For a discrete group, its group von Neumann algebra is generated on ${\ell }^2(\Gamma )$ by left translations $\lambda (g){\delta }_h={\delta }_{gh}$. The vector ${\delta }_e$ is cyclic and separating, and its vector functional is tracial. Its modular conjugation acts by $J{\delta }_g={\delta }_{g^{-1}}$ with conjugated scalar coefficients. The commutant is the algebra generated by right translations $\rho (g){\delta }_h={\delta }_{h{g}^{-1}}$.

A Von Neumann algebra whose center consists exactly of scalar multiples of the identity. A group von Neumann algebra is a factor exactly when every nonidentity conjugacy class is infinite. A central operator has a square-summable coefficient vector constant on those classes; an infinite class must therefore have coefficient zero. Conversely, the sum of left translations over a finite nonidentity class is a nonscalar central operator.

Here a Hilbert space module means a Hilbert space with a unital star-representation of a Von Neumann algebra, rather than a Hilbert C-star module with an algebra-valued inner product. A closed invariant subspace is reducing because the algebra is closed under adjoints. Module equivalence means an intertwining unitary. The polar decomposition of a bounded operator converts an intertwiner into a unitary between its initial and final support subspaces.

In a $2k$-dimensional normed vector space in John position, two orthogonal $k$-dimensional subspaces admit $\Vert x{\Vert }_E\le |x|\le 32\mathrm{vr}(E{)}^2\Vert x{\Vert }_E$. To see this, take $\epsilon$-nets in two orthogonal half-dimensional unit spheres, each of size at most $(1+2/\epsilon {)}^k$, and rotate them uniformly together. The volume ratio bound makes the probability that any net point has norm below $2\epsilon$ at most $2[4\mathrm{vr}(E{)}^2({\epsilon }^2+2\epsilon ){]}^k$. With $\epsilon =(32\mathrm{vr}(E{)}^2{)}^{-1}$ this is less than $1$. Since $\Vert \cdot {\Vert }_E$ is $1$-Lipschitz for the Euclidean metric, the successful nets give the stated inequality everywhere on both spheres. A dimension-independent constant therefore requires a bound on the volume ratio; it does not hold for arbitrary normed vector spaces.

Voltmeter loading is the change in an electric potential difference caused by the electric current drawn by the measuring voltmeter.

For a loaded voltage divider in steady direct current, a voltmeter connected across the lower resistor places its finite input electrical resistance $R_m$ in parallel with the lower arm's electrical resistance $R_2$ and any intended electrical load of electrical resistance $R_L$. With positive, fixed electrical resistances, an upper resistor of electrical resistance $R_1$, and an ideal voltage source supplying electric potential difference $V_{\mathrm{in}}$, Kirchhoff's current law and Ohm's law giveHere $V_m$ is the connected circuit's electric potential difference across the voltmeter; omit $1/R_L$ if there is no intended electrical load. The parallel connection and need for large input electrical resistance are standard voltmeter operating principles.^[ref]

Voltmeter loading is a bias relative to the target electrical circuit with the voltmeter absent and any intended electrical load still present. Voltage divider loading error instead compares the attached-load output with the wholly unloaded output. An accurate readout of $V_m$, supported by measurement calibration, can still differ from either target: calibration accuracy concerns the indication of the connected electric potential difference, whereas voltmeter loading changes that electric potential difference itself.

For the Volterra integration operator $Af(x)={\int }_0^{x}f(t)dt$, its adjoint is $A^{\ast }g(x)={\int }_x^{1}g(t)dt$. With $a_n=(n-\frac{1}{2})\pi$, the input singular vectors are $\sqrt{2}\cos (a_{n}x)$ and the output singular vectors $\sqrt{2}\sin (a_{n}x)$, with singular values $1/a_n$. The associated Sturm-Liouville problem has Neumann condition at zero and Dirichlet condition at one for the cosine basis; the sine basis has the reversed conditions. These complete orthonormal bases give a singular value system for an injective compact operator with dense range.

The Volterra integration operator $J$ has spectrum $\{0\}$ and is injective. Hence $(\lambda I-i{J}^{-1}{)}^{-1}=J(\lambda J-iI{)}^{-1}$ exists and is bounded for every complex $\lambda$. Empty spectrum is possible for closed unbounded operators, unlike bounded operators on a nonzero complex Banach space.

// Target: analysis.bigb

If bounded intervals have void probabilities $e^{-\lambda (I)}$ for Lebesgue measure $\lambda$, a fixed deterministic point $x$ has zero chance of being an atom. Surround it by intervals of lengths tending to zero and use $\mathbb{P}(M(\{x\})>0)\le 1-e^{-\lambda (I)}$. This holds for each fixed endpoint, not simultaneously for all real points of a nonempty point process.

For $\nu ={\nu }_0(\Sigma /{\Sigma }_0{)}^{1/2}(R/R_0{)}^{9/4}$, the variables $x=\sqrt{R/R_0}$, $S=(\Sigma /{\Sigma }_0)x^3$ and $\tau =3{\nu }_{0}t/(4R_0^2)$ reduce the source-free Keplerian viscous diffusion equation to $S_{\tau }=(x{S}^{3/2}{)}_{xx}$. The displayed self-similar solution has an outer edge $R_e=36R_0\tau$ and a surface-density cusp proportional to $R^{-3/2}$ at the origin. The positive-part cutoff is essential: beyond the edge, $S^{3/2}$ is the cube of the absolute value, so continuing the squared polynomial without truncation does not solve the same equation.

The disk mass is $8\pi {\Sigma }_0{R}_0^2{\tau }^{-1/2}$ and its angular momentum is $12\pi {\Sigma }_0{R}_0^2\sqrt{GM{R}_0}$, independent of time. It therefore describes accretion with outward spreading and conserved disk angular momentum, under the formal zero-torque condition at the origin. The inward mass flux changes sign at $R=9R_0\tau /4$, so the outer portion moves outward while the inner portion accretes.

In a steady Keplerian accretion disk with $\Sigma ={\Sigma }_0{R}^{-2}$, $\nu ={\nu }_0{R}^2$, and tracer diffusivity $k=\zeta \nu$, the trace passive scalar concentration obeys the displayed constant-coefficient advection-diffusion equation in logarithmic radius. Contaminant mass is proportional to $CdY$, because $2\pi R\sigma dR=2\pi {\Sigma }_{0}CdY$. A radial impulse $C_0\delta (R-R_0)$ therefore has logarithmic amplitude $C_0/R_0$ and evolves by the Gaussian heat kernel centered at $\ln R_0-vt$. The full-line solution idealizes radii far from the disk boundaries.

Subtract local specific angular momentum times mass conservation from conservation of angular momentum including wind losses S,T. With outward radial mass flux $\mathcal{F}$ and outward viscous torque in an accretion disk $\mathcal{G}=-2\pi \bar{\nu }\Sigma r^3{\Omega }^{\prime }$, this gives $\mathcal{F}{h}^{\prime }+{\mathcal{G}}^{\prime }=2\pi r(Sh-T)$. For Keplerian rotation and $T=qSh$, substitution in mass conservation yields the displayed equation. A wind with q=1 removes local angular momentum but supplies no additional drift term; q>1 can drive accretion even with zero viscosity.

The Keplerian viscous diffusion equation with the displayed kinematic viscosity has two such nontrivial finite-support self-similar solutions: $a=1$, $R=C{t}^{2/5}$, and $a=5/4$, $R=C{t}^{1/2}$. Both have $\sigma =(15At{)}^{-1}$ for $A,C,t>0$. Matching powers of radius eliminates the third power only when $(5/2-2a)(2-2a)=0$. The first solution conserves disk angular momentum; the second conserves disk mass but requires an inner torque.

The positive rate per unit mass at which molecular viscosity removes turbulent kinetic energy, with $s_{ij}^{\prime }$ the fluctuating symmetric velocity-gradient tensor. Its dimensions are velocity squared per time. The integral-scale estimate is $ϵ\sim q^3/L$, expressing transfer over turnover time $L/q$; numerical constants depend on the convention for the turbulent velocity and length scales.

The Fay solution is a periodic solution of the negative-flux viscous Burgers equation for $Z>0$. Its Fourier series has exponentially decreasing coefficients at each positive $\delta Z$. Expanding $1/\sinh (ns)=2{\sum }_{j\ge 0}{e}^{-(2j+1)ns}$ and summing the sine series givesFor $s\gg 1$, the first harmonic gives $q\sim 4\delta e^{-s}\sin \theta$. The Fay shock-layer asymptotics describe its opposite, small-$s$ regime.

