undergrad.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: 'vector_space'
    product of vector spaces: 'prod.semimodule'
    vector subspace: 'subspace'
    quotient space: 'submodule.quotient'
    sum of subspaces: 'submodule.complete_lattice'
    direct sum: '' # this is supposed to be the internal direct sum
    complementary subspaces: 'submodule.exists_is_compl'
    linear independence: 'linear_independent'
    generating sets: 'submodule.span'
    bases: 'is_basis'
    existence of bases: 'exists_is_basis'
    linear map: 'linear_map'
    range of a linear map: 'linear_map.range'
    kernel of a linear map: 'linear_map.ker'
    algebra of endomorphisms of a vector space: 'module.endomorphism_algebra'
    general linear group: 'linear_map.general_linear_group'
  Duality:
    dual vector space: 'module.dual'
    dual basis: 'is_basis.dual_basis'
    transpose of a linear map: 'module.dual.transpose'
    orthogonality: ''
  Finite-dimensional vector spaces:
    finite-dimensionality : 'finite_dimensional'
    isomorphism with $K^n$: 'module_equiv_finsupp'
    rank of a linear map: 'rank'
    rank of a set of vectors: ''
    rank of a system of linear equations: ''
    isomorphism with bidual: 'vector_space.eval_equiv'
  Multilinearity:
    multilinear map: 'multilinear_map'
    determinant of vectors: 'is_basis.det'
    determinant of endomorphisms: ''
    special linear group: ''
    orientation of a $\R$-valued vector space: ''
  Matrices:
    commutative-ring-valued matrices: 'matrix'
    field-valued matrices: 'matrix'
    matrix representation of a linear map: 'linear_map.to_matrix'
    change of basis: ''
    rank of a matrix: ''
    determinant: 'matrix.det'
    invertibility: 'matrix.nonsing_inv'
    elementary row operations: ''
    elementary column operations: ''
    Gaussian elimination: ''
    row-reduced matrices: ''
  Endomorphism polynomials:
    annihilating polynomials: ''
    minimal polynomial: 'minpoly'
    characteristic polynomial: 'char_poly'
    Cayley-Hamilton theorem: 'aeval_self_char_poly'
  Structure theory of endomorphisms:
    eigenvalue: 'module.End.has_eigenvalue'
    eigenvector: 'module.End.has_eigenvector'
    diagonalization: ''
    triangularization: ''
    invariant subspaces of an endomorphism: ''
    generalized eigenspaces: 'module.End.generalized_eigenspace'
    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: 'group'
    group morphism: 'monoid_hom'
    direct product of groups: 'prod.group'
    subgroup: 'subgroup'
    subgroup generated by a subset: 'subgroup.closure'
    order of an element: 'group_theory/order_of_element.html'
    normal subgroup: 'normal_subgroup'
    quotient group: 'quotient_group.quotient.group'
    group action: 'mul_action'
    stabilizer of a point: 'mul_action.stabilizer'
    orbit: 'mul_action.orbit'
    quotient space: 'mul_action.orbit_equiv_quotient_stabilizer'
    class formula: ''
    conjugacy classes: 'conj_classes'
  Abelian group:
    cyclic group: 'is_cyclic'
    finite type abelian groups: ''
    complex roots of unity: 'complex.mem_roots_of_unity'
    primitive complex roots of unity: 'complex.is_primitive_root_iff'
  Permutation group:
    permutation group of a type: 'equiv.perm'
    decomposition into transpositions: 'equiv.perm.trunc_swap_factors'
    decomposition into cycles with disjoint support: 'equiv.perm.trunc_cycle_factors'
    signature: 'equiv.perm.sign'
