ANALYTICAL AND STOCHASTIC MATHEMATICAL METHODS FOR ENGINEERING 2025-2026 - INP5070357
Section outline
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Definition of \(\sigma-\)algebra of sets. Examples. \(\sigma-\)algebra generated by a family of sets. Definition of abstract measure. Dirac measure.
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Counting measure. Basic properties of abstract measures. Continuity properties. Definition of Lebesgue's outer measure.
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Exercises on abstract measures. Properties of Lebesgue's outer measure, Vitali's example of non countable additivity.
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Lebesgue class. Basic properties. Measurable functions: definition and first examples.
The video recording has been interrupted by a blackout.
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Continuous functions on \(\Bbb R^k\) are Lebesgue measurable. Complete measures. Algebraic composition of measurable functions. Chain rule. Point-wise limit of measurable functions.
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Exercises on Lebesgue's measure. Lebesgue's definition of abstract integral.
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Pointwise approximation of positive measurable functions through simple functions. Definition of integral for real and complex valued functions, and main properties. Chebyshev inequality.
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Lebesgue's integral on \(\Bbb R^k\) and connections with Riemann and generalized integrals. Examples. Fubini's theorem.
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Exercises on abstract integrals. Beppo-Levi's and monotone convergence theorems. Examples.
Unfortunately the recording suffered of technical problems and interrupts after 1/3 of the class.
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Monotone convergence for decreasing sequences. Examples. Lebesgue's dominated convergence theorem. Exercise.
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Exercises on dominated convergence. Integrals depending on parameters, continuity and differentiability theorems.
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Fourier transform of the Gaussian distribution. Exercises on differentiaiblity theorem.
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Stronger and equivalent norms. Finite dimensional norms are always equivalent. Examples with infinite dimensional norms. \( L^p\) spaces.
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Exercises on normed spaces. L^2 norm: Cauchy-Schwarz inequality. Essentially bounded functions, \( L^\infty \) norm.
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Limits in normed spaces: definition, examples, Cauchy property. Uniform convergence.
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Exercises on convergence in norm. Completeness of \(B(X)\) and \(\mathscr C(K)\), \(K\subset\Bbb R^d\) compact. Continuity of uniform limit. Convergence in \(L^p\) norm.
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Exercises on convergence. Convolution: definition, Young's inequality, example.
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Approximate units, mollification theorem. Scalar and hermitian products. Examples. Canonical norm of an inner product space, abstract Cauchy-Schwarz inequality. Orthogonal vectors, Pythagorean theorem, parallelogram identity. Exercises.
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Orthogonal projection: existence, uniqueness and characterization. Examples and exercises.
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Orthogonal complement: exercises. Series of vectors: Weierstrass' test and convergence of orthogonal series. Orthonormal bases.
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Orthonormal basis for \( L^2([a,b]) \) , classical Fourier series. \( L^1\) Fourier Transform: definition, remarks and first examples.
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FT of rectangle. Basic properties of FT: continuity, boundedness, Riemann-Lebesgue lemma. FT of derivative and derivative of FT. FT maps Schwarz functions space into itself. FT of convolution product. Exercises.
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Exercise. Inversion formula. FT of Cauchy distribution. Exercise.
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Injectivity of FT, Fourier original and inverse FT. Exercises
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Submissions opened: Friday, 31 October 2025, 6:00 PMSubmissions closed: Saturday, 1 November 2025, 11:59 PM
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Submissions opened: Tuesday, 4 November 2025, 8:00 PMSubmissions closed: Friday, 7 November 2025, 8:00 PMAssessments opened: Friday, 7 November 2025, 8:01 PMAssessments closed: Sunday, 9 November 2025, 12:00 PM
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Submissions opened: Monday, 10 November 2025, 8:00 PMSubmissions closed: Friday, 14 November 2025, 8:00 PMAssessments opened: Friday, 14 November 2025, 8:00 PMAssessments closed: Saturday, 15 November 2025, 8:00 PM
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Submissions opened: Saturday, 15 November 2025, 8:00 PMSubmissions closed: Monday, 17 November 2025, 8:00 PMAssessments opened: Monday, 17 November 2025, 8:00 PMAssessments closed: Sunday, 23 November 2025, 8:00 PM
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Random variables, expected value, variance and covariance, linear correlation. Law of a random variable. Examples. Cumulative distribution function.
