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A convex combination is a finite weighted average of points in a real vector space. For $n\ge 1$, it has the formThe nonnegative coefficients forbid subtracting a component, and their unit sum keeps the result on the same affine scale as the inputs. With two points it parametrizes their line segment; equal coefficients give their centroid. A convex set contains every finite convex combination of its points. Conversely, the two-point case already gives the defining property of a convex set.^[ref]

The distinction from a linear combination is concrete. Combining $(0,0)$ and $(2,0)$ with coefficients $-1$ and $2$ gives $(4,0)$, outside their line segment, despite the unit coefficient sum. Using two coefficients both equal to one gives $(2,0)$ here, but their sum is two, so it is not a convex combination with those coefficients. A numerical output may happen to coincide with a point admitting a different convex representation; the definition applies to the specified coefficients.

The nonnegative orthant consists of vectors whose coordinates are all nonnegative. Its interior is the positive orthant, where every coordinate is positive.

It is the closed orthant obtained by choosing the nonnegative sign for every coordinate.

Let $C$ be a nonempty closed convex set in a real finite-dimensional Euclidean space, and let $F$ be a differentiable convex function. A projected gradient descent step isThe Euclidean projection onto a convex set is single-valued under these assumptions and restores feasibility after the unconstrained gradient descent step. Closedness matters: the nearest point to zero in the convex set $(0,1)$ does not exist.

This is the proximal gradient method with nonsmooth term equal to the indicator functional of a constraint set, whose proximal operator is ${\Pi }_C$. If $\nabla F$ is $L$-Lipschitz continuous, a standard choice $0<\eta \le 1/L$ gives the usual convex objective-error bound relative to an existing minimizer.^[ref]

For $C=[0,1]$, ${\Pi }_C(z)=\max (0,\min (z,1))$. Minimizing $F(x)=(x-2{)}^2/2$ from $x_0=0$ with $\eta =1$ gives $x_1={\Pi }_C(2)=1$, the constrained optimum.

For a real matrix $P$, exactly one of the following holds: there is $\varphi$ with $P\varphi \ge 0$ and $P\varphi \ne 0$, or there is $q>0$ with $P^{T}q=0$. Separating $\mathrm{Im}P$ from the standard simplex proves the result; $q$ may then be normalized to sum to one.

This is an alternative related to Farkas' lemma. It is not the strict-primal Gordan alternative: for the column matrix $(1,0{)}^T$, a nonnegative nonzero primal vector exists, while a componentwise strictly positive one does not.

For a convex function $f$, the subdifferential at $x$ isThe point $x$ minimizes $f$ exactly when $0\in \partial f(x)$.

The swap test estimates the overlap of two pure states without requiring their classical descriptions. Prepare an auxiliary qubit in $|0⟩$, apply a Hadamard gate, a Fredkin gate controlled by that auxiliary qubit, and another Hadamard gate. Then measure the auxiliary qubit in the computational basis.

For target states $|a⟩,|b⟩$, the auxiliary-zero branch has unnormalized target vector $(|a⟩|b⟩+|b⟩|a⟩)/2$. Its squared norm is thereforeIdentical pure states give $p_0=1$, while pure states with zero inner product give $p_0=1/2$. A single zero outcome does not certify equality; repeated trials estimate the probability. The calculation is a direct projection onto the symmetric target subspace using the SWAP gate.

The Hadamard test converts an overlap $⟨\psi |U|\psi ⟩$ into an ancilla measurement probability. Prepare an ancilla in $|+⟩$, apply controlled $U$ to the data state $|\psi ⟩$, apply a Hadamard to the ancilla, and measure it. ThenInterference between the ancilla's two branches reveals the real part of the overlap. One run supplies one binary sample; repeated preparations estimate the probability and hence the expectation value. A single outcome does not reveal a general real number exactly. A Hermitian Pauli operator is a useful choice of $U$, while the imaginary-part Hadamard test changes the reference phase to obtain the other component.

The controlled-NOT gate is a two-qubit unitary operator that flips a target qubit when its control is in $|1⟩$. In the ordered labels below, $a$ denotes the control, $b$ the target, and $\oplus$ is addition modulo two.

The controlled-NOT gate maps $|a,b⟩$ to $|a,b\oplus a⟩$.

The basis rule extends linearly to quantum superpositions. For example, it sends $|+⟩|0⟩$ to $(|00⟩+|11⟩)/\sqrt{2}$, a Bell state. The control is not first measured: its quantum coherence can become entanglement between the two qubits. Applying the controlled-NOT gate twice returns the original state. ^[ref]

Every set lying between a connected set and its closure is connected.

An arbitrary product topology of connected spaces is connected. For two nonempty factors, fix $y_0\in Y$ and use the connected subsetsThey cover $X\times Y$ and share the connected horizontal slice. This proves the two-factor case, and induction proves every finite-factor case.

