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.
1
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Consider a vector $x = (x_1, \dots, x_n)$. The 1-norm of $x$ is the sum of the absolute values of its entries:
\[\norm{x}_1 = \abs{x_1} + \dots + \abs{x_n}.\]
A
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Consider a function $f : [a, b] \to \reals$ and finite collections of disjoint subintervals $(x_1, y_1), \dots, (x_n, y_n)$ of $[a, b]$. The function $f$ is absolutely continuous if for every $\varepsilon > 0$, there exists $\delta > 0$ such that whenever such a collection has total length
\[\sum_{k=1}^n (y_k - x_k) < \delta,\]the total change of $f$ satisfies
\[\sum_{k=1}^n \abs{f(y_k) - f(x_k)} < \varepsilon.\]Consider a function $f : [a, b] \to \reals$. The function $f$ is absolutely continuous if there is a Lebesgue integrable function $g$ such that
\[f(x) = f(a) + \int_a^x g(t) \mathrm{d}t\]for every $x \in [a, b]$.
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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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Consider a control system whose states must satisfy some constraints, such as a minimum distance to an obstacle. The admissible set is the set of states that satisfy the state constraints.
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A number $a$ is an algebraic number if $a$ is a solution to a polynomial equation whose coefficients are all integers.
Consider a real or complex number $a$. The number $a$ is an algebraic number if $a$ is a root of a nonzero polynomial with integer coefficients.
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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 the time it takes to repeat varies, so that a shift by any fixed time $T > 0$ changes it.
A sequence of times $t_1$, $t_2$, $t_3$, … is called aperiodic if there does not exist a constant $P > 0$ such that
\[t_{i + 1} - t_{i} = P \quad \text{for each } i = 1, 2, 3, \dots.\] -
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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Think of a curved path, like a winding road. The arc length of the curve is how long it would be if it were straightened out, like measuring the path with a piece of string.
Consider a curve in the plane or in space. The arc length of the curve is its length, as if it were straightened out.
Consider a parametrized curve $\gamma : [a, b] \to \reals^n$ whose derivative is continuous. The arc length of $\gamma$ is
\[L = \int_a^b \norm{\gamma^{\prime}(t)} \mathrm{d}t.\] -
The area of a rectangle is the product of its base times its height.
Consider a region in the plane, or a surface in space. Its area is its two-dimensional size. The area of a surface in space, such as a sphere, is called its surface area.
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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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A way of combining numbers is associative if, when you combine three numbers in a row, it doesn’t matter which pair you combine first, such as $(2 + 3) + 4 = 2 + (3 + 4)$.
Consider a set $S$ with a binary operation $\ast$. The operation $\ast$ is associative if
\[(a \ast b) \ast c = a \ast (b \ast c) \quad \text{for all } a, b, c \in S.\] -
Functions $f$ and $g$ are asymptotically equivalent as $x \to \infty$ if the ratio $f(x) / g(x)$ approaches $1$, so that $f(x)$ and $g(x)$ 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.
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Think of a list of numbers, like the heights of the students in a class. An average of the list is a single number that represents a typical value of it. Usually, “average” means the arithmetic mean.
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An axiom is a statement that is accepted without proof as a starting point, from which other statements are proved.
Consider a mathematical theory, such as set theory or the theory of groups. An axiom of the theory is a statement that is taken as given, rather than proved, and from which the theory’s theorems are derived.
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:
\[\mathbb{B} = \setdef{x \suchthat d(x, c) \leq r}.\]If boundary is excluded, then it is called an open ball:
\[\mathbb{B} = \setdef{x \suchthat d(x, c) < r}.\] -
A Banach space is a normed space that is complete: every Cauchy sequence converges.
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Consider a space of vectors, such as the plane, in which vectors can be added and stretched. A basis of the space is a set of vectors from which every vector in the space can be built in exactly one way, by stretching each of them and adding the results.
Consider a vector space $V$. A basis of $V$ is a set of vectors in $V$ that is linearly independent and spans $V$.
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Consider a random experiment that either succeeds or fails, such as a coin flip that succeeds if the coin lands heads. A random variable $X$ that equals $1$ for success and $0$ for failure has a Bernoulli distribution with success probability $p$ if
\[P(X = 1) = p \quad \text{and} \quad P(X = 0) = 1 - p.\] -
Consider a vector space $V$. The function $B : V \times V \to \reals$ is a bilinear form on $V$ if $B$ is linear in each argument when the other is held fixed. It need not be symmetric or positive.
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A binary operation is a way of combining two numbers to get another number.
A binary operation on a set $S$ is a function $\ast : S \times S \to S$. The output must be in the same set $S$ (the set is “closed” under the operation).
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Consider $n$ independent trials, each of which succeeds with probability $p$. The number $X$ of successes has a binomial distribution with parameters $n$ and $p$:
\[P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \quad \text{for } k = 0, 1, \dots, n.\] -
Consider a shape drawn on a piece of paper using a continuous line that forms a closed loop. There is a portion of the paper that is inside the shape and a portion that is outside. The boundary of the shape is curve that separates the inside from the outside.
Consider a set $A \subseteq \reals^n$ and a point $x$. The point $x$ is in the boundary of $A$ if every open ball around $x$ contains both a point of $A$ and a point not in $A$.
Consider a subset $A$ of a topological space, with its closure $\overline{A}$ and its interior $\operatorname{int}(A)$. The boundary of $A$ is
\[\partial A = \overline{A} \setminus \operatorname{int}(A).\] -
A set of numbers is bounded if it does not go on forever in either direction: there is some number that every number in the set is smaller than in size.
A set $S$ of real numbers is bounded if there is a number $M$ such that $\abs{x} \leq M$ 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.
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Calculus of variations is the study of finding the best curve or function among all possible curves or functions, where “best” is measured by a number computed from the whole curve, like the length of the curve or the time it takes to slide down the curve.
Calculus of variations is the study of problems of finding a function $y : [a, b] \to \reals^n$, with fixed end values $y(a) = y_a$ and $y(b) = y_b$, that minimizes (or maximizes) a functional of the form
\[J[y] = \int_a^b L\bigl(t, y(t), y^{\prime}(t)\bigr) \, \mathrm{d}t.\] -
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 $A$ and $B$. This extends counting to infinite sets.
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Cascade control is a way of controlling a system with two controllers, one inside the other. The outer controller decides what an intermediate quantity should be, and the inner controller makes that quantity follow the decision. For example, to control the temperature of a room, an outer controller chooses how much hot water should flow, and an inner controller adjusts a valve until the water flow matches.
Consider a plant in which an intermediate variable $y_2$, which can be measured, lies between the control input $u$ and the output $y_1$ to be controlled. Cascade control uses two nested feedback loops:
- the outer loop compares $y_1$ with its reference $r_1$ and computes a reference $r_2$ for the intermediate variable, and
- the inner loop compares $y_2$ with $r_2$ and computes the control input $u$.
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Think of a quantity that goes up or down, like the temperature during a day. The change in the quantity is how much it went up or down: its final value minus its starting value.
Consider a quantity $x$ with an initial value and a final value. The change in $x$ is the final value minus the initial value:
\[\Delta x = x_{\text{final}} - x_{\text{initial}}.\]It is negative when the quantity decreases.
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A chart is a picture that presents information.
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Consider two points $p = (p_1, \dots, p_n)$ and $q = (q_1, \dots, q_n)$ in $\reals^n$. The Chebyshev distance between $p$ and $q$ is the largest difference in any one coordinate:
\[d_{\infty}(p, q) = \max_i \abs{p_i - q_i}.\] -
Consider a symmetric matrix $A$ whose eigenvalues are all positive (that is, $A$ is positive definite). The Cholesky decomposition of $A$ is the factorization
\[A = L L^{\top},\]where $L$ is lower triangular with positive diagonal entries.
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A circle is the round curve made of all the points at a given distance from a single point. The distance is called the radius and the point at the middle is called the center.
A circle is only the curve, not the region inside it.
Consider a center $c$ in the plane and a radius $r > 0$. The circle with center $c$ and radius $r$ is the set of points in the plane at distance $r$ from $c$. It is a sphere in $\reals^2$ (the $1$-sphere $S^1$).
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Consider a function $\alpha : [0, a) \to [0, \infty)$. The function $\alpha$ is of class-$\mathcal{K}$ if
- $\alpha$ is continuous
- $\alpha$ is strictly increasing and
- $\alpha(0) = 0$.
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Consider a class-$\mathcal{K}$ function $\alpha$ defined on all of $[0, \infty)$. The function $\alpha$ is of class-\(\mathcal{K}_\infty\) if $\alpha(r) \to \infty$ as $r \to \infty$.
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Think of a machine that is trying to reach a goal, like a heater warming a room. In closed-loop control, the machine checks the result and adjusts its action based on what it measures.
Consider a controller that chooses the input to a system. In closed-loop control, the controller measures the result and adjusts the input to correct errors.
Consider a control system $\dot x = f(x, u)$. In closed-loop control, the input $u$ is computed from measurements of the output (or state), forming a feedback loop. The closed-loop system is the plant together with its controller.
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For all real numbers $a$ and $b$ such that $a < b$, the interval of numbers between $a$ and $b$ including $a$ and $b$ is called an closed interval, and is written $[a, b]$.
Consider a set $F$ of points on the number line, in the plane, or in space. The set $F$ is closed if $F$ contains all of its boundary points.
Consider a set $F$ in a metric space. The set $F$ is closed if $F$ contains the limit of every convergent sequence of points of $F$.
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A collection is a group of objects considered together.
Collection is an informal word, used as a synonym for “set,” to produced for variety (e.g., “a collection of subsets”) or for classes that are too large to be sets (“the collection of all sets”).
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A way of combining two numbers is commutative if the order of the numbers doesn’t matter, such as $2 + 3 = 3 + 2$.
Consider a set $S$ with a binary operation $\ast$. The operation $\ast$ is commutative if
\[a \ast b = b \ast a \quad \text{for all } a, b \in S.\] -
Think of two shapes cut out of paper. The shapes are congruent if both shapes have the same form and the same size, so that one could be placed exactly on top of the other (possibly after flipping it over).
Consider two figures in the plane, and the rigid motions that move one of them: rotations, reflections, and translations. The figures are congruent if such motions can move one exactly onto the other: the figures have the same shape and the same size.
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Consider a mechanical system moving under forces. The system is conservative if the total energy of the system stays constant as the system moves.
More precisely, for a system with total energy
\[E = U + K\]where $U$ is the potential energy and $K$ is the kinetic energy, the value of $E$ is constant over time:
\[\frac{d}{dt} E = 0.\]Consider a mechanical system with generalized coordinates $q = (q_1, \dots, q_n)$. The system is conservative if every force on the system comes from a potential energy $V(q)$ that depends only on the positions:
\[F = -\nabla V(q).\]A dynamical system $\dot x = f(x)$ is called conservative if there exists a real-valued function $x \mapsto V(x)$ such that
\[f(x) = -\nabla V(x)\] -
Think of a function $y = f(x)$. The graph of $f(x)$ is a plot that shows the value of $y$ for each values of $x$ by plotting the value of $x$ on a horizontal axis and the value $y = f(x)$ on the vertical axis. The function is continuous if you can draw its graph without lifting your pencil.
Consider a function $f$. The function $f$ is continuous if
\[\lim_{x \to a} f(x) = f(a)\]for every $a$ in the domain of $f$.
Consider a function $f : X \to Y$ between metric spaces. The function $f$ is continuous if for every $x \in X$ and every $\varepsilon > 0$, there exists $\delta > 0$ such that
\[d(f(x), f(y)) < \varepsilon \quad \text{for every } y \in X \text{ with } d(x, y) < \delta.\]The choice of $\delta$ may depend on both $\varepsilon$ and $x$.
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A function $f$ is called continuously differentiable if the derivative $f^{\prime}$ exists at each point in the domain of $f$ and the derivative is itself a continuous function.
Consider a function $f : U \to \reals^m$ on an open set $U \subseteq \reals^n$. The function $f$ is continuously differentiable if every partial derivative of $f$ exists and is continuous on $U$.
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Consider a system that you want to steer, like a car. A control input is a quantity you choose in order to steer the system.
In a control system, a control input is an input $u$ chosen by the controller at each time to change the behavior of the system.
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A covector is a rule that takes a vector and returns a number, in a way that respects adding vectors and scaling them. In coordinates, it is written as a row of numbers, which multiplies a column vector.
Consider a vector space $V$ and its scalars. A covector on $V$ is a linear map from $V$ to its scalars; an element of the dual space $V^{\ast}$. In coordinates, covectors are often written as row vectors, which act on column vectors by matrix multiplication.
Consider a manifold $M$ and a point $x$ of $M$, with its cotangent space $T_x^{\ast} M$. A covector at $x$ is an element of $T_x^{\ast} M$. The differential $\mathrm{d}f$ of a function is a covector field (a one-form).
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Consider a flow, like water moving in a pond, and imagine placing a tiny paddle wheel at a point in it. The curl of the flow at that point measures how much the flow swirls around the point: how fast the paddle wheel would spin.
Consider a vector field $F = (F_1, F_2, F_3) : \reals^3 \to \reals^3$. The curl of $F$ is the vector field
\[\nabla \times F = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right).\]It measures the rotation of $F$ around each point.
D
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Consider a linear system $\dot{x} = A x + B u$ with output $y = C x$, and its unobservable modes: the parts of the state that do not affect the output. The system is detectable if every unobservable mode is asymptotically stable.
Consider a system $\dot{x} = f(x)$ with output $y = h(x)$. The system is (zero-state) detectable if every solution such that $y(t) = 0$ for all $t \geq 0$ converges to the origin.
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The diameter of the circle is the distance across a circle, measured as the length of a straight line drawn through its center.
Consider a circle or sphere $S$. The diameter of $S$ is the length of every line segment through the center of $S$ with both endpoints on the boundary of $S$.
Consider a set $A$ in a metric space with metric $d$. The diameter of $A$ is the largest distance between two points of $A$:
\[\operatorname{diam}(A) = \sup \setdef{d(x, y) \suchthat x, y \in A}.\] -
Think of two numbers, one subtracted from the other. The difference of the two numbers is the result of the subtraction: $7 - 3 = 4$, so the difference of $7$ and $3$ is $4$.
Consider two numbers $a$ and $b$. The difference of $a$ and $b$ is the result $a - b$ of subtraction. It depends on the order and can be negative: $3 - 7 = -4$.
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Consider a function $f$ and a point $a$ in its domain. The function $f$ is differentiable at $a$ if the limit
\[f^{\prime}(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}\]exists. It is differentiable if this holds at every point of its domain.
Consider a function $f$ on an open subset of $\reals^n$ and a point $a$ in its domain. The function $f$ is differentiable at $a$ if there is a linear map $Df(a)$ with
\[\lim_{h \to 0} \frac{\norm{f(a + h) - f(a) - Df(a) h}}{\norm{h}} = 0.\] -
Consider a differentiable function $f$ and a point $x$ in its domain. The differential of $f$ at $x$ is the linear map $\mathrm{d}f_x : v \mapsto Df(x) v$ that best approximates $f(x + v) - f(x)$ for small $v$. For $f : \reals^n \to \reals$,
\[\mathrm{d}f_x(v) = \frac{\partial f}{\partial x_1}(x) v_1 + \dots + \frac{\partial f}{\partial x_n}(x) v_n.\]Consider a smooth function $f : M \to \reals$ on a manifold $M$. The differential of $f$ is the one-form $\mathrm{d}f$, which assigns to each point $x$ the covector $\mathrm{d}f_x$ on the tangent space $T_x M$.
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The dimension of an object is the number of perpendicular directions you can move within that space. Think of the directions you can move in a space: along a line, you can only go back and forth but not side to side or up and down, so a line has one dimension.
Consider a space, such as a line, a plane, or the space around us, and coordinates that describe each point in it. The dimension of the space is the number of coordinates needed to describe a point.
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Consider a function of several variables, like the height of a hill at each point of a map. The directional derivative of the function is its vertical rate of change as you move horizontally in a particular direction.
Consider a function $f$, a point $x$, and a direction $v$. The directional derivative of $f$ at $x$ in the direction $v$ is the rate of change of $f$ in that direction:
\[D_v f(x) = \lim_{t \to 0} \frac{f(x + t v) - f(x)}{t}.\] -
Consider a set $X$. The discrete metric on $X$ is the metric $d$ such that $d(p, q) = 1$ for $p \neq q$ and $d(p, p) = 0$.
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Consider two sets $A$ and $B$ that have no elements in common. The union $A \cup B$ is a union of disjoint sets.
Consider sets $A$ and $B$ such that $A \cap B = \emptyset$. A union of disjoint sets is the ordinary union $A \cup B$ in this case. Writing $A \sqcup B$ (or $A \mathbin{\dot{\cup}} B$) for it asserts that $A$ and $B$ are disjoint.
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Consider two sets $A$ and $B$, which may overlap. The disjoint union of $A$ and $B$ is a set that combines $A$ and $B$ while keeping the elements of $A$ apart from the elements of $B$:
\[A \sqcup B = (\setdef{1} \times A) \cup (\setdef{2} \times B),\]so each element is tagged with the set it came from.
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Think of a moving object, like a ball rolling across a floor. The displacement of the object is how far, and in which direction, it ended up from where it started.
Consider an object that moved from a starting point $P$ to an ending point $Q$. The displacement of the object is its change in position, as a vector from $P$ to $Q$.
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Think of two objects, like two houses on a street. The distance between the two houses is how far apart the houses are. No distance is negative.
Consider two points, either on the number line or in the plane. On the number line, the distance between $a$ and $b$ is $\abs{a - b}$. In the plane, the distance between $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$.
Consider a metric space with metric $d$, and points $x$ and $y$ in it. The distance between $x$ and $y$ is the value $d(x, y)$.
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Consider a system that is trying to follow a course, like a drone flying to a target. A disturbance is an outside influence that pushes the system off course.
Consider a control system $\dot{x} = f(x, u, d)$, with inputs from outside. A disturbance is an unwanted exogenous input, such as $d$, that affects the dynamics of the system. It changes the actual state of the system.
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Consider a flow, like air moving in a room. The divergence of the flow measures how much it spreads out from each point (positive) or gathers in toward it (negative).
Consider a vector field $F = (F_1, \dots, F_n) : \reals^n \to \reals^n$. The divergence of $F$ is the scalar function
\[\nabla \cdot F = \frac{\partial F_1}{\partial x_1} + \dots + \frac{\partial F_n}{\partial x_n}.\]It measures how much $F$ spreads out from a point.
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Consider two vectors $x = (x_1, \dots, x_n)$ and $y = (y_1, \dots, y_n)$ in $\reals^n$. The dot product of $x$ and $y$ is
\[x \cdot y = x_1 y_1 + x_2 y_2 + \dots + x_n y_n.\]Consider the vector space $\reals^n$, with vectors written as columns. The dot product is the standard inner product on $\reals^n$,
\[x \cdot y = x^{\top} y.\]
E
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Imagine a network drawn as dots connected by lines, like a map of cities and roads. Each line connecting two dots in the network is called an edge.
Consider a graph, made of a set of vertices and a set of edges. An edge of the graph is an element of its edge set. In a simple undirected graph, an edge is an unordered pair $\setdef{u, v}$ of distinct vertices.
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A polygon is a 2D shape with straight sides that form a loop. Each side is called an edge of the polygon.
In 2D, an edge of the polygon is a line segment between two adjacent vertices (one of its sides). In 3D, an edge of the polyhedron is a segment where two of its faces meet, such as the 12 line segments separating the faces of a cube.
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Consider a square matrix $A$. An eigendecomposition of $A$ is a factorization
\[A = P D P^{-1},\]where $P$ is invertible and $D$ is diagonal. The diagonal entries of $D$ are eigenvalues of $A$, and the columns of $P$ are corresponding eigenvectors.
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Consider a square matrix $A$, which stretches and turns vectors when it multiplies them. An eigenvalue of $A$ is a number $\lambda$ such that $A$ stretches some nonzero vector $v$ by the factor $\lambda$ without turning it off its line:
\[A v = \lambda v.\]Consider a square matrix $A$. An eigenvalue of $A$ is a number $\lambda$ such that
\[A v = \lambda v\]for some nonzero vector $v$, called an eigenvector.
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Think of two amounts, like the number of apples in two baskets. The amounts are equal if the amounts have the same value.
Consider two mathematical objects $a$ and $b$. The statement $a = b$ means that $a$ and $b$ are equal: $a$ and $b$ are the same object.
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Consider a set $X$ and a relation $\sim$ on $X$. The relation $\sim$ is an equivalence relation if, for all $a, b, c \in X$,
- $a \sim a$ (reflexive),
- $a \sim b$ implies $b \sim a$ (symmetric), and
- $a \sim b$ and $b \sim c$ imply $a \sim c$ (transitive).
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Consider two objects and some way of comparing them that has been chosen. The objects are equivalent if the objects are the same in that particular sense.
Consider an equivalence relation $\sim$ on a set: a relation that is reflexive ($a \sim a$), symmetric ($a \sim b$ implies $b \sim a$), and transitive ($a \sim b$ and $b \sim c$ imply $a \sim c$). Two objects $a$ and $b$ are equivalent if $a \sim b$.
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Think of an estimate or a measurement of some quantity, like the length of a table. The error of the estimate is how far it is from the true value.
Consider an approximate or measured value $\hat{x}$ of a true value $x$. The error of $\hat{x}$ is how far it is from $x$. The absolute error is $\abs{\hat{x} - x}$, and the relative error is $\abs{\hat{x} - x} / \abs{x}$.
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Consider an unknown quantity and some data related to it. An estimator is a rule for guessing the unknown quantity from the data.
Consider an unknown quantity, such as a state or a parameter, and the data available about it. An estimator is any rule for computing an estimate of the quantity from the data.
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Consider two points $p = (p_1, \dots, p_n)$ and $q = (q_1, \dots, q_n)$ in $\reals^n$. The Euclidean distance between $p$ and $q$ is the length of the straight segment from $p$ to $q$:
\[d_2(p, q) = \sqrt{(p_1 - q_1)^2 + \dots + (p_n - q_n)^2}.\]Consider two points $p = (p_1, \dots, p_n)$ and $q = (q_1, \dots, q_n)$ in $\reals^n$. The Euclidean distance between $p$ and $q$ is
\[d_2(p, q) = \norm{p - q}_2\]where $\norm{\cdot}$ indicates the Euclidean norm.
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Consider a vector $x = (x_1, \dots, x_n)$, drawn as an arrow from the origin. The Euclidean norm of $x$ is the length of the arrow:
\[\norm{x}_2 = \sqrt{x_1^2 + \dots + x_n^2}.\] -
Consider a random experiment, such as rolling two dice, whose outcomes are pairs of values, such as $(1, 3)$. An event is a set of outcomes, such as “the sum of the dice is $4$,” which is the set $A = \setdef{(1, 3), (2, 2), (3, 1)}$. The event occurs if the outcome of the experiment is in the set.
Consider a probability space $(\Omega, \mathcal{F}, P)$. An event $A$ is a subset of the sample space $\Omega$ such that $A$ is an element of the σ-algebra $\mathcal{F}$. For each event, the probability measure $P$ assigns the probability that $A$ occurs in each experiment.
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Consider a system that changes over time. An exogenous input is any signal that comes into the system from outside it, such as a control input or a disturbance.
Consider a control system. An exogenous input is any signal that enters the system from outside, rather than being determined by the system’s own state. This includes control inputs, reference signals, disturbances, and noise.
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Consider events that happen independently of each other at an average rate of $\lambda$ per unit of time, such as customers arriving at a shop. The waiting time $T$ until the first event has an exponential distribution with rate $\lambda$, which has the probability density
\[f(t) = \lambda e^{-\lambda t} \quad \text{for } t \geq 0.\] -
An extended Kalman filter is a version of the Kalman filter for systems that are not linear. At each step, it approximates the system by a linear one near the current estimate, then applies the Kalman filter’s predict-and-correct steps to the approximation.
Consider a discrete-time system with state $x_k$, input $u_k$, and measured output $y_k$, given by
\[x_{k+1} = f(x_k, u_k) + w_k, \qquad y_k = h(x_k) + v_k,\]where $f$ and $h$ are differentiable, and the noises $w_k$ and $v_k$ are independent, zero-mean, and white, with covariances $Q$ and $R$. The extended Kalman filter for this system is the filter that computes a state estimate $\hat{x}_k$ and an approximate error covariance $P_k$ by repeating two steps. First, it predicts
\[\hat{x}_k^- = f(\hat{x}_{k-1}, u_{k-1}), \qquad P_k^- = F_{k-1} P_{k-1} F_{k-1}^\top + Q,\]where $F_{k-1}$ is the Jacobian of $f$ with respect to $x$ at $(\hat{x}{k-1}, u{k-1})$. Then, it corrects the prediction using the measurement $y_k$:
\[\hat{x}_k = \hat{x}_k^- + K_k \bigl(y_k - h(\hat{x}_k^-)\bigr), \qquad P_k = (I - K_k H_k) P_k^-,\]where $H_k$ is the Jacobian of $h$ at $\hat{x}_k^-$ and $K_k = P_k^- H_k^\top (H_k P_k^- H_k^\top + R)^{-1}$.
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Think of a shape drawn on paper, with an inside, an edge, and an outside. The exterior of the shape is everything outside it, not including its edge.
Consider a set $A \subseteq \reals^n$ and a point $x$. The point $x$ is in the exterior of $A$ if there exists an open ball around $x$ that does not contain any points in $A$.
F
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Think of a system that can see the results of what it does, like a person riding a bike. Feedback is when the system watches its own results and adjusts.
Consider a system with an input and an output. Feedback is using information about the system’s output to decide its input.
Consider a system with an input $u$ and an output $y$ (or state $x$). Feedback is the use of measurements of the output (or state) to determine the input, as in $u = k(y)$.
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Consider a system that is affected by a disturbance that you can see coming. Feedforward control reacts to the disturbance before it affects the result.
Consider a control system with a measured disturbance or a known reference signal. Feedforward is a control action computed from the disturbance or reference, applied before the disturbance or the change in the reference affects the output.
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Consider a quantity that is measured over time, with noisy measurements. In estimation, a filter estimates the current value of the quantity from the measurements made up to now.
Consider a system with state $x(t)$ and noisy measurements of it. In estimation, a filter computes an estimate of the current state $x(t)$ from the measurements up to time $t$.
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A set of items is finite if you could count its elements completely, finishing at some number.
Consider a set $S$. The set $S$ is finite if $S$ is empty or in bijection with $\setdef{1, 2, \dots, n}$ for some $n \in \nats$.
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Think of a function as a machine that changes its input into an output. A fixed point of the function is an input that the function leaves unchanged: The output equals the input.
Consider a function $f$ whose inputs and outputs are the same kind of thing. A fixed point of $f$ is a point $x$ such that $f(x) = x$: the function leaves it unchanged.
Consider a function $f : X \to X$. A fixed point of $f$ is a point $x \in X$ such that $f(x) = x$.
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A functional takes a whole function as its input and returns a number.
Consider a vector space, often a space of functions, and its scalars. A functional is a map from the vector space to its scalars.
G
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Consider two points $p$ and $q$ on a surface, such as two cities on the Earth. The geodesic distance between $p$ and $q$ is the length of the shortest path from $p$ to $q$ that stays on the surface.
Consider a connected Riemannian manifold $M$ and two points $p, q \in M$. The geodesic distance between $p$ and $q$ is the infimum of the lengths of the piecewise smooth curves in $M$ from $p$ to $q$:
\[d(p, q) = \inf \setdef{\int_0^1 \norm{\gamma^{\prime}(t)} \mathrm{d}t \suchthat \gamma(0) = p, \gamma(1) = q}.\] -
Consider a sequence of independent trials, each of which succeeds with probability $p$. The number $N$ of trials up to and including the first success has a geometric distribution with parameter $p$:
\[P(N = k) = (1 - p)^{k - 1} p \quad \text{for } k = 1, 2, 3, \dots.\] -
Consider a differentiable function $f : \reals^n \to \reals$. The gradient of $f$ is the vector of partial derivatives
\[\nabla f(x) = \left( \frac{\partial f}{\partial x_1}(x), \dots, \frac{\partial f}{\partial x_n}(x) \right).\]Consider a smooth function $f$ on a Riemannian manifold, with inner product $\ip{\cdot}{\cdot}$ on each tangent space. The gradient of $f$ at $x$ is the vector $\nabla f(x)$ such that
\[\ip{\nabla f(x)}{v} = \mathrm{d}f_x(v) \quad \text{for every tangent vector } v.\] -
Consider a set $G$ with a binary operation $\ast$. The set $G$ is a group if
- $\ast$ is associative: $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a, b, c \in G$,
- $G$ has an identity element $e$ such that $e \ast g = g \ast e = g$ for all $g \in G$, and
- every element $g \in G$ has an inverse $g^{-1}$ such that $g \ast g^{-1} = g^{-1} \ast g = e$.