    alternating group: 'alternating_subgroup'
  Classical automorphism groups:
    general linear group: 'linear_map.general_linear_group'
    special linear group: 'matrix.special_linear_group'
    orthogonal group: 'matrix.orthogonal_group '
    special orthogonal group: ''
    unitary group: 'matrix.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: 'ring'
    subrings: 'is_subring'
    ring morphisms: 'ring_hom'
    ring structure $\Z$: 'int.comm_ring'
    product of rings: 'pi.ring'
  Ideals and Quotients:
    ideal of a commutative ring: 'ideal'
    quotient rings: 'ideal.quotient'
    prime ideals: 'ideal.is_prime'
    maximal ideals: 'ideal.is_maximal'
    Chinese remainder theorem: 'ideal.quotient_inf_ring_equiv_pi_quotient'
  Algebra:
    algebra over a commutative ring:
    associative algebra over a commutative ring: 'algebra'
  Divisibility in integral domains:
    irreducible elements: 'irreducible'
    invertible elements: 'invertible'
    coprime elements: 'is_coprime'
    unique factorisation domain (UFD): 'unique_factorization_monoid'
    greatest common divisor: 'gcd_monoid.gcd'
    least common multiple: 'gcd_monoid.lcm'
    $A[X]$ is a UFD when $A$ is a UFD: 'polynomial.unique_factorization_monoid'
    principal ideal domain: 'submodule.is_principal'
    Euclidean rings: 'euclidean_domain'
    Euclid's' algorithm: 'nat.xgcd'
    $\Z$ is a euclidean ring: 'int.euclidean_domain'
    congruence in $\Z$: 'int.modeq'
    prime numbers: 'prime'
    Bezout's identity: '' # todo: prove this for PIDs https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity
    $\Z/n\Z$ and its invertible elements: 'zmod.unit_of_coprime'
    Euler's totient function ($\varphi$): 'nat.totient'
  Polynomial rings:
    $K[X]$ is a euclidean ring when $K$ is a field: 'polynomial.euclidean_domain'
    irreducible polynomial: 'irreducible'
    cyclotomic polynomials in $\Q[X]$: polynomial.cyclotomic'
    Eisenstein's criterion: 'polynomial.irreducible_of_eisenstein_criterion'
    polynomial algebra in one or several indeterminates over a commutative ring: 'mv_polynomial'
    roots of a polynomial: 'polynomial.roots'
    multiplicity: 'polynomial.root_multiplicity'
    relationship between the coefficients and the roots of a split polynomial:
    Newton's identities:
    polynomial derivative: 'polynomial.derivative'
    decomposition into sums of homogeneous polynomials: 'mv_polynomial.sum_homogeneous_component'
    symmetric polynomials: 'mv_polynomial.is_symmetric'
  Field Theory:
    fields: 'field'
    characteristic of a ring: 'ring_char'
    characteristic zero: 'char_zero'
    characteristic p: 'char_p'
    Subfields: 'field_theory/subfield.html'
    Frobenius morphisms: 'frobenius'
    field $\Q$ of rational numbers: 'rat.division_ring'
    field $\R$ of real numbers: 'real.division_ring'
    field $\C$ of complex numbers: 'complex.field'
    $\C$ is algebraically closed: 'complex.exists_root'
    field of fractions of an integral domain: 'fraction_map'
    algebraic elements: 'is_algebraic'
    transcendental elements:
    algebraic extensions: 'algebra.is_algebraic'
    algebraically closed fields: 'is_alg_closed'
    rupture fields: 'adjoin_root'
    splitting fields: 'polynomial.splitting_field'
    finite fields: 'field_theory/finite.html'
    rational fraction fields with one indeterminate over a field:
    $\R(X)$-partial fraction decomposition:
    $\C(X)$-partial fraction decomposition:

# 4.
Bilinear and Quadratic Forms Over a Vector Space:
  Bilinear forms:
    bilinear forms: 'bilin_form'
    alternating bilinear forms: 'alt_bilin_form.is_alt'
    symmetric bilinear forms: 'sym_bilin_form.is_sym'
    nondegenerate forms: 'bilin_form.nondegenerate'
    matrix representation: 'bilin_form.to_matrix'
    change of coordinates: 'bilin_form.to_matrix_comp'
    rank of a bilinear form:
  Quadratic forms:
    quadratic form: 'quadratic_form'
    polar form of a quadratic: 'quadratic_form.polar'
  Orthogonality:
    orthogonal elements: 'bilin_form.is_ortho'
    adjoint endomorphism: 'bilin_form.left_adjoint_of_nondegenerate'
    Sylvester's law of inertia:
    real classification:
    complex classification:
    Gram-Schmidt orthogonalisation:
  Euclidean and Hermitian spaces:
    Euclidean vector spaces: 'inner_product_space'
    Hermitian vector spaces: 'inner_product_space'
    dual isomorphism in the euclidean case: 'inner_product_space.isometric.to_dual'
    orthogonal complement: 'submodule.orthogonal'
    Cauchy-Schwarz inequality: 'inner_mul_inner_self_le'
    norm: 'inner_product_space.of_core.to_has_norm'
    orthonormal bases: 'maximal_orthonormal_iff_is_basis_of_finite_dimensional'
  Endomorphisms:
    orthogonal group:
    unitary group:
    special orthogonal group:
    special unitary group:
    self-adjoint endomorphism: 'bilin_form.is_self_adjoint'
    normal endomorphism:
    diagonalization of a self-adjoint endomorphism:
    diagonalization of normal endomorphisms:
    simultaneous diagonalization of two real quadratic forms, with one positive-definite:
    decomposition of an orthogonal transformation as a product of reflections:
    polar decompositions in $\mathrm{GL}(n, \R)$:
    polar decompositions in $\mathrm{GL}(n, \C)$:
  Low dimensions:
    cross product:
    triple product:
    classification of elements of $\mathrm{O}(2, \R)$:
    classification of elements of $\mathrm{O}(3, \R)$:

# 5.
Affine and Euclidean Geometry:
  General definitions:
    affine space: 'add_torsor'
    affine function: 'affine_map'
    affine subspace: 'affine_subspace'
    barycenter: 'finset.affine_combination'
    affine span: 'span_points'
    equations of affine subspace:
    affine groups:
    affine property:
    group generated by homotheties and translations:
    transformations fixing a basis of directions: # "affinité", no good English translation
  Convexity:
    convex subsets:
    convex hull of a subset of an affine real space:
    extreme point:
  Euclidean affine spaces:
    isometries of a Euclidean affine space:
    group of isometries of a Euclidean affine space:
    isometries that do and do not preserve orientation:
    direct and opposite similarities of the plane:
    classification of isometries in two and three dimensions:
    angles between vectors: 'inner_product_geometry.angle'
    angles between planes:
    inscribed angle theorem:
    cocyclicity:
    group of isometries stabilizing a subset of the plane or of 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:
    focus:
    eccentricity:
    quadric surfaces in 3-dimensional Euclidean affine spaces:

# 6.
Single Variable Real Analysis:
  Real numbers:
    definition of $\R$: 'real'
    field structure: 'real.division_ring'
    order: 'real.linear_order'
  Sequences of real numbers:
    convergence: 'filter.tendsto'
    limit point: 'map_cluster_pt'
    recurrent sequences: 'nat'
    limit infimum and supremum: 'order/liminf_limsup.html'
    Cauchy sequences: 'cauchy_seq'
  Topology of R:
    metric structure: 'real.metric_space'
    completeness of R: 'real.complete_space'
    Bolzano-Weierstrass theorem: 'tendsto_subseq_of_bounded'
    compact subsets of $\R$: 'metric.compact_iff_closed_bounded'
    connected subsets of $\R$: 'set_of_is_preconnected_eq_of_ordered'
    additive subgroups of $\R$: 'real.subgroup_dense_or_cyclic'
  Numerical series:
    Convergence of real valued-series:
    Geometric series: 'tsum_geometric_of_norm_lt_1'
    convergence of $p$-series for $p>1$: 'nnreal.summable_one_rpow_inv'
    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: 'continuous'
    limits: 'filter.tendsto'
    intermediate value theorem: 'intermediate_value_Icc'
    image of a segment: 'real.image_Icc'
    continuity of monotonic functions: 'order_iso.continuous'
    continuity of inverse functions: 'order_iso.to_homeomorph'
  Differentiability:
    derivative at a point: 'has_deriv_at'
    differentiable functions: 'has_deriv_at'
    derivative of a composition of functions: 'deriv.comp'
    derivative of the inverse of a function: 'has_strict_deriv_at.of_local_left_inverse'
    Rolle's theorem: 'exists_deriv_eq_zero'
    mean value theorem: 'exists_ratio_deriv_eq_ratio_slope'
    higher order derivatives of functions: 'iterated_deriv'
    $C^k$ functions: 'times_cont_diff'
    piecewise $C^k$ functions:
    Leibniz formula: 'deriv_mul'
  Taylor-like theorems:
    Taylor's theorem with little-o remainder:
    Taylor's theorem with integral form for remainder:
    Taylor's theorem with Lagrange form for remainder:
    Taylor series expansions:
  Elementary functions (trigonometric, rational, $\exp$, $\log$, etc):
    polynomial functions: 'polynomial.eval'
    rational functions:
    logarithms: 'real.log'
    exponential: 'real.exp'
    power functions: 'real.rpow'
    trigonometric functions: 'real.sin'
    hyperbolic trigonometric functions: 'real.sinh'
    inverse trigonometric functions: 'real.arcsin'
    inverse hyperbolic trigonometric functions: 'real.arsinh'
  Integration:
    integral over a segment of piecewise continuous functions:
    antiderivatives:
    Riemann sums:
    antiderivative of a continuous function: 'interval_integral.integral_has_strict_deriv_at_of_tendsto_ae_right'
    change of variable:
    integration by parts: 'interval_integral.integral_mul_deriv_eq_deriv_mul'
    improper integrals:
    absolute vs conditional convergence of improper integrals:
    comparison test for improper integrals:
  Sequences and series of functions:
    pointwise convergence:
    uniform convergence: 'tendsto_uniformly'
    normal convergence:
    continuity of the limit: '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: 'convex_on'
    continuity and differentiability of convex functions:
    characterizations of convexity: 'convex_on_of_deriv2_nonneg'
    convexity inequalities: 'analysis/mean_inequalities.html'

# 7.
Single Variable Complex Analysis:
  Complex Valued series:
    radius of convergence: 'formal_multilinear_series.radius'
#    properties of sums of complex valued series on their disks of convergence: (this is a heading?)
    continuity: 'has_fpower_series_on_ball.continuous_on'
    differentiability with respect to the complex variable: 'has_fpower_series_on_ball.differentiable_on'
    antiderivative:
    complex exponential: 'complex.exp'
    extension of trigonometric functions to the complex plane: 'complex.sin'
    power series expansion of elementary functions:
  Functions on one complex variable:
    holomorphic functions:
    Cauchy-Riemann conditions:
    contour 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:
    principle of analytic continuation:
    maximum principle:
    isolated singularities:
    Laurent series:
    meromorphic functions:
    residue theorem:
    sequences and series of holomorphic functions:
    holomorphic stability under uniform convergence:

