Dictionary of Fine (Mathematical) Distinctions
In mathematics, many terms are similar but not interchangeable. This dictionary groups easily confused terms, defines each one, and explains the distinction with examples to illustrate the difference. This work was inspired by the Dictionary of Fine Distinctions: Nuances, Niceties, and Subtle Shades of Meaning by Eli Burnstein.
- Distinctions: groups of easily confused terms, with the differences between them.
- Definitions: definitions of each term at several levels, from elementary to graduate.
- Examples: examples and non-examples of the terms.
Distinctions
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Grids of numbers appear throughout linear algebra and computing, but a grid can be just data, a matrix with its own arithmetic, or a way of writing a linear map. An array is a grid of data. A matrix is a grid of numbers with matrix arithmetic. A linear map is a function between vector spaces. A matrix represents a linear map as an array of scalars with respect to a given vector basis.
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Consider a binary operation, such as addition, that combines two inputs, such as two numbers, into one result. Associativity and commutativity both involve changing an order, but they change the order of different things. An associative operation gives the same result when the order in which the operations are performed changes, while the inputs stay in place, as in $(a \ast b) \ast c = a \ast (b \ast c)$: on the left, $a \ast b$ is computed first; on the right, $b \ast c$ is computed first. A commutative operation gives the same result when the order of the inputs changes, as in $a \ast b = b \ast a$: $a$ and $b$ trade places. An operation can have either property without the other.
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The calculus of variations and optimal control seek a function of time that minimizes a cost given by an integral. In the calculus of variations, the unknown is a curve $y(t)$, and the cost depends on $y$ and its derivative $y^{\prime}$, which can be chosen freely. In optimal control, the unknown is an input $u(t)$, which may be constrained to a set $U$, and the state $x(t)$ is determined by the input through the dynamics $\dot{x} = f(x, u)$.
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Lipschitz continuity and continuous differentiability both rule out functions that are too steep or too wild, but in different ways. A Lipschitz continuous function has a bounded slope but may have corners. A continuously differentiable function has no corners, but its slope may be unbounded on an unbounded domain.
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“Convex” describes both sets and functions, and the two meanings are linked. A convex set contains the segment between every two of its points, and a convex function lies on or below the segment between every two points on its graph. A function is convex exactly when the region on and above its graph is a convex set.
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For a function of a real variable, the derivative $f^{\prime}(x)$ measures the slope of the graph at each point $x$. A differentiable function has a derivative at every point, but the derivative can oscillate wildly as $x$ changes causing the derivative to be discontinuous. A continuously differentiable function has a derivative at every point, and the derivative is also a continuous function of $x$.
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“Disjoint union” can mean an ordinary union of sets that happen to be disjoint, or a construction that tags each element with the set it came from, so that copies of overlapping sets are kept apart.
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Bilinear forms, inner products, and the dot product are all ways of multiplying two vectors to get a number, from the most general to the most specific. The dot product is one particular inner product on $\reals^n$. An inner product is a bilinear form that is also symmetric and positive definite.
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In first-order logic, statements about objects are written as strings of symbols with variables and quantifiers. A formula is every such string that is built by the rules of the language, such as $x > 3$ or $\forall x \, (x^2 \geq 0)$, and the variables in a formula can be free or bound by quantifiers. A sentence is a formula in which a quantifier binds every variable, so a sentence is true or false once its symbols are interpreted, without assigning values to variables.
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“Function” and “map” (or “mapping”) mean exactly the same thing. “Map” is more common when the function preserves some structure or acts on geometric objects, but every statement about functions holds for maps, and vice versa.
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Mathematics and science both have words for statements that are proposed, tested, or established, and the words are often confused. A conjecture is a statement believed true but not yet proved, and it belongs to mathematics, where statements are settled by proof. In science, where statements are tested by evidence, a hypothesis is a testable proposal, and a theory is a well-supported explanation of many observations. A law describes a pattern that holds in every tested case, often as an equation such as $F = G m_1 m_2 / r^2$, without explaining why the pattern holds; explaining it is the job of a theory. In everyday speech, “theory” means a guess, but in science a theory is an a principle or body of principles that explains observed data and phenomenon.
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A list of vectors can point in “different directions” in two senses. A set of linearly independent vectors have no redundant vectors: no vector in the list is a linear combination of the others. Orthogonal vectors are pairwise at right angles: every two different vectors in the list have inner product $0$. Linear independence is defined in every vector space, but orthogonality requires an inner product.
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A list of vectors can avoid redundancy as a whole or one pair at a time. A list of linearly independent vectors has no vector that is a linear combination of the others, which rules out redundancy among three or more vectors as well as among pairs. A list of pairwise linearly independent (pairwise non-parallel) vectors has no two vectors on the same line through the origin.
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Whether the natural numbers include $0$ depends on the author. Set theory, logic, and computer science usually include it, while number theory and analysis often start at $1$.
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“Whole numbers” and “natural numbers” are used synonymously by some others to indicate all nonnegative integers, $0, 1, 2, 3, \dots$. “Whole numbers” is the usual name in primary schools, and “natural numbers” (or “nonnegative integers”) is the usual name in mathematics. “Natural numbers” is, however, used by some authors to indicate the non-negative integers, $1, 2, 3, \dots$ and “whole numbers” is used to indicate all integers, “$\dots, -2, -1, 0, 1, 2, \dots$.
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Norms, metrics, and inner products measure size, distance, and angle, and each one gives the next in a natural way. A norm measures the lengths of vectors, a metric measures distances between points, and an inner product measures both lengths and angles. An inner product gives a norm, and a norm gives a metric, but not the reverse.
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Observability, reconstructibility, and detectability describe how much the output of a system reveals about its state. An observable system’s initial state can be determined from its output, whereas a reconstructible system’s current state can be determined. A detectable system may have unobservable parts but those parts must that decay to zero so that an observer’s error still converges.
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A controller can decide its input with or without measuring what the system is doing. Open-loop control computes the input without measuring the output, and closed-loop (feedback) control uses the measured output to correct errors. Feedforward acts on a measured disturbance or a known reference before it affects the output, and is usually combined with feedback.
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In a dynamical system, a solution can return to its starting state and go around the same loop again and again. A periodic orbit is such a loop. A limit cycle is a periodic orbit that is isolated, with no other periodic orbits nearby, so that nearby solutions spiral toward the limit cycle or away from it.
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Consider events that happen independently of each other at a constant average rate $\lambda$, such as customers arriving at a shop (a Poisson process). The Poisson distribution describes the number of events in a fixed length of time, which is a whole number. The exponential distribution describes the time until the next event, which is a positive real number.
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Positive definiteness and semidefiniteness describe the sign of the quadratic form $x^{\top} A x$ of a symmetric matrix, and they appear in optimization, stability, and statistics. A symmetric matrix $A$ is positive semidefinite if $x^{\top} A x \geq 0$ for every vector $x$, and positive definite if $x^{\top} A x > 0$ for every nonzero $x$. Equivalently, the eigenvalues of $A$ are nonnegative or positive.
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In logic, statements are classified by whether they include enough information to determine whether they are true or false. A proposition is a statement that is true or false, such as “$7 > 3$.” A predicate is a statement with variables, such as “$x > 3$,” that becomes a proposition once its variable $x$ is given a value, such as “$7 > 3$” (true) or bound by quantifiers, such as “$\forall x,\, x > 3$” (false).
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In logic, statements are combined with logical connectives such as “and” ($\land$), “or” ($\lor$), and “not” ($\neg$). A proposition is a particular statement that is true or false, such as “$7$ is prime.” A propositional formula is a pattern, such as $p \land \neg q$, built from propositional variables that stand for propositions, and the formula has a truth value only once each variable is assigned a truth value. Substituting a proposition for each variable in a propositional formula gives a proposition.
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A feedback controller is designed to meet a goal, and the goals have different names. Stabilization makes an equilibrium asymptotically stable. Regulation holds an output at a constant set point despite disturbances. Tracking makes an output follow a reference that changes over time.
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Symbols placed between two objects, such as $+$ and $<$, do different kinds of things. A binary operation takes two objects and produces a new object of the same type. A binary relation takes two objects and says whether they are related (true or false).
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Right angles, perpendicular lines, orthogonal vectors, and normal vectors all describe things that meet “squarely.” A right angle is an angle of $90^\circ$, so the term describes an angle. Perpendicular describes lines or planes that intersect and form right angles. Orthogonal describes vectors whose inner product is $0$, which extends the idea to vectors that do not meet, such as directions of skew lines, to the zero vector, and to vectors that are not arrows, such as functions. A normal vector is perpendicular to a line, curve, or surface at a point, so the term describes a vector relative to a shape.
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In logic, a statement can be studied as a string of symbols or as a claim that is true or false. A sentence is a formula of first-order logic, a string of symbols in which a quantifier binds every variable. A proposition is a claim that is true or false. A sentence expresses a proposition only once its symbols are interpreted, for example by choosing the set that the variables range over, and the same sentence can express different propositions under different interpretations.
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Formal logic has formulas at two levels: propositional logic (zeroth-order logic), whose variables stand for whole statements, and first-order logic, whose variables stand for objects such as numbers. A sentence is a formula of first-order logic in which a quantifier, such as “for all” ($\forall$) or “there exists” ($\exists$), binds every variable, and the sentence is true or false once its symbols are interpreted, for example as statements about real numbers. A propositional formula is a formula of propositional logic, built from propositional variables with logical connectives such as $\land$ and $\neg$, and the formula is true or false once each variable is assigned a truth value.
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Eigenvalues and singular values both describe how a matrix stretches vectors, but they agree only for special matrices. An eigenvalue of a square matrix $A$ is a factor $\lambda$ by which $A$ scales a particular nonzero vector $x$, called an eigenvector: $Ax = \lambda x$. A singular value of a matrix $A$, square or not, is a factor $\sigma$ by which $A$ stretches vectors along some direction, allowing them to turn. Singular values are real and nonnegative, but eigenvalues can be negative or complex.
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A feedback controller computes its input from information about the system, which may be more or less complete. State feedback uses the full state, which is rarely measured directly. Output feedback uses only the measured output, and it may need an observer or other dynamics to do what state feedback can.
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In logic, the expressions of a formal language, which are the well-formed strings of symbols, come in two kinds. A term, such as $x^2 + 1$, names an object, so a term has a value, such as a number. A formula, such as $x^2 + 1 > 3$, makes a claim about objects, so a formula has a truth value. Applying a relation symbol, such as $>$ or $=$, to terms gives a formula.
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Mathematics has several kinds of “a bunch of things,” which differ in whether order and repetition matter. A set ignores order and repetition. A multiset keeps repetition. A sequence keeps order and repetition. “Collection” is informal, and “group” is a technical term for a set with an operation.
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There are many ways to describe the continuity of a function. On a compact interval, each is a special case of the next, from continuously differentiable to continuous.
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Many quantities compare two things by “how far apart” they are. A difference is the result of subtraction and can be negative. No distance is negative. An error compares an approximation with the truth, a change compares the end with the start, and a set difference removes elements.
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A metric measures how far apart two points are, and the same two points can be at different distances under different metrics. The Euclidean distance is the straight-line distance, the taxicab distance travels along the axes, and the Chebyshev distance takes the largest difference in one coordinate. The geodesic distance is the length of the shortest path that stays on a surface, such as the great-circle distance on the Earth. The discrete metric only says whether two points are equal.
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A control system is affected by signals that it does not generate itself, and these signals have different names depending on their role. Exogenous inputs are any inputs from outside the system. A control input is chosen; a disturbance acts on the dynamics; measurement noise only corrupts what is measured. A perturbation is a change to the system itself, which may or may not be modeled as an input.
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Infinity appears in several different senses: as the size of a set, as unboundedness, and as the limit of a divergent sequence. An infinite set has infinitely many elements, but it can be bounded. Infinite sets come in different sizes: countable and uncountable. A divergent sequence does not converge, but it doesn’t necessarily go to infinity.
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A matrix decomposition writes a matrix in terms of simpler matrices. Most are products: LU, PLU, and Cholesky decompositions use triangular factors, QR and singular value decompositions use orthogonal factors, and eigendecompositions, Schur decompositions, and Jordan normal forms are similarities $A = P M P^{-1}$ to a simpler matrix $M$ (diagonal, triangular, or made of Jordan blocks). In contrast, the Jordan–Chevalley decomposition is a sum of commuting diagonalizable and nilpotent parts, and the Toeplitz decomposition is a sum $H + K$ of a symmetric matrix $H$ and a skew-symmetric matrix $K$.
