Latest

A maximal solution of an ordinary differential equation is a solution that is not the restriction of a solution on a larger interval. Under a local Lipschitz condition, each initial value problem has exactly one maximal solution. See the definition.

The flux of a vector field through an oriented surface measures how much of the field passes through the surface. It counts flow in the direction of the chosen normal. It is the surface integral of the normal component of the field. For a velocity field of a fluid, it is the volume of fluid that crosses the surface per unit time.

The continued fraction expansion of a real number is the simple continued fraction found by taking integer parts and reciprocals again and again. The definition of the continued fraction expansion gives the precise form.

An autonomous system is a system of ordinary differential equations $x' = f(x)$ whose right side does not depend explicitly on the independent variable. When the independent variable is time, such a system is also called time-invariant. See the definition.

Multiplication of natural numbers is the operation that sends two natural numbers $a$ and $b$ to their product $a \cdot b$. The product $a \cdot b$ is the sum of $b$ copies of $a$.

Take a real-valued function of several real variables on a convex set. The function is convex when its graph lies on or below each chord between two points of the graph. Convex functions are important in optimisation because each local minimum of a convex function is a global minimum.

Let $f$ be a function on an interval of $\mathbb{R}$. The phase line of the scalar autonomous equation $x' = f(x)$ is a diagram of the line of states. It marks the equilibrium points, which are the zeros of $f$. It puts an arrow on each interval between them: up where $f > 0$ and down where $f < 0$. The arrows show the direction in which each solution moves.

The power series method looks for a solution of a differential equation in the form $y = \sum_{k \ge 0} a_k (x - x_0)^k$. Consider $y'' + p(x) y' + q(x) y = 0$, where $p$ and $q$ are analytic at $x_0$: they equal convergent power series near $x_0$. One puts the series into the equation and compares the coefficients of each power $(x - x_0)^k$. This gives a recurrence for the $a_k$, with $a_0$ and $a_1$ free. The resulting series converge at least on each disc around $x_0$ where the series of $p$ and $q$ converge, and they give two independent solutions.

Gauss–Jordan elimination is the variant of Gaussian elimination that continues until the matrix is in reduced row echelon form. After the forward phase gives a row echelon form, it scales each pivot row to make the pivot equal to $1$. Then it adds multiples of each pivot row to the rows above it, to make every other entry in each pivot column zero. Applied to the augmented matrix of a system, the result shows the solutions directly, with no back substitution.

A partial derivative of a function of several variables is its derivative with respect to one variable, while all other variables stay fixed. It measures the rate of change of the function along one coordinate axis. For $f(x, y) = x^2 y$, the partial derivatives are $\partial f / \partial x = 2xy$ and $\partial f / \partial y = x^2$.

A higher partial derivative of a function of several variables is a partial derivative of a partial derivative, taken one or more times. A second partial derivative such as $\partial^2 f / \partial y \, \partial x$, in which the two variables differ, is called a mixed partial derivative.

A function has an infinite limit at a point when its values become arbitrarily large, positive or negative, near the point. Infinite limits describe vertical asymptotes. The node infinite limit gives the precise definition.

The work done by a force field $F$ on a particle that moves along a curve $\gamma$ is the line integral $W = \int_\gamma F \cdot d\mathbf{r}$. Only the component of the force along the direction of motion contributes, positively when it points forward and negatively when it points backward.

The partition function $p(n)$ gives the number of integer partitions of $n$. It is a classical object of additive number theory, with a product generating function and deep congruences. The node partition function gives the precise definition.

An improper multiple integral extends the Riemann integral to unbounded regions or unbounded functions. It is a limit of integrals over bounded regions where the function is bounded. For a non-negative function $f$ on $\mathbb{R}^n$ that is Riemann integrable on each cube $[-a, a]^n$, the improper integral is $\int_{\mathbb{R}^n} f = \lim_{a \to \infty} \int_{[-a, a]^n} f$, when this limit exists.

Casting out nines is a method to check a computation with integers by the remainders modulo $9$. It uses the fact that a natural number is congruent modulo $9$ to the sum of its decimal digits. To check $a \cdot b = c$, one compares the digit sum of $c$ with the product of the digit sums of $a$ and $b$, all modulo $9$. The same works for sums and differences.

A forest is a graph without cycles, that is, a disjoint union of trees. The spanning forests of a graph generalize spanning trees to graphs that are not connected. The node forest gives the precise definition.

A function of several variables is differentiable at a point when, near that point, it is close to a linear function plus a constant. The error is small compared with the distance to the point. The linear function is the total derivative of the function at the point.

The limit of a function of several variables at a point $a$ is the value that $f(x)$ approaches when $x$ approaches $a$ in Euclidean space. The approach can be from any direction and along any route. The distance in $\mathbb{R}^n$ measures how close $x$ is to $a$.

The complement of a graph has the same vertices and exactly the edges that the graph does not have. It swaps cliques and independent sets. The node complement graph gives the precise definition.

A Poisson point process is a random set of points in which the numbers of points in disjoint regions are independent Poisson random variables. On the time axis, it models events that occur at random times with a constant rate. The node Poisson point process gives the precise definition of the homogeneous process on $[0, \infty)$.

The cross product is a binary operation on the vectors of three-dimensional space. It takes two vectors and gives a vector that is perpendicular to both. The length of the result is the area of the parallelogram with the two vectors as sides.

A singular solution of a differential equation is a solution along which uniqueness of solutions fails. It is often the envelope of a family of solutions, and the general solution does not contain it. See the definition.

The prime-counting function $\pi(x)$ gives the number of primes that are less than or equal to a real number $x$. The definition of the prime-counting function gives the precise form.

Addition of natural numbers is the operation that sends two natural numbers $a$ and $b$ to their sum $a + b$. The sum $a + b$ is the number that you get when you start at $a$ and apply the successor function $b$ times.

A refinement of a partition is a partition that contains all the points of the given one, and possibly more. Two partitions always have a common refinement. The node refinement of a partition gives the precise definition.

An acyclic graph is a graph without cycles. Acyclic graphs are the forests, and their connected components are trees. The node acyclic graph gives the precise definition.

A Fermat pseudoprime to a base $a$ is a composite number $n$ that satisfies the congruence $a^{n-1} \equiv 1 \pmod n$ of Fermat's little theorem. The definition of a Fermat pseudoprime gives the precise form.

A subgraph of a graph is a graph formed by some of its vertices and some of its edges. Subgraphs are how one speaks of a structure, such as a cycle or a tree, inside a graph. The node subgraph gives the precise definition.

A subgroup of a group is a subset that is itself a group under the same operation.

Matrix addition is the operation that adds two matrices of the same size entry by entry.

The arcsine is the inverse of the sine function restricted to $[-\pi/2, \pi/2]$. It is defined on $[-1, 1]$. The node arcsine gives the precise definition.

A constructive proof of an existence statement $\exists x \, P(x)$ gives a specific object $a$, or a method to find it, and proves $P(a)$. A non-constructive proof shows that an object exists without giving it, for example by proof by contradiction.

A one-form is a differential form of degree one. At each point it gives a linear function of tangent vectors. One-forms are the natural integrands of line integrals, and the differential of a function is the basic example.

The hyperbolic cosine is the average of $e^x$ and $e^{-x}$. It is an even function with minimum value $1$ at $0$. The node hyperbolic cosine gives the precise definition.

The sample mean of a sample is the arithmetic mean of its values. It is the usual estimator of the mean of the population. The node sample mean gives the precise definition.

A closed curve is a parametrised curve on a closed interval whose end point is equal to its initial point. Its image can be seen as the image of a circle under a continuous map.

A field is a commutative ring in which you can divide by every nonzero element. The rational numbers, the real numbers and the complex numbers are fields. The definition of a field gives the axioms.

A finitely generated module is a module in which every element is a linear combination of a fixed finite set of elements.

A Noetherian ring is a ring in which every ideal is finitely generated, or equivalently, every increasing chain of ideals is eventually constant. The definition of a Noetherian ring gives the conditions.