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Applying the amplitude damping channel to the Bloch vector form of a qubit density operator gives an affine map. Transverse components contract by $\sqrt{1-p}$; the longitudinal component contracts by $1-p$ and shifts toward the ground-state north pole. Reading the entries of the output density operator proves the map. The displacement distinguishes this channel from centered depolarization.

The Kraus formula for entanglement fidelity applies to any input, including mixed qubit states. For the amplitude damping channel, the diagonal Kraus trace is $[1+s_z+\sqrt{1-p}(1-s_z)]/2$ and the jump Kraus trace has squared modulus $p(s_x^2+s_y^2)/4$. Summing them gives the formula. Ground-state inputs have unit entanglement fidelity; excited-state inputs have fidelity $1-p$.

For a local basis $e_a$ of the tangent bundle, an affine connection has components defined by ${\nabla }_{e_c}{e}_b={\Gamma }^a{}_{bc}{e}_a$. If $V=V^b{e}_b$, then $({\nabla }_{e_c}V{)}^a=e_c(V^a)+{\Gamma }^a{}_{bc}{V}^b$. The connection components depend on the chosen basis and have an inhomogeneous transformation law; they are not themselves a tensor field. In a coordinate basis, a torsion-free connection has symmetry in the two lower indices.

In a commutative diagram of two kernel rows, a monic arrow between the final objects makes the square between their kernels and middle objects a pullback in a category. The kernel universal property factors any categorical cone uniquely. Conversely, if the square between middle and final objects is a pullback, the induced arrow between their kernels in a category is an isomorphism, since both kernels have the same universal property after pullback.

Let $B=0$ with probability $\pi$, and otherwise let $B$ have a Gamma distribution with mean one and variance $\tau$. Given $B$, counts $Y_j$ are conditionally independent Poisson random variables with means $B{\mu }_j$. For $q=1-\pi$, the law of total variance and law of total covariance giveEach component has a zero-inflated negative binomial distribution, but the components are not independent after marginalizing $B$. The zero component is shared by the whole profile. At $\tau =0$, interpret the positive Gamma component as a point mass at one.

For a vertex $(g/c_W){\gamma }^{\mu }(c_V-c_A{\gamma }^5)$ into one massless color-singlet particle-antiparticle pair, the massive-vector spin average and two-body Lorentz-invariant phase space give $\Gamma =g^2{m}_Z(c_V^2+c_A^2)/(12\pi c_W^2)$. With Standard Model neutral-current vector and axial couplings, one neutrino flavor has ${\Gamma }_{\nu }=g^2{m}_Z/(96\pi c_W^2)$ and one charged-lepton flavor has ${\Gamma }_{\ell }={\Gamma }_{\nu }(1-4s_W^2+8s_W^4)$. Chiral projectors already remove inactive neutrino helicities. These are tree-level massless results.

For a Young tableau $t$, let $r_t$ sum its row permutations and $c_t$ be the signed sum of its column permutations in the group algebra of the symmetric group. The product $h_t=c_t{r}_t$ is a Young symmetrizer. The other order is another conventional realization. It satisfies $h_t^2=H_{\lambda }{h}_t$, where $H_{\lambda }$ is the hook product of a partition, so division by that scalar gives a primitive idempotent in characteristic zero. Acting on a tensor power constructs a Schur module.

The color-contracted density ${ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }^a{F}_{\rho \sigma }^a$ is even under charge conjugation and odd under parity. In matrix form the two charge-conjugation signs cancel and transposition leaves the trace of the product unchanged. Parity contributes its orientation-reversing determinant to the four-index Levi-Civita symbol. Thus the density is CP odd. A generic fixed coefficient violates CP symmetry; being a local total derivative does not eliminate effects from nontrivial gauge topology.

Start traps at a unit-intensity Poisson random measure in Euclidean space and attach mutually independent Brownian motions, independent of the initial measure. For a deterministic target path $f$, independently thin initial positions according to whether their marked path comes within radius one of the target by time $t$. The retention probability $p_t(x)$ integrates, by the Tonelli theorem, to the expected volume of the sausage along $f-B$. Symmetry of Brownian motion makes this the expected volume of the sausage along $f+B$. The Poisson zero-count probability proves the displayed survival formula. This derives the formula directly from independent thinning, without assuming a separate marking or displacement theorem.

For four independent odd Grassmann variables $a,b,c,d$, $(bc-ad{)}^2=-2abcd$, whereas $(\psi \psi )(\chi \chi )=(-2cd)(-2ab)=4abcd$. This proves the displayed two-component identity and fixes its sign. It is not valid with the same interpretation for commuting spinors, whose self-contractions vanish.

For the normalized velocity-reset collision operator on the weighted Hilbert space, $L=\Pi -I$ vanishes on the equilibrium direction and equals $-I$ on its orthogonal complement. The spectral gap is exactly one. Conservation of the projected component and the displayed energy identity give exact exponential decay of the remaining component; the full solution need not decay to zero.

If two nonzero entire functions have identical zeros and orders, their quotient extends to a nowhere-zero entire function. Its logarithmic derivative has a primitive on the simply connected plane, producing a holomorphic logarithm $h$ and the displayed relation. Any two choices differ by one constant integer multiple of $2\pi i$. On a multiply connected domain the logarithm can fail to exist globally.

An entire function of finite order factors as its canonical zero product times the exponential of a polynomial. For order at most one and nonzero value at zero, it has the form $f(z)=f(0)e^{Bz}{\prod }_j(1-z/a_j)e^{z/a_j}$. Genus-one factors converge when ${\sum }_j|a_j{|}^{-2}<\infty$. A zero-free function of this order is therefore the exponential of an affine polynomial.

At fixed total particle number, $N_0=N-{\sum }_{k\ne 0}{b}_k^{†}{b}_k$. Expanding the condensate interaction energy subtracts $n{V}_0$ from the normal excited-particle coefficient, cancelling its direct Hartree term. The resulting quadratic coefficients are $A_k={ϵ}_k+n{V}_k$ and $B_k=n{V}_k$, with an opposite-momentum anomalous pair term. Replacing $N_0$ by $N$ before making this correction loses the cancellation.

The stable quadratic spectrum is $E_k=\sqrt{{ϵ}_k({ϵ}_k+2n{V}_k)}$. If $V_k\to V_0>0$, its low-momentum form is $\hslash |k|\sqrt{n{V}_0/m}$, a linear sound branch. For contact repulsion the healing length separates the phonon and particle regimes. General momentum-dependent interactions can change the higher-energy shape.

The quasiparticle vacuum contains physical particles at nonzero momentum because the Bogoliubov transformation mixes creation and annihilation operators. Their occupation is $v_k^2=(A_k/E_k-1)/2$, and the depletion density is its momentum sum divided by the volume. At nonzero temperature an additional $(A_k/E_k)n_B(E_k)$ contribution is required; quantum depletion alone is a zero-temperature expression.

Conservation of particle number corresponds to global circle group phase rotations of a nonrelativistic Bose field. A positive-density uniform mean-field saddle chooses a phase, with degenerate saddle manifold a circle. The associated Goldstone boson is the long-wavelength phase mode. This is a thermodynamic symmetry-selection statement; an exact finite-volume number eigenstate does not itself choose a phase.

In units $\hslash =1$, polar fields rewrite the coherent-state action asafter removing the periodic total density derivative. The potential has positive-density minimum ${\rho }_0=\mu /g$ for $\mu >0$. Density curvature gives a finite static amplitude cost, while phase changes cost only derivatives. Number-phase conjugacy couples their dynamics; the two fields do not automatically represent two independent excitation branches.

On càdlàg paths on a compact time interval, convergence in this topology means that increasing continuous bijections of the time interval can be chosen converging uniformly to the identity, while the corresponding time-changed paths converge uniformly to the limit. At a continuous limiting path this reduces to uniform convergence. It is a standard topology for weak limits of accelerated discrete-time chains, whose paths are step functions.

A scaling limit of random curves is a weak convergence of probability measures after spatial rescaling, usually modulo monotone reparameterization. Tightness specifies the topology and yields subsequential limits; identification of their laws gives a unique continuum model. SLE is a central family of such limiting curve laws.