With convention $t^{-1}at=\varphi (a)$, choose representatives of the right cosets of $A$ and $B$ in the base group, with identity representing each subgroup. Normal strings have the form $g_0{t}^{{ϵ}_1}{r}_1\cdots t^{{ϵ}_n}{r}_n$, where $r_i$ represents a coset of $B$ for ${ϵ}_i=1$ and of $A$ for ${ϵ}_i=-1$, and $r_i\ne 1$ when adjacent exponents are opposite. Left multiplication by base elements or stable letters defines permutations of these strings. They satisfy the HNN relations. Applying the induced action to the empty string distinguishes different normal strings, proving uniqueness without presupposing Britton's lemma.

// Target: geometry-and-topology.bigb

For normal $x$, the closed unital C-star subalgebra generated by $x,x^{\ast }$ is commutative. When the ambient spectrum is real, its complement is connected. Membership of the resolvent in this subalgebra is both open and closed on that complement, and holds at large modulus by the Neumann series. Thus the subalgebra spectrum is the same real set. Its isometric Gelfand transform takes $x$ to a real-valued function and $x^{\ast }$ to its conjugate, proving equality. Normality is necessary: the nonzero matrix $\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$ has spectrum $\{0\}$ but is not self-adjoint.

The unital continuous functional calculus has image $C^{\ast }(1,x)$, the norm closure of star polynomials including constants. The smallest closed star subalgebra $C^{\ast }(x)$ need not contain the ambient identity: for $x=0$ it is zero. In a nonunital algebra, apply the unital calculus in the C-star unitization and retain functions vanishing at zero to recover $C^{\ast }(x)$ inside the original algebra.

Since a normal C-star element commutes with its adjoint, words in those two generators can be rearranged into the displayed finite sum. Including the constant term generates the unital star algebra; omitting it generates the nonunital star algebra. Under continuous functional calculus, such a sum is the ordinary function ${\sum }_{j,k}{c}_{jk}{z}^j{\overline{z}}^k$ on the spectrum. The Stone-Weierstrass theorem makes these functions uniformly dense.

Choose an orientation of a vector space on $T_{q}N$. The quotient identification for a regular value gives a smoothly oriented normal bundle along $F=f^{-1}(q)$. An oriented normal frame followed by a frame of $TF$ defines the latter's orientation through the ambient orientation. Changing normal lifts adds only tangent components, so the determinant sign is unchanged. No orientation on all of $N$ is required.

A Continuity correction shifts a discrete Probability boundary by half a lattice spacing before using a continuous normal approximation. For an upper-tail event $T\ge t$ on a unit lattice, the approximation is $1-\Phi ((t-\frac{1}{2}-E(T))/\sqrt{\mathrm{Var}(T)})$. Lower tails use the boundary $t+\frac{1}{2}$. The correction is used in approximate Wilcoxon signed-rank tests and Wilcoxon rank sum tests; tied ranks also require the appropriate variance adjustment.

For a twice continuously differentiable regular curve describing motion, let $v=\Vert r^{\prime }(t)\Vert >0$, $T=r^{\prime }/v$, and let $N$ be the principal normal vector where the curvature $\kappa$ is positive. Differentiating the velocity vector $r^{\prime }=vT$ givesThe tangential component changes the speed; the normal component changes the direction of the velocity vector. A particle can therefore have nonzero acceleration at constant speed, as in circular motion with constant speed. The formula follows from $dT/dt=(dT/ds)(ds/dt)=\kappa vN$ and the product rule. When $\kappa =0$, the normal contribution vanishes even though the principal normal vector is undefined there. ^[ref]

In dimensionless coordinates, take $r(t)=(t,t^2)$. At $t=1$,The resulting vector components areBoth increasing speed and turning contribute to the same acceleration.

For mutually orthogonal projectors summing to the identity, a pinching map discards coherences between their ranges while preserving each diagonal block. It is a quantum channel with Kraus operators $P_i$, and a unital quantum channel. A state is fixed by pinching exactly when it commutes with every projector.

In Synchronous gauge in cosmology with coordinates comoving with cold dark matter, the linearized Einstein field equations and matter conservation give the displayed density equation. Primes denote conformal time derivatives. The radiation term contains its pressure contribution and cannot be dropped on superhorizon scales merely because radiation oscillates after cosmological horizon entry.

Diagonalize two density operators with eigenvalues $r_i,s_j$, and put $q_{ij}=|⟨i|j⟩{|}^2$. This overlap matrix has row and column sums one. The scalar inequality $u\ln (u/v)\ge u-v$ yields $D(\rho \Vert \sigma )\ge {\sum }_{ij}{q}_{ij}(r_i-s_j)=0$ in natural units; division by $\ln 2$ gives bits. Equality forces $r_i=s_j$ whenever $q_{ij}>0$, so $\sigma |i⟩=r_i|i⟩$ and $\rho =\sigma$. Zero eigenvalues use limits, and failure of support inclusion gives infinite quantum relative entropy.

