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.

Write $f={\sum }_k{\beta }_{k,n}{b}_{k,n}$ in the Bernstein basis. Representing the same polynomial at degrees $n$ and $n+1$ gives the displayed convex combinations for internal indices; the endpoint coefficients are copied. Therefore the minimum coefficient cannot decrease under degree elevation. In the unnormalized basis $x^k(1-x{)}^{n-k}$, the elevated coefficients are $c_k^{\prime }=c_{k-1}+c_k$ with the corresponding endpoint convention, so nonnegativity certificates are preserved.

If a polynomial $f$ is strictly positive on $[0,1]$, take $g_n=B_n^{-1}f$. By inverse Bernstein approximation on a fixed-degree polynomial space, eventually $g_n>0$ on the interval. Thereforehas strictly positive coefficients. The converse implication to nonnegativity follows immediately because each basis term is nonnegative. Strict positivity is essential for the general existence result: a nonzero polynomial vanishing at an interior point cannot have a nonnegative-coefficient representation in this basis, whose individual terms are positive throughout the open interval.

Let $R$ be the number of rejections by the Benjamini-Hochberg procedure at level $\alpha$, and let $R_i$ be the rejection count after replacing $p_i$ by zero. Then $R_i\ge 1$ andIf $i$ was rejected, lowering its p-value does not change any ordered p-value beyond rank $R$, so $R_i=R$. Conversely, if $p_i\le \alpha R_i/m$, the other $R_i-1$ rejected values together with $p_i$ satisfy the original threshold; monotonicity gives $R=R_i$. For a valid true-null p-value independent of the others, conditioning on $R_i$ bounds the expectation of the right side by $\alpha /m$. Summing over true nulls proves false discovery rate at most $m_0\alpha /m$.

Suppose an unknown has prior distribution $\Pi$ and its conditional observation law $P_u$ has jointly measurable Radon-Nikodym derivative $L(u,m)$ relative to a fixed observation law $P_0$. If $L>0$ for $\Pi (du)P_0(dm)$-almost every pair, thendefines the posterior distribution for almost every observation. Indeed, $\int L(u,m)P_0(dm)=1$ for almost every $u$, so Tonelli theorem gives $\int Z(m)P_0(dm)=1$. Hence $Z$ is finite almost everywhere, and positivity of $L$ and Fubini's theorem give $Z>0$ almost everywhere. The data law is $Z{P}_0$, and integration of the proposed posterior against this data law recovers the joint law. This proves the conditional distribution property and also shows that the data law is an equivalent probability measure to $P_0$.

The nonoscillatory thermal branch of the radially stratified shearing sheet spectrum. Its decoupled small-buoyancy limit is a diffusing scalar perturbation.

A stationary axisymmetric magnetic field in ideal purely rotational flow requires constant angular velocity along the magnetic field lines of its poloidal magnetic field. Otherwise axisymmetric magnetic winding generates a time-dependent toroidal component.

For Keplerian rotation, initial $B_R\propto R^{-1}$ and thermal pressure $p\propto R^{-2}$, axisymmetric magnetic winding gives $B_{\varphi }=-(3/2)\Omega t{B}_R$. If the initial plasma beta is ${\beta }_p>1$, the formal equality of magnetic pressure and gas pressure occurs atThe magnetic-to-thermal pressure ratio decreases with radius, so magnetic pressure dominates inside this radius. In a disk with a finite radial range, an equality radius exists in the disk only when this formal radius lies within that range.

For the usual semi-grey irradiated atmosphere with downward thermal optical depth $\tau$, an inversion occurs where $f_r{T}_{\mathrm{irr}}^4({\gamma }^2-1)e^{-\sqrt{3}\gamma \tau }>T_{\mathrm{int}}^4$, with $\gamma$ the visible-to-thermal opacity ratio. At negligible intrinsic flux the threshold reduces to $\gamma >1$.

A symmetric triangle with direct calls and two-link overflow routes has Erlang fixed point approximation equation $B=E(\lambda [1+2B(1-B)],C)$. For $C=1000$ and $\lambda =950$, this has at least three solutions, certified by the intermediate value theorem using signs at $B=0,0.01,0.1,0.4$.

A simplified conservative model with a fully convective donor gives long-lived stable Roche-lobe overflow only for donor-to-accretor mass ratio $q<2/3$. This follows from the Roche-lobe radius response exponent and the donor's adiabatic stellar radius response exponent $-1/3$. The boundary is model-dependent; radiative donors, evolved core structure and mass or angular-momentum loss change it. Van Rensbergen and collaborators' Algol population study includes reported Algol mass ratios above $0.7$ and discusses observational uncertainties.

During binary mass-ratio reversal, an initially more massive donor star loses enough material that its companion becomes the more massive member. The stars retain their different prior nuclear evolