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The negative part of a real-valued function $f$ is the nonnegative function $f_{-}(x)=\max (0,-f(x))$. It records the magnitude of the negative values, rather than retaining their sign: if $f(x)=-3$, then $f_{-}(x)=3$. Together with the positive part, it gives $f=f_{+}-f_{-}$ and $|f|=f_{+}+f_{-}$. This decomposition defines the Lebesgue integral of a signed function and makes the integrability condition explicit.

If the observed samples are $y_i=f(x_i)+{\epsilon }_i$, the resulting quadrature rule changes by $\Delta Q={\sum }_i{w}_i{\epsilon }_i$. Bounded sample perturbations $|{\epsilon }_i|\le \delta$ therefore satisfyFor ordinary numerical integration with nonnegative quadrature weights exact on constant functions, the bound is $(b-a)\delta$, independent of panel refinement. A common bias ${\epsilon }_i=\delta$ attains it: the quadrature error due to the function approximation may vanish while this sample error persists. Interpolating a fixed data set onto a finer grid likewise does not create new observations or remove its uncertainty.

With new samples whose errors are independent random variables of zero expectation and common variance ${\sigma }^2$, a different model givesFor equal panels this is ${\sigma }^2(b-a{)}^2/n$ for the composite midpoint quadrature rule and ${\sigma }^2{h}^2(n-1/2)$ for the composite trapezoidal rule. These variance reductions require the stated independence and unbiasedness assumptions; a common bias or correlated sample errors need not shrink. The two displayed perturbation formulas follow directly from the triangle inequality and the variance additivity for independent random variables, respectively.

A function is nonconstant when two inputs in its domain of a function have different outputs. For example, $f(x)=x^2$ on the real line is nonconstant, whereas the same rule restricted to $\{-1,1\}$ is a constant function.

A real-valued function $f:D\to \mathbb{R}$ is a nondecreasing function if every pair $x<y$ in its domain of a function satisfies $f(x)\le f(y)$. Flat intervals are allowed, unlike for a strictly increasing function. For example,is a continuous nondecreasing function, but it repeats its output on each flat interval.

A nondecreasing function can also be discontinuous. The step function $F(x)=0$ for $x<0$ and $F(x)=1$ for $x\ge 0$ has a jump discontinuity at zero without ever decreasing. It is the cumulative distribution function of a random variable that equals zero with probability one; quantile functions handle such jumps through a generalized inverse. Nondecreasing allocation functions permit critical-value payments in single-parameter mechanisms.

A critical point of a real function with continuous second partial derivatives is nondegenerate when its Hessian matrix is an invertible matrix there, equivalently when none of its eigenvalues is zero. A positive-definite matrix gives a strict local minimum, a negative definite Hessian gives a strict local maximum, and opposite-sign eigenvalues give a saddle point of a scalar function. A nondegenerate critical point need not be an extremum: $x^2-y^2$ has Hessian matrix $\mathrm{diag}(2,-2)$, which is an invertible matrix.

Such critical points are the local building blocks of Morse theory.

A nondeterministic finite automaton allows a set of possible destinations for each automaton state and input symbol. Without the extra moves permitted in an Epsilon-NFA, replace the DFA function by $\Delta :Q\times \Sigma \to \mathcal{P}(Q)$, where $\mathcal{P}(Q)$ is the power set. A word is accepted when at least one path consumes all its symbols and ends in an accepting automaton state. These choices express existential quantification, not probability: no chance of choosing a particular path is assigned. One successful path suffices even if other paths fail.

Consider a three-state NFA accepting exactly the words of binary digits ending in $01$. Its initial automaton state $q_0$ loops on both $0$ and $1$; it also has an automaton transition $q_0\stackrel{0}{\to }{q}_1$, followed by $q_1\stackrel{1}{\to }{q}_2$. Only $q_2$ is accepting, and all unlisted automaton transitions are absent. Staying at $q_0$ skips an arbitrary beginning; leaving it guesses that the current zero starts the final two symbols.

The word $01$ accepts along $q_0\to q_1\to q_2$. The word $1101$ accepts along $q_0\to q_0\to q_0\to q_1\to q_2$. The word $010$ rejects: a path reaching $q_2$ after $01$ cannot consume the remaining zero, while the other paths end outside $q_2$. Thus finding $01$ somewhere is insufficient; it must occur at the end.

To simulate the NFA deterministically, retain the set $S$ of all currently possible automaton states. The powerset construction updates $S$ by taking the union of the destinations of its members. Here only $\{q_0\}$, $\{q_0,q_1\}$, and $\{q_0,q_2\}$ are reachable, although the full power set has eight members. Reading $010$ gives $\{q_0\}\to \{q_0,q_1\}\to \{q_0,q_2\}\to \{q_0,q_1\}$, which rejects. In general, an $N$-state NFA yields a DFA with at most $2^N$ automaton states. Nondeterminism can simplify a description, while the powerset construction keeps the memory finite.

The second generator checks all 511 words of up to eight binary digits against the direct test of whether their final two symbols are $01$. Exactly 127 accept, and only the three listed sets of automaton states occur. In particular, the empty word rejects for this NFA.

Michael O. Rabin and Dana Scott's 1959 paper studied finite-state machines as recognizers with finite internal memory. Its section 5 defines NFAs through successful paths, and theorem 11 proves their equivalence to DFAs by the powerset construction. This gives a historical source for the distinction between many possible paths and greater recognition power: ^[ref].

A nonlinear ordinary differential equation depends nonlinearly on the unknown function or its derivatives: it cannot be written in the form defining a linear differential equation. For example, $y^{\prime }=y^2$ contains a square of the unknown function, and $y^{\prime }=y(1-y)$ is a logistic differential equation. A coefficient such as $t^2$ depending only on the independent variable does not make an ordinary differential equation nonlinear. This distinction concerns dependence on the unknown, rather than the shape of its function graph.

For a concrete initial value problem, consider $y^{\prime }=y^2$, $y(0)=y_0>0$. Its solution of a differential equation isTaking its derivative gives $y^{\prime }=y_0^2/(1-y_{0}t{)}^2=y^2$, and evaluating at zero checks the initial condition. The stated interval contains zero and ends at finite-time blowup: $y$ tends to positive infinity as $t$ approaches $1/y_0$ from below. Increasing the positive initial condition brings finite-time blowup closer. This is a simple mathematical example with dimensionless $t$ and $y$, rather than a calibrated physical model.

Choose $y_0=1$ and $y_0=2$, giving $y_1=1/(1-t)$ and $y_2=2/(1-2t)$. At $t=1/4$ they have values $4/3$ and $4$, so their sum $s=y_1+y_2$ has value $16/3$. On the common interval $0\le t\le 0.4$, both are valid solutions of a differential equation, but their sum fails the same ordinary differential equation:Already at $t=0$, $s=3$ and its derivative is $1+4=5$, whereas $y^{\prime }=y^2$ would require $9$. At $t=1/4$, the defect $s^{\prime }-s^2$ is $-32/3$. The cross term explains the failure of the superposition principle, even though each individual solution of a differential equation is a smooth function.

The first panel shows $y_1$, $y_2$ and their invalid sum only up to $t=0.4$, before either finite-time blowup. The second measures the defect in the ordinary differential equation, including the computed value at $t=1/4$. For a homogeneous linear differential equation, the corresponding defect of a sum is zero. Nonlinear ordinary differential equations can describe state-dependent growth such as the logistic differential equation, whose negative quadratic term limits growth instead of accelerating it without bound. Recognizing the distinction tells us when we may assemble solutions of a differential equation from simpler pieces and when we must account for their interaction. See MIT's worked counterexample to nonlinear superposition.

Nonparametric statistics studies statistical inference without restricting the unknown probability distribution or target function to a fixed finite-dimensional parametric statistical model. In density estimation and nonparametric regression, the number of features that an estimator can represent may grow with the sample size. Rank tests offer another approach: the probability distribution of their test statistic under the null hypothesis can be obtained without specifying the common continuous probability distribution.

"Nonparametric" does not mean free of assumptions or tuning choices. A kernel density estimator still requires a kernel for density estimation and a smoothing bandwidth, and its interpretation depends on assumptions about the statistical sample. The freedom is useful when a prescribed shape, such as a single normal distribution, would hide features that the observations might reveal.

A normalizing constant makes a nonnegative integrable function into a probability density function: if $0<Z=\int h<\infty$, then $h/Z$ is a probability density function.

