Definitions
Each term in the Dictionary of Fine Distinctions is defined at one or more levels, from elementary to graduate. See also the list of all examples. Choose a level below to see the terms defined at or below that level, each at the most advanced of those levels. Hover over (or tap) a level next to a term to see the term's definition at that level.
A
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Addition is a way to combine the number of items in two groups of items to find the total number of items.
Addition is a binary operation, written $+$, that is associative and commutative and has an identity element $0$, as on numbers, vectors, matrices, and functions.
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We say that something occurs aperiodically if the length of time between each time it happens is not always the same.
A function, signal, or sequence of events is aperiodic if it is not periodic.
A sequence of times $t_1$, $t_2$, $t_3$, … is called aperiodic if there does not exist a constant $P > 0$ such that $t_{i + 1} - t_{i} = P$ for each $i = 1, 2, 3, \dots$.
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To append something to a list is to add it at the end.
To append $x$ to a list $(a_1, \dots, a_n)$ is to form the list $(a_1, \dots, a_n, x)$, with $x$ added at the end.
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An array is an arrangement of objects in rows and columns.
An array is a set of items arranged in rows and columns (or in a list, or in a block with more directions).
An array is a collection of elements indexed by tuples of integers $(i_1, \dots, i_k)$ in a rectangular index range. The number $k$ of indices is the number of axes (or dimensions) of the array.
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Functions $f$ and $g$ are asymptotically equivalent as $x \to \infty$ if their ratio $f(x) / g(x)$ approaches $1$, so that they are nearly equal in relative terms for large $x$.
Functions $f$ and $g$ are asymptotically equivalent as $x \to \infty$, written $f(x) \sim g(x)$, if \(\lim_{x \to \infty} f(x) / g(x) = 1.\)
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A stationary object is said to be in an asymptotically stable position if, after a small push, the object returns to its initial resting position.
An equilibrium is asymptotically stable if solutions that start close to it stay close and eventually approach it.
An equilibrium is asymptotically stable if it is both stable and attractive.
B
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A ball is the set of all points within a distance $r$ of a center $c$. It is consists of the sphere together with everything inside it.
A ball is the set of all points within a distance $r$ of a center $c$: the sphere together with everything inside it. If the boundary of the ball is included in the set, then the ball is a closed ball:
\[\mathbb{B} = \{x \mid d(x, c) \leq r\}.\]If boundary is excluded, then it is called an open ball:
\[\mathbb{B} = \{x \mid d(x, c) < r\}.\] -
A Banach space is a normed space that is complete: every Cauchy sequence converges.
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A set of numbers is bounded if it does not go on forever in either direction: there is some number that all of them are smaller than in size.
A set $S$ of real numbers is bounded if there is a number $M$ with $\lvert x \rvert \leq M$ for every $x \in S$.
A subset of a metric space is bounded if it is contained in some ball. A sequence or function is bounded if its set of values is bounded.
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A bug is a mistake in a program that makes it behave incorrectly.
C
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A cardinal number is a number that tells how many.
The cardinal number of a set is its cardinality. Two sets have the same cardinal number if there is a bijection between them.
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The cardinality of a set is how many elements it has.
The cardinality of a finite set is its number of elements.
Sets $A$ and $B$ have the same cardinality if there is a bijection between them. This extends counting to infinite sets.
P
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To prepend something to a list is to add it at the beginning.
To prepend $x$ to a list $(a_1, \dots, a_n)$ is to form the list $(x, a_1, \dots, a_n)$, with $x$ added at the beginning.
S
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State feedback is a control law that uses the full state, $u = k(x)$ instead of only a partial measurement of the state.
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A feedback law is static (or memoryless) if the input depends only on the current measurement: $u(t) = k(y(t))$.
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The steady state of a system is the behavior it settles into after its initial transients die out. A steady state can a constant value or a steady oscillation.
The steady-state response of a stable system to an input is what remains of its response as $t \to \infty$, after the transient response (which depends on the initial state) decays.
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A set $A$ is a subset of a set $B$ if everything in $A$ is also in $B$. For example, the set of golden retrivers is a subset of the set of dogs, and dogs are a subset of all animals.
$A$ is a subset of $B$, written $A \subseteq B$, if every element of $A$ is also an element of $B$.
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A function is surjective if every element of the codomain is an output for some input.
A function $f : X \to Y$ is surjective if for every $y \in Y$ there exists $x \in X$ such that $f(x) = y$.
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A dynamical system is called a switched system if it exhibits instantaneous switches between several sets of rules for how it changes over time.
A switched system is a family of continuous-time systems $\dot{x} = f_{p}(x)$, $p \in \lbrace 1, \dots, m \rbrace$, together with a switching signal $\sigma(t)$ that selects the active one: $\dot{x} = f_{\sigma(t)}(x)$.
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A symmetric matrix is a square matrix that is unchanged when flipped across its main diagonal: the entry in row $i$ and column $j$ equals the entry in row $j$ and column $i$.
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A relation $R$ is symmetric if $x \mathrel{R} y$ implies $y \mathrel{R} x$.
T
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In a sum, each number being added is a term.
In the summation of two or more values, each value is called a term of the sum.
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For a function $f(t, x)$ evaluated along a path $x(t)$, the total derivative is
\[\frac{\mathrm{d}}{\mathrm{d}t} f(t, x(t)) = \frac{\partial f}{\partial t} + \nabla_x f \cdot \dot{x}.\] -
In multivariable calculus, the total derivative of $f : \reals^n \to \reals^m$ at $x$ is the linear map $Df(x)$ that best approximates $f(x + h) - f(x)$ for small $h$.
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Tracking is making a quantity follow a changing target, as a driver steers a car to follow a road.
In control theory, tracking is a control problem with the goal of making the output $y(t)$ follow a time-varying reference $r(t)$, so that the tracking error $e(t) = r(t) - y(t)$ converges to zero (or stays small).
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A transcendental number is a number that does not solve any polynomial equation with integer coefficients.
A transcendental number is a (real or complex) number that is not algebraic.
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A transform turns a function into another function, often to reveal hidden information or make a problem easier.
A transform is an operator, usually an integral operator, that converts a function of one variable into a function of another variable.
V
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A vector is an arrow with a length and a direction.
A vector is a quantity with a magnitude and a direction. Graphically, a vector is drawn as an arrow. In coordinates, it is written as a list of numbers.
A vector is an element of a vector space: a set with vector addition and scalar multiplication satisfying the vector space axioms. Vectors are defined by what you can do with them, not by what they look like.
W
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A weighted average is an average in which some values count more than others.
The weighted average of values $x_i$ with nonnegative weights $w_i$ is
\[\frac{\sum_i w_i x_i}{\sum_i w_i}.\]Values with larger weights count more.
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The whole numbers are $0, 1, 2, 3, \dots$, the numbers with no fractional or decimal part.
Z
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The Z-transform of a sequence $x[0], x[1], x[2], \dots$ is
\[X(z) = \sum_{n=0}^{\infty} x[n] z^{-n},\]for the complex numbers $z$ where the series converges. It is the discrete-time analog of the Laplace transform.
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A solution of a hybrid system is Zeno if it has infinitely many jumps while its ordinary time stays bounded.
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Consider a box that could contain some items. If there are no items in the box, then number of items is zero.
The number zero is the additive identity of real and complex numbers.
No definitions match these filters.