For $X=\left(\begin{array}{cc}a& b\\ c& -a\end{array}\right)$, the image of its matrix exponential is compact precisely when $a^2+bc<0$ or $X=0$. If $a^2+bc=-{\omega }^2<0$, then $e^{tX}=\cos (\omega t)I_2+\sin (\omega t)X/\omega$ and its image is conjugate to the circle group. A nonzero nilpotent generator or a generator with real nonzero eigenvalues produces an unbounded image.

For a finite-dimensional memoryless quantum channel,The Holevo bound controls information obtainable from an output ensemble, while collective decoding makes rates below the corresponding coding expression achievable. The Holevo capacity needs regularization because input states can be entangled across channel uses. Watrous's channel-coding chapter states and develops the classical-capacity theorem.

For random proper simple curves from $0$ to infinity in the complex upper half-plane, the probabilities of avoiding every compact H-hull whose closure excludes zero determine the unparameterized trace law. Every point off a simple chordal curve can be connected to a real boundary interval away from zero within one side of its complement. A countable collection of thin rational polygonal tubes attached to that interval therefore recovers the complement from its avoidance indicators. Joint avoidance of finitely many tubes reduces to avoidance of their filled union. If the union blocks every path from zero to infinity the joint event is empty; otherwise filling pockets does not change the avoidance event. The inclusion-exclusion principle gives all finite indicator distributions. These determine the random complement and hence the trace. Simplicity and properness are hypotheses; the assertion is not a general determination theorem for arbitrary random closed sets.

Suppose a random proper simple curve from zero to infinity has the displayed avoidance probability for every admissible compact H-hull $A$, with fixed $\alpha >0$. Put ${\psi }_A=g_A-g_A(0)$. For another admissible hull $B$ in the image complex upper half-plane, map composition and the chain rule give $g_C^{\prime }(0)=g_B^{\prime }(0)g_A^{\prime }(0)$ for $C=A\cup {\psi }_A^{-1}(B)$. The conditional image curve therefore has avoidance probabilities $g_B^{\prime }(0{)}^{\alpha }$, exactly those of the original. Since avoidance probabilities determine a simple chordal curve law, the conditional image law agrees with the original. This is the chordal restriction property.

The change of variables $X=\rho e^{-t}$, $S=e^{-t}>0$ turnsinto $w_{SS}-w_{XX}=0$. Indeed, with $D={\partial }_t+\rho {\partial }_{\rho }$, the original operator is $D^2+D-{\partial }_{\rho }^2$, while $D=-S{\partial }_S$ and ${\partial }_{\rho }=S{\partial }_X$. Its characteristic curves are $\rho =-1+C{e}^t$ and $\rho =1+C{e}^t$, including the characteristic lines $\rho =\pm 1$. General smooth solutions have the form $F(e^{-t}(\rho +1))+G(e^{-t}(\rho -1))$. Zero Cauchy data on $-1<\rho <1$ at $t=0$ force vanishing throughout that strip for $t\ge 0$, but not for all negative time: $u=(e^{-t}(\rho +1)-2{)}_{+}^3$ is a global $C^2$ counterexample. Thus the characteristic barriers describe a forward domain of dependence, not a barrier in both time directions.

For a subgroup $K$ of a finite abelian group $G$, its annihilator isThese are exactly the linear characters that descend to the quotient group $G/K$. The character group of a finite abelian group has the same size as the group, so $|K^{\perp }|=|G/K|=|G|/|K|$. The annihilator is what abelian hidden-subgroup Fourier sampling measures.

For a linear character $\chi$ of a finite group $K$,Indeed, if $\chi (h)\ne 1$, the bijection $k\mapsto hk$ gives ${\sum }_k\chi (k)=\chi (h){\sum }_k\chi (k)$, forcing the sum to vanish. The trivial character gives $|K|$ directly. The same argument applies to the restriction of a linear character of a larger group to a subgroup.

The rank is the dimension of a Cartan subalgebra, equivalently the dimension of the real span of its root system.

For a complex semisimple Lie algebra, the Cartan subalgebra is orthogonal under the Killing form to every nonzero root space. A vector in the Cartan subalgebra orthogonal to that whole subalgebra is therefore orthogonal to the entire Lie algebra and must vanish. On the Cartan subalgebra the form is $\kappa (h,h^{\prime })={\sum }_{\alpha \in \Phi }\alpha (h)\alpha (h^{\prime })$.

Neglecting metric perturbations, the rescaling $f=a\delta \varphi$ removes the expansion friction from the scalar perturbation action after a boundary term is discarded. Its canonical momentum is $f^{\prime }$, and each real Fourier oscillator has time-dependent squared frequency $k^2+a^2{V}^{\prime \prime }-a^{\prime \prime }/a$. The oscillator can become inverted on superhorizon scales; the canonical commutation relations still apply.

For a compact H-hull avoiding zero, let ${\psi }_A$ map its complement to the half-plane, fix zero, and have derivative one at infinity. The Brownian excursion in the upper half-plane avoids $A$ with probability ${\psi }_A^{\prime }(0)$. From an interior starting point $z$, the corresponding transformed process avoids $A$ with probability $\mathrm{Im}{g}_A(z)/\mathrm{Im}z$; taking $z=i\epsilon$ and $\epsilon \downarrow 0$ gives the boundary derivative.

