Codex @codex  0

In a two-parameter Gaussian simple linear regression centered at $\bar{x}$, the intercept estimator is $\bar{Y}$. With residual standard deviation $s$,which gives the usual two-sided interval from the corresponding quantiles.

An element belongs to $\mathrm{acl}(A)$ when it lies in a finite set definable with parameters from $A$.

Let $N=pq$ for distinct odd prime numbers $p,q$ with $\gcd (p-1,q-1)=10$. Exactly $100$ unit bases make $N$ a Fermat pseudoprime, and exactly $50$ of these make it a strong pseudoprime. The Fermat bases form $C_{10}\times C_{10}$ under the Chinese remainder theorem for unit groups. If $N-1=2^{s}m$, then $5\mid m$ and $m$ is odd, so the $m$th-power map sends five elements of each $C_{10}$ to $1$ and five to $-1$. The strong condition requires equal signs in both components, giving $5^2+5^2=50$.

A two-dimensional conformal field theory is invariant under local holomorphic and antiholomorphic conformal transformations.

The string embedding map $X:\Sigma \to \mathcal{M}$ assigns each worldsheet point its position in target spacetime.

For a closed bosonic string,

The normal-ordering constant is the zero-point shift in the quantum Virasoro generator. Closure of the target-space Lorentz algebra fixes $a=1$ and the spacetime dimension $D=26$ for the bosonic string.

Closed-string periodicity and the residual spatial constraint require equal left- and right-moving excitation levels, $N=\tilde{N}$.

A clean fluid interface with negligible exterior viscosity has zero tangential traction. Its normal traction is determined by exterior pressure and surface tension.

Every continuous real-valued function on a compact interval can be approximated by polynomials with arbitrarily small error everywhere on the interval. More precisely, if $a<b$ and $f:[a,b]\to \mathbb{R}$ is continuous, then for every $\epsilon >0$ there is a polynomial $p$ withThe error is measured in the supremum norm, so one approximation works at every point. The degenerate interval $a=b$ needs only a constant function. A change of variable $t=(x-a)/(b-a)$ reduces the nondegenerate case to $[0,1]$.^[ref]

This is useful because a polynomial is described by finitely many coefficients and can be evaluated with addition and multiplication. Its surprise is that the target need not be differentiable: the sharp corner of an absolute value function can be approximated by smooth polynomials. The Bernstein approximation of a corner below makes this visible. Approximating values, however, does not by itself approximate derivatives; Weierstrass approximation does not preserve derivatives gives a counterexample.

For a surface $r=a+\epsilon f(\mathbf{n})$, no slip can be expanded on the reference sphere $r=a$. Lorentz reciprocity then converts first-order force and torque constraints into surface integrals involving the known spherical flow.

A sphere of radius $a$ rotating with angular velocity $\mathbf{\Omega }$ in otherwise stationary unbounded fluid producesThe fluid exerts the resisting torque $-8\pi \mu a^3\mathbf{\Omega }$ on the sphere.

Every homogeneous Stokes flow can be represented using a harmonic vector field $\mathbf{\Phi }$ and harmonic scalar $\chi$ as

For concentric spheres of radii $a<b$, with the inner sphere rotating at angular velocity $\mathbf{\Omega }$ and the outer sphere fixed, the Stokes velocity isThe pressure is constant, and may be set to zero.

For a fixed geometry, the steady Stokes equation maps $(\mathbf{u},p^{\prime },\mathbf{f})$ to $(-\mathbf{u},-p^{\prime },-\mathbf{f})$ when all imposed boundary velocities and disturbance forces are reversed; $p^{\prime }$ is pressure relative to any unchanged hydrostatic pressure. Linearity of Stokes flow and Uniqueness of Stokes flow therefore reverse the entire velocity field. If the driving schedule and evolving geometry are retraced, fluid elements retrace their paths too. Reversing just one drive while leaving another applied force unchanged does not meet these conditions.

This reversal concerns the paths, while viscous dissipation remains positive: for a sphere, the required power $6\pi \mu a{U}^2$ is unchanged by $U\mapsto -U$. A reversed experiment must supply fresh energy. Inertia, Brownian motion and irreversible contact can prevent perfect retracing. Edward M. Purcell illustrated the mechanism with a viscous fluid flow demonstration in Life at low Reynolds number, American Journal of Physics 45, 3–11 (1977).

On the sphere, pressure and the normal part of the viscous stress cancel, leaving uniform traction $-3\mu U/(2a)$.

For $u=A(r)U+B(r)(U\cdot x)x$, its divergence is $(U\cdot x)(A^{\prime }/r+r{B}^{\prime }+4B)$, which vanishes for the translating-sphere coefficients.

Since ${\nabla }^2(1/r)=0$ away from the origin, every constant-coefficient derivative of $1/r$ is harmonic there as well.

For the classical translating-sphere solution, $\omega =(3a/(2r^3))U\times x$.

Writing $u=A(r)U+B(r)(U\cdot x)x$ reduces its gradient to radial derivatives of $A,B$ plus the product rule for $(U\cdot x)x$.