For a real integrable random variable $X$, a conditional expectation $Y=\mathbb{E}[X\mid \mathcal{G}]$ with the same probability distribution as $X$ must equal $X$ almost surely. No second moment is needed. For any real $c$, the nonnegative random variable $D_c=|X-c|-\mathrm{sign}(Y-c)(X-c)$ has expected value zero, by conditioning and equality of the two absolute moments. Thus $Y>c$ implies $X\ge c$, $Y<c$ implies $X\le c$, and $Y=c$ implies $X=c$, outside a null set. Using a countable family of rational $c$ separates any unequal real values and proves the claim. Equality of the probabilities of $X=c$ and $Y=c$ also gives equality of their zero sets and hence equality of the two sign functions for every fixed $c$.

Prepare a qubit in $|+⟩$, apply a phase gate $P(\varphi )=\mathrm{diag}(1,e^{i\varphi })$, and correlate its computational basis states with normalized environment states $|e_0⟩,|e_1⟩$. Write $⟨e_0|e_1⟩=v{e}^{i\alpha }$. After a final Hadamard gate, the environment amplitude accompanying output zero is $(|e_0⟩+e^{i\varphi }|e_1⟩)/2$. Its squared norm is $P_0=[1+v\cos (\varphi +\alpha )]/2$. Thus $v$ is the interference visibility, while $\alpha$ shifts the fringe. This quantum decoherence map commutes with the phase gate, so placing it immediately before that gate gives the same probabilities.

A regular conditional distribution of a random variable $X$ given $Y$ is a Markov kernel $K$ such that, for all measurable sets $A,B$,For each $y$, $K(y,\cdot )$ is a probability measure; for each $B$, $K(\cdot ,B)$ is measurable. Thus it assigns an actual probability distribution to a specified observation even when $P(Y=y)=0$. Such kernels exist for standard Borel state spaces, including real-valued variables, and their uniqueness is only up to $P_Y$-null sets. With a joint density and positive marginal density, the ratio of the two densities defines the usual conditional density kernel. Values outside the marginal support or on a null set are not uniquely determined by the joint law.

A conditionally convergent real series can be rearranged to converge to any prescribed real value or to diverge. Its positive and negative subseries have infinite total magnitudes: successively add positive terms to cross a desired value and negative terms to cross back. Since the terms tend to zero, the overshoots shrink to zero.

Talagrand's concentration inequality gives strong concentration in product spaces using convex distance. If $A$ has positive product probability, its convex-distance enlargement has Gaussian tails. A Talagrand concentration inequality for certifiable functions follows by bounding this distance using bounded differences and small certificates.

Equip the symmetric group $S_n$ with its uniform probability measure and unnormalized Hamming distance. If $f$ has Lipschitz constant $K$, it obeys the displayed two-sided bound, and each one-sided tail has the same exponential without the factor two. Reveal a uniform permutation one image at a time. Two possible next images have completion sets paired by swapping those two values, changing exactly two coordinates. Thus the corresponding conditional means differ by at most $2K$. Each increment of the Doob exposure martingale has conditional range length at most $2K$. Applying the conditional Hoeffding lemma gives $\mathbb{E}{e}^{\lambda (f-\mathbb{E}f)}\le e^{n{\lambda }^2{K}^2/2}$; optimizing the exponential Markov bound proves the bound. A mere absolute-increment bound of $2K$ gives a weaker constant, so using the range length matters.

For a mixture of density operators, the lower bound follows from $S(\bar{\rho })-{\sum }_i{p}_{i}S({\rho }_i)={\sum }_i{p}_{i}D({\rho }_i\Vert \bar{\rho })\ge 0$, with equality precisely when all positive-weight states coincide. For the upper bound, resolve each state into pure eigenvectors and form the Gram matrix of the resulting weighted vectors. Its nonzero eigenvalues are those of $\bar{\rho }$, while its diagonal has probabilities $p_i{\lambda }_{ia}$. Averaging diagonal unitary conjugations deletes its off-diagonal entries; Concavity of Von Neumann entropy therefore gives the bound. Equality holds exactly when positive-weight component supports are mutually orthogonal.

For $0<b<1$ and distinct density operators, the Von Neumann entropy of $\rho =b{\rho }_1+(1-b){\rho }_2$ is strictly larger than the corresponding average. The difference is $bD({\rho }_1\Vert \rho )+(1-b)D({\rho }_2\Vert \rho )$, positive by nonnegativity of quantum relative entropy unless both states coincide. On the closed interval $0\le b\le 1$, equality holds exactly when $b=0$, $b=1$, or ${\rho }_1={\rho }_2$.

The lower conjugate uses an infimum, unlike the supremum defining a convex conjugate. For any $F$, $F_{\ast }(\tau )=-(-F{)}^{\ast }(-\tau )$ and $F(e)\le \tau :e-F_{\ast }(\tau )$. The inequality is informative only when the lower conjugate is finite. If the infimum is attained uniquely and the dual is differentiable, $D{F}_{\ast }(\tau )$ is its minimizing argument. This sign convention is useful for upper comparison-energy bounds.

The nine-qubit Shor code encodes one logical qubit with basis vectors $|0_L⟩=|G_{+}{⟩}^{\otimes 3}$ and $|1_L⟩=|G_{-}{⟩}^{\otimes 3}$, where $|G_{\pm }⟩=(|000⟩\pm |111⟩)/\sqrt{2}$. Six adjacent $Z$-pair stabilizer generators locate a single bit flip within a block, while $X_1\cdots X_6$ and $X_4\cdots X_9$ diagnose which block has had its relative phase changed. Together they correct every single-qubit Pauli operator, hence every single-qubit error by the Pauli expansion of a quantum error.

A Pauli Z gate on qubit four has phase-check eigenvalues $(-1,-1)$ and leaves all six $Z$-pair checks positive. Applying $Z_4$ restores the state. The same phase syndrome arises from $Z_5$ or $Z_6$; these have the same action on the code since their pairwise products are stabilizers. This is a degeneracy, not a failure of correction.

There is a two-colour computable colouring of pairs with no infinite computably enumerable set homogeneous for it. Enumerate these sets as $W_e$. In stages, give each eligible requirement two markers from its enumerated set, avoiding markers of higher priority, and cancel all lower-priority markers whenever a new assignment is made. Colour $(x,y)$ by the marker status of $x$ at stage $y$: first markers have colour zero, second markers colour one, and unmarked points colour zero. Each requirement is injured only finitely often by induction on its index. If $W_e$ is infinite it eventually has permanent distinct markers $a_e,b_e$; for every sufficiently large $y\in W_e$, the pairs $(a_e,y)$ and $(b_e,y)$ have different colours. Simulating finitely many stages computes each pair's colour, proving both effectiveness and the obstruction.

The sign quotient of $S_5$ controls its cyclic composition factors, while its unique nontrivial proper normal subgroup $A_5$ supplies the nonabelian factor. There are four composition chains in $C_2\times S_5$: the three maximal normal subgroups are the kernels of the nonzero maps to $C_2$, and only $C_2\times A_5$ has two choices for its next term. In $S_5\times S_5$ there are eight chains: six pass through $A_5\times A_5$ and one of its two simple direct factors; two instead pass through a single $S_5$ direct factor. The equal-sign subgroup gives two of the six chains. It has no quotient $A_5$, since its odd diagonal element acts by an outer automorphism on either alternating factor.

Use a fixed right-handed Cartesian basis in three dimensions and active rotations in three dimensions through $\pi /2$ radians. The right-hand rule gives the following rotation matrices about the positive $x$ and $z$ axes:For the vector $v=(1,1,0{)}^T$, the two orders giveBoth outputs have Euclidean norm $\sqrt{2}$, but they point in different directions. Each arrow below is a vector from the origin; the curved paths track its endpoint during each rotation in three dimensions.

The axes in this example remain fixed throughout. A command to turn about an object's own current axis describes a different operation because that axis can move after the first turn. With an object's orthonormal basis represented by columns of $Q$, a fixed-axis rotation matrix $R$ acts by $RQ$, whereas the corresponding rotation expressed in the object's own basis acts by $QR$.^[ref]

For $c\ne 0$ and real $r$, this equation defines a straight line. Writing $z=x+iy$ and $c=u+iv$ gives $2ux+2vy+r=0$, so $c$ supplies a normal direction. Conversely every real affine line $Ax+By+C=0$ has this representation with $c=(A+iB)/2$ and $r=C$. The condition $c\ne 0$ excludes an empty set or the entire complex plane.

Represent Cartesian coordinates $(x,y)$ in the plane by the complex number $z=x+iy$, whose real part and imaginary part recover them. Multiplication by $e^{i\theta }$ rotates the coordinates through angle $\theta$, so a rotating-frame calculation becomes a complex exponential change of variable.

Given a compact planar set $K$ and an open neighborhood $V$, a finite square-grid construction gives a compact polygonal neighborhood inside $V$ with finitely many complementary components. Remove thin polygonal corridors disjoint from $K$ joining holes that lie in the same component of $\mathbb{C}\setminus K$, and joining holes in its unbounded component to the exterior. Corridors can also include a prescribed point from each of finitely many specified bounded components. There remains a neighborhood whose bounded holes lie in distinct bounded components of $\mathbb{C}\setminus K$, each containing its chosen point. Standard small polygonal rounding gives a finite union of planar surfaces with boundary. The index map from its first integral homology group to one integer per bounded hole is an isomorphism. Triangulate a surrounding disc: a one-cycle bounds a two-chain whose coefficients are constant in each complementary component. Zero hole coefficients give a filling supported inside the neighborhood; the face chain of each hole supplies a dual cycle. The cycles can be sums of boundary curves in different components. This uses finite triangulation, not the general Jordan curve theorem.

For a proper map with fixed complex normal orientation, apply the Thom isomorphism in complex cobordism, pull back by the collapse of a tubular neighborhood of a graph embedding, and desuspend the auxiliary Euclidean coordinates. The degree increase is the relative real codimension. Changing the orientation can change this map.

If the right-hand side and initial data are jointly holomorphic, the local solution is jointly holomorphic in its independent variable and parameters. On a smaller closed polydisc, bound $|A|$ by $M$ and $|{\partial }_{w}A|$ by $L$. Choose $r$ so that $rM$ stays inside the dependent-variable disc and $rL<1$. The integral contraction mapping has jointly holomorphic iterates converging uniformly, so locally uniform convergence of holomorphic functions proves the assertion. Shifting the dependent variable turns initial values into parameters.

A singular location is fixed if its position is determined by the coefficients, independently of the initial data. Not every solution must actually be singular there. For a rational first-order equation, coefficient poles and exceptional fibres where the reduced numerator and denominator have common zeros are included among the possible fixed locations.

A solution has a movable singularity if its location varies with the initial data. For example $w^{\prime }=w^2$ has solutions $w=-1/(z-c)$, whose pole moves with $c$. The Painlevé determinateness theorem restricts movable singularities of first-order equations rational in the dependent variable to poles and algebraic branch points.