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.

Volatility clustering means that large absolute observations tend to be followed by further large absolute observations, and small ones by small ones. In an ARCH process, persistence is in the conditional variance rather than necessarily in the signed observations.

For ${\alpha }_0>0$ and $0<{\alpha }_2<1$, the model $X_t=\sqrt{{\alpha }_0+{\alpha }_2{X}_{t-2}^2}{\epsilon }_t$ splits into independent even-time and odd-time chains in its stationary causal solution. With standard normal noise, $\mathbb{E}{X}_t^2={\alpha }_0/(1-{\alpha }_2)$, and its fourth moment is finite exactly when $3{\alpha }_2^2<1$.

With symmetric driving noise whose sign is independent of its magnitude, changing the sign of one driving variable changes the corresponding $X_t$ but leaves future conditional scales unchanged. Consequently $\mathrm{Cov}(X_t,f(X_{t+h}))=0$ for $h>0$ and square-integrable $f(X_{t+h})$. This conclusion uses symmetry, beyond the martingale difference sequence property.

An automatic structure for a group consists of a finite generating alphabet $A$, a surjective representative regular language $L\subseteq A^{\ast }$, and finite-state automata accepting the padded convolutions of words $(u,v)$ in $L$ satisfying $\overline{u}x=\overline{v}$ for each $x\in A\cup \{\epsilon \}$. Symmetry of $A$ and uniqueness of representatives are additional conditions, not part of this definition.