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A norm measures the size of a vector, and $\reals^n$ has several common norms that measure size in different ways. The Euclidean norm is the straight-line length, the 1-norm adds the sizes of the entries, and the infinity norm takes the largest entry. Each is a norm, so each satisfies the triangle inequality, and for every vector, $\norm{x}_{\infty} \leq \norm{x}_2 \leq \norm{x}_1$.
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Mathematics has many ways to say that two objects are “the same.” Equal objects are the same object. The other words describe objects that are different but “the same in some respect,” and each one depends on what respect is meant.
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The directional derivative, the Jacobian, the gradient, the differential, and the Lie derivative all package the partial derivatives of a function differently: as a vector (gradient), a matrix (Jacobian), a linear map (differential), or a single number in a chosen direction (directional derivative) or along a vector field (Lie derivative).
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Several terms describe a set that is “small” or “negligible,” even when it has infinitely many points. “Nowhere dense” means small in a topological sense (and a meager set is a countable union of nowhere dense sets), and “measure zero” means small in a size sense. Neither implies the other. An event of probability zero is a set of measure zero for a probability measure.
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When the state of a system is not measured directly, it can be estimated in several ways. An estimator is any rule for estimating an unknown quantity. An observer is a dynamical system whose state estimate converges to the true state. A filter estimates the current state from noisy measurements, and a predictor estimates a future state.
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The mean, median, and mode are different ways to summarize the middle of data, and they can be far apart. “Average” usually means the mean. The geometric mean averages growth factors and ratios, and the expected value is the mean of a probability distribution rather than of data.
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A function from $\reals^n$ to $\reals^m$ has several derivatives in vector calculus, each combining its partial derivatives differently. The Jacobian of a vector field contains all of its first partial derivatives. The divergence (a scalar) and curl (a vector) summarize parts of it. The Laplacian is a second-order operator on scalar functions.
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The spread of data is measured by the average squared distance of the data from their mean. The variance of the data divides the sum of the squared distances by the number $n$ of data points, which describes the data themselves. The sample variance divides by $n - 1$ instead, which gives an unbiased estimate of the variance of the population that the data were sampled from.
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Points and vectors are both written as lists of coordinates in $\reals^n$, but they behave differently. A point is a location. A vector is a displacement, defined by the operations of addition and scaling. The difference of two points is a vector. Two vectors can be added, and a vector can be added to a point, but the sum of two points is not meaningful.
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Vectors in a vector space are often described by lists of numbers, which depend on a choice of basis. A vector is an element of a vector space. Its coordinate vector is a list of numbers (coordinates) that describes it relative to a chosen basis. Changing the basis changes the coordinates, not the vector.
Definitions
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Lipschitz continuous High schoolUndergraduate
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T₁ Graduate
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absolutely continuous UndergraduateGraduate
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addition ElementaryUndergraduate
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admissible set Graduate
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algebraic number High schoolUndergraduate
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aperiodic ElementaryHigh schoolUndergraduate
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append ElementaryUndergraduate
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arc length ElementaryHigh schoolUndergraduate
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area ElementaryHigh school
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array ElementaryHigh schoolUndergraduate
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shape High schoolUndergraduate
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associative ElementaryUndergraduate
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asymptotically equivalent High schoolUndergraduate
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asymptotically stable ElementaryHigh schoolUndergraduate
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average Elementary
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axiom High schoolUndergraduate
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ball High schoolUndergraduate
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Banach space Graduate
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Bernoulli distribution High school
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bilinear form Undergraduate
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binary operation ElementaryUndergraduate
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binomial distribution High school
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boundary ElementaryUndergraduateGraduate
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bounded ElementaryHigh schoolUndergraduate
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bug Elementary
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calculus of variations High schoolUndergraduate
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cardinal number ElementaryGraduate
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cardinality ElementaryHigh schoolGraduate
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cascade control High schoolUndergraduate
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change ElementaryHigh school
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chart Elementary
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Chebyshev distance High school
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Cholesky decomposition Undergraduate
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circle ElementaryHigh school
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class-$\mathcal{K}$ Graduate
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class-$\mathcal{K}_\infty$ Graduate
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closed loop ElementaryHigh schoolUndergraduate
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closed set ElementaryHigh schoolUndergraduateGraduate
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collection ElementaryGraduate
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commutative ElementaryUndergraduate
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congruent ElementaryHigh school
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conservative system High schoolUndergraduateGraduate
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continuous ElementaryHigh schoolUndergraduateGraduate
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continuously differentiable High schoolUndergraduate
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control input High schoolUndergraduate
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coordinate vector High schoolUndergraduate
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covector High schoolUndergraduateGraduate
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curl High schoolUndergraduate
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detectable UndergraduateGraduate
