The reduced BCS pairing Hamiltonian retains scattering of opposite-momentum, opposite-spin Cooper pairs:A mean-field approximation introduces the BCS anomalous average and pairing field ${\Delta }_k=-{\sum }_{k^{\prime }}{V}_{k,k^{\prime }}⟨b_{k^{\prime }}⟩$. The subtraction $-\sum V_{k,k^{\prime }}⟨b_k^{†}⟩⟨b_{k^{\prime }}⟩$ corrects the double counting of the mean-field interaction energy.

The BCS anomalous average is the pair expectation $⟨b_k⟩=⟨c_{-k\downarrow }{c}_{k\uparrow }⟩$. In the BCS ground state it is $u_k{v}_k={\Delta }_k/(2E_k)$ for real gap at zero temperature. At temperature $T$ it becomes ${\Delta }_k\tanh (E_k/(2k_{B}T))/(2E_k)$. Substitution into the pairing-field definition gives the BCS gap equation.

For real gap and the particle-hole block $\left(\begin{array}{cc}{\xi }_k& -{\Delta }_k\\ -{\Delta }_k& -{\xi }_k\end{array}\right)$, the BCS coherence factors obey $u_k^2=(1+{\xi }_k/E_k)/2$, $v_k^2=(1-{\xi }_k/E_k)/2$ and $u_k{v}_k={\Delta }_k/(2E_k)$, where $E_k=\sqrt{{\xi }_k^2+{\Delta }_k^2}$. They diagonalize the Bogoliubov--de Gennes Hamiltonian and give the empty/pair amplitudes of the BCS ground state. Their relative sign depends on the pairing convention.

For an original check, write $x={\xi }_k/\Delta$ with constant real $\Delta >0$. At $x=0$, the BCS coherence factors give $u_k^2=v_k^2=1/2$. At $x=1$, they give $(u_k^2,v_k^2)=(0.853553,0.146447)$; at $x=-1$ these values exchange places. Thus the zero-temperature occupation $v_k^2$ rounds the normal Fermi sea's step over an energy interval of order $\Delta$. This mixing is strongest at the Fermi surface, where the BCS quasiparticle spectrum has its minimum.

The BCS ground state is the normalized product of empty and occupied pair states ${\prod }_k(u_k+v_k{b}_k^{†})|0⟩$. For $u_k^2+v_k^2=1$, each pair factor is normalized. The inverse Bogoliubov transformation annihilates it, so it is the vacuum of positive-energy quasiparticles. Although individual factors use opposite momenta, $k$ runs over the relative pair momenta; it is not restricted to $k=0$.

A Cooper pair is a correlated pair of fermions associated with superconductivity. The conventional BCS ground state uses opposite-momentum, opposite-spin pair creation operators $c_{k\uparrow }^{†}{c}_{-k\downarrow }^{†}$. Their total momentum is zero for every relative momentum $k$; zero total momentum does not restrict $k$ to zero.

Data augmentation enlarges a posterior distribution with latent variables whose removal by marginalization recovers the original posterior distribution. Convenient full conditional distributions then permit a Gibbs sampler on the enlarged space.

Variational inference approximates a posterior distribution by minimizing a Kullback-Leibler divergence over a chosen family of probability distributions. For an unnormalized posterior density $h$ and a trial probability density function $q$, minimizing ${\mathbb{E}}_q[\log q-\log h]$ is equivalent to minimizing $\mathrm{KL}(q\Vert h/\int h)$.

BUGS specifies a probabilistic model through stochastic nodes, deterministic nodes and observed data, then uses Markov chain Monte Carlo to simulate its Bayesian posterior. Its normal sampling notation uses a precision parameter rather than a variance, and its gamma notation uses shape and rate.

A Bayesian Nash equilibrium consists of type-dependent strategies such that every type maximizes its conditional expected payoff given the other players' strategies and the type distribution. It is a Nash equilibrium in the strategic description whose actions are whole contingent strategy functions. Auction incentive comparisons can test a deviation by imitating another type's equilibrium bid.

For Lasso loss $\Vert Y-X\beta {\Vert }_2^2/(2n)+\lambda \Vert \beta {\Vert }_1$ and $Y=X{\beta }^0+\eta$, stationarity with a penalty subgradient gives $\Vert X\delta {\Vert }_2^2/n\le {\eta }^{T}X\delta /n+\lambda (\Vert {\beta }^0{\Vert }_1-\Vert \hat{\beta }{\Vert }_1)$, $\delta =\hat{\beta }-{\beta }^0$. This follows by multiplying the Karush-Kuhn-Tucker conditions by $\delta$ and using $z^T\hat{\beta }=\Vert \hat{\beta }{\Vert }_1$ and $z^T{\beta }^0\le \Vert {\beta }^0{\Vert }_1$.