H
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Consider a shape, such as a segment, a square, or a cube. Scaling the shape up by a factor of $2$ multiplies its “size” by $2^d$, where $d$ is its dimension: by $2$ for a segment, $4$ for a square, and $8$ for a cube. The Hausdorff dimension makes this precise.
Consider a subset $A$ of a metric space and a number $s \geq 0$. The $s$-dimensional Hausdorff measure of $A$ adds up the diameters of small sets that cover $A$, each raised to the power $s$:
\[\mathcal{H}^s(A) = \lim_{\delta \to 0} \inf \setdef{\sum_{i = 1}^{\infty} \operatorname{diam}(U_i)^s \suchthat A \subseteq \bigcup_{i = 1}^{\infty} U_i, \ \operatorname{diam}(U_i) < \delta}.\]The Hausdorff dimension of $A$ is the number at which $\mathcal{H}^s(A)$ jumps from $\infty$ to $0$:
\[\dim_H(A) = \inf \setdef{s \geq 0 \suchthat \mathcal{H}^s(A) = 0}.\] -
A hypothesis is a guess about how something works that can be checked with an experiment.
A hypothesis is a proposed explanation or prediction that can be tested by observation or experiment, and that the evidence could show to be false.
I
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Consider two objects that are not literally equal, but correspond to each other through a natural bijection. The objects are identified when we agree to treat the objects as the same.
Consider some points of a topological space that we want to treat as one point. To identify those points is to pass to a quotient space in which they become a single point.
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Consider an operation that combines two things, like addition or multiplication. An identity element for the operation is one that leaves everything unchanged.
Consider a set $S$ with a binary operation $\ast$. An identity element for $\ast$ is an element $e$ such that
\[e \ast a = a \ast e = a \quad \text{for all } a \in S.\]For addition, $e$ is called the additive identity and is usually written as $0$. For multiplication, $e$ is the multiplicative identity and is usually written $1$.
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Consider an index set $I$ and a set of values. An indexed family is a function $i \mapsto x_i$ from $I$ to the set, written \((x_i)_{i \in I}\). Unlike a set, an indexed family can repeat elements. A sequence is an indexed family with $I = \nats$.
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A list of items is infinite if it never runs out: no matter how many elements you list, there are more.
Consider a set $S$. The set $S$ is infinite if the number of elements in $S$ is not any natural number.
Consider a set $S$, and assume the axiom of choice. The set $S$ is infinite exactly when $S$ is in bijection with a proper subset of itself (Dedekind infinite).
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Consider a vector $x = (x_1, \dots, x_n)$. The infinity norm of $x$ is the largest absolute value of its entries:
\[\norm{x}_{\infty} = \max_i \abs{x_i}.\] -
A function $f$ is injective if every two different inputs $x$ and $y$ give different outputs: $f(x) \neq f(y)$.
Consider a function $f : X \to Y$. The function $f$ is injective if, for all $x_1, x_2 \in X$,
\[f(x_1) = f(x_2) \implies x_1 = x_2.\] -
An inner product is a way of multiplying two vectors to get a number, generalizing the dot product, that gives a way to measure lengths and angles.
Consider a real vector space $V$ and a function $\ip{\cdot}{\cdot} : V \times V \to \reals$. The function is an inner product if
- it is linear in its first argument: $\ip{a x + y}{z} = a \ip{x}{z} + \ip{y}{z}$ for all $x, y, z \in V$ and $a \in \reals$,
- it is symmetric: $\ip{x}{y} = \ip{y}{x}$ for all $x, y \in V$, and
- it is positive definite: $\ip{x}{x} > 0$ for for all $x \neq 0 in $V$.
Consider a vector space $V$ over $\mathbb{F}$, where $\mathbb{F}$ is $\reals$ or $\mathbb{C}$, and a sesquilinear form $\ip{\cdot}{\cdot}$ on $V$: a function $V \times V \to \mathbb{F}$ that is linear in its first argument and conjugate-linear in its second. The form is an inner product if, for all $x, y \in V$,
- it is Hermitian: $\ip{x}{y} = \overline{\ip{y}{x}}$, and
- it is positive definite: $\ip{x}{x} > 0$ for $x \neq 0$.
Over $\reals$, a sesquilinear form is a bilinear form, and Hermitian means symmetric.
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An integer is a whole number or the negative of a whole number:
\[\dots, -3, -2, -1, 0, 1, 2, 3, \dots\]An integer is an element of $\mathbb{Z}$, which can be constructed from the natural numbers as equivalence classes of pairs $(a, b)$ of natural numbers, where $(a, b)$ represents $a - b$ and $(a, b) \sim (c, d)$ if $a + d = b + c$.
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The interior of a shape is the part on its inside, not including its edge.
A point $x$ is in the interior of $A$ if there exists an open ball around $x$ that is contained in $A$.
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Consider two sets $A$ and $B$. The intersection of $A$ and $B$, written $A \cap B$, is the set of every element that is in both $A$ and $B$.
Consider two sets $A$ and $B$. The intersection of sets $A$ and $B$ is
\[A \cap B = \setdef{x \suchthat x \in A \text{ and } x \in B}.\] -
An invariant is a quantity that stays the same while something else changes.
An invariant is a quantity or property that does not change under a given transformation or along solutions of a given system.
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For addition of numbers, for every number $a$, there is another number $-a$ such that adding $a + (-a)$ is zero. We call $a$ the additive inverse $a$. For example, the additive inverse of $5$ is $-5$.
The multiplicative inverse (or reciprocal) of a nonzero number $a$ is the number $\frac{1}{a}$ that you multiply by $a$ to get $1$:
\[a \times \frac{1}{a} = 1.\]For example, the multiplicative inverse of $5$ is $\frac{1}{5}$. The number $0$ has no multiplicative inverse, because $0$ times every number is $0$.
Consider a way of combining two things, such as addition of numbers, that has an element that changes nothing, such as $0$ for addition. An inverse of an element $a$ is an element that undoes $a$: combining $a$ with the inverse gives the element that changes nothing. For example, the additive inverse of $5$ is $-5$, because $5 + (-5) = 0$.
Consider a set $S$ with a binary operation $\ast$ and an identity element $e$. An inverse of an element $a \in S$ is an element $b \in S$ such that
\[a \ast b = b \ast a = e.\] -
An irrational number is a number that cannot be written as a fraction of whole numbers (or negative whole numbers).
An irrational number is a real number that cannot be written as a fraction of integers.
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Two structures are isomorphic if the two structures are the same except for the names of their elements.
J
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The Jacobian of a function with several inputs and several outputs is the table of all of its partial derivatives, with one row per output and one column per input.
The Jacobian of $f : \reals^n \to \reals^m$ is the $m \times n$ matrix of partial derivatives, whose entry in row $i$ and column $j$ is $\frac{\partial f_i}{\partial x_j}$.
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Consider a function $f(x, y)$ that maps each point of the $xy$-plane to a new point in the plane. Near a given point, $f$ stretches or shrinks tiny regions, and it may also flip them over, like a mirror. The Jacobian determinant of $f$ at that point is the factor by which $f$ scales the areas of those tiny regions, made negative if $f$ flips them over.
Consider a differentiable function $f : \reals^n \to \reals^n$ (which has as the same number of outputs as inputs, so its Jacobian matrix $Df(x)$ at each point $x$ is square). The Jacobian determinant of $f$ at $x$ is the determinant of that matrix, $\det Df(x)$.
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A Jordan block is a square matrix with a number $\lambda$ on its diagonal, $1$ on the diagonal just above it, and $0$ elsewhere, such as
\[\begin{bmatrix} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{bmatrix}.\]Consider a square complex matrix $A$. A Jordan normal form of $A$ is a factorization
\[A = P J P^{-1},\]where $P$ is invertible and $J$ is block diagonal with Jordan blocks on its diagonal.
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The Jordan–Chevalley decomposition of a square matrix $A$ writes it as a sum $A = D + N$, where $D$ is diagonalizable, $N$ is nilpotent ($N^k = 0$ for some $k$), and $D N = N D$.
The Jordan–Chevalley decomposition of a linear map $T$ on a finite-dimensional vector space writes it as $T = S + N$, where $S$ is semisimple, $N$ is nilpotent, and $S$ and $N$ commute.
K
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A Kalman filter is a method for estimating a changing quantity, such as the position of a car, from noisy measurements made over time. At each step, it predicts the quantity from a model of how the quantity changes, then corrects the prediction toward the new measurement, trusting whichever of the two is less uncertain more.
Consider a discrete-time linear system with state $x_k$, input $u_k$, and measured output $y_k$, given by
\[x_{k+1} = A x_k + B u_k + w_k, \qquad y_k = C x_k + v_k,\]where the process noise $w_k$ and the measurement noise $v_k$ are independent, zero-mean, Gaussian, and white, with covariances $Q$ and $R$, and the initial state $x_0$ has mean $\hat{x}0$ and covariance $P_0$. The _Kalman filter for this system is the filter that computes a state estimate $\hat{x}_k$ and its error covariance $P_k$ by repeating two steps. First, it predicts
\[\hat{x}_k^- = A \hat{x}_{k-1} + B u_{k-1}, \qquad P_k^- = A P_{k-1} A^\top + Q.\]Then, it corrects the prediction using the measurement $y_k$:
\[\hat{x}_k = \hat{x}_k^- + K_k (y_k - C \hat{x}_k^-), \qquad P_k = (I - K_k C) P_k^-,\]where $K_k = P_k^- C^\top (C P_k^- C^\top + R)^{-1}$ is the Kalman gain.