# 8.
Topology:
  Topology and Metric Spaces:
    topology of a metric space: 'metric.is_open_iff'
    induced topology: 'topological_space.induced'
    finite product of metric spaces: 'metric_space_pi'
    limits of sequences: 'metric.tendsto_at_top'
    cluster points: 'cluster_pt'
    continuous functions: 'continuous'
    homeomorphisms: 'homeomorph'
    compactness in terms of open covers (Borel-Lebesgue): 'compact_iff_finite_subcover'
    sequential compactness is equivalent to compactness (Bolzano-Weierstrass): 'metric.compact_iff_seq_compact'
    connectedness: 'connected_space'
    connected components: 'connected_component'
    path connectedness: 'is_path_connected'
    Lipschitz functions: 'lipschitz_with'
    uniformly continuous functions: 'metric.uniform_continuous_iff'
    Heine-Cantor theorem: 'compact_space.uniform_continuous_of_continuous'
    complete metric spaces: 'metric.complete_of_cauchy_seq_tendsto'
    contraction mapping theorem: 'contracting_with.exists_fixed_point'
  Normed vector spaces on $\R$ and $\C$:
    topology on a normed vector space: 'semi_normed_space.has_continuous_smul'
    equivalent norms:
    Banach open mapping theorem: 'open_mapping'
    equivalence of norms in finite dimension: 'linear_equiv.to_continuous_linear_equiv'
    norms $\lVert\cdot\rVert_p$ on $\R^n$ and $\C^n$: 'pi_Lp.normed_space'
    absolutely convergent series in Banach spaces: 'summable_of_summable_norm'
    continuous linear maps: 'continuous_linear_map'
    norm of a continuous linear map: 'linear_map.mk_continuous'
    uniform convergence norm (sup-norm): 'emetric.tendsto_uniformly_on_iff'
    normed space of bounded continuous normed-space-valued functions: 'bounded_continuous_function.normed_space'
    its completeness: 'bounded_continuous_function.complete_space'
    Heine-Borel theorem (closed bounded subsets are compact in finite dimension): 'finite_dimensional.proper'
    Riesz' lemma (unit-ball characterization of finite dimension):
    Arzela-Ascoli theorem: 'bounded_continuous_function.arzela_ascoli'
  Hilbert Spaces:
    Hilbert projection theorem: 'exists_norm_eq_infi_of_complete_convex'
    orthogonal projection onto closed vector subspaces: 'orthogonal_projection_fn'
    dual space: 'normed_space.dual.normed_space'
    Riesz representation theorem: 'inner_product_space.to_dual'
    inner product spaces $l^2$ and $L^2$: 'measure_theory.L2.inner_product_space'
    their completeness: 'measure_theory.Lp.complete_space'
    Hilbert (orthonormal) bases (in the separable case): 'exists_is_orthonormal_dense_span'
    Hilbert basis of trigonometric polynomials:
    Hilbert bases of orthogonal polynomials:
    Lax-Milgram theorem:
    $H^1_0([0,1])$ and its application to the one-dimensional Dirichlet problem:

# 9.
Multivariable calculus:
  Differential Calculus:
    differentiable functions on an open subset of $\R^n$: 'differentiable_on'
    differentials (linear tangent functions): 'fderiv'
    directional derivative:
    partial derivatives:
    Jacobian matrix:
    gradient vector:
    Hessian matrix:
    chain rule: 'fderiv.comp'
    mean value theorem: 'exists_ratio_deriv_eq_ratio_slope'
    differentiable functions: 'differentiable'
    $k$-times continuously differentiable functions: 'times_cont_diff'
    $k$-th order partial derivatives:
    partial derivatives commute:
    Taylor's theorem with little-o remainder:
    Taylor's theorem with integral form for remainder:
    local extrema: 'is_local_min.fderiv_eq_zero'
    convexity of functions on an open convex subset of $\R^n$: 'convex_on'
    diffeomorphisms: 'structomorph'
    inverse function theorem: 'has_strict_deriv_at.to_local_inverse'
    implicit function theorem: 'implicit_function_data.implicit_function'
  Differential equations:
    Cauchy-Lipschitz Theorem:
    maximal solutions:
    Grönwall lemma: 'norm_le_gronwall_bound_of_norm_deriv_right_le'
    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 $> 1$:
  Submanifolds of $\R^n$:
    local graphs:
    local parameterization:
    local equation:
    tangent space:
    position with respect to the tangent plane:
    gradient:
    line integral:
    curve length:
    Lagrange multipliers:

