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For a fixed centered design matrix $X$, independent errors of variance ${\sigma }^2$, and objective $\Vert Y-X\beta {\Vert }^2+a\Vert \beta {\Vert }^2$, put $G=X^{T}X$ and $M=G+aI$. The ridge regression estimator hasThus its standard error alone does not justify centering a confidence interval for $\beta$ at the biased estimator. A penalty chosen from the responses also changes its sampling distribution.

A sphere of radius $a$ at gap $aϵ$ translates at $U{\mathbf{e}}_x$ and rotates at $\omega {\mathbf{e}}_y$ above a stationary wall in a right-handed coordinate system. At leading order its gap is $h=aϵ+r^2/(2a)$. In its translating frame the lower and upper tangential velocities are $-U{\mathbf{e}}_x$ and $-a\omega {\mathbf{e}}_x$. The Reynolds lubrication equation givesIntegrating the upper-boundary traction, including pressure on the sloping surface, and its moment about the sphere centre gives the sphere-on-fluid force and torqueThe off-diagonal symmetry is required by the Lorentz reciprocal theorem for Stokes flow. A torque-free sphere therefore has $a\omega =U/4$ and leading drag $3\pi \mu aU\log (1/ϵ)$. Compare the rigid-wall terms of Bertin et al., equations (4.4)–(4.5); those forces act on the sphere and have the opposite sign.

For $s\ge 2$ and $\delta ={\delta }_s(A)<1/3$, noisy basis pursuit has error $\Vert \hat{x}-x_0{\Vert }_2\le C_1\eta +C_2{\sigma }_s(x_0)/\sqrt{s}$, with constants depending only on $\delta$. One admissible pair isThese constants follow from Theorem 3.3 of the sharp restricted-isometry recovery analysis, with the actual noise bounded by the tolerance $\eta$. The restriction $s\ge 2$ is necessary: equal unit columns give ${\delta }_1=0$ without unique recovery. The coefficient of the approximation term cannot universally be one.

For a transitive set $X$, use $L_0(X)=X$, $L_{\alpha +1}(X)=\mathrm{Def}(L_{\alpha }(X))$ and unions at nonzero limit ordinals. The successor operation is the definable power set. Each level is transitive and the levels increase; moreover $X\in L_1(X)$. Their union is $L(X)$, an inner model of ZF. The initial-set convention differs from a hierarchy using a predicate, or one starting at $X\cup \{X\}$. Axiom of choice need not hold for an arbitrary base without an internally available well-order.

An inner model $W$ of ZF containing all ambient subsets of $\omega$ computes the same countable ordinals as the ambient universe. Every ambient countably infinite ordinal has a well-order code coded by a subset of $\omega$, hence that code belongs to $W$; finite ordinals are already shared. The Mostowski collapse theorem inside $W$ recovers the actual ordinal and its countability witness. Conversely any countability witness in $W$ remains one outside. If $W$ also contains the full power set of $\omega$ as a set, its Continuum hypothesis would give an ambient bijection from ${\omega }_1$ onto that same power set. Thus an ambient failure of this well-orderable formulation of the Continuum hypothesis is preserved in such an inner model. No internal axiom of choice is required for this argument.

During matter domination, the growing mode under a reduced clustering strength $1-f$ isIt grows more slowly than the standard $t^{2/3}$ mode whenever $0<f<1$.

The closed cone of curves is the closure in $N_1(X{)}_{\mathbb{R}}$ of the convex cone generated by classes of integral curves. A nef divisor pairs nonnegatively with this entire closed cone. Its boundary can contain limiting classes which are not represented by one curve.

The nef cone is the closed convex cone of nef divisor classes. It is dual to the closed cone of curves. Its interior is the ample cone on a projective scheme.

A connected locally finite graph is recurrent if its simple random walk returns to its starting vertex almost surely. In that case the walk hits every fixed vertex almost surely from every starting vertex.

On a $d$-dimensional Hilbert space, let $X|j⟩=|j+1\mathrm{mod}d⟩$ and $Z|j⟩=e^{2\pi ij/d}|j⟩$. The $d^2$ unitaries $W_{k,m}=X^k{Z}^m$ form an orthogonal operator basis under the Hilbert-Schmidt inner product.

The Gaussian model $Y=X\beta +Zb+\epsilon$ with $b\sim N(0,D)$ and independent $\epsilon \sim N(0,{\sigma }^{2}I)$ has marginal law $N(X\beta ,ZD{Z}^T+{\sigma }^{2}I)$. Its group coefficients are integrated out of the marginal likelihood function. The special random-slope model uses predictor values in the grouping columns of $Z$.

A binary star of total mass $M$, reduced mass $\mu$ and separation vector $\mathbf{a}$ has far-field potential functionwhere $\theta$ is the angle between $\mathbf{x}$ and $\mathbf{a}$. The dipole term vanishes in centre of mass coordinates. Averaging a circular coplanar binary over its fast phase gives $\overline{\Phi }=-GM/x-G\mu a^2/(4x^3)$.

For continuous semimartingales and $Q\ll P$, their quadratic covariation has the same version under the two probability measures, up to $Q$-indistinguishability of stochastic processes. The dyadic product sums have the same limit by uniform convergence on compacts in probability under an absolutely continuous measure change. Uniqueness of a limit in probability identifies that limit with the bracket under $Q$. Semimartingale stability under an absolutely continuous measure change supplies the correct class under $Q$: a local martingale need not remain a local martingale. Taking the two processes equal proves the same statement for quadratic variation without a separate assumption.

Two continuous local martingales are orthogonal when their quadratic covariation is zero. With zero initial values, the martingale product identity says that their product is a local martingale. For two Brownian motions in a common filtration, orthogonality and the Lévy characterization of multidimensional Brownian motion make the pair a two-dimensional Brownian motion and imply independence. Merely having the two marginal Brownian motion laws on the same space does not imply orthogonality.

For a proposal $Y=AX+Z$, with isotropic $Z\sim N(0,I)$ and an independent random orthogonal matrix $A$, invariance of the distribution of $A$ under transpose makes the proposal distribution symmetric. Indeed $|y-Ax|=|x-A^{T}y|$, and averaging the isotropic Gaussian density over an inversion-invariant law yields $q(x,y)=q(y,x)$. The Metropolis–Hastings algorithm therefore accepts with the target density ratio alone.

If an abelian subgroup $A$ has index $p^a$, the centralizer of any $1\ne g\in A$ contains $A$, so the class size of $g$ divides $p^a$. The prime-power conjugacy-class obstruction to simplicity rules this out. If $A=1$, the group is a p-group and has nontrivial center of a group, also ruling out nonabelian simplicity.

Every proper ideal in a Noetherian ring is a finite intersection of primary ideals. This existence theorem does not give uniqueness of the individual components: embedded primary components may vary.

The inner Shakura--Sunyaev thin disk is supported mainly by radiation pressure and has nearly constant electron-scattering opacity. Equating radiative diffusion with local dissipation and imposing vertical hydrostatic equilibrium gives the characteristic half-thicknessThe radiation-supported height approaches a plateau away from the inner boundary. The formal zero at that boundary is outside the reliable radiation-dominated approximation. At an Eddington accretion rate defined using efficiency $0.1$, this height can make the inner flow only marginally thin.

For an unstable Taylor–Goldstein equation mode in a finite channel, take $a=1$ in the power-transformed Taylor–Goldstein energy identity. Its imaginary part givesThus the real phase velocity lies strictly between the extremes of a nonconstant smooth shear profile. Equality would force the regular eigenfunction to vanish on an interval, hence everywhere by uniqueness for its ordinary differential equation.