L
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A scientific law is a statement, often an equation, that describes a pattern that holds in every tested case, without explaining why.
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Consider a function $f$ and a number $a$, and let $x$ approach $a$ from below (from the left). The left limit of $f$ at $a$, written $\lim_{x \to a^{-}} f(x)$, is the number that $f(x)$ approaches.
Consider a function $f$ defined on an interval to the left of $a$. We write $\lim_{x \to a^{-}} f(x) = L$, and call $L$ the left limit of $f$ at $a$, if for every $\varepsilon > 0$ there is a $\delta > 0$ such that
\[\abs{f(x) - L} < \varepsilon \quad \text{whenever } a - \delta < x < a.\] -
Consider a function $V$ and a vector field $F$, with solutions of $\dot{x} = F(x)$. The Lie derivative of $V$ along $F$ is
\[L_F V(x) = \nabla V(x) \cdot F(x).\]It is the rate of change of $V$ along those solutions.
Consider a function $V$ and a vector field $F$ on a manifold, where $\phi_t$ is the flow of $F$. The Lie derivative of $V$ along $F$ is
\[L_F V(x) = \frac{\mathrm{d}}{\mathrm{d}t} V(\phi_t(x)) \big\rvert_{t = 0}.\]Lie derivatives are also defined for vector fields and differential forms.
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Consider a statistical model that gives a probability (or probability density) $p(x \mid \theta)$ of data $x$ for each value of a parameter $\theta$, and observed data $x$. The likelihood of $\theta$ is the probability of the observed data, viewed as a function of the parameter:
\[L(\theta) = p(x \mid \theta).\] -
Consider a function $f$ and a number $a$, and let $x$ get close to, but not equal to, $a$. The limit of $f(x)$ as $x$ approaches $a$ is the number that $f(x)$ gets close to.
The limit of $f$ is written as
\[\lim_{x \to a} f(x).\]Consider a function $f$ defined near $a$, except possibly at $a$. We write
\[\lim_{x \to a} f(x) = L\]and call $L$ the limit of $f$ at $a$, if for every $\varepsilon > 0$ there is a $\delta > 0$ such that
\[\abs{f(x) - L} < \varepsilon \quad \text{whenever } 0 < \abs{x - a} < \delta.\] -
Consider a function $f$ from $\reals$ to $\reals$. The function $f$ is a linear map if $f$ has the form $f(x) = m x$ for some number $m$. The graph of a linear map is a straight line that passes through the origin.
Consider vector spaces $V$ and $W$, and a function $T : V \to W$. The function $T$ is called a linear map if, for all vectors $u, v$ and scalars $c$,
- $T(u + v) = T(u) + T(v)$ (additivity), and
- $T(c v) = c T(v)$ (homogeneity).
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Consider a function $f$ and the lines through pairs of points on its graph. The function $f$ is Lipschitz continuous if there is a number $L$ such that every one of those lines has a slope between $-L$ and $L$.
Consider a function $f : X \to Y$ between metric spaces. The function $f$ is Lipschitz continuous if there exists $L \geq 0$ such that
\[d(f(x), f(y)) \leq L d(x, y) \quad \text{for all } x, y \in X.\]The smallest such $L$ is the Lipschitz constant of $f$.
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Consider a function $f$ and the lines through pairs of points on its graph. The function $f$ is locally Lipschitz continuous if each point has a surrounding interval in which the slopes of those lines are bounded, though the bound can be different for different intervals.
Consider a function $f : X \to Y$ between metric spaces. The function $f$ is locally Lipschitz continuous if every point $x_0 \in X$ has a neighborhood $U$ and a number $L_U \geq 0$ such that
\[d(f(x), f(y)) \leq L_U \, d(x, y) \quad \text{for all } x, y \in U.\] -
Consider a square matrix $A$. An LU decomposition of $A$ is a factorization $A = L U$, where $L$ is lower triangular (usually with ones on its diagonal) and $U$ is upper triangular.
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A Luenberger observer estimates the unmeasured quantities of a system by running a copy of the system’s model alongside it and nudging the copy toward the real system whenever their measurements disagree.
Consider a linear system $\dot{x} = A x + B u$ with state $x$, input $u$, and measured output $y = C x$. A Luenberger observer for this system is the dynamical system
\[\dot{\hat{x}} = A \hat{x} + B u + L (y - C \hat{x}),\]where $\hat{x}$ is the state estimate and the matrix $L$ is a constant gain, called the observer gain.
M
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Think of numbers arranged in a grid, like the seats in a theater. A matrix is a rectangular grid of numbers arranged in rows and columns.
A matrix is a rectangular grid of numbers arranged in rows and columns. A matrix can be multiplied by a scalar, two matrices $A$ and $B$ of the same size can be added ($A + B$), and two matrices $A$ and $B$ can be multiplied ($AB$) if $A$ and $B$ have compatible sizes.
An $m \times n$ matrix over a field $F$ (such as $\reals$) is an array $A = (a_{ij})$ with entries in $F$, together with the operations of scalar multiplication, matrix addition, and matrix multiplication.
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The Lebesgue measure assigns a size to (measurable) subsets of $\reals^n$, extending length ($n = 1$), area ($n = 2$), and volume ($n = 3$) to complicated sets. More generally, a measure is a countably additive function from a collection of sets to $[0, \infty]$.
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Consider a set $A \subseteq \reals$. The set $A$ has measure zero if for every $\varepsilon > 0$, it can be covered by countably many intervals with total length less than $\varepsilon$.
Consider a measure $\mu$ on a space (by default, Lebesgue measure) and a measurable set $A$. The set $A$ has measure zero if $\mu(A) = 0$.
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A metric is a way of generalizing the concept of distance. Consider a set of points, and a rule that gives a “distance” between every two of them. The rule is a metric if
- no distance is negative,
- the distance is zero only from a point to itself,
- the distance is the same in both directions, and
- no distance is longer than a detour through a third point (the triangle inequality).
Consider a set $X$ and a function $d : X \times X \to [0, \infty)$. The function $d$ is a metric on $X$ if, for all $x, y, z \in X$,
- $d(x, y) = 0$ if and only if $x = y$,
- $d(x, y) = d(y, x)$ (symmetry), and
- $d(x, z) \leq d(x, y) + d(y, z)$ (the triangle inequality).
The last condition is the triangle inequality. The set $X$ need not be a vector space.
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A metric space is a set of points with a way of measuring the distance between every two points.
A metric space is a set $X$ together with a metric $d$ on $X$ that defines the distance between any two points.
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Consider a set, in which each element appears only once. A multiset is like a set, except that an element can appear more than once.
A multiset is a collection in which order does not matter but repetition does. Formally, a function from a set to the positive integers that gives the multiplicity of each element.
N
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The natural numbers are the numbers $0, 1, 2, 3, \dots$ that can be used to answer questions about “how many?” objects there are.
The natural numbers are the set $\setdef{0, 1, 2, 3, \dots}$ of nonnegative integers, often written $\nats$, \(\nats_0\), or \(\mathbb{Z}_{\geq 0}\).
The natural numbers are the smallest set that contains $0 = \emptyset$ and contains $n \cup \setdef{n}$ whenever it contains $n$ (the von Neumann construction). Each natural number is the set of all smaller natural numbers.
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The natural numbers are the counting numbers $1, 2, 3, 4, \dots$
The natural numbers are the set $\setdef{1, 2, 3, \dots}$ of positive integers, often written $\nats$, \(\mathbb{Z}_{> 0}\), or \(\nats_{+}\).
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A vector is arrow between two points. Such a vector has a direction and a length. The norm of the vector is its length: how long the arrow is.
Consider a vector $x = (x_1, \dots, x_n)$. The norm of $x$ is its length:
\[\norm{x} = \sqrt{x_1^2 + \dots + x_n^2}.\]Consider a vector space $V$ and a function $\norm{\cdot} : V \to [0, \infty)$. The function is a norm on $V$ if
- $\norm{x} = 0$ only for $x = 0$
- $\norm{c x} = \abs{c} \norm{x}$, and
- $\norm{x + y} \leq \norm{x} + \norm{y}$ (the triangle inequality).
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A random variable has a normal distribution if its values cluster symmetrically around a mean $\mu$, following a bell-shaped curve whose width is set by the standard deviation $\sigma$.
A random variable $X$ has a normal distribution with mean $\mu$ and variance $\sigma^2 > 0$, written $X \sim \mathcal{N}(\mu, \sigma^2)$, if $X$ has the probability density
\[f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\left(-\frac{(x - \mu)^2}{2 \sigma^2}\right).\] -
Consider a point on a curve. A normal vector at the point is a vector that is perpendicular to the curve at that point. More precisely, a vector $v$ is normal to a curve at $P$ if the line spanned by $v$ from $P$ is perpendicular to the tangent line of the curve at $P$.
Consider a smooth surface $S$ in $\reals^n$ and a point $p \in S$. A vector $n$ is called a normal vector to $S$ at $p$ if $n$ is orthogonal to every tangent vector of $S$ at $p$:
\[\ip{n}{v} = 0 \quad \text{for every tangent vector } v \text{ of } S \text{ at } p.\] -
Consider a set $A$ in a topological space, with its closure $\overline{A}$. The set $A$ is nowhere dense if the interior of its closure is empty:
\[\operatorname{int}(\overline{A}) = \emptyset.\]
O
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Consider a system whose starting state is unknown, but whose input and output you can see. The system is observable if you can figure out its starting state by watching its output (and knowing its input) for a while.
Consider a system with state, input, and output. The system is observable if its initial state is uniquely determined by its input and output over some finite time interval.