# 10.
Measures and integral Calculus:
  Measure theory:
    measurable spaces: 'measurable_space'
    sigma-algebras: 'measurable_space'
    product of sigma-algebras: 'measurable_space.pi'
    Borel sigma-algebras: 'borel_space'
    positive measure: 'measure_theory.measure'
    counting measure: 'measure_theory.measure.count'
    Lebesgue measure: 'measure_theory.measure_space'
    product measure: 'measurable_space.pi'
    measurable functions: 'measurable'
    approximation by step functions: 'measure_theory/simple_func_dense.html'
  Integration:
    integral of positive measurable functions: 'measure_theory.lintegral'
    monotone convergence theorem: 'measure_theory.lintegral_infi_ae'
    Fatou's lemma: 'measure_theory.lintegral_liminf_le'
    integrable functions: 'measure_theory.integrable'
    dominated convergence theorem: 'measure_theory.tendsto_integral_of_dominated_convergence'
    finite dimensional vector-valued integrable functions: 'measure_theory.integrable'
    continuity of integrals with respect to parameters:
    differentiability of integrals with respect to parameters:
    $\mathrm{L}^p$ spaces where $1 ≤ p ≤ ∞$: 'measure_theory.Lp'
    Completeness of $\mathrm{L}^p$ spaces: 'measure_theory.Lp.complete_space'
    Holder's inequality: 'nnreal.lintegral_mul_le_Lp_mul_Lq'
    Fubini's theorem: 'measure_theory.integral_prod'
    change of variables for multiple integrals:
    change of variables to polar co-ordinates:
    change of variables to spherical co-ordinates:
    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 probability space:
    probability measure: 'measure_theory.probability_measure'
    events:
    independent events: 'probability_theory.Indep_set'
    sigma-algebra: 'measurable_space'
    independent sigma-algebra: 'probability_theory.Indep'
    $0$-$1$ law:
    Borel-Cantelli lemma (easy direction): 'measure_theory.measure_limsup_eq_zero'
    Borel-Cantelli lemma (difficult direction):
    conditional probability:
    law of total probability:
  Random variables and their laws:
    discrete law: 'pmf'
    absolute continuity of probability laws:
    probability density function:
    law of joint probability:
    independence of random variables: 'probability_theory.Indep_fun'
    mean and variance of a real-valued random variable:
    transfer theorem:
    moments:
    Bernoulli law: 'pmf.bernoulli'
    binomial law:
    geometric law:
    Poisson law:
    uniform law:
    exponential law:
    Gaussian law:
    characteristic function:
    probability generating functions:
    applications of probability generating functions to sums of independent random variables:
  Convergence of series of random variables:
    convergence in probability:
    $\mathrm{L}^p$ convergence:
    almost surely convergence: 'measure_theory.measure.ae'
    Markov inequality:
    Tchebychev inequality:
    Levy's theorem:
    weak law of large numbers:
    strong law of large numbers:
    central limit theorem:

# 12.
Distribution calculus:
  Spaces $\mathcal{D}(\R^d)$:
    smooth functions with compact support on $\R^d$:
    stability by derivation:
    stability by multiplication by a smooth function:
    partitions of unity:
    constructing approximations of probability density functions in spaces of common functions (trig, exp, rational, log, etc):
  Distributions on $\R^d$:
    definition of distributions:
    locally integrable functions:
    dirac measures:
    Cauchy principal values:
    multiplication by a smooth function:
    probability distribution function from a dataset:
    convergent distribution series:
    support of a distribution:
  Spaces $\mathcal{S}(\R^d)$:
    Schwartz space of rapidly decreasing functions:
    stability by derivation:
    stability by multiplication by a slowly growing smooth function:
    gaussian functions:
    Fourier transforms on $\mathcal{S}(\R^d)$:
    convolution of two functions of $\mathcal{S}(\R^d)$:
  Tempered distributions:
    definition:
    derivation of tempered distributions:
    multiplication by a function $C^\infty$ of slow growth:
    $L^2$ functions and Riesz representation:
    $L^p$ functions:
    periodic functions:
    Dirac comb:
    Fourier transforms:
    inverse Fourier transform:
    Fourier transform and derivation:
    Fourier transform and convolution product:
  Applications:
    Poisson’s formula:
    using convolution and Fourier-Laplace transform to solve one dimensional linear differential equations:
    weak solution of partial derivative equation:
    fundamental solution of the Laplacian:
    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: