 1 ≤ p ≤ ∞:
  Completeness of $\mathrm{L}^p$ spaces:
  Holder's inequality:
  Fubini's theorem:
  Change of variables for multiple integrals:
    Polar coordinate case:
    Spherical coordinate case:
  Convolution:
  Regularization and approximation by convolution:
Fourier Analysis:
  Fourier series of locally integrable periodic real-valued functions:
  Riemann-Lebesgue lemma:
  convolution product of periodic functions:
  Dirichlet theorem:
  Fejer theorem:
  Parseval theorem:
  Fourier transforms on $\mathrm{L}^1(\R^d)$ and $\mathrm{L}^2(R^d)$:
  Plancherel’s theorem:

# 11.
Probability Theory:
  Definitions of a probabilistic space:
    Events:
    measure of probability:
    independent events:
    sigma-algebra:
    independent sigma-algebra:
    0-1 law:
    Borel-Cantelli lemma:
    conditional probability:
    Law of total probability:
  Random variables and their laws:
    Discrete law:
    Absolute continuity of probability laws:
    probability density function:
    law of joint probability:
    independence of random variables:
    mean and variance of a real-valued random variable:
    transfer theorem:
    moments:
    examples:
      Bernoulli law:
      Binomial law:
      Geometric law:
      Poisson law:
      Uniform law:
      Exponential law:
      Gaussian law:
    characteristic function:
    generating functions:
      generating functions:
      applications to sums of independent random variables:
  Convergence of series of random variables:
    Probabilistic convergence:
    L^p convergence:
    Almost surely convergence:
    Markov inequality:
    Tchebychev inequality:
    Levy's theorem:
    Law of large numbers:
      Strong form:
      Weak form:
    central limit theorem:

# 12.
Distribution calculus:
  Integration by parts:
  Distributions on Rd:
    vector spaces on C with compact support:
    stability by derivation:
    stability by multiplication by a function on $\mathcal{C}^{\infty}$:
    partitions of unity:
    constructing approximations of probability density functions in spaces of common functions (trig, exp, rational, log, etc):
    Distributions:
      Examples of distributions:
      locally integrable functions:
      dirac measures:
      Cauchy principal values:
      multiplication by a function in C-infinity:
      probability distribution function from a dataset:
      convergent distribution series:
      support for a distribution:
    spaces S(Rd) and S’(Rd):
      Schwartz space S (Rd) of rapidly decreasing functions as well as all their derivatives:
      gaussian functions are their derivatives:
      stability by derivation:
      stability by multiplication by a function C-infinity of slow growth:
      Fourier transforms on S(Rd):
      convolution of two functions of S(Rd):
  Tempered distribution spaces in Spaces S(Rd):
  Linear forms of T on S(Rd) such that there exists C > 0 and k ∈ N such that |hT|φi| ≤ C sup{|x^α ∂^βφ(x)|, x ∈ Rd, |α| ≤ k, |β| ≤ k} forall φ ∈ S (Rd):
  Examples of tempered distributions:
  $L^2$ functions and Riesz representation:
  $L^p$ functions:
  Periodic case:
  Dirac comb:
  Derivation of tempered distributions:
  Multiplication by a function $C^\infty$ of slow growth:
  Fourier transforms on S^0(Rd):
  inverse formula:
  Fourier transform and derivation:
  Fourier transform on a product of convolution:
Applications:
  derivative and the Fourier transform of a distribution:
  Poisson’s formula:
  using convolution and Fourier-Laplace transform to solve one dimensional linear differential equations:
  Notion of constant coefficient differential operator elementary solution (laplacien case):
  weak solution of partial derivative equation:
  solving the laplace equations:
  heat equations:
  wave equations:

# 13.
Numerical Analysis:
  Solving systems of linear inequalities:
    conditioning:
    Gershgorin-Hadamard theorem:
    Gauss’s pivot:
    LU decomposition:
  Iterative methods:
    Jacobian:
    gauss-seidel:
    convergence analysis:
    spectral ray:
    singular value decomposition:
    example of discretisation matrix by finite differences of the laplacian in one dimension:
  Iterative methods of solving systems of real and vector valued equations:
    linear systems case:
    proper element search:
    brute force method:
    optimization of convex function in finite dimension:
    gradient descent square root:
    nonlinear problems with real and vector values:
    bisection method:
    Picard method:
    Newton’s method:
    rate of convergence and estimation of error:
  Numerical integration:
    Rectangle method:
    error estimation:
    Monte-Carlo method:
    rate of convergence:
    application to the calculation of multiple integrals:
  Approximation of numerical functions:
    Lagrange interpolation:
    Lagrange polynomial of a function at (n + 1) points:
    estimation of the error:
  Ordinary differential equations:
    Numerical aspects of Cauchy's problem:
    explicit Euler method:
    consistency:
    stability:
    convergence:
    order:
  Fourier transform:
    Discrete Fourier transform on a finite abelian group:
    Fast Fourier transform: