Place one vertex of a tetrahedron at the origin and denote the other vertex vectors by $a,b,c$. Orthogonality of two pairs of opposite edges of a tetrahedron forces $a\cdot b=a\cdot c=b\cdot c$. It follows that the third pair is orthogonal too. The three sums of squared lengths of opposite edges all equal $|a{|}^2+|b{|}^2+|c{|}^2-2a\cdot b$. This gives a dot product test for the special geometry without solving for the face planes.

For a stationary open migration process with local total departure rate ${\varphi }_i(n_i)$ and traffic rates ${\alpha }_i$, adjacent product-weight ratios give the displayed reversed parameters. The traffic equations of an open migration process make each reversed routing row sum to one, while total external arrival rates agree in the two directions. Thus the reversed process has the same local departure functions and is again an open migration process.

An open is positive when its positivity truth value holds. Positivity gives a constructive witness of inhabitedness and must not be replaced by double-negated nonemptiness.

A positive open locale is totally connected when every positive open of a locale is connected. Under classical logic its positive opens form a completely prime filter, hence specify a dense point. Positivity of the whole locale is essential: the empty locale satisfies the condition on positive opens vacuously but has no point.

For $j=o(U)$, the dense monomorphism classifier consists of the truth values above $u$, so has least element $u$. Conversely, if it has least element $u$, density of a monomorphism is exactly containment of the pulled-back subterminal $U$. For a subobject $S\subseteq X$, the mono $S\to \overline{S}$ is dense, so $U\cap \overline{S}\subseteq S$ and $j(p)\le u\Rightarrow p$. Since $j(u)=\top$, applying $j$ to $u\wedge (u\Rightarrow p)\le p$ gives $u\Rightarrow p\le j(p)$. Hence equality holds.

Choose $N$ points of $S^d$ such that every $d+1$ of their position vectors are linearly independent. Any equatorial hyperplane then contains at most $d$ points. If $N-d\ge 2r-1$, at least one of each opposite pair of open hemispheres contains $r$ points. Such configurations exist: take distinct real parameters $t_i$ and normalize $(1,t_i,\dots ,t_i^d)$; the Vandermonde determinant proves the required independence. This weaker one-of-two guarantee already suffices for the topological lower bound in Lovász theorem on Kneser graphs.

Differentiate the planar rotation matrix at angle zero:The even and odd powers in the matrix exponential therefore giveThis exact formula gives the whole one-parameter subgroup. Taking $X=J$ means unit angular velocity: at $t=\pi /2$, $\exp (tJ)=J$ sends $(1,0{)}^T$ to $(0,1{)}^T$, a counterclockwise quarter-turn. At $t=\pi$, it equals $-I$; at $t=2\pi$, it returns to $I$. Choosing $X=2J$ reaches the quarter-turn at $t=\pi /4$. Changing the generator changes the speed, while preserving the circular path.

For a moving vector $v(t)=R_{t}v(0)$, the ordinary differential equation $v^{\prime }(t)=Jv(t)$ says its velocity is perpendicular to its position. The beautiful compression is that one constant matrix specifies every angle of this endless family of rotations.

For $PD(\theta )$, $\mathbb{E}{\sum }_i\phi (p_i)=\int \phi d{\alpha }_{\theta }$ whenever the absolute integral is finite. Apply the Mecke formula for a Poisson random measure to a selected jump $u$ and the remaining total $t$, whose density is $t^{\theta -1}{e}^{-t}/\Gamma (\theta )$. Changing variables to $s=u+t$ and $x=u/s$ factors out a Gamma function and gives the displayed measure. Weighting it by $x$ gives a probability density $\theta (1-x{)}^{\theta -1}$, the Beta distribution of a size-biased component.

This density is supported where all $x_i>0$ and ${\sum }_i{x}_i<1$, and gives the expected sum over ordered distinct components of $PD(\theta )$. Apply the iterated Mecke formula for a Poisson random measure to $m$ selected jumps $u_i$ and the remaining gamma total $t$. With $s=t+{\sum }_i{u}_i$ and $x_i=u_i/s$, the Jacobian is $s^m$ and the integral in $s$ is $\Gamma (\theta )$, giving the density. It describes joint counts of large components, beyond the first-moment measure.

For $1/2\le a\le 1$, at most one component exceeds $a$, so $1-F_{\theta }(a)=\theta {\int }_a^1{x}^{-1}(1-x{)}^{\theta -1}dx$. Below one half several components can exceed the threshold. Put $N_a=\#\{i:p_i>a\}$ and use the finite identity $1_{\{N_a=0\}}={\sum }_m(-1{)}^m(N_a{)}_m/m!$, with the factorial moment densities of Poisson-Dirichlet components. It gives the full probability distribution as finite inclusion-exclusion over $x_i>a$, ${\sum }_i{x}_i<1$. In contrast, a size-biased component is beta-distributed.

With $A_D=S_D{\Lambda }^{ϵ}/(2\pi {)}^D$, the one-loop renormalization-group beta function is ${\beta }_g=-ϵg+(n-2)A_D{g}^2$. For $n>2$ and $ϵ>0$, its two nonnegative renormalization-group fixed points are zero and $g_{\ast }=ϵ/[(n-2)A_D]$. Differentiating the beta function gives the displayed thermal eigenvalues, so the nonzero point has correlation-length critical exponent $\nu =1/ϵ$ to leading order. At $ϵ=0$ the zero eigenvalue requires nonlinear analysis: $g$ is marginally relevant for $n>2$, but the $n=2$ perturbative spin-wave theory has a line of fixed points and omits vortices.

For $d=4-ϵ$, the scalar one-loop determinant has the displayed local divergent potential. If $V$ is a quartic polynomial, then $V^{\prime \prime }$ has degree at most two and $(V^{\prime \prime }{)}^2$ has degree at most four. Its subtraction therefore uses the original polynomial couplings and a vacuum-energy term. A polynomial potential of degree $r>4$ instead produces degree $2r-4>r$, requiring new couplings. This is a direct potential-level illustration of power counting in quantum field theory and renormalizability.

At quadratic order, $\mu d\lambda /d\mu =-ϵ\lambda +3{\lambda }^2/(16{\pi }^2)$ for $ϵ>0$. Solving the linear equation for $1/\lambda$ gives the displayed expression. If $0<\lambda ({\mu }_0)<{\lambda }_{\ast }$, the ultraviolet limit is zero and the infrared limit is ${\lambda }_{\ast }$. Initial data above ${\lambda }_{\ast }$ reach a Landau pole at a finite ultraviolet scale. The interacting value is the Wilson-Fisher fixed point along the quartic coupling direction; the full scalar theory also has a thermal relevant direction at the Wilson-Fisher fixed point, so a critical infrared limit requires mass tuning.

For a positive weak quartic scalar field theory coupling in four dimensions, the one-loop renormalization-group beta function is $\beta =a{\lambda }^2$, $a=3/(16{\pi }^2)>0$. Integration gives the displayed inverse-coupling law, so $\lambda$ tends to zero logarithmically as $\mu \to 0$ and grows toward the ultraviolet. The interacting Wilson-Fisher fixed point of $4-ϵ$ dimensions merges with the Gaussian fixed point at $ϵ=0$. This describes the quartic direction in the massless or critically tuned infrared regime, not disappearance of every finite-energy interaction in a massive theory. The ultraviolet Landau pole signals breakdown of that perturbative extrapolation; one loop alone does not prove a nonperturbative triviality theorem.

With interaction $-{\lambda }_0{\varphi }^4/4!$, the tadpole diagram has symmetry factor one half. After Wick rotation and a rotationally invariant Euclidean cutoff, its self-energy is $({\lambda }_0/2){\int }_{|k|<\Lambda }{d}^{4}k(2\pi {)}^{-4}(k^2+m_0^2{)}^{-1}$. The sphere area $2{\pi }^2$ and radial integral ${\int }_0^{\Lambda }{k}^{3}dk/(k^2+m_0^2)=\frac{1}{2}[{\Lambda }^2-m_0^2\log (1+{\Lambda }^2/m_0^2)]$ give the displayed shift. It is independent of external momentum, so no one-loop momentum-dependent wave-function renormalization arises from this graph. Different regulator schemes can treat power divergences differently while defining the same chosen pole mass.

With equatorial qubit measurement vectors $(|0⟩+(-1{)}^s{e}^{i\eta }|1⟩)/\sqrt{2}$, the gate in one-bit teleportation is $U(\eta )=H\mathrm{diag}(1,e^{-i\eta })=J(-\eta )$. Direct multiplication gives $U(\eta )Z=XU(\eta )$ and $U(\eta )X=e^{-i\eta }ZU(-\eta )$. Moving a Pauli frame through this gate therefore exchanges its $X$ and $Z$ exponents and reverses the angle if the old $X$ exponent is one, as displayed up to global phase. A measurement outcome $s$ adds a further $X^s$ factor. This calculation supplies adaptive angle choices and the final classical output-bit correction of a linear cluster state computation.

For circular Couette flow with $v=Ar+B/r$, the rotating-frame reduction of circular viscoelastic shear gives shear rate $g=-2B/r^2$. The total shear stress is $\mu g$, and the azimuthal minus radial normal-stress difference is $2(\mu -{\mu }_r)\tau g^2$ in the stress convention where ${\sigma }_{rr}=-p$. Radial force balance without inertia is $p^{\prime }=-2(\mu -{\mu }_r)\tau g^2/r$, which integrates to the displayed pressure. The pressure constant is undetermined by these equations.

In an inertialess cone-and-plate rheometer, reverse a steady torque from T to -T at time zero. The polymer shear stress remains continuous while the positive solvent viscosity allows the rate to jump. The linear stress equation gives ${\lambda }_2={\mu }_0\tau /({\mu }_0+G_0\tau )$ and the displayed velocity for t>0, where ${\Omega }_{+}=3\alpha T/[2\pi a^3({\mu }_0+G_0\tau )]$. The transient backward speed exceeds the final backward speed because stored polymer stress initially retains its former sign.

Put $S={\sigma }^d-2{\mu }_{r}D$ in the Oldroyd-A model. It obeys ${\mathcal{D}}_{l}S+S/\tau =2(\mu -{\mu }_r)D/\tau$. Pulling back as $F^{T}SF$ converts this to an ordinary linear relaxation equation. With decayed remote-past initial memory, its solution isAt a finite initial time, include the homogeneous transported term $e^{-(t-t_0)/\tau }F(t{)}^{-T}F(t_0{)}^{T}S(t_0)F(t_0)F(t{)}^{-1}$. The integral-only law is not equivalent to all arbitrary initial-value solutions without a memory condition.

For constant simple shear flow rate $s$, the nonpressure stress in the component-first gradient convention $L_{12}=s$ isThus the first normal-stress difference is $N_1=2(\mu -{\mu }_r)\tau s^2$, while the second is $N_2=-N_1$. Adding the incompressibility pressure changes neither difference.