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Examples of cdf. Absolutely continuous random variables, density function. Examples. Standard random variables: exponentials, gaussian, Gamma, Cauchy, log-normal. Change of density.
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Exercises on random variables, cdfs, densities. Multivariate random variable, mean and covariance matrix.
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Multivariate random variables: law, cdf, absolutely continuous r.v. and densities. Multivariate Gaussian.
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Exercises on multivariate distributions. Mapping multivariate random variables, exercise.
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Exercises on multivariate random variables. Fourier Transform of a Borel measure. FT of a multivariate Gaussian distribution.
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Characteristic function of a random variable, examples and main properties. Uniqueness theorem. Exercises.
(no video sorry)
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Independent events and random variables. Characterization of independence through joint cdf and densities. Exercise.
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Exercises on independence. Kac theorem on characteristic function of independent random variables.
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Conditional expectation: case of \( L^2 \) random variables through orthogonal projection, extension to \( L^1 \) random variables, main properties.
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Conditional expectation on finitely generated \(\sigma-\)algebras. Jensen's inequality for conditional expectation. Conditional density and formula for \(\Bbb E[X\mid Y]\).
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Convergence in \(p\) mean (\(L^p\) convergence. Convergence with probability 1. Borel-Cantelli's Lemmas. Example.
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Exercises on \(L^p\) and \(\Bbb P=1\) limits. Convergence in probability, relations with \(L^p\) and a.s. convergence. Example.
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Convergence in distribution. Exercises on convergence of random variables.
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Exercises on convergence of sequences of random variables.
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Weak law of large numbers: Chebishev bound, applications to monte-Carlo method and Bernstein-Weierstrass polynomial approximation.
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Exercises on WLLN. Strong law: \(L^4\) SLLN, \(L^1\) SLLN.
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Exercise CLT. Definition of Brownian motion. The Levy-Ciesielskii construction.
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SCHEDULES
ANALYTICAL METHODS
- Nov. 20, 2025, 9am, Room RL Vallisneri Building
- Feb. 10, 2026, 9am, Room H7, HUB
- Jun. 29, 2026, 9am, Room H1, HUB
- Sept. 7, 2026, 9am, Room H1, HUB
PROBABILISTIC METHODS
- Jan. 19, 2026, 9am, Room H12, HUB
- Feb. 18, 2026, 9am, Room H12, HUB
- Jul. 2, 2026, 9am, Room RL, Vallisneri
- Sept. 10, 2026, 9am, Room RL Vallisneri
VALIDATION LIST (to validate your exam, if eligible)
- Jan 23, 2026
- Feb 20, 2026
- Jul. 4, 2026
- Sept. 14, 2026
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Submissions opened: Monday, 15 December 2025, 10:00 PMSubmissions closed: Saturday, 20 December 2025, 8:00 PMAssessments opened: Saturday, 20 December 2025, 8:01 PMAssessments closed: Monday, 22 December 2025, 8:00 PM
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Submissions opened: Tuesday, 23 December 2025, 8:58 AMSubmissions closed: Sunday, 28 December 2025, 11:59 PMAssessments opened: Monday, 29 December 2025, 12:00 AMAssessments closed: Tuesday, 30 December 2025, 11:59 PM
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Submissions opened: Wednesday, 31 December 2025, 12:00 AMSubmissions closed: Monday, 5 January 2026, 11:59 PMAssessments opened: Tuesday, 6 January 2026, 12:00 AMAssessments closed: Thursday, 8 January 2026, 11:59 PM
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Submissions opened: Wednesday, 14 January 2026, 12:00 AMSubmissions closed: Tuesday, 20 January 2026, 11:59 PMAssessments opened: Wednesday, 21 January 2026, 12:00 AMAssessments closed: Thursday, 22 January 2026, 11:59 PM
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