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diameter ElementaryHigh schoolUndergraduate
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difference ElementaryHigh school
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differentiable High schoolUndergraduate
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differential UndergraduateGraduate
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dimension (of a vector space) ElementaryHigh schoolUndergraduate
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directional derivative High schoolUndergraduate
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discrete metric Undergraduate
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disjoint union (tagged) Undergraduate
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displacement ElementaryHigh school
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distance ElementaryHigh schoolUndergraduate
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disturbance High schoolUndergraduate
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divergence High schoolUndergraduate
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dot product High schoolUndergraduate
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edge (of a graph) ElementaryUndergraduate
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edge (of a polygon or polyhedron) ElementaryHigh school
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eigendecomposition Undergraduate
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eigenvalue High schoolUndergraduate
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equal ElementaryUndergraduate
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equivalence relation Undergraduate
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equivalent High schoolUndergraduate
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error ElementaryHigh school
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estimator High schoolUndergraduate
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Euclidean distance High schoolUndergraduate
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Euclidean norm High school
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event High schoolUndergraduate
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exogenous input High schoolUndergraduate
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exponential distribution Undergraduate
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extended Kalman filter High schoolUndergraduate
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exterior ElementaryUndergraduateGraduate
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feedback ElementaryHigh schoolUndergraduate
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feedforward High schoolUndergraduate
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filter (estimation) High schoolUndergraduate
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finite High schoolUndergraduate
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fixed point ElementaryHigh schoolUndergraduate
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functional High schoolUndergraduate
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geodesic distance High schoolGraduate
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geometric distribution High school
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gradient UndergraduateGraduate
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group Undergraduate
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Hausdorff dimension High schoolGraduate
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hypothesis (in science) ElementaryHigh school
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identified UndergraduateGraduate
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identity element High schoolUndergraduate
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indexed family Undergraduate
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infinite ElementaryUndergraduateGraduate
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infinity norm High school
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injective High schoolUndergraduate
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inner product High schoolUndergraduateGraduate
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integer ElementaryUndergraduate
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interior ElementaryUndergraduateGraduate
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invariant High schoolUndergraduate
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inverse element ElementaryHigh schoolUndergraduate
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irrational number ElementaryHigh school
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isomorphic High schoolUndergraduate
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Jacobian High schoolUndergraduate
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Jacobian determinant High schoolUndergraduate
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Jordan–Chevalley decomposition UndergraduateGraduate
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Jordan normal form Undergraduate
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Kalman filter High schoolUndergraduate
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measure Undergraduate
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left limit High schoolUndergraduate
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Lie derivative UndergraduateGraduate
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likelihood Undergraduate
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limit (of a function) High schoolUndergraduate
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linear map High schoolUndergraduate
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locally Lipschitz continuous High schoolUndergraduate
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LU decomposition Undergraduate
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Luenberger observer High schoolUndergraduate
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matrix ElementaryHigh schoolUndergraduate
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measure zero UndergraduateGraduate
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metric High schoolUndergraduate
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metric space High schoolUndergraduate
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multiset High schoolUndergraduate
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natural numbers (from 1) ElementaryUndergraduate
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natural numbers (from 0) ElementaryUndergraduateGraduate
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norm ElementaryHigh schoolUndergraduate
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normal distribution High schoolUndergraduate
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normal vector High schoolUndergraduate
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nowhere dense Graduate
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observable High schoolUndergraduate
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observer High schoolUndergraduate
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1-norm High school
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open ball High schoolUndergraduate
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open loop ElementaryHigh schoolUndergraduate
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open map UndergraduateGraduate
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open set ElementaryHigh schoolUndergraduateGraduate
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operator High schoolUndergraduate
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optimal control High schoolUndergraduate
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ordinal number ElementaryHigh schoolGraduate
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ordinary derivative High school
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orthogonal High schoolUndergraduate