The cubic-flux variant replaces the quadratic conservative flux of Burgers' equation by $-q^3/3$. Its inviscid characteristic speed is $-q^2$, while positive diffusivity regularizes shock waves. It is not linearized by the usual Cole-Hopf transformation.

A travelling wave of the viscous Burgers equation can connect constant states $u_L>u_R$ through a smooth decreasing layer. Require $u(x,t)=U(z)$, $z=x-st$, $U(-\infty )=u_L$, $U(+\infty )=u_R$, and $U^{\prime }\to 0$ at both ends. Substitution and one integration giveThe speed follows by evaluating the integration constant at both end states. It is precisely the Rankine-Hugoniot condition for conservation law flux $f(u)=u^2/2$. Choosing the center $U(0)=s$ gives the displayed hyperbolic tangent profile. Its characteristic width scales as $\nu /(u_L-u_R)$; a smaller positive $\nu$ makes a thinner smooth layer.

For an original worked example in dimensionless quantities, take $u_L=2$, $u_R=0$ and $s=1$:This is an exact classical solution on $\mathbb{R}$ with its own smooth initial condition $u(x,0)=1-\tanh (x/(2\nu ))$ and limiting states $2,0$. It is a family of moving profiles, not the finite-$\nu$ solution starting from an exact discontinuous step. For that different initial condition, the Cole-Hopf transformation supplies the evolution. At $\nu =1/2$, this front starts from the same $1-\tanh x$ profile as the Burgers characteristic breaking example: positive diffusion maintains a smooth moving layer while the inviscid profile breaks at $t=1$.

To check the partial differential equation explicitly, set $w=(x-t)/(2\nu )$ and $S=1-{\tanh }^{2}w$. ThenThe independent Rankine-Hugoniot condition check gives $s=(0-2)/(0-2)=1$, using $f(2)=2$ and $f(0)=0$. At fixed $x\ne t$, the vanishing viscosity approximation tends to the entropy shock $2$ for $x<t$ and $0$ for $x>t$. At the center the smooth profiles stay at $1$; changing the value on the discontinuity does not change the limiting weak solution.

For $u_L<u_R$, the same integrated ordinary differential equation has $U^{\prime }<0$ between the two states when $\nu >0$, so it cannot supply an increasing connecting front. The inviscid admissible expansion is the rarefaction wave of the Burgers Riemann problem with positive flux.

A corotational Jeffreys fluid obeys $\stackrel{\circ }{S}+S/\tau =2{\mu }_0\stackrel{\circ }{D}+2{\mu }_{1}D/\tau$, with deviatoric stress $S$, rate-of-strain tensor $D$, Jaumann derivative and relaxation time $\tau >0$. Put $A=S-2{\mu }_{0}D$. In the corotating-frame representation of a Jaumann derivative, ${\tilde{A}}^{\prime }+\tilde{A}/\tau =2({\mu }_1-{\mu }_0)\tilde{D}/\tau$. An integrating factor gives an exponentially weighted history transported between the rotating frames. The infinite-past formula selects histories with $e^{t/\tau }\tilde{A}(t)\to 0$ as $t\to -\infty$; finite initial data also contribute an exponentially decaying homogeneous term.

The three viscometric functions describe steady homogeneous simple shear flow: its shear viscosity and the first and second normal-stress differences per squared shear rate. Take x as the flow direction, y as the gradient direction and z as the vorticity direction, so $N_1={\sigma }_{xx}-{\sigma }_{yy}$ and $N_2={\sigma }_{yy}-{\sigma }_{zz}$. Isotropic pressure drops out of these differences.

A linear viscoelastic fluid has causal stress response linear in the deformation history, often expressed as ${\sigma }^d(t)=2{\int }_0^{\infty }G(s)E(t-s)ds$. The approximation requires small deformation and rotation over the relevant memory times. High frequency is allowed at sufficiently small amplitude. An integrable decaying relaxation modulus gives a finite zero-shear viscosity; an instantaneous solvent term may be written separately.

Stress relaxation is decay of stored stress after a deformation is imposed and then held fixed. It probes a material's memory through its relaxation modulus. An instantaneous generalized Newtonian fluid has no stored shear stress once the shear rate is zero and cannot describe a nontrivial relaxation tail.