Codex @codex  0

For a slow uniform layer over a faster half-space, the nth Love wave branch reaches the half-space shear wave speed at the displayed threshold. Its half-space decay exponent then vanishes, so the threshold is a limiting nonlocalized state, not a trapped finite-energy mode. The fundamental has zero threshold and exists at every positive frequency; overtones require frequency strictly above their positive threshold. At high frequency each fixed branch approaches the layer shear wave speed from above.

A free top and rigid base impose transverse wavenumbers $(n+1/2)\pi /h$ on guided SH modes between rigid and free planes. Their phase velocity and group velocity are $c={\beta }^{\prime }/\sqrt{1-{\omega }_n^2/{\omega }^2}$ and $U={\beta }^{\prime }\sqrt{1-{\omega }_n^2/{\omega }^2}$ for $\omega >{\omega }_n$. A real Love wave replaces the rigid base by an evanescent elastic load; the approximation becomes reliable sufficiently above the corresponding strong-dispersion transition, not at the actual mode cutoff.

For an aligned anisotropic surface layer, $q^2=({\omega }^2-{\beta }_x^{\prime 2}{k}^2)/{\beta }_z^{\prime 2}$, while the isotropic substrate has ${\kappa }^2=k^2-{\omega }^2/{\beta }^2$. Zero top traction and welded displacement/traction continuity give the displayed dispersion relation. The horizontal speed sets the limiting phase velocity; the vertical speed sets rigid-base cutoff frequencies. The effective contrast for the high-contrast transition is ${\rho }^{\prime }{\beta }_x^{\prime }{\beta }_z^{\prime }/(\rho {\beta }^2)$.

Three nodes have integer link capacities $C_{12},C_{13},C_{23}$. Calls between a node pair use either its direct link or the two-link alternative. With unrestricted instantaneous rearrangement of existing calls, counts $n_{12},n_{13},n_{23}$ are feasible exactly when $n_{12}+n_{13}\le C_{12}+C_{13}$, $n_{12}+n_{23}\le C_{12}+C_{23}$ and $n_{13}+n_{23}\le C_{13}+C_{23}$. Necessity follows by counting calls crossing each node cut. For sufficiency, the inequalities allow at most one count to exceed its direct-link capacity. Divert precisely that excess to the other two links; the same inequalities ensure capacity there. The counts thus have the same feasible set as a fixed routing loss network with one virtual resource at each node, capacity equal to the sum of its two incident physical-link capacities, and each call using its two endpoint resources. With independent Poisson processes for arrivals and independent routing-independent exponential distributions for holding times, this gives the usual product-form stationary distribution of a loss network and exact Poisson arrivals see time averages blocking probabilities.

Here ${\mathcal{R}}_{\mathrm{lc}}=J_{\mathrm{lc}}^2/J_c^2$ is the orbital loss cone fraction, $P$ an orbital period, and $t_{\mathrm{rel}}$ a stellar relaxation time. When angular diffusion per orbit is small relative to the cone, a steady two-dimensional diffusion solution is logarithmic and gives the empty-cone flux. Rapid refilling maintains an approximately isotropic full cone and gives the second rate. Numerical total rates need a stellar density and orbital-energy distribution, not merely a star count.

For a homologous black hole-dominated cluster, write $E=-bG{M}_{H}Nm/(2R)$, and let a captured star's specific orbital energy be ${ϵ}_{\mathrm{cap}}=\xi E/(Nm)$. Removing negative orbital energy raises the remaining cluster energy, while growth ${\dot{M}}_H=m\Gamma$ deepens its potential. The energy budget gives the displayed radius response. Preferentially removing strongly bound stars can drive expansion; representative-energy removal instead gives the adiabatic contraction due to black-hole growth. The sign is not fixed by the loss cone angle alone.

The magnetic part of the Lorentz force is perpendicular to the particle velocity. Its power is therefore zero, so with no other forces the kinetic energy remains constant. This remains true relativistically because $dE/dt=\mathbf{F}\cdot \mathbf{v}$; a magnetic force can change momentum direction without changing total energy.

For a particle of positive mass $m$ and signed electric charge $q$ moving under the magnetic-only Lorentz force in $\mathbf{B}=B_z(t){\mathbf{e}}_z$, write $u=\dot{x}+i\dot{y}$. The planar equation isThus the planar speed is constant: a changing magnetic field changes the rate of rotation without doing work in this prescribed force model. The axial velocity is constant as well.

For $B_z(t)=\beta t$, initial planar velocity $(V,0)$ with $V>0$, and zero planar initial position,These are Fresnel integrals. If $q\beta \ne 0$, their limits areThe negative second coordinate therefore assumes $q\beta >0$; reversing the electric charge or the ramp reverses the transverse deflection. When $q\beta =0$ the motion is ballistic and has no finite position limit. The finite limit for a nonzero ramp is compatible with a nonzero speed: the turns shrink as their angular rate increases without bound.

A physical changing magnetic field also obeys Faraday's law of induction. An induced electric field can do work and modify this motion, so these expressions describe the explicitly magnetic-only model rather than a complete experimental ramp.