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output feedback Undergraduate
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pairwise linearly independent High schoolUndergraduate
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partial derivative High schoolUndergraduate
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particle filter High schoolUndergraduate
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periodic High schoolUndergraduate
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perpendicular ElementaryHigh schoolUndergraduate
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PLU decomposition Undergraduate
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point ElementaryUndergraduate
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Poisson distribution High school
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pole High schoolUndergraduate
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polynomial function High schoolUndergraduate
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positive definite function Graduate
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positive definite matrix Undergraduate
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positive semidefinite Undergraduate
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posterior High schoolUndergraduate
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postulate High school
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predicate High schoolUndergraduate
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predictor High schoolUndergraduate
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preimage High schoolUndergraduate
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prepend ElementaryUndergraduate
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prior High schoolUndergraduate
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probability ElementaryHigh schoolUndergraduate
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probability density Undergraduate
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probability zero High schoolUndergraduate
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proper subset ElementaryHigh school
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proposition Elementary
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propositional formula Undergraduate
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QR decomposition Undergraduate
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random variable High schoolUndergraduate
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rational number ElementaryHigh schoolUndergraduate
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reachable High schoolUndergraduate
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real number High schoolUndergraduate
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reconstructible Undergraduate
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relation High schoolUndergraduate
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right angle ElementaryHigh school
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right limit High schoolUndergraduate
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Schur decomposition UndergraduateGraduate
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law (in science) High school
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sentence Undergraduate
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sequence ElementaryHigh schoolUndergraduate
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set ElementaryGraduate
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set difference High school
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intersection High schoolUndergraduate
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union High schoolUndergraduate
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σ-algebra UndergraduateGraduate
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singular value Undergraduate
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singular value decomposition Undergraduate
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small gain theorem UndergraduateGraduate
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smooth High schoolUndergraduate
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span High schoolUndergraduate
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uniform (in space) High school
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sphere ElementaryHigh schoolUndergraduate
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stabilization High schoolUndergraduate
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state feedback Undergraduate
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static feedback Undergraduate
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statistical independence High schoolUndergraduateGraduate
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steady state High schoolUndergraduate
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subset ElementaryHigh school
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surjective High schoolUndergraduate
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switched system High schoolUndergraduate
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symmetric matrix High schoolUndergraduate
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symmetric relation Undergraduate
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Taxicab distance High schoolUndergraduate
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term Undergraduate
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term ElementaryHigh school
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theory (in science) High school
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Toeplitz decomposition Undergraduate
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topology High schoolUndergraduate
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total derivative (along a path) Undergraduate
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total derivative (linear map) Undergraduate
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tracking High schoolUndergraduate
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transcendental number High schoolUndergraduate
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transform High schoolUndergraduate
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zero (of a transfer function) High schoolUndergraduate
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uncountable High schoolUndergraduate
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uniform convergence Undergraduate
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uniform distribution High school
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uniformly continuous High schoolUndergraduate
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uniformly stable Graduate
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disjoint union (of disjoint sets) High schoolUndergraduate
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unit ElementaryHigh school
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unit vector High schoolUndergraduate
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unscented Kalman filter High schoolGraduate
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vector ElementaryHigh schoolUndergraduate
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basis High schoolUndergraduate
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viability kernel Graduate
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volume ElementaryUndergraduate
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weighted average ElementaryHigh school
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whole numbers Elementary
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Z-transform Undergraduate
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Zeno (solution) Graduate
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zero ElementaryUndergraduate
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zero vector ElementaryHigh schoolUndergraduate
Examples
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$(0, 0) \in \reals^2$, for vector additionHigh school