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Consider a system with some quantities that you can measure and others that you cannot. An observer is a calculation that estimates the quantities you cannot measure directly from the ones you can.
Consider a plant with input $u$, output $y$, and an unknown state $x(t)$. An observer is a dynamical system that takes $u$ and $y$ and produces a state estimate $\hat{x}(t)$ such that $\hat{x}(t) - x(t) \to 0$.
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Consider a center $c$ and a radius $r$. The open ball with center $c$ and radius $r$ is a ball without its surface: all of the points at distance strictly less than $r$ from $c$.
Consider a metric space with metric $d$, a center $c$, and a radius $r$. The open ball of radius $r$ around $c$ is the set of points at distance strictly less than $r$ from $c$:
\[B(c, r) = \setdef{x \suchthat d(x, c) < r}.\]It does not include the sphere $S(c, r)$.
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Think of a machine that is trying to reach a goal, like a toaster making toast. In open-loop control, the machine acts without checking the result.
Consider a controller that chooses the input to a system. In open-loop control, the controller chooses the input without measuring the result.
Consider a control system $\dot x = f(x, u)$. In open-loop control, the input $u$ is computed in advance, or from the reference alone, and does not depend on measurements of the system’s output.
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A function $f : \reals^n \to \reals^m$ is called open if the image of every open set $U$ in $\reals^n$ is an open set $f(U)$ in $\reals^m$.
Consider topological spaces $S$ and $T$. A function $f : S\to T$ is called open if the image of every open set $U$ in $S$ is an open set $f(U)$ in $T$.
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For all real numbers $a$ and $b$ such that $a < b$, the interval of numbers strictly between $a$ and $b$ is called an open interval, and is written $(a, b)$.
Consider a set $U$ of points on the number line, in the plane, or in space. The set $U$ is open if every point of $U$ has a small ball around it that lies entirely inside $U$.
Consider a set $U$ in a metric space. The set $U$ is open if, for every point $x \in U$, there is an $\varepsilon > 0$ such that
\[B(x, \varepsilon) \subseteq U,\]where $B(x, \varepsilon)$ is the open ball of radius $\varepsilon$ around $x$.
Consider a set $X$ with a topology $\mathcal{T}$. A set $U \subseteq X$ is open if $U \in \mathcal{T}$.
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An operator is a function that takes a function as its input and gives a function as its output.
An operator is a map from a vector space to a vector space, usually linear and often between spaces of functions.
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Optimal control is the study of choosing the inputs to a system over time to make the behavior of the system as good as possible, measured by a cost, while the system obeys its equations of motion.
Consider a control system $\dot{x} = f(x, u)$ with state $x(t) \in \reals^n$, input $u(t)$ in a set $U$ of allowed inputs, and initial state $x_0$. Optimal control is the study of problems of the form
\[\begin{aligned} \operatorname*{minimize}_{u(\cdot)} \quad & \varphi\bigl(x(T)\bigr) + \int_0^T \ell\bigl(x(t), u(t)\bigr) \, \mathrm{d}t \\ \text{subject to} \quad & \dot{x}(t) = f\bigl(x(t), u(t)\bigr) \\ & x(0) = x_0 \\ & u(t) \in U \text{ for all } t, \end{aligned}\]where the unknown is the input $u$ as a function of time.
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An ordinal number is a number that tells position of an item in a list, such the first, second, and third runner in a race.
An ordinal number is a number that tells position of an item in a list, such the first, second, and third.
An ordinal number is the order type of a well-ordered set. The finite ordinals are the natural numbers, followed by infinite ordinals $\omega, \omega + 1, \omega + 2, \dots$
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Consider a function $f$ of one variable. The derivative of $f$ is
\[f^{\prime}(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}:\]the slope of the graph of $f$ at $x$, or the rate at which $f$ changes.
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Two vectors in the plane or in space are orthogonal if the vectors meet at a right angle, which happens exactly when their dot product is $0$.
Consider an inner product space $V$. Two vectors $u, v \in V$ are orthogonal if
\[\ip{u}{v} = 0.\]A list of vectors is orthogonal if every two different vectors in the list are orthogonal.
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Consider a control system with state $x$ and measured output $y$. Output feedback is a control law that uses only $y$, either directly, as in $u = k(y)$, or through a dynamic controller.
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A list of vectors is pairwise linearly independent if every two vectors in the list point along different lines through the origin.
Consider a list $v_1, \dots, v_k$ of vectors in a vector space. The list is pairwise linearly independent if, for every two different indices $i$ and $j$, the two vectors $v_i$ and $v_j$ are linearly independent.
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Consider a function $f(x, y)$ of two variables. The partial derivative of $f$ with respect to $x$ is the derivative of $f$ found by treating $y$ as a constant.
Consider a function $f$ of several variables $x_1, \dots, x_n$. The partial derivative $\frac{\partial f}{\partial x_i}$ is the derivative of $f$ with respect to $x_i$ while the other variables are held fixed:
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A particle filter estimates a changing quantity from noisy measurements by keeping many guesses of it, called particles. At each step, it moves each guess according to a model of how the quantity changes, with some randomness, gives more weight to the guesses that agree better with the new measurement, and then draws a new set of guesses that favors the heavily weighted ones.
Consider a discrete-time system whose state $x_k$ evolves randomly according to a model $p(x_k \mid x_{k-1})$, and whose measured output $y_k$ has likelihood $p(y_k \mid x_k)$. A particle filter for this system is a filter that approximates the probability distribution of $x_k$, given the measurements $y_1, \dots, y_k$, by $N$ samples $x_k^1, \dots, x_k^N$, called particles, with weights $\omega_k^1, \dots, \omega_k^N$. At each step, it draws each particle $x_k^i$ from $p(x_k \mid x_{k-1}^i)$, multiplies each weight by $p(y_k \mid x_k^i)$, rescales the weights to sum to $1$, and, when a few particles carry most of the weight, resamples: it draws $N$ new particles from the old ones with probabilities equal to their weights, and gives each new particle the weight $1 / N$.
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Consider a function $f$ of time $t$, and a number $T > 0$. The function $f$ is periodic with period $T$ if
\[f(t + T) = f(t) \quad \text{for all } t.\]Periodic events happen at regular intervals of length $T$.
Consider a solution $x(t)$ of a dynamical system. The solution is periodic if, for some $T > 0$,
\[x(t + T) = x(t) \quad \text{for all } t.\] -
Two lines are perpendicular if the lines cross and make square corners where they cross.
Two lines in a plane or in space are perpendicular if the lines intersect and form a right angle at their intersection.
Shapes are perpendicular at at a point $P$ if they intersect at $P$ at a right angle.
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Consider a square matrix $A$. A PLU decomposition of $A$ is a factorization
\[A = P L U,\]where
- $P$ is a permutation matrix (the identity matrix with its rows reordered),
- $L$ is lower triangular with ones on its diagonal, and
- $U$ is upper triangular.
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A point is an exact location, drawn as a dot. It has no width or length.
Consider an affine space, which has locations but no chosen origin. A point is an element of an affine space. The difference between two points is a vector (the displacement between them), and adding a vector to a point gives a point, but two points cannot be added.
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Consider events that happen independently of each other at an average rate of $\lambda$ per unit of time, such as customers arriving at a shop. The number $X$ of events in one unit of time has a Poisson distribution with mean $\lambda$:
\[P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \quad \text{for } k = 0, 1, 2, \dots.\] -
Consider a fraction of polynomials, after canceling common factors. A pole of the fraction is a value where the denominator is zero, so that the function blows up near that point.
Consider a transfer function $G(s)$, a function of the complex variable $s$. A pole of $G$ is a value of $s$ at which $G(s)$ is infinite: for a rational function, a root of the denominator after common factors are canceled.
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A polynomial function is a function whose output is given by a polynomial in its input.
Consider a field $F$ and a function $f : F \to F$. The function $f$ is a polynomial function if, for some $a_0, \dots, a_n \in F$,
\[f(x) = a_0 + a_1 x + \dots + a_n x^n \quad \text{for all } x \in F.\]Two polynomial functions are equal when they give the same output for every input.
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Consider a function $V : D \to \reals$ whose domain $D$ contains $0$. The function $V$ is positive definite if $V(0) = 0$ and $V(x) > 0$ for all $x \neq 0$ in $D$.
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Consider an unknown quantity, such as the chance that a coin lands heads, and a prior belief about the quantity. The posterior is a description of what you believe about the quantity after you see the data, found by updating the prior with the data. In the case of the coin, you may have a prior belief that the coin is fair: It lands on heads 50% of the time. But if you flip the coin 1000 times and it comes up heads 900 times, your posterior belief may become that the probability of the coin landing on heads is around 90%.
Consider a statistical model $p(x \mid \theta)$ with parameter $\theta$, a prior $p(\theta)$, and observed data $x$. The posterior is the conditional probability distribution of the parameter given the data, which is given by Bayes’ rule:
\[p(\theta \mid x) = \frac{p(x \mid \theta) \, p(\theta)}{p(x)}.\] -
A postulate is a statement accepted without proof as a starting point, especially in geometry, as in Euclid’s postulates.
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Consider a true/false statement with a variable in it, such as “$x$ is even.” The statement is a predicate: it becomes true or false once a value is chosen for the variable.
Consider a set $X$ of objects. A predicate on $X$ is a function $P$ that assigns to each $x \in X$ a proposition $P(x)$; for example, the predicate “$x$ is prime” assigns to $7$ the proposition “$7$ is prime.” Each proposition is either true or false, so the predicate $P$ determines the function
\[P : X \to \setdef{\text{true}, \text{false}}\]that sends each $x$ to the truth value of $P(x)$. Logic books usually identify the predicate with this function.