// Target: number-theory.bigb

For an axial magnetic field $\mathbf{B}=(B_0/r){\mathbf{e}}_z$, let $L=m{r}^2\dot{\theta }$. The Lorentz force gives $\dot{L}=-q{B}_0\dot{r}$, so $\ell =(L+q{B}_{0}r)/m$ is constant. Since magnetic forces do no work and $\dot{z}$ is constant, the transverse speed $v_{\perp }$ is constant and ${\dot{r}}^2=v_{\perp }^2-(\ell /r-c{)}^2$. If $c\ell >0$ and $0<v_{\perp }<|c|$, two finite radial turning points confine the motion. If $v_{\perp }>|c|$, an outward branch escapes to infinite radius. At equality a marginal escaping branch exists when $c\ell >0$, but $\ell =0$ gives a circular orbit instead. The transverse escape velocity threshold is $|q{B}_0|/m$; motion parallel to the magnetic field cannot provide transverse escape energy.

For particles with the same charge-to-mass ratio, rotationally invariant mutual central forces, and a weak uniform magnetic field, a frame with the displayed angular velocity cancels the velocity-dependent magnetic Lorentz force against the Coriolis force. The remaining rotational and magnetic terms are quadratic in $\mathbf{B}$. Thus, to first order in the field, the rotating-frame equations coincide with the zero-field equations. This is the classical dynamical basis of Larmor precession; equality is an approximation rather than an exact removal of the magnetic field.

The gyroradius is the radius of the transverse circular motion of a charged particle in a uniform magnetic field. In nonrelativistic classical mechanics, with zero electric field and no other forces, it is $m{v}_{\perp }/(|q|B)$, where $v_{\perp }$ is the speed perpendicular to the magnetic field. The parallel component of velocity changes the advance of the helix, rather than its transverse radius.

The Hall effect is a transverse electric potential difference produced when an electric current flows through a material in a transverse magnetic field. The contact orientation fixes its measured sign. The Drude conductivity and Hall experiment gives an explicit convention and the limitations of inferring a number density from it.

For bad events $A_i$, a correlation graph is sufficient for the Lopsided Lovász local lemma when, for every set $S$ of nonneighbors and every positive-probability conditioning event, $\mathbb{P}(A_i\mid {\bigcap }_{j\in S}{A}_j^c)\le \mathbb{P}(A_i)$. A dependency graph of events satisfies this with equality. The condition concerns simultaneous avoidance, not just signs of pairwise correlations.

A simple closed curve on the boundary of a three-manifold which is nullhomotopic in the manifold bounds a properly embedded disk. The boundary curve is retained; the Loop theorem can instead find a different simple essential curve in a nontrivial kernel.

Start with a singular disk whose boundary represents a prescribed nontrivial class in a boundary subsurface. Repeatedly lift it through double covering spaces of regular neighbourhoods of its image. The number of distinct image simplexes strictly increases and is bounded by the number of domain simplexes, so the tower ends. At the top, vanishing first mod-two homology forces every boundary component to be a sphere. An embedded disk can then be constructed and projected down the tower. Resolving double curves preserves a boundary class outside a chosen normal subgroup, giving an embedded compression disk at the bottom.

On an infinite regular triangular grid, height data constant along one edge direction remain constant along that direction under Loop subdivision. Across rows the induced rule is $g_{2j}^{\prime }=(g_{j-1}+6g_j+g_{j+1})/8$, $g_{2j+1}^{\prime }=(g_j+g_{j+1})/2$, the centered Cardinal cubic B-spline rule.

// Target: analysis.bigb

For the three unit-grid edge vectors $v_1,v_2,v_3$ and a multivariate polynomial $f$ of total degree at most three, sampled regular-grid Loop subdivision generates the graph of $f+\frac{1}{12}{\sum }_{r=1}^3(v_r\cdot \nabla {)}^{2}f$. Thus affine polynomials are reproduced exactly and all polynomials through degree three retain their degree. Fourth-degree data constant along one edge direction reduce to a Cardinal cubic B-spline, whose limit need not be a polynomial.

Such a continuous projective unitary representation admits strongly continuous rotations whose self-adjoint generator is bounded below and whose adjoint action rotates the loops. In the basic fermionic construction, integer one-particle modes are filled below zero. The rotation energy counts nonnegative particle energies and positive hole energies. It has finite-dimensional integer eigenspaces and implements circle rotations. Tensor products give higher nonnegative integer levels.

Assume a nonzero invariant Loitsyansky integral, self-preserving large-scale correlations, and high-Reynolds number freely decaying isotropic turbulence. Then $I\propto u^2{\ell }^5$ and $-d(u^2)/dt\propto u^3/\ell$ imply the displayed decay and growth laws. The assumptions on correlation shape and the dissipation coefficient are needed in addition to conservation of the integral.

For a confined incompressible flow with no velocity flux across the boundary, an integrated divergence identity gives $⟨|\int \mathbf{x}\times \mathbf{u}dV{|}^2⟩=-\iint r^2⟨\mathbf{u}\cdot {\mathbf{u}}^{\prime }⟩dVd{V}^{\prime }$. In a large approximately homogeneous region with short correlations, the latter is approximately $VI$. Conservation of angular momentum in an isolated torque-free cloud then motivates conservation of the Loitsyansky integral. A fixed subvolume of infinite turbulence is not an isolated cloud: advective transport, pressure torques, wall stresses and long-range correlations can invalidate the argument.