An irreducible Markov chain is recurrent exactly when every nonnegative superharmonic function for a Markov chain is constant. In the recurrent case, absorb the chain at $j$. The inequality $g_i\ge ({\tilde{P}}^{n}g{)}_i\ge g_j{\mathbb{P}}_i({\tau }_j\le n)$ and almost-sure hitting give $g_i\ge g_j$; exchange $i,j$. In the transient case, the hitting probability of a state $j$ is one there and satisfies the superharmonic inequality, but its one-step average at $j$ is the strictly smaller return Probability. It is a nonconstant example.

For $\mathrm{Re}s>1$ this Eisenstein series also equals ${\sum }_{\gamma \in {\Gamma }_{\infty }\backslash S{L}_2(\mathbb{Z})}(\mathrm{Im}\gamma \tau {)}^s$, where ${\Gamma }_{\infty }$ consists of translations and their negatives. It is modular invariant. The nonholomorphic Eisenstein series summed over all nonzero integer pairs instead equals $2\zeta (2s)E(\tau ,s)$: separate each pair into a positive integer times a primitive pair. This normalization matters in Rankin–Selberg unfolding.

This modular-invariant function, initially defined for $\Re s>1$, has Fourier expansionwhere $\Xi (u)={\pi }^{-u/2}\Gamma (u/2)\zeta (u)$ is the completed Riemann zeta function, without the entire-xi polynomial factor. Poisson summation and the integral representation of the modified Bessel function of the second kind prove the expansion. It gives meromorphic continuation, $\mathcal{E}(z,s)=\mathcal{E}(z,1-s)$, and simple poles at zero and one with residues $-1/2,1/2$. Apparent poles at $s=1/2$ cancel between the two constant terms.

Parametrize one ellipse as $\gamma (t)=c+A(\cos t,\sin t)$ and substitute it into the quadratic equation $Q=0$ of the other. At a transverse intersection, $g(t)=Q(\gamma (t))$ has a simple zero, so its sign changes. Periodicity makes the number of zeros even. The function is a trigonometric polynomial of degree at most two, so multiplication by $z^2$, with $z=e^{it}$, gives a nonzero polynomial of degree at most four. Hence there are at most four distinct intersection points. Coincident ellipses are excluded by transversality.

For $Q\sim {\chi }_m^2(\lambda )$ with $m>2$, completing the square in its Gaussian representation gives $\mathbb{E}{e}^{-tQ}=(1+2t{)}^{-m/2}{e}^{-\lambda t/(1+2t)}$. Integrate the identity $1/q={\int }_0^{\infty }{e}^{-tq}dt$ and use Tonelli theorem to obtain the displayed reciprocal moment. It is strictly decreasing in $\lambda \ge 0$, since its positive integrand is strictly decreasing for every $t>0$. At $\lambda =0$ the integral equals $1/(m-2)$, proving finiteness and the sharp maximum over noncentrality parameters.

If an autoregressive model has white noise of positive variance, a unique weakly stationary process solving its equation on all integer times exists precisely when its autoregressive polynomial has no root on the unit circle. In that case $1/A(z)$ has an absolutely summable Laurent series on an annulus containing the unit circle; its coefficients define a bilateral linear filter of a stationary time series. Applying that inverse also proves uniqueness among bounded second-moment solutions. A root on the unit circle forces the candidate spectral density of a stationary process to have a nonintegrable singularity. The stronger causality root criterion for an autoregressive model requires every root outside the closed unit disk.

The real matrix $B_M=\left(\begin{array}{cc}1& M\\ 0& 1\end{array}\right)$ has both eigenvalues equal to one, whereas $x^T{B}_{M}x/(x^{T}x)=1+M/2$ for $x=(1,1{)}^T$. Hence no universal constant bounds this quotient by the maximum eigenvalue of every real matrix with real eigenvalues.

Extend the interior grid vector by zero. With axial weights $2/(3h^2)$ and diagonal weights $1/(6h^2)$, the negative central coefficient equals the total neighbour weight. Expanding the edge-square sum gives the displayed quadratic form. Equality forces all axial differences to vanish, and connection to the zero exterior forces the vector to be zero. Simultaneous row and column permutations preserve symmetry and negative definiteness.

The Jacobi method has axial neighbour weights $1/5$ and diagonal weights $1/20$. Interior row sums are one, boundary-adjacent sums are smaller, and the positive-weight graph is connected. If an eigenvalue had modulus one, a maximal-modulus eigenvector component would force all its neighbours to have the same modulus. Propagation reaches a deficient row, a contradiction. This proves strict spectral-radius control even though the infinity norm of the iteration matrix can equal one.

Let ${\Gamma }_9$ have center weight $-10/3$, axial weights $2/3$ and diagonal weights $1/6$, and let $M_h$ have center weight $2/3$ and axial weights $1/12$. Taylor expansion gives $h^{-2}{\Gamma }_{9}u=\Delta u+h^2{\Delta }^{2}u/12+O(h^4)$ and $M_{h}u=u+h^2\Delta u/12+O(h^4)$. On a smooth solution of the constant-coefficient Helmholtz equation, the second-order term becomes $h^2\Delta (\Delta u+\lambda u)/12=0$. Thus the normalized PDE residual is fourth order, while the original unscaled stencil residual is sixth order. This local cancellation alone does not prove uniform global accuracy at or near a resonance.