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$0$, for the set $S = (0, 1]$High school
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$0$, for $x \mapsto x^2$High school
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$0 < 0.5 < 1$, for the interval $[0, 1]$High school
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$1$ and $\setdef{1, 2}$Elementary
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$1$, for $S = [1, 2]$High school
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$1$, for $\dot{x} = f(x) = 1 - x$High school
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$(2, 3) \in \reals^2$High school
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$2$, for $f(x) = x^2 - 4$High school
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$3$ and $10$, as integers modulo $7$High school
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$3$, for $f(x) = x^2 + 3$High school
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$3 - 7 = -4$Elementary
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$3x + 2$High school
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"$7$ is prime"High school
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"$7$ is prime and $4$ is not prime"Undergraduate
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$\reals^2$ and $\mathbb{C}$, as real vector spacesUndergraduate
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$\reals^3$High school
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A bar graph of monthly rainfallElementary
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A bouncing ballUndergraduate
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A clock that chimes every hourElementary
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A continuous map from a sphere to the planeHigh school
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$A = \pi r^2$High school
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A "fat" Cantor setGraduate
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A flowchartElementary
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A gust of wind hitting a quadcopter droneHigh school
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A heated rod with insulated endsHigh school
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A heated rod with its ends held in ice waterHigh school
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A Kalman filterUndergraduate
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A lineElementary
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A linear map $T : \reals^2 \to \reals^2$Undergraduate
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A low-pass filter on a sensor signalHigh school
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A Luenberger observerUndergraduate
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A particle filterUndergraduate
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A pie chartElementary
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A flat surface (a plane)Elementary
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A sampled-data control systemUndergraduate
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A scatter plot of height vs. weightElementary
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A solid bowling ballHigh school
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A squareElementary
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A thermostatElementary
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A thermostat setting of 20 degreesHigh school
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A toaster that runs for a fixed timeElementary
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A velocity, in meters per secondHigh school
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A weather forecast for tomorrowHigh school
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$\abs{3 - 7} = 4$Elementary
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$f(x) = \abs{x}$High school
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The absolute value of a numberHigh school
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Addition $+$ on $\reals$Elementary
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An extended Kalman filterUndergraduate
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An unscented Kalman filterGraduate
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Balancing an inverted pendulum uprightHigh school
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The Borel sets of $\reals$Graduate
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$[0, 1]$High school
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$x \mapsto 5$High school
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$d(x, y) = 1$ for $x \neq y$, and $d(x, x) = 0$Undergraduate
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The data $1, 1, 2, 3, 100$Elementary
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$\emptyset$ and $\setdef{1, 2}$Elementary
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The empty set $\emptyset$High school
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$=$, $<$, and $\subseteq$High school
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$e$High school
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$\exists x \, (x^2 = 2)$Undergraduate
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$\exists y \, (y > x)$Undergraduate
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$f \mapsto f(0)$Undergraduate
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$f(x) = x^2 \sin(1/x)$ for $x \neq 0$, $f(0) = 0$Undergraduate
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$f(x) = x^2$, $g(x) = x$Undergraduate
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The roll of a fair dieHigh school
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Goldbach's conjectureHigh school
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$[0, 1)$, as a subset of $\reals$High school
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The $3 \times 3$ identity matrixHigh school
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$x \mapsto x$, for composition of functionsHigh school
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The integers $\mathbb{Z}$High school
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$\mathbb{Z}$ with additionHigh school
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The interval $(0, 1]$High school
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An invertible linear map $V \to V$Undergraduate
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"I have a theory about who ate the cake"Elementary
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The relation $<$ on $\reals$Undergraduate
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A linear map $\reals^2 \to \reals^3$Undergraduate
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Multiplication of $2 \times 2$ real matricesUndergraduate
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$-5$, in $-5x$Elementary
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Multiplication $\times$ on $\reals$Elementary
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$\nats$High school
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$\nats$ with additionHigh school
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$f(x) = -\abs{x}$High school
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$1$Elementary
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$\frac{1}{2}$ and $\frac{2}{4}$Elementary
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The $1000 \times 1000$ matrix of onesHigh school
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The $3 \times 4$ matrix of onesHigh school
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$(0, 1)$ and $(1, 2)$Graduate
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$(0, 1)$High school
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$p \land \neg q$Undergraduate
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The tautology $p \lor \neg p$Undergraduate
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$P \implies Q$ and $\neg Q \implies \neg P$High school
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The parabola $y = x^2$High school