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Consider a quantity that changes over time, and the data about it available now. A predictor estimates a future value of the quantity from that data.
Consider a system with state $x(t)$, measured up to time $t$, and a time $\tau > 0$ into the future. A predictor estimates the future state $x(t + \tau)$ from the measurements up to time $t$.
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Consider a function $f$ and a set $B$ of possible outputs. The preimage of $B$ under $f$ is the set of all inputs $x$ whose output $f(x)$ is in $B$.
Consider a function $f : X \to Y$ and a set $B \subseteq Y$. The preimage of $B$ is
\[f^{-1}(B) = \setdef{x \in X \suchthat f(x) \in B}.\] -
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.
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Consider an unknown quantity, such as the chance that a coin lands heads. The prior is a description of what you believe about the quantity before you see the data, given by a probability for each possible value. In the case of the coin, you may have a prior belief that the coin is fair: It lands on heads 50% of the time.
Consider a statistical model $p(x \mid \theta)$ of data $x$ for each value of a parameter $\theta$. A prior is a probability distribution $p(\theta)$ on the parameter that is chosen before the data are observed.
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The probability of something happening is a number from $0$ to $1$ that says how likely it is: $0$ means it cannot happen, and $1$ means it is certain.
Consider a random experiment, and collect every possible outcome of the experiment into a set, called the sample space. Think of the probability as a way of measuring the size of sets of outcomes, like area measures the size of regions, such that the whole sample space has size $1$. An event is a set of outcomes, and the probability of an event is the size of the event. For example, if you roll a fair die, then the sample space is $\setdef{1, 2, 3, 4, 5, 6}$, each outcome has size $\tfrac{1}{6}$, and the probability of the event “the roll is even,” which is $\setdef{2, 4, 6}$, is the size $\tfrac{1}{6} + \tfrac{1}{6} + \tfrac{1}{6} = \tfrac{1}{2}$.
Consider a probability space $(\Omega, \mathcal{F}, P)$. The probability of an event $A \in \mathcal{F}$ is the number $P(A)$ that the probability measure $P$ assigns to it.
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Consider a random variable $X$ with real values. A function $f \geq 0$ is a probability density of $X$ if probabilities are areas under the graph of $f$:
\[P(a \leq X \leq b) = \int_a^b f(x) \mathrm{d}x \quad \text{for all } a \leq b.\] -
Consider an event in a random experiment. The event has probability zero if its probability is $0$.
Consider a probability space with probability measure $P$, and an event $E$. The event $E$ has probability zero if $P(E) = 0$.
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Think of two sets, $A$ and $B$, where everything in $A$ is also in $B$. The set $A$ is a proper subset of $B$ if $B$ has at least one thing that $A$ does not.
Consider two sets $A$ and $B$ such that $A \subset B$. The set $A$ is a proper subset of $B$ if $A \neq B$, so $B$ has at least one element that is not in $A$.
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A proposition is a statement that is either true or false.
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A propositional formula is an expression built from propositional variables $p, q, r, \dots$ using logical connectives such as $\neg$, $\land$, $\lor$, and $\implies$. It has a truth value only once each variable is assigned true or false.
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Consider an $m \times n$ matrix $A$. A QR decomposition of $A$ is a factorization $A = Q R$, where the columns of $Q$ are orthonormal and $R$ is upper triangular.
R
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A random variable is a number whose value depends on the outcome of a random experiment, such as the number shown by a rolled die.
Consider a probability space $(\Omega, \mathcal{F}, P)$. A random variable is a measurable function $X : \Omega \to \reals$. For each outcome $\omega$ of an experiment, $X$ produces a real number $X(\omega).$
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A rational number is a number that can be written as a fraction of whole numbers, or the negative of one.
A rational number is a number that can be written as $p/q$, where $p$ and $q$ are integers and $q \neq 0$.
A rational number is an element of $\mathbb{Q}$, constructed as equivalence classes of pairs $(p, q)$ of integers such that $q \neq 0$, where $(p, q) \sim (r, s)$ if $p s = q r$.
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Consider a control system with an input that you choose. The system is reachable if you can steer it from the origin to every state you choose in a finite amount of time.
Consider a system with state $x$ and input $u$, starting from the origin. The system is reachable if every state can be reached in finite time by some input.
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A real number is any number on the number line. Every real number can be written as a (possibly infinite) decimal.
A real number is an element of $\reals$, the unique complete ordered field.
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Consider a system with state, input, and output, observed up to the present time. The system is reconstructible if its current state is uniquely determined by its past input and output over some finite time interval.
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A relation is a rule that determines whether whether two objects are related.
Consider sets $X$ and $Y$ and their product $X \times Y$. A binary relation between $X$ and $Y$ is a subset $R \subseteq X \times Y$. For $x \in X$ and $y \in Y$, the statement $x \mathrel{R} y$ (“$x$ is related to $y$”) is true if $(x, y) \in R$ and false if $(x, y) \notin R$.
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A right angle is an angle that is a quarter of a full turn, like the corner of a square or of a sheet of paper.
A right angle is an angle that measures
\[90^\circ = \frac{\pi}{2} \text{ radians}.\] -
Consider a function $f$ and a number $a$, and let $x$ approach $a$ from above (from the right). The right limit of $f$ at $a$, written $\lim_{x \to a^{+}} f(x)$, is the number that $f(x)$ approaches.
Consider a function $f$ defined on an interval to the right of $a$. We write $\lim_{x \to a^{+}} f(x) = L$, and call $L$ the right limit of $f$ at $a$, if for every $\varepsilon > 0$ there is a $\delta > 0$ such that
\[\abs{f(x) - L} < \varepsilon \quad \text{whenever } a < x < a + \delta.\]
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Consider a square real matrix $A$ whose eigenvalues are all real. A Schur decomposition of $A$ is a factorization
\[A = Q T Q^{\top},\]where $Q$ is orthogonal and $T$ is upper triangular.
Consider a square complex matrix $A$. A Schur decomposition of $A$ is a factorization $A = Q T Q^$, where $Q$ is unitary ($Q^ Q = I$) and $T$ is upper triangular.
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Consider a formula of first-order logic, such as $\forall x \, (x^2 \geq 0)$ or $x > 3$. The formula is a sentence if it has no free variables: every variable in it is bound by a quantifier.
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A sequence is a list of things in a particular order.
A sequence is an ordered list of terms $a_1, a_2, a_3, \dots$ Both order and repetition matter: $(1, 2)$, $(2, 1)$, and $(1, 1, 2)$ are different sequences.
A sequence is a function from $\nats$ (an infinite sequence) or from $\setdef{1, \dots, n}$ (a finite sequence, or $n$-tuple) to a set.
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A set is a collection of different objects, called its elements. Order and repetition do not matter: $\setdef{1, 2}$, $\setdef{2, 1}$, and $\setdef{1, 1, 2}$ are the same set.
A set is an object of a set theory such as ZFC. By the axiom of extensionality, a set is determined entirely by its elements.
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Consider1 two sets $A$ and $B$. The set difference $A \setminus B$ is the set of elements of $A$ that are not in $B$.
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Consider an array of numbers, which has some length in each direction. The shape of the array is the list of those lengths.
Consider an array with one or more axes. The shape of the array is the tuple of its lengths along each axis.
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Consider an $m \times n$ real matrix $A$. The singular values of $A$ are the square roots of the eigenvalues of the $n \times n$ matrix $A^{\top} A$, which are nonnegative.
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Consider an $m \times n$ real matrix $A$. A singular value decomposition of $A$ is a factorization
\[A = U \Sigma V^{\top},\]where $U$ and $V$ are orthogonal matrices and $\Sigma$ is an $m \times n$ diagonal matrix whose diagonal entries $\sigma_1 \geq \sigma_2 \geq \dots \geq 0$ are the singular values of $A$.
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Consider two stable systems $H_1$ and $H_2$ connected in a feedback loop, so that the output of each system is the input of the other. The small gain theorem says that if the loop gain is less than $1$, that is,
\[\gamma_1 \gamma_2 < 1,\]where $\gamma_1$ and $\gamma_2$ are the gains of $H_1$ and $H_2$, then the feedback loop is stable.
Consider two input-to-state stable systems connected in feedback, whose states $x_1$ and $x_2$ satisfy ISS bounds with gain functions $\gamma_1$ and $\gamma_2$ of class $\mathcal{K}$, where $x_2$ is the input of the first system and $x_1$ is the input of the second system. The nonlinear small gain theorem says that the interconnection is globally asymptotically stable if
\[\gamma_1\bigl(\gamma_2(r)\bigr) < r \quad \text{for every } r > 0.\] -
Consider a function, its derivative, the derivative of that, and so on. The function is smooth if you can take as many derivatives of it as you like, and each derivative is continuous.
Consider a function on an open set, and its derivatives of each order. The function is smooth if it has continuous derivatives of every order.
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Consider a list of vectors $v_1, \dots, v_k$. The span of the vectors is the set of all vectors that can be built from them by stretching and adding them.
Consider vectors $v_1, \dots, v_k$ in a vector space $V$. The span of the vectors is the set of their linear combinations:
\[\operatorname{span}(v_1, \dots, v_k) = \setdef{c_1 v_1 + \dots + c_k v_k \suchthat c_1, \dots, c_k \text{ scalars}}.\]The vectors span $V$ if
\[\operatorname{span}(v_1, \dots, v_k) = V.\] -
Think of a ball, like a basketball. A sphere is the surface of the ball, like the skin of the basketball. A sphere is hollow: it does not include the inside.
Consider a point $c$ in space (the center) and a distance $r$ (the radius). The sphere with center $c$ and radius $r$ is the set of all points in space at distance $r$ from $c$.
