De Morgan’s Laws

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 \(l\) onto a floor with parallel lines a distance \(t\) apart. The probability that the needle will cross a line is \(P = \frac{2l}{\pi t}\) (for \(l \le t\)). This provides a physical experiment to estimate \(\pi\).

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

A fundamental solution of a linear partial differential operator \(L\) is a solution to the equation \(Lu = delta(x)\), where \(delta(x)\) 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 \(Lu = f(x)\) can be found by convolution: \(u(x) = (G * f)(x)\), where \(G\) is the fundamental solution.

A differentiable manifold is a topological space that is locally similar to Euclidean space, allowing calculus to be applied. Each point has a neighborhood that is homeomorphic to an open subset of \(\mathbb{R}^n\). These local coordinate systems, called charts, are related by smooth transition functions, forming an atlas that defines the manifold's differentiable structure.

Digital electronics is based on Boolean algebra, a mathematical system of logic introduced by George Boole. It uses two values, typically 0 and 1 (or false and true), and three basic operations: AND (conjunction), OR (disjunction), and NOT (negation). These operations directly correspond to the logic gates that form the building blocks of all digital circuits.

The Prime Number Theorem describes the asymptotic distribution of prime numbers among the integers. It states that the prime-counting function \(\pi(x)\), which gives the number of primes less than or equal to \(x\), is asymptotically equivalent to \(x / \ln(x)\). Formally, \(\lim_{x \to \infty} \frac{\pi(x)}{x/\ln(x)} = 1\). This provides a fundamental link between primes and the natural logarithm.

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 \(\chi = V - E + F\), where V, E, and F are the number of vertices, edges, and faces, respectively. For a sphere, \(\chi = 2\), while for a torus, \(\chi = 0\).

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

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.

The Gauss-Bonnet theorem connects the geometry of a compact two-dimensional surface to its topology. It states that the integral of the Gaussian curvature \(K\) over the entire surface \(M\) is equal to \(2\pi\) times the Euler characteristic \(\chi(M)\) of the surface. The formula is \(\int_M K \, dA = 2\pi \chi(M)\).

Accuracy refers to the closeness of a measured value to a standard or known true value. Precision refers to the closeness of two or more measurements to each other. A measurement system can be precise but not accurate, accurate but not precise, neither, or both. These two concepts are independent of each other in measurement theory.

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds—smooth manifolds equipped with a Riemannian metric. This metric is a collection of inner products on the tangent spaces, varying smoothly from point to point. It allows for the definition of local geometric notions like angle, length of curves, surface area, and volume, leading to a generalized notion of curvature.

A homeomorphism is a continuous function between two topological spaces that has a continuous inverse function. Two topological spaces are called homeomorphic if such a function exists. From a topological viewpoint, homeomorphic spaces are identical. This concept captures the idea that an object can be stretched, bent, or deformed into another without tearing or gluing, like a coffee mug into a donut.

This theorem states that for any continuous function \(f\) mapping a compact convex set to itself, there is a point \(x_0\) such that \(f(x_0) = x_0\). This point is called a fixed point. Informally, if you take a map of a country, crumple it up, and place it inside the country's borders, there will always be at least one point on the map directly above its corresponding real-world location.