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A current-measuring coil and compass arranged so that the coil's central magnetic field is perpendicular to the horizontal terrestrial field. With unchanged coil geometry and terrestrial field, the electric current ratio is the ratio of the tangents of the needle deflections. A small needle and controlled alignment make the central-field approximation useful.

The set of database attributes determined by $X$ under given functional dependencies. Start from $X$ and repeatedly add the right-hand side of each functional dependency whose left-hand side is already included.

A full ternary tree whose leaves all have the same depth. At depth $j$ there are $3^j$ vertices, so a tree with $h+1$ levels has $N_h={\sum }_{j=0}^h{3}^j=(3^{h+1}-1)/2$ vertices and $3^h$ leaves, by the finite geometric series formula. For a rose tree representation, recursively flattening children and joining their lists with ordinary list concatenation copies each label once per ancestor. The work is therefore $\Theta ((h+1)3^h)=\Theta (N_h\log N_h)$ when the complete output is demanded. Passing an already constructed suffix through the recursion instead emits each label once, giving $\Theta (N_h)$ work.

Let $f$ be a Borel measurable function in L1 space on $\mathbb{R}$, and let $a,b$ be measurable real functions of $x$ with $b(x)>0$ and $W={\int }_{\mathbb{R}}|a(x)|/b(x)dx<\infty$. The joint integrand $G(x,y)=a(x)f(b(x)y)$ is measurable, and Tonelli's theorem followed by Lebesgue substitution on an interval givesFor each fixed $x$, substitute $z=b(x)y$ in the inner $y$ integral. The absolute Jacobian is $1/b(x)$, leaving $|a(x)|\Vert f{\Vert }_1/b(x)$. Integrating in $x$ proves the formula and absolute integrability. Fubini's theorem then permits both orders of signed integration, givingThe one-dimensional sections are integrable for almost every value of the other variable; the theorem does not assert integrability of every section.

Define the displayed function away from the origin and set $f(0,0)=0$. Every section in either coordinate is integrable on $[-1,1]$ and is an odd function, so both orders of iterated integrals on $[-1,1{]}^2$ are zero. However, on the annulus $\epsilon <\sqrt{x^2+y^2}<1$, polar integration givesThus the function is not Lebesgue integrable jointly. Reflection interchanges its positive part and negative part, whose integrals are both infinite. Existence and even agreement of finite iterated integrals do not establish the absolute integrability hypothesis of Fubini's theorem.

The function $(x-y)/(x+y{)}^3$ on the unit square is not Lebesgue integrable. On $0<y<x/2$, its absolute value is at least $4/(27x^2)$, whose area integral diverges logarithmically. Its two signed iterated integrals nevertheless exist, ignoring null boundary sections, and equal $1/2$ and $-1/2$. This illustrates why absolute integrability is required for Fubini's theorem.

For the real two-by-two matrix space, the Frobenius inner product is $\mathrm{tr}(A^{t}B)={\sum }_{i,j}{a}_{ij}{b}_{ij}$. The displayed four matrices form an orthonormal basis. The first spans the identity direction, the next two span the traceless symmetric matrices, and the last spans the antisymmetric matrices. Symmetric and antisymmetric matrices are orthogonal because transposition changes the sign of their trace pairing; pairing with the identity gives the trace. This basis therefore displays the orthogonal direct sum into these three natural pieces.

For a symmetric uniformly elliptic divergence-form operator with zero conormal flux and zero reaction, Lax-Milgram theorem constructs the shifted inverse $K:L^2\to H^1$. The compact Rellich-Kondrachov theorem embedding makes it compact on $L^2$, and symmetry makes it self-adjoint. If the homogeneous kernel consists of constants, then $\ker (I-K)$ is the constant line and $K1=1$. The Fredholm alternative makes solvability equivalent to $Kf$ being orthogonal to constants, or $⟨Kf,1⟩=⟨f,K1⟩=\int f=0$. The weak solution is unique up to an additive constant. For a disconnected domain the identical argument gives one compatibility condition per connected component.

Let $X_i$ be independent and identically distributed random variables taking positive integer values with $\mathbb{P}(X_i\le k)=k/(k+1)$, and put $M_n={\max }_{i\le n}{X}_i$. For $x>0$, independence givesIndeed $n/(\lfloor nx\rfloor +1)\to 1/x$ and $\log (1-u)/u\to -1$ as $u\downarrow 0$. For $x\le 0$ the probability is zero. The limiting cumulative distribution function is continuous, including at zero, and is a Fréchet distribution with shape one. Its scale grows linearly with sample size even though the individual random variables have infinite expectation.

Theoremhood is formal derivability with no undischarged assumptions. A logical formula is a theorem of a calculus exactly when it has an assumption-free formal proof.

The significant-digit factor $m$ in a floating-point representation $(-1{)}^{s}m{\beta }^e$, where $s$ is the sign bit, $\beta$ the radix and $e$ the exponent. For a normal positive binary significand, $1\le m<2$, and the leading one need not be stored. In a subnormal number, the leading digit is zero at the minimum normal exponent scale. The significand is a numerical factor, distinct from the stored trailing fraction bits that encode part of it.

A fixed point of a function on a partially ordered set that is greater than or equal to every other fixed point. For the order-preserving function $H(X)=A\cap \{s:\mathrm{Post}(s)\subseteq X\}$ on the power set of a finite transition system with no terminal states, its greatest fixed point is the set of states from which every infinite path in a transition system remains in $A$. Iteration from the whole state space decreases to this set; every strict decrease removes at least one state.

A first-order theory whose consequences can be decided by an algorithm that terminates on every input first-order sentence. Decidability of a chosen axiom set alone does not ensure decidability of its consequences.

In the language containing one binary equivalence relation $E$, require all classes to have size at most two and infinitely many classes of each of the two sizes. This is a complete theory: every countable model has countably many singleton classes and countably many doubleton classes, giving a class-by-class isomorphism; Downward Lowenheim-Skolem theorem then rules out a sentence separating arbitrary models. It is also an aleph-zero-categorical theory.

It is not finitely axiomatizable. Any finite fragment of the explicit infinitude schemes has a finite model with sufficiently many classes of each size; compactness theorem would make such a finite fragment suffice if any finite axiomatization existed. It is not model-complete: adjoining a new singleton preserves the theory, and subsequently giving that new point a partner also preserves the theory, while changing the existential partner formula at that point. Naming the singleton-class predicate removes this particular obstruction to elementary embeddings.

A set of first-order sentences is satisfiable when it has a nonempty structure in which all its sentences are true. Truth of a sentence is independent of assignment in a first-order structure. The soundness theorem for first-order logic gives consistency from satisfiability, and Gödel's completeness theorem gives the converse for the standard classical deductive system.

The existential theory of a structure is the set of existential sentences with quantifier-free matrices true in it. Its witnesses persist in extensions because quantifier-free truth is preserved by structure embeddings. It differs from an existential diagram, which additionally uses names for Elements.

No finite theory in the full language containing one function symbol for each scalar of an infinite field can axiomatize exactly its nonzero vector spaces. It mentions only finitely many symbols; changing an unused scalar function to violate preservation of zero leaves those finite sentences unchanged. Relative to the vector-space axiom schema, nonzeroness is instead the single axiom $\exists x(x\ne 0)$. Absolute finite axiomatizability and finite additional axiomatizability must be distinguished.

An arithmetic theory is bounded-complete if it proves every true closed formula with only bounded quantifiers. It then proves every true Sigma-1 sentence by choosing a standard witness. If it is also consistent, it cannot prove a false Pi-1 sentence: a standard bounded counterexample supplies a provable contradiction to that universal theorem. This arithmetic condition is distinct from bounded completeness of a statistical model or family.

A first-order model class is finitely axiomatizable in a fixed language if a finite set of sentences, equivalently one finite conjunction, has exactly that model class. Relative finite axiomatizability permits a fixed background theory and asks for only finitely many additional axioms. The distinction matters in an infinite scalar language, where even vector-space structure itself needs infinitely many named-symbol constraints.