An ultrafilter $U$ on $P_{\kappa }(\lambda )$ is normal when each function with $f(x)\in x$ on a $U$-large set is constant on a $U$-large subset. This is the small-subset version of normality; together with fineness and kappa-completeness it gives the measure characterization of a supercompact cardinal.

Numerical integration approximates a definite integral using finitely many sampled function values. For an integrand $f$ on a finite interval $[a,b]$, $a<b$, a quadrature rule has the formThe quadrature nodes $x_j$ tell us where to evaluate $f$; the quadrature weights $w_j$ turn those values into an approximation to the accumulated signed area. The exact definite integral depends on the integrand throughout the interval, whereas the quadrature rule sees only these samples. Integrating a chosen interpolant exactly still generally approximates the original integrand's integral.^[ref]

The fundamental theorem of calculus evaluates $I(f)$ as $F(b)-F(a)$ when an antiderivative $F$ is available. Numerical integration is useful when no convenient antiderivative is known, or each evaluation of the integrand requires an expensive computation. The task then becomes choosing informative quadrature nodes and controlling the quadrature error with as few evaluations as possible. Fixed data also permit a quadrature rule, but refinement cannot supply new information between observations.^[ref]

For an exact comparison, take the polynomial $f(x)=x^2$ on $[0,1]$. Its antiderivative $x^3/3$ gives $I(f)=1/3$. On a partition of an interval into $n\ge 1$ equal panels, the composite midpoint quadrature rule uses $n$ midpoint samples, each with quadrature weight $1/n$; the composite trapezoidal rule uses $n+1$ endpoint samples, with outer quadrature weights $1/(2n)$ and interior quadrature weights $1/n$. The existing derivations under these two quadrature rules giveIn particular, $M_1=1/4$ and $T_1=1/2$ already bracket $1/3$; two panels give $M_2=5/16$ and $T_2=3/8$. The convex function $x^2$ explains the opposite signs. A midpoint rectangle can cross the curve within its panel, so its total signed area, rather than pointwise ordering, determines the underestimate.

Define the signed approximation error as $E_n=Q_n(f)-I(f)$. In this example the absolute errors are $1/(12n^2)$ and $1/(6n^2)$: doubling $n$ reduces each by exactly four. A plot with both axes on a logarithmic scale therefore has slope $-2$, expressing order two with respect to panel refinement. This is an exact calculation for this polynomial, including $n=1$, rather than an observed rate extrapolated from a few computed answers.

For a general integrand, the composite midpoint quadrature rule and composite trapezoidal rule give second-order quadrature error bounds when $f$ has two continuous derivatives on $[a,b]$ and a known bound on $|f^{\prime \prime }|$. These hypotheses describe behavior between samples. A formal error formula containing an unknown derivative is not a numerical accuracy certificate or a justified sample count. Finite samples do not certify quadrature accuracy makes the missing information explicit. Errors in the evaluations and roundoff error must also be considered; noise sensitivity of a quadrature rule treats the former.

More elaborate choices exploit other information: Gaussian quadrature obtains high polynomial exactness from carefully chosen quadrature nodes, and Simpson's rule uses a quadratic interpolant. Adaptive numerical integration compares approximations and concentrates samples where they disagree, often saving evaluations. Such a difference is an error estimate, not a certified bound without further assumptions: both approximations can miss the same feature.^[ref]

An omitted value of a function $f:D\to Y$ is a point $w\in Y$ outside its image: the equation $f(z)=w$ has no solution in $D$. Both $D$ and $Y$ matter. The complex exponential function on the complex plane has one omitted value in $\mathbb{C}$, namely zero, and two omitted values when regarded as a map into the Riemann sphere, namely zero and infinity. Restricting $D$ can create additional omitted values.

An ordinary differential equation relates an unknown function of one independent variable to its ordinary derivatives. Its order is the highest derivative appearing. Thus $y^{\prime }=-k(y-T_a)$ is first order, while the harmonic oscillator equation $y^{\prime \prime }+{\omega }^{2}y=0$ is second order. A first-order equation in explicit form isHere $t$ need not be time, and $y$ can be a vector, giving a system of coupled ordinary differential equations. A classical solution of a differential equation is a sufficiently differentiable function on an interval which satisfies the equation at every point. For the explicit first-order equation, if $f$ is continuous, this solution of a differential equation has a continuous derivative.

For Newton's law of cooling, the state is a single temperature; for a second-order differential equation, a first-order system also records its first derivative. This is why a starting position alone does not specify a harmonic oscillator trajectory: its starting velocity matters too.

The power of an ordinary differential equation is that one rule describes a whole family of histories. An initial value problem chooses one history when uniqueness holds. An autonomous differential equation has $f$ independent of $t$; its equilibrium points satisfy $f(y_{\ast })=0$ and give constant solutions of a differential equation. Equations with explicit time dependence can instead respond to a changing environment.

For real monic orthogonal polynomials $p_j$ with squared weighted norms $h_j$ under a positive weight $w$, the functions $p_j\sqrt{w/h_j}$ form an orthonormal set for the unweighted reference measure. Their finite-rank projection kernel turns the squared weighted Vandermonde determinant into a kernel determinant. Normalization on the full labelled configuration space is $1/m!$.

The orthogonal projection $P_F$ onto a closed subspace of a Hilbert space $F$ sends $x$ to its component in the orthogonal decompositionThe residual $r$ is orthogonal to every permitted direction. For any $z\in F$, the Pythagorean theorem givesThus $P_{F}x$ uniquely minimizes the distance to $F$. It is a linear map, is idempotent, and is nonexpansive: $\Vert P_{F}x-P_{F}y\Vert \le \Vert x-y\Vert$; taking $y=0$ gives $\Vert P_{F}x\Vert \le \Vert x\Vert$.

In ${\mathbb{R}}^2$, project $v=(1,3)$ onto the line spanned by $u=(2,1)$. The inner product givesThe check $⟨r,u⟩=-2+2=0$ confirms orthogonality, and $\Vert v{\Vert }^2=10=5+5=\Vert P_{F}v{\Vert }^2+\Vert r{\Vert }^2$. Projecting again changes nothing, since the result already lies on the line.

For a finite orthonormal basis $e_1,\dots ,e_m$ of $F$, the same operation is $P_{F}x={\sum }_j⟨x,e_j⟩e_j$, using the convention linear in the first argument. Replacing coordinate vectors by functions gives exactly the same approximation principle. For $f(x)=x^2$ in the real L2 space on $[-1,1]$ and $F=\mathrm{span}\{1,x\}$, the directions are orthogonal, with squared norms $2$ and $2/3$. Their inner products with $f$ are $2/3$ and zero, soDirect integration gives the squared error $\Vert x^2-1/3{\Vert }^2=8/45$. For any competing affine polynomial, the orthogonal residual yieldsEvery deviation from $a=1/3$, $b=0$ adds a nonnegative error. The best function need not pass through any selected data point: it minimizes an integrated error across the whole interval.

Two vectors are orthogonal when their inner product is zero.

For real Euclidean vectors, the inner product is the dot product, so $(3,4)$ and $(4,-3)$ are orthogonal: $3(4)+4(-3)=0$. For nonzero Euclidean vectors, this means their angle is $90^{\circ }$. The zero vector is orthogonal to every vector, although it has no direction or defined angle.^[ref]

In an abstract inner product space, orthogonality always refers to its chosen inner product. For example, the weighted inner product $⟨u,v{⟩}_w=4u_1{v}_1+u_2{v}_2$ makes $(1,1)$ and $(1,-4)$ orthogonal, since $4-4=0$. Their ordinary dot product is $-3$, so their arrows are not perpendicular in the usual Cartesian coordinate system. The algebraic definition also applies to functions, where arrows need not be available.

For a real function $g$ on $X\times Y\times Z$ and functions $a(y,z),b(x,z),c(x,y)$ bounded in absolute value by one,First apply Cauchy-Schwarz inequality in $(y,z)$ to remove $a$, duplicating $x$. Apply it in $(x_0,x_1,z)$ to remove $b(x_0,z)b(x_1,z)$, duplicating $y$. Finally apply it in $(x_0,x_1,y_0,y_1)$ to remove the four $c$ factors, duplicating $z$. The resulting eighth-power bound is exactly the eight-factor average defining the three-dimensional box norm. Functions depending on just one coordinate, or constants, can be incorporated in any of the three pair factors.