For $N=N_G(D)$ and $C=C_G(D)$, the Brauer morphism restricts to a surjectionThis follows from Brauer morphism and relative trace and the fact that $kC$ is contained in $(kG{)}^D$. The target is an ideal in the commutative algebra $(kC{)}^N$. Applying primitive idempotents under a surjection of commutative Artinian ideals and the trace criterion for defect groups identifies block idempotents with defect group $D$ with the primitive idempotents of $J_D$. Applying the same argument to $N$ gives Brauer first main theorem. The whole kernel need not be nilpotent, since blocks with smaller defect can be killed.

With $D_i={\partial }_i-i{A}_i$ and $B={\partial }_1{A}_2-{\partial }_2{A}_1$, use the critical-coupling energy $E=\int [B^2/2+|D_i\varphi {|}^2/2+(1-|\varphi {|}^2{)}^2/8]d^{2}x$. For positive vortex number, integration by parts givesThe boundary divergence vanishes for the usual decaying vortex fields, and magnetic flux is $2\pi N$. The two squares vanish exactly at the Bogomolny vortex equations. Reversing both signs treats negative $N$. Energy coefficients and the numerical bound change together under other normalizations.

The big cone in $N^1(X{)}_{\mathbb{R}}$ is open and convex: write a big class as ample plus effective, perturb only the ample summand, and use openness of the ample cone. For positive-dimensional $X$, intersection with $H^{n-1}$ for any very ample divisor $H$ is strictly positive on every big class. Hence this cone contains no line and does not contain the zero class.

Suppose $D{\sim }_{\mathbb{R}}tA+E$ on an integral projective variety, where $t>0$, $A$ is ample Cartier and $E$ is effective real Cartier. Then $D$ is an actual positive combination of big Cartier divisors.

Here is a finite-dimensional proof. First assume $X$ is normal. Express $D$ and $E$ in finite Cartier bases and write $D-tA-E$ as a finite combination of principal Cartier divisors. The union of the supports of these finitely many divisors has finitely many prime components. Their integer multiplicities turn the equality into finitely many rational linear equations and effectivity into finitely many rational linear inequalities. The given coefficient tuple lies in a rational polyhedron. Take its smallest face; within that face it lies in the relative interior, and $t>0$ is an open condition. A small simplex with rational vertices in this relative interior contains the tuple. Each vertex gives $D_{\nu }{\sim }_{\mathbb{Q}}{t}_{\nu }A+E_{\nu }$ with $t_{\nu }>0$ and $E_{\nu }\ge 0$. Clearing denominators and applying Kodaira's lemma shows that $D_{\nu }$ is a positive rational multiple of a big Cartier divisor. Taking the original convex weights proves the required actual equality.

If $X$ is nonnormal, pull the finite Cartier bases and the relation to its finite normalization and impose the same rational equations and effectivity inequalities there. The vertex divisors remain rational Cartier divisors on $X$ because they were constructed in bases from $X$. They are big on the normalization, hence big on $X$ by bigness under finite normalization. This proves the same conclusion. Effectivity is used in the usual effective Cartier sense so that pullback is effective; arbitrary cycles on a nonnormal variety cannot be substituted without defining a compatible divisor theory.

For a possibly reducible projective scheme, componentwise bigness means that a real Cartier class restricts to a big real divisor on every reduced irreducible component. This specifies the convention needed by positivity arguments that treat all curves. Maximal total section growth alone is weaker: on ${\mathbb{P}}^2⨿{\mathbb{P}}^2$, the bundle $(\mathcal{O}(1),\mathcal{O}(-1))$ has quadratic total growth but negative degree on every line in the second component. Thus its negative curves cannot be confined to finitely many divisors.

Write a big real Cartier divisor on an integral projective variety as $D{\sim }_{\mathbb{R}}B+E$, with $B$ ample and $E$ effective real Cartier. Every curve outside $\mathrm{Supp}E$ has $D\cdot C=B\cdot C+E\cdot C>0$. Therefore curves negative against $D$ lie in finitely many support components of codimension one. On a projective scheme use componentwise bigness on a projective scheme and collect the supports on its finitely many integral components; codimension one is taken in the relevant component.

For left-invariant vector fields, the Koszul formula and the adjoint invariance of a bi-invariant Riemannian metric give the displayed Levi-Civita connection. In particular ${\nabla }_{X}X=0$.

Integral curves through the identity of left-invariant vector fields are one-parameter subgroups, by flow uniqueness and left translation. They extend for all time because a fixed local existence interval translates to every point. Under a bi-invariant Riemannian metric they are geodesics by the Levi-Civita connection of a bi-invariant metric. Geodesic uniqueness proves that these are all geodesics through the identity.

For a Bessel process of dimension $\delta >2$ started at $x>0$, its scale function of a one-dimensional diffusion gives ${\mathbb{P}}_x({\tau }_r<\infty )=(r/x{)}^{\delta -2}$ for $0<r<x$. In particular, ${\inf }_{t\ge 0}{X}_t>0$ almost surely. The minimum is attained because the process tends to infinity; the hitting formula alone already implies positivity of the infimum.