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$f(x, y, z) = x^2 + y^2$Undergraduate
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Euclid's parallel postulateHigh school
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The swinging pendulum of a grandfather clockHigh school
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A pendulum swinging on its ownHigh school
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A phone numberElementary
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$\pi$Elementary
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$x^2 - 1 = 0$High school
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$f(x) = x^2 - 1$, for $x \in \reals$High school
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$x^2 - 1$High school
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The set of polynomials of degree at most $2$Undergraduate
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All subsets of a set $X$ (its power set)Undergraduate
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The set of all subsets of $\nats$Undergraduate
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The set of rational numbers, $\mathbb{Q}$High school
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$\mathbb{Q}$, as a subset of $\reals$Undergraduate
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$\mathbb{Q} \cap [0, 1]$High school
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The set of real numbers, $\reals$High school
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$\reals^n$ with the Euclidean normUndergraduate
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$\reals$ with the standard topologyUndergraduate
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Rotation of the plane by $90°$High school
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$\setdef{1, 2, 2}$Elementary
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$\setdef{1, 2, 3}$High school
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$\setdef{1, 2}$ and $\setdef{1, 2}$Elementary
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The set $\setdef{3, 5, 8}$Elementary
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$\setdef{(x, x^2) \suchthat x \in \reals}$High school
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$(x, y) \mapsto x y + 1$ on $\reals^2$Undergraduate
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$\sin^2 x + \cos^2 x = 1$High school
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$f(x) = \frac{\sin x}{x}$, at $x = 0$High school
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$\sqrt{2}$High school
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Subtraction on $\nats$High school
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Subtraction $a - b$ on $\reals$Elementary
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$f(t, x) = t x^2$Undergraduate
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The "23" on a jerseyElementary
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The "3" in "3 apples"Elementary
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The "3rd" in "finished 3rd"Elementary
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The boundary of a squareHigh school
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The Cantor function on $[0, 1]$Undergraduate
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The Cantor setUndergraduate
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The class of all setsGraduate
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The closed unit ball of $\ell^2$Graduate
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The data $1, 2, 3, 4$High school
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The line with two originsGraduate
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The location of ParisElementary
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The motor commands of a quadcopter droneHigh school
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The polynomial $p(x) = 3 + 2x$High school
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The sequence $1, 2, 3, \dots$Elementary
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The surface of a basketballHigh school
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The surface of a cubeHigh school
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The theory of evolution by natural selectionHigh school
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$G(s) = \frac{1}{s + 2}$High school
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$G(s) = \frac{s - 1}{(s + 2)(s + 3)}$High school
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$G(s) = \frac{s + 1}{(s + 1)(s + 2)}$High school
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Tricomi equation $y u_{xx} + u_{yy} = 0$Undergraduate
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$(1, 2, 2)$High school
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$[0, 1] \cup [2, 3]$High school
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$2.5$Elementary
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$u = -K x$, with the full state measuredHigh school
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$u = -K y$, when $y = x$High school
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$\cup$ on the subsets of a set $X$High school
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The vector $x = (1, 1) \in \reals^2$High school
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The vector $x = (3, -4) \in \reals^2$High school
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The vector $x = (5, 0) \in \reals^2$High school
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The vectors $(1, 0)$ and $(0, 0)$ in $\reals^2$Undergraduate
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Walking $3$ m east and then $3$ m westElementary
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Wave equation $u_{tt} = u_{xx}$Undergraduate
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$x = 0$, for $f(x) = x^2 + 3$High school
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"$x > 3$"High school
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$x_{k+1} = 0 \cdot x_k + 0 \cdot u_k$Undergraduate
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$x_{k+1} = 0 \cdot x_k$, $y_k = 0$Undergraduate
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$f(x) = x^2$ on $\reals$High school
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$x^2 + 1$High school
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$f(x, y) = x^2 + 3y$Undergraduate
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$x \mapsto \sin(1/x)$ on $(0, 1]$Undergraduate
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$x \mapsto \sin x$High school
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$x \mapsto \sqrt{x}$ on $[0, 1]$Undergraduate
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$x \mapsto \abs{x_1} + \abs{x_2}$ on $\reals^2$Undergraduate
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$x \mapsto x \abs{x}$High school
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$x \mapsto x^3$, from $\reals$ to $\reals$Undergraduate
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$x \mapsto \abs{x}$ on $[-1, 1]$High school
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$x \mapsto x^2$ on $[0, 1]$High school
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$(x, y) \mapsto 2 x_1 y_1 + x_2 y_2$ on $\reals^2$Undergraduate
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$(x, y) \mapsto x_1 y_1 - x_2 y_2$ on $\reals^2$Undergraduate
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$(x, y) \mapsto x_1 y_1 + x_2 y_2$ on $\reals^2$Undergraduate
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$\dot{x} = -x^3$High school
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$\dot{x} = -x + \sin t$High school
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$\dot{x} = -x + u$High school
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$\dot{x} = -x$, $y = 0$Undergraduate
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$\dot{x} = x + 0 \cdot u$High school
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$\dot{x} = x$, $y = x$Undergraduate
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$0$Elementary