A basic feasible solution is degenerate when one or more of its basic variables is zero. A simplex algorithm pivot may then change the basis without moving the feasible point or changing the objective. Assignment networks force many zero-flow basic tree arcs.

A coasting fluid has pressure $p=-\rho /3$. The cosmological perfect-fluid continuity equation makes its density scale as $a^{-2}$. If it alone fills a flat expanding universe, the scale factor grows linearly in physical time. Its background density has the same scale-factor dependence as a spatial-curvature term, although the fluid does not itself create spatial curvature.

The invertible elements of a unital Banach algebra form an open multiplicative group. For $a\in G(A)$, every $b$ satisfying $\Vert a^{-1}\Vert \Vert b-a\Vert <1$ is invertible, because $b=a(1+a^{-1}(b-a))$ and the second factor has a Neumann series inverse.

For an algebra of complex-valued functions with a complete algebra norm, every point value $f(t)$ belongs to the spectrum of an element of $f$ in the artificial unitization. Indeed $(h,\lambda )\mapsto h(t)+\lambda$ is an algebraic unital character of an algebra, and it vanishes on $f-f(t)1$. The spectrum bound gives $|f(t)|\le \Vert f\Vert$ without first assuming that evaluation is continuous.

A commutative Banach algebra is semisimple when its Jacobson radical is zero. For a complex unital Banach algebra this is equivalent to injectivity of the Gelfand transform. This condition concerns the Jacobson radical; it does not assert a finite product decomposition as in finite-dimensional semisimple algebra theory.

For three vectors $a,b,c$ in an inner product space, expansion of each squared norm provesTherefore the second-order backward differentiation formula $3U^{n+2}-4U^{n+1}+U^n=2kL{U}^{n+2}$ with a dissipative operator $L$ decreases the discrete energy ${\mathcal{E}}_n=\Vert U^{n+1}{\Vert }^2+\Vert 2U^{n+1}-U^n{\Vert }^2$. This quadratic form is equivalent to the product norm of the two time levels with constants independent of $k$ and $L$. It proves uniform stability even when amplification roots of a polynomial coalesce inside the unit disk, where an eigenvector-separation proof may lose its bound.

For equal-width time bins and infections assigned to bin-end time $t_i$, let $q_{\ell }=\mathrm{Pr}((\ell -1)\Delta \le T<\ell \Delta )$. The next onset-bin mean is ${\mu }_k={\sum }_{i<k}{h}_i{q}_{k-i}$. Assigning infections instead to bin starts shifts the index by one. Bin probabilities integrate a delay density; they are not the density itself.

If $V_{\lambda }$, for an infinite cardinal number $\lambda$, satisfies Axiom schema of replacement for every external functional class, then an external cofinal function from an ordinal below $\lambda$ cannot exist: its range would have rank of a set $\lambda$ and would have to belong to $V_{\lambda }$. Similarly, a surjection $\mathcal{P}(\theta )\to \lambda$ with $\theta <\lambda$ is impossible. Infinity therefore makes $\lambda$ an uncountable regular cardinal and a strong limit cardinal. Full semantics for second-order logic, allowing arbitrary external functional relations, is essential.

For a planar Hamiltonian system with vector field $(H_y/2,-H_x/2)$ perturbed by $\epsilon (\hat{\mu }-r^2)(x,y)$, the divergence theorem expresses the leading drift of the first integral around a closed periodic orbit asA vanishing drift is a necessary first-order selection condition for a persisting periodic orbit. A simple zero with outward drift on its inner side and inward drift on its outer side yields an attracting limit cycle. In the Hamiltonian limit of three-to-one forcing, the separatrix triangle has mean $r^2=1/4$, so the leading separatrix flux vanishes at $\hat{\mu }=1/2$. A homoclinic or heteroclinic transition still requires the separatrix splitting and higher-order corrections to be controlled.

For minimal causal and invertible models, the partial autocorrelation function cuts off after the order of a pure autoregressive model, whereas the autocorrelation function cuts off after the order of a pure moving-average model. Estimated cutoffs suggest candidate orders; finite-sample noise and residual diagnostics still matter.