For the arbitrary-family case, if the Cartesian product is empty it is already connected. Otherwise fix one tuple $a$. For every finite index set $F$, let $X_F$ consist of tuples that equal $a$ outside $F$. Each $X_F$ is a finite product of connected spaces, and all contain $a$, so their union $D$ is connected. Every nonempty basic open set restricts only finitely many coordinates, and therefore meets $D$. Thus $D$ is dense, and the closure of a connected set proves that the whole Cartesian product is connected.

If connected subsets cover a space and every two of them intersect, then their union is connected. Equivalently, any continuous integer-valued function is constant on each member, and the intersections force all those constants to agree.

A topological space is connected exactly when every continuous map from it to the discrete space $\mathbb{Z}$ is constant. A disconnection produces a nonconstant indicator map, while the continuous image of a connected space in $\mathbb{Z}$ must be a singleton.

If $f:X\to Y$ is a continuous map and $X$ is a connected space, then its image $f(X)$, with the subspace topology from $Y$, is a connected subset of $Y$. The whole codomain $Y$ need not be connected; the assertion concerns the values actually attained. No injectivity is required.^[ref]

For example, the continuous function $f(x)=x^2$ maps the connected closed interval $[-2,2]$ onto the connected closed interval $[0,4]$, even though opposite inputs have the same output. The same continuous function maps the disconnected union $[-2,-1]\cup [1,2]$ onto $[1,4]$. Thus a connected image does not imply a connected source: a continuous map can merge different connected components.

For a continuous function $r:[0,1]\to \mathbb{R}$ representing a continuous reading, $r(0)=0$ and $r(1)=1$ force some input with $r(t)=1/2$. Its image is a connected subset of the real line, hence a real interval containing both endpoint readings. For the synthetic model $r(t)=t^2$, the missing-value test is explicit: $r(1/\sqrt{2})=1/2$. A step reading equal to $0$ for $t<1/2$ and $1$ for $t\ge 1/2$ skips $1/2$, but fails continuity at $t=1/2$.

Proof outline: a separation of a topological space of $f(X)$ into nonempty open subsets pulls back under $f$ to nonempty open subsets of $X$. Their intersection is empty and their union is $X$, contradicting connectedness. This also shows why a continuous map sends each connected component of its source into a single connected component of its target.

A connected component of a topological space $X$ is a nonempty connected subset that is maximal under subset inclusion. If $C$ is a connected component and $C\subseteq D\subseteq X$ with $D$ a connected subset, then $D=C$. Maximality means that no larger connected subset contains $C$; it does not mean largest cardinality among all connected subsets.

For example, the connected components of $[-2,2]\setminus \{0\}$ in the usual subspace topology of the real line are $[-2,0)$ and $(0,2]$. Each is a real interval, and no connected subset of the real line can meet both without containing the missing point $0$. For the finite set $\{-2,0,2\}$ in the same subspace topology, the three connected components are its singleton sets.

Every point $x\in X$ lies in exactly one connected component, constructed by taking the unionAll members contain $x$, so their union is connected by connected union with a connected hub. The union includes every connected subset through $x$, so it is maximal. Equivalently, defineThis is an equivalence relation: singleton sets give reflexivity, symmetry is immediate, and two connected subsets sharing the middle point have a connected union, giving transitivity. The equivalence classes are precisely the connected components. They form a partition of a set: they cover $X$ and any two are either equal or disjoint.^[ref]

Every nonempty connected subset lies within one connected component. Moreover, every connected component is closed in $X$: closure of a connected set makes its closure connected, and maximality forces that closure to equal the original connected component. A connected component need not be open. For example, in $S=\{0\}\cup \{1/n:n=1,2,\dots \}$ with its usual subspace topology from the real line, all connected components are singleton sets, but $\{0\}$ is not open: every neighbourhood of $0$ in $S$ contains some $1/n$.^[ref]

The map $z\mapsto \cosh (\pi z)$ is a conformal bijection fromonto the upper half-plane. Its three boundary intervals map respectively to $(1,\infty )$, $(-1,1)$ and $(-\infty ,-1)$.

If a holomorphic function $f$ satisfies $f^{\prime }(z_0)\ne 0$, then it has a holomorphic local inverse near $z_0$ and is conformal there. Its possible failures of local conformality are therefore its critical points.

Two plane domains are conformally equivalent when there is a bijective holomorphic map between them whose inverse is holomorphic.

In the simply typed lambda calculus, types are generated from base types by the arrow constructor $A\to B$. A context $\Gamma$ assigns types to variables. Its basic rules areThus lambda abstraction creates a function type and lambda application consumes its input type. For example, $\lambda x.x$ has type $A\to A$, and $\lambda f.\lambda x.fx$ has type $(A\to B)\to A\to B$.

The untyped self-application $xx$ would require one finite type to satisfy $A=A\to B$, which is impossible in this type grammar. This excludes the familiar looping term $(\lambda x.xx)(\lambda x.xx)$; more generally the Strong normalization theorem for simply typed lambda calculus says that every reduction sequence from a well-typed term terminates. The typing rules also correspond to implication rules through the Curry-Howard correspondence.

Disjoint sets $A,B\subseteq \mathbb{N}$ are recursively inseparable when no recursive set $C$ satisfies $A\subseteq C$ and $B\cap C=\varnothing$.