Consider a metric space with metric $d$, a center $c$, and a radius $r$. The sphere of radius $r$ around $c$ is
\[S(c, r) = \setdef{x \suchthat d(x, c) = r}.\]In $\reals^{n+1}$ with the Euclidean metric, the unit sphere is written $S^n$; it is $n$-dimensional.
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Consider a system that does not stay at a desired resting state by itself, like a broom balanced on your hand. Stabilization is designing a controller that makes the system settle at that resting state and stay near it.
Consider a control system with an equilibrium (or a set) that is to be made stable. Stabilization is the control problem of finding a feedback law that makes the equilibrium of the closed-loop system asymptotically stable.
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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:
\[u(t) = k(y(t)).\] -
Consider two events, such as “the first coin lands heads” and “the second coin lands heads.” The events are independent if the probability of each event is the same whether or not the other event occurs.
Consider a probability space $(\Omega, \mathcal{F}, P)$. Two events $A$ and $B$ are independent if
\[P(A \cap B) = P(A) \, P(B).\]Two random variables $X$ and $Y$ are independent if the events $\setdef{X \in A}$ and $\setdef{Y \in B}$ are independent for every pair of Borel sets $A$ and $B$.
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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 $\dot{x} = f_{p}(x)$, $p \in \setdef{1, \dots, m}$, together with a switching signal $\sigma(t)$ that selects the active one:
\[\dot{x} = f_{\sigma(t)}(x).\] -
A square matrix $A$ is called symmetric if $A$ is unchanged when flipped across the main diagonal: for every $i$ and $j$, the entry in row $i$ and column $j$ of $A$ equals the entry in row $j$ and column $i$ of $A$.
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A relation $R$ is symmetric if $x \mathrel{R} y$ implies $y \mathrel{R} x$.
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Consider two points $p = (p_1, \dots, p_n)$ and $q = (q_1, \dots, q_n)$ in $\reals^n$. The taxicab distance between $p$ and $q$ is the distance traveled from $p$ to $q$ when moving parallel to one axis at a time, as along the streets of a grid:
\[d_1(p, q) = \abs{p_1 - q_1} + \dots + \abs{p_n - q_n}.\]Consider two points $p = (p_1, \dots, p_n)$ and $q = (q_1, \dots, q_n)$ in $\reals^n$. The taxicab distance between $p$ and $q$ is defined as
\[d_1(p, q) = \sum_{i=1}^n \abs{p_i - q_i}.\] -
Consider a first-order language, with variables, constant symbols, and function symbols. A term is a string of symbols built by the rules of the language that names an object, such as $x$, $3$, or $x^2 + 1$. Terms are built in two steps:
- every variable and every constant symbol is a term, and
- if $f$ is a function symbol with $n$ inputs and $t_1, \dots, t_n$ are terms, then $f(t_1, \dots, t_n)$ is a term.
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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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A scientific theory is a well-tested explanation of a broad range of observations, supported by a large body of evidence and used to make predictions.
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Consider a square real matrix $A$. The Toeplitz decomposition of $A$ is a sum
\[A = H + K\]where $H$ is a symmetric matrix and a $K$ is skew-symmetric.
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A topology on a set is a choice of which subsets count as “open.” It describes which points are near which, without requiring a definition of “distance”.
Consider a set $X$ and a collection $\mathcal{T}$ of subsets of $X$. The collection $\mathcal{T}$ is a topology on $X$ if
- $\emptyset$ and $X$ are in $\mathcal{T}$,
- every union of sets in $\mathcal{T}$ is in $\mathcal{T}$, and
- every intersection of finitely many sets in $\mathcal{T}$ is in $\mathcal{T}$.
Each set in a topology is defined as an open set.
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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.
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Consider a topological space $X$. The space $X$ is T₁ if every point of $X$ is a closed set.
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Consider an infinite set $S$. The set is uncountable if it is impossible to list the elements of $S$ in an (infinite) list $a_1, a_2, a_3, \dots$. The set is so large that every such list must leave out some elements of $S$.
Consider an infinite set $S$. The set $S$ is uncountable if there does not exist a surjection from $\nats$ onto $S$.
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Consider a quantity that varies over a region, such as the density of a material or the strength of an electric field. The quantity is uniform if it has the same value at every point of the region.
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Consider a function $f$ and a sequence of functions $f_n$ on a set $X$. The sequence $f_n$ converges uniformly to $f$ if, for every $\varepsilon > 0$, there exists $N \in \nats$ such that
\[\abs{f_n(x) - f(x)} < \varepsilon \quad \text{for all } n \geq N \text{ and all } x \in X.\] -
Consider a random experiment with outcomes in an interval $[a, b]$ (or in a finite set). The outcome has a uniform distribution if all outcomes are equally likely: for an interval, the probability of landing in a subinterval is proportional to its length,
\[P(c \leq X \leq d) = \frac{d - c}{b - a} \quad \text{for } a \leq c \leq d \leq b.\] -
A function $f : \reals \to \reals$ is uniformly continuous if for every $\varepsilon > 0$, there exists $\delta > 0$ such that
\[\abs{f(x) - f(y)} < \varepsilon \quad \text{for all } \abs{x - y} < \delta.\]The choice of $\delta$ depends only on $\varepsilon$, so the same $\delta$ works at every point.
Consider a function $f : X \to Y$ between metric spaces. The function $f$ is uniformly continuous if for every $\varepsilon > 0$, there exists $\delta > 0$ such that
\[d(f(x), f(y)) < \varepsilon \quad \text{for all } x, y \in X \text{ with } d(x, y) < \delta.\]The choice of $\delta$ depends only on $\varepsilon$, so the same $\delta$ works at every point.
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Consider a time-varying system $\dot{x} = f(t, x)$ with an equilibrium at $0$, and its solutions $x(t)$ starting at $x(t_0)$ at the initial time $t_0$. The equilibrium is uniformly stable if, for every $\varepsilon > 0$, there is a $\delta > 0$, independent of $t_0$, such that
\[\norm{x(t_0)} < \delta \implies \norm{x(t)} < \varepsilon \quad \text{for all } t \geq t_0 \text{ and every } t_0 \geq 0.\] -
Consider two sets $A$ and $B$. The union of $A$ and $B$, written $A \cup B$, is the set of every element that is in $A$, in $B$, or in both.
Consider two sets $A$ and $B$. The union of sets $A$ and $B$ is
\[A \cup B = \setdef{x \suchthat x \in A \text{ or } x \in B}.\]More generally, the union of a family of sets \((A_i)_{i \in I}\) is the set of all $x$ such that $x \in A_i$ for some $i \in I$.
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A unit is a standard amount used for measuring, such as “1 second” which gives a standard way to measure time.
Consider a physical quantity of a given dimension, such as a length. A unit is a fixed quantity of that dimension, used as a reference. A measurement is a number times a unit.
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A unit vector is a vector of length $1$, such as $(1, 0)$ or $\left(\frac{3}{5}, \frac{4}{5}\right)$ in the plane.
Consider a vector space $V$ with a norm $\norm{\cdot}$. A unit vector is a vector $u \in V$ such that
\[\norm{u} = 1.\] -
An unscented Kalman filter is a version of the Kalman filter for systems that are not linear. Instead of approximating the system by a linear one, it represents the current estimate and its uncertainty by a few carefully chosen sample points, passes each point through the system’s model, and computes the new estimate and uncertainty from the results.
Consider a discrete-time system with state $x_k \in \reals^n$, input $u_k$, and measured output $y_k$, given by
\[x_{k+1} = f(x_k, u_k) + w_k, \qquad y_k = h(x_k) + v_k,\]where the noises $w_k$ and $v_k$ are independent, zero-mean, and white, with covariances $Q$ and $R$. The unscented Kalman filter for this system is the filter that alternates prediction and correction steps like the Kalman filter, but computes each mean and covariance by the unscented transform. To pass a mean $\hat{x}$ and covariance $P$ through a function $g$, the unscented transform chooses $2 n + 1$ points, called sigma points, whose weighted mean is $\hat{x}$ and whose weighted covariance is $P$, applies $g$ to each point, and returns the weighted mean and covariance of the results.
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.
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Consider a control system $\dot x = f(x, u)$ and a set of state constraints $K$. The viability kernel of $K$ is a subset of $V \subset K$ such that for every initial state $x_0 \in V$, there exists at least one choice of admissible inputs $u(t)$ such that a solution starting from $x_0$ and controlled by $u(t)$ stays in $K$ forever.
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Think of a solid object, like a box or a ball. The volume of the solid is the amount of space it takes up, measured in cubic units.
Consider a region $R \subseteq \reals^n$. The volume of $R$ is its three-dimensional (or $n$-dimensional) size:
\[\operatorname{vol}(R) = \int_R 1 \mathrm{d}x.\]
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.
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Consider a fraction of polynomials, after canceling common factors. A zero of the fraction is a value where the numerator is zero, so that the function is zero.
Consider a transfer function $G(s)$, a function of the complex variable $s$. A zero of $G$ is a value of $s$ at which $G(s) = 0$: for a rational function, a root of the numerator after common factors are canceled.
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Think of vectors as arrows, which can be long or short. The zero vector is the arrow with length zero. It does not point in any direction and adding it using the tip-to-tail rule does not change the vector it is added to.
The zero vector is the vector whose components are all zero.
Consider a vector space $V$, with its vector addition. The zero vector of $V$ is its additive identity: the vector $\mathbf{0}$ with
\[v + \mathbf{0} = v \quad \text{for every } v \in V.\]In a space of functions, it is the function that is $0$ everywhere.
Σ
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A σ-algebra is a collection of the subsets of a set that can be measured, such as the events whose probabilities are defined. If a set can be measured, then so can everything outside it, and so can every combination of countably many measurable sets.
No definitions match these filters.