Método de Mínimos Cuadrados Ordinarios (MCO)

A fundamental theorem in topology and geometry stating that for any convex polyhedron, the number of vertices (V), edges (E), and faces (F) are related by the formula [latex]V - E + F = 2[/latex]. This value, 2, is the Euler characteristic of a sphere, revealing a deep topological property independent of the polyhedron's specific shape.

Bayes' theorem describes the probability of an event based on prior knowledge of conditions that might be related to the event. It is a fundamental concept in probability theory and statistics. Mathematically, it is stated as [latex]P(A|B) = \frac{P(B|A)P(A)}{P(B)}[/latex], where A and B are events and [latex]P(B) \neq 0[/latex]. It relates the conditional and marginal probabilities of two random events.

A second-order linear elliptic partial differential equation that describes systems in a steady-state or equilibrium condition. It is written as [latex]nabla^2 u = 0[/latex] or [latex]Delta u = 0[/latex], where [latex]nabla^2[/latex] (or [latex]Delta[/latex]) is the Laplace operator. Solutions, called harmonic functions, are the smoothest possible functions and represent potentials in fields like electrostatics, gravitation, and fluid flow.

A fundamental second-order linear parabolic partial differential equation describing heat distribution or other diffusion processes. Its canonical form is [latex]\frac{partial u}{partial t} = \alpha \nabla^2 u[/latex], where [latex]u(\vec{x},t)[/latex] is temperature, [latex]t[/latex] is time, and [latex]\alpha[/latex] is thermal diffusivity. Solutions model how an initial temperature distribution evolves, smoothing out irregularities over time and approaching a steady state.

Los coeficientes de la serie de Fourier de una función [latex]s(x)[/latex] con período [latex]P[/latex] se calculan utilizando fórmulas integrales. El componente de CC es [latex]a_0 = frac{2}{P} int_{P} s(x) , dx[/latex]. Los coeficientes del coseno son [latex]a_n = frac{2}{P} int_{P} s(x) cosleft(frac{2pi nx}{P}right) , dx[/latex], y los coeficientes del seno son [latex]b_n = frac{2}{P} int_{P} s(x) sinleft(frac{2pi nx}{P}right) , dx[/latex] para [latex]n ge 1[/latex].

A fundamental solution of a linear partial differential operator [latex]L[/latex] is a solution to the equation [latex]Lu = delta(x)[/latex], where [latex]delta(x)[/latex] is the Dirac delta function. It represents the response of the system to a point source or impulse. Once known, the solution to the inhomogeneous equation [latex]Lu = f(x)[/latex] can be found by convolution: [latex]u(x) = (G * f)(x)[/latex], where [latex]G[/latex] is the fundamental solution.

A second-order linear hyperbolic partial differential equation that governs the propagation of various types of waves. In its simplest form, it is written as [latex]\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u[/latex], where [latex]u(\vec{x},t)[/latex] is the amplitude of the wave, [latex]c[/latex] is the wave speed, and [latex]\nabla^2[/latex] is the Laplace operator. It models phenomena like vibrating strings, sound waves, and light waves.

The Euler characteristic is a topological invariant, a number that describes a topological space's structure or shape regardless of how it is bent. For polyhedra, it is defined by the formula [latex]\chi = V - E + F[/latex], where V, E, and F are the number of vertices, edges, and faces, respectively. For a sphere, [latex]\chi = 2[/latex], while for a torus, [latex]\chi = 0[/latex].

One of the earliest problems in geometric probability, it is considered a precursor to the Monte Carlo method. It involves dropping a needle of length [latex]l[/latex] onto a floor with parallel lines a distance [latex]t[/latex] apart. The probability that the needle will cross a line is [latex]P = \frac{2l}{\pi t}[/latex] (for [latex]l \le t[/latex]). This provides a physical experiment to estimate [latex]\pi[/latex].

El teorema de Parseval relaciona la energía total de una señal (la integral de su cuadrado sobre un período) con la suma de las energías al cuadrado de sus componentes de la serie de Fourier. Para una función [latex]s(x)[/latex] con período [latex]P[/latex], el teorema establece: [latex]frac{1}{P} int_P |s(x)|^2 , dx = sum_{n=-infty}^{infty} |c_n|^2[/latex], donde [latex]c_n[/latex] son ​​los coeficientes complejos de Fourier.

La inferencia bayesiana es un método estadístico que utiliza el teorema de Bayes para actualizar la probabilidad de una hipótesis a medida que se dispone de más evidencia o información. Es un principio fundamental de la estadística bayesiana. La idea central se expresa como: la probabilidad posterior es proporcional al producto de la probabilidad previa y la verosimilitud, [latex]p(theta|D) propto p(D|theta)p(theta)[/latex], donde [latex]theta[/latex] es el parámetro y D son los datos.

Una serie de Fourier descompone cualquier función o señal periódica en una suma de funciones oscilantes simples, a saber, senos y cosenos. Para una función [latex]s(x)[/latex] con periodo [latex]P[/latex], la serie viene dada por [latex]s(x) approx frac{a_0}{2} + sum_{n=1}^{infty} left[ a_n cosleft(frac{2pi nx}{P}right) + b_n sinleft(frac{2pi nx}{P}right)right][/latex]. Los términos [latex]a_n[/latex] y [latex]b_n[/latex] son ​​los coeficientes de Fourier.

Theorema Egregium (Latin for "Remarkable Theorem") states that the Gaussian curvature of a surface is an intrinsic property. This means it depends only on how distances are measured on the surface itself, not on how the surface is embedded in three-dimensional space. A flat sheet of paper can be rolled into a cylinder but not a sphere without stretching.

For a Fourier series to converge to the function's value, the function must satisfy the Dirichlet conditions over one period. These are: (1) the function must be absolutely integrable, (2) it must have a finite number of extrema (maxima and minima), and (3) it must have a finite number of finite discontinuities.