A function is operator concave when its negative is operator convex. Equivalently, the defining operator-convexity inequality is reversed.

If $\mathbf{P}\ne \mathbf{NP}$, then NP contains a decision problem that is neither in P nor NP-complete.

Quadratic-equation satisfiability over ${\mathbb{F}}_2$ asks whether a finite system of polynomial equations of degree at most two over the finite field ${\mathbb{F}}_2$ has a common solution.

The Boolean satisfiability problem asks whether a Boolean formula has an assignment making it true. A complete assignment is a Boolean valuation: it gives every Boolean variable a truth value, consistently at all occurrences. A satisfying assignment, also called a model of a Boolean formula, makes the entire Boolean formula true. A solver's answer SAT asserts that at least one satisfying assignment exists; UNSAT asserts that the input is an unsatisfiable Boolean formula. These are statements about existence, rather than the truth value under one particular Boolean valuation.

For conjunctive normal form, or CNF, writeEach Boolean literal $\ell$ is a Boolean variable or its logical negation. Each Boolean clause uses inclusive logical disjunction: one true Boolean literal suffices. The outer logical conjunction requires every Boolean clause to succeed under the same Boolean valuation. Choosing a different Boolean valuation for each Boolean clause would answer a different question.

Consider the original three-Boolean variable exampleAt the Boolean valuation $(x,y,z)=(0,1,0)$, the Boolean literals in $C_1$ have values $(0,1)$, those in $C_2$ have values $(1,0)$, and those in $C_3$ have values $(0,1)$. Every Boolean clause therefore has truth value one, so $F$ has truth value one: this is a satisfying assignment.

The complete truth table has $2^3=8$ Boolean valuations. Exactly two are satisfying assignments: $(0,1,0)$ and $(1,0,1)$. In particular, $(1,1,1)$ satisfies the first two Boolean clauses but falsifies the third, so satisfying most Boolean clauses does not suffice for SAT. That different objective belongs to maximum satisfiability.

A short proof of completeness avoids listing all eight rows. If $x=0$, the Boolean clause $C_1$ forces $y=1$, after which $C_3$ forces $z=0$; $C_2$ is already true. If $x=1$, $C_2$ forces $z=1$, after which $C_3$ forces $y=0$; $C_1$ is already true. These two cases exhaust the truth values of $x$ and produce precisely the two satisfying assignments above.

Verifying a satisfying assignment and finding a satisfying assignment are different tasks. Given a candidate, evaluate the Boolean formula once; finding a candidate requires choosing compatible truth values across all its Boolean clauses. The Cook-Levin theorem says that general SAT is NP-complete, even though a proposed satisfying assignment is efficiently checkable. Its proof encodes a bounded nondeterministic computation as a Boolean formula whose satisfying assignments describe accepting computations. Stephen A. Cook's 1971 paper established this connection between computational complexity theory and propositional logic.^[ref]

A concrete use is checking finite configuration rules. Let $x$ mean that an optional feature is enabled, $y$ that a fallback is selected, and $z$ that the feature's required component is installed. Require a feature or fallback, require the component whenever the feature is enabled, and forbid using the fallback with that component. Logical implication rewrites the second rule $x\to z$ as $\neg x\vee z$; De Morgan's laws rewrite the third rule $\neg (y\wedge z)$ as $\neg y\vee \neg z$. Their logical conjunction is exactly $F$, so its two satisfying assignments give the two allowed configurations. The usefulness of SAT lies in expressing local rules uniformly while demanding one globally consistent Boolean valuation. The conclusion concerns exactly these encoded rules.

Ordinary least squares minimizes the unweighted sum of squared residuals between observed and fitted responses.

If $Z_1,Z_2$ are independent $N(0,1)$ random variables, their product has characteristic function $(1+t^2{)}^{-1/2}$ and density $K_0(|x|)/\pi$, where $K_0$ is a modified Bessel function. A scale multiple $a{Z}_1{Z}_2$ has the corresponding scaled distribution.

Gaussian noise is an additive random error with a normal distribution, often assumed independent across observations and centered at zero.

A Gaussian random variable is a random variable whose probability distribution is a normal distribution. For $\sigma >0$, write $X=\mu +\sigma Z$ with a standard normal random variable $Z$; conversely, standardization of a normal random variable recovers $Z=(X-\mu )/\sigma$. For example, $X\sim N(100,15^2)$ has mean $100$, standard deviation $15$, and variance $225$. The outcome $x=115$ corresponds to standard score $z=1$, one standard deviation above the mean.

Some definitions also allow a Gaussian random variable with $\sigma =0$. This means $X=\mu$ almost surely, with variance zero and probability distribution equal to the Dirac measure ${\delta }_{\mu }$. It has no probability density function with respect to Lebesgue measure: $\mathbb{P}(X=\mu )=1$. The positive-scale density formula and division by $\sigma$ apply only when $\sigma >0$. MIT's constructive definition explicitly includes the constant case. ^[ref].

For an oriented submanifold of half the dimension of its ambient manifold, the self-intersection number is the intersection number with a transverse perturbation. It equals the Euler number of the normal bundle.

A CAT(0) cube complex is a simply connected nonpositively curved cube complex.

For the normalization $(T_{p}f)(\tau )=p^{-1}(f(p\tau )+{\sum }_{b\mathrm{mod}p}f((\tau +b)/p))$, one has

The Shannon scaling function is $\varphi (x)=\sin (\pi x)/(\pi x)$ and has a rectangular Fourier transform.

The Meyer-Mallat theorem associates an orthonormal wavelet basis to every orthonormal multiresolution analysis.

Attaching a two-cell to $S^1$ by a map of degree $n$ gives the Moore space $M(\mathbb{Z}/n,1)$. Its positive-dimensional cellular chain complex isso $H_1\cong \mathbb{Z}/n$ and all its other reduced integral homology groups vanish.

The ray class group modulo $\mathfrak{m}$ is $I_K(\mathfrak{m})/P_K(\mathfrak{m})$, where the numerator consists of fractional ideals prime to ${\mathfrak{m}}_0$ and the denominator consists of $(\alpha )$ with $\alpha \equiv 1$ at the finite modulus and positive at every real place in ${\mathfrak{m}}_{\infty }$.

A Dirichlet process mixture model uses a Dirichlet process as a random mixing distribution, allowing the number of occupied mixture components to be inferred from the data.

A finite mixture model draws each observation from one of finitely many component distributions according to a categorical latent label.

A squirmer is an idealized spherical microswimmer propelled by a prescribed tangential surface slip velocity. The Lorentz reciprocal theorem for Stokes flow determines its rigid translation and rotation without solving its complete exterior flow.

A pseudometric satisfies the axioms of a metric except that distinct points may have distance zero.

Sanov's theorem is the large deviation principle for empirical distributions of independent identically distributed observations. On a finite alphabet with sampling law $Q$, its rate function is $D_e(R\Vert Q)$.