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Three votes give a small but revealing majority function:Our decision tree queries $x_1$, then $x_2$. If these bits agree, return their common value: two equal votes already force the majority function. If they disagree, query $x_3$ and return its value. The first two queries have a fixed order, but whether a third query occurs is adaptive.

Each internal label in the diagram names the next bit to inspect. Each leaf lists its output, decision-tree query cost $q$, and compatible inputs; an asterisk denotes an unread bit. The two shallow leaves each cover two inputs, while each deeper leaf covers one. Reading every bit would also compute the majority function, but would miss these exact shortcuts.

All eight inputs give the following exhaustive check. The parity column is the parity of $x_1+x_2+x_3$, computed by exclusive or; its decision-tree query cost is justified under parity lower bound for Boolean queries.

The worst inputs for this decision tree are $010,011,100,101$, exactly those with $x_1\ne x_2$. Its actual depth is three. Every exact decision tree for this majority function has depth at least three: after the first query, an adversary can answer the second distinct query with the opposite bit, leaving the last bit decisive. Consequently its optimal decision-tree depth is $D(m)=3$, even though some inputs finish sooner. Every input has a two-bit query certificate, namely two agreeing votes; locating those votes can require a third query. This separates having a short query certificate from finding one.

Let the input alphabet be $\{0,1,2\}$ and let the output be $Y=X+Z$, where independent noise is uniform on two distinct integers. Translating the output alphabet reduces the noise to $\{0,d\}$, $d\ge 1$. The reachable output alphabet has size $3+\min (d,3)$: it is the union of two three-element intervals shifted by $d$.

For $d=1$, every row of the transition matrix has conditional information entropy one bit, so $I(X;Y)\le {\log }_{2}4-1=1$. Input probabilities $(1/2,0,1/2)$ make the output uniform and attain one bit per use. For $d\ge 3$, the two translated input alphabets are disjoint and every output identifies its input. A uniform input then attains ${\log }_{2}3$ bits per use.

For $d=2$, reflection symmetry and concavity of information entropy permit restricting to input probabilities $(s/2,1-s,s/2)$. Conditional on output parity, outputs whose labels are even integers have probabilities $(1/4,1/2,1/4)$ and outputs whose labels are odd integers have probabilities $(1/2,1/2)$. Consequently $I(X;Y)=h_2(s)+s/2$. Its strictly negative second derivative and stationary equation ${\log }_2((1-s)/s)+1/2=0$ give $s=\sqrt{2}/(1+\sqrt{2})$ and channel capacity ${\log }_2(1+\sqrt{2})$. All three values follow from overlap of translated input alphabets, rather than from output alphabet size alone.

Three identical quadrupole quanta occupy the third symmetric power of the five-dimensional spin-two space. Its dimension is $\left(\genfrac{}{}{0ex}{}{7}{3}\right)=35$, and its irreducible representations have $J=0,2,3,4,6$, each once. All have positive parity. This counts allowed angular states in the ideal harmonic nuclear quadrupole vibration; anharmonic interactions can split their energies.

Consider an original hypothetical atom with a lower angular momentum multiplet $F=1$, $m=-1,0,+1$, and one opposite-parity excited level $F^{\prime }=0,m^{\prime }=0$. Stipulate that all spontaneous emission returns to these three lower quantum states, with no other decay channel. This is an isolated four-state model, not the actual sodium or rubidium D-line structure. In particular, the physical rubidium cell above contains multiple hyperfine structure levels.

The electric-dipole selection rule permits $\Delta F=-1$ here; ${\sigma }^{+}$ absorption with $\Delta m=+1$ excites only $m=-1$. Decay from $F^{\prime }=0$ has three equal integrated branching probabilities $b_{-}=b_0=b_{+}=1/3$: rotational symmetry gives equal squared Clebsch-Gordan coefficients. The emitted photons carry the three permitted angular momentum projections; equal integrated branching does not imply equal intensity in every observation direction. Thus $m=0$ and $m=+1$ are both dark. The chosen beam cannot prepare a unique pure $m=+1$ state.

Let $R\ge 0$ be the absorption rate of an atom in $m=-1$, and let $\Gamma$ be the excited-state spontaneous emission rate, both in inverse seconds. Assume $R,\gamma \ll \Gamma$, negligible stimulated emission, and no maintained quantum coherence between sublevels. Eliminating the short-lived excited population gives normalized lower populations $p_{-}+p_0+p_{+}=1$ to this approximation. At each scattering event the atom leaves $m=-1$ with probability $2/3$; the remaining $1/3$ simply returns it there.

Model depolarizing collisions and other population relaxation by resets to the unpolarized distribution at rate $\gamma \ge 0$. The rate equations areThis relaxation prescription assumes that Zeeman splitting is small compared with thermal energy, so the unpolarized distribution approximates thermal equilibrium. It is not an energy-independent relaxation law at arbitrarily low temperature.

Every transfer has a nonnegative rate: optical transfers $-\to 0$ and $-\to +$ each have rate $R/3$, and relaxation transfers between each distinct pair have rate $\gamma /3$. These define a continuous-time Markov chain. The three derivatives sum to zero, and at $p_i=0$ the corresponding derivative cannot be negative. Hence the model conserves population and preserves nonnegative populations. Optical cycles that return to $m=-1$ contribute to scattering even though they cause no net population transfer.

A finite graph has a perfect matching if and only if the displayed inequalities hold, where $o(H)$ counts the connected components of a graph $H$ with an odd integer number of vertices. The theorem is associated with William Thomas Tutte.^[ref]

Necessity has a direct counting proof: each such component of $G-S$ must send at least one edge of a perfect matching to $S$, and different components must use different vertices of $S$. The sufficiency direction is the substantive content of the theorem. The criterion is useful because vertex degree and parity estimates can prove existence without constructing the perfect matching, as in the perfect matching theorem for a cubic graph with at most two bridges.

absorption of two photons whose summed energy matches an atomic transition. In the weak-field regime its rate scales with the product of the two intensities. Two electric-dipole interactions preserve the initial parity.

The vector-minus-axial-vector charged weak current contains the left-chiral projector $P_L=(1-{\gamma }^5)/2$. It violates parity because parity interchanges left and right chiral fields. For a massless particle chirality fixes helicity, but a massive charged fermion's helicity need not be pure even when the vertex selects a chiral field. Nuclear beta decay uses the nonrelativistic reduction of vector and axial nucleon currents, giving Fermi and Gamow-Teller operators in the allowed limit.

A spin-one parent cannot decay into two identical spin-zero particles in the same internal state while preserving rotational symmetry and Bose exchange symmetry. Their final total angular momentum is orbital $L=1$, which is antisymmetric under interchange. Bose symmetry requires even $L$ instead. An energetically allowed decay with matching parity can therefore remain forbidden independently of the interaction's coupling strength.

A meson with total angular momentum one and negative parity. In the ordinary quark model, a quark-antiquark pair with orbital angular momentum zero and total spin one has these quantum numbers; its neutral self-conjugate states have charge conjugation $C=-1$. The light-flavour tensor product gives an octet and a singlet, forming a vector nonet, just as spin-zero negative-parity pairs form a pseudoscalar meson nonet. Nonzero orbital excitations can also produce vector states.

The Wigner-Eckart theorem separates the dependence of a spherical tensor operator on angular momentum projections from its dynamical content. In one standard convention,The reduced matrix element on the right is independent of $m,m^{\prime },q$. The Clebsch-Gordan coefficients impose $m^{\prime }=m+q$ and $|j-k|\le j^{\prime }\le j+k$; additional selection rules, including parity, can make the reduced matrix element vanish.

To see why the factorization holds, combine the rank-$k$ transformation law of the operator with the spin-$j$ transformation law of the initial state. This gives an intertwining operator from the tensor product representation $j\otimes k$ to the final representation $j^{\prime }$. The Clebsch-Gordan decomposition contains each allowed $j^{\prime }$ once. Therefore the corresponding intertwining operator is unique up to an overall scalar, which is the reduced matrix element after the chosen normalization. Ratios within the same pair of multiplets depend only on Clebsch-Gordan coefficients, whenever the common reduced matrix element is nonzero.

For a smooth symmetric confining potential energy with four separated simple classical turning points $-x_2<-x_1<x_1<x_2$, defineThe Airy turning-point connection formula applied to a decaying exterior wavefunction gives a central WKB approximation proportional to ${\chi }^{-1/2}[2\cos \theta e^K+\sin \theta e^{-K}]$, where $\chi =\sqrt{2m(V-E)}$ and $K(x)={\hslash }^{-1}{\int }_x^{x_1}\chi dx$. Since ${\chi }^{\prime }(0)=0$ and $2K(0)=\varphi$, the even Neumann boundary condition gives $\tan \theta =2e^{\varphi }$, while the odd Dirichlet boundary condition gives $\tan \theta =-2e^{\varphi }$. For $\varphi \gg 1$, $\theta =(n+\frac{1}{2})\pi \mp \frac{1}{2}{e}^{-\varphi }+O(e^{-3\varphi })$, with the minus correction for even parity. Since ${\theta }^{\prime }=\mathcal{T}/(2\hslash )$ for the single-well classical oscillation period $\mathcal{T}$, the leading double-well tunneling splitting is $2\hslash e^{-\varphi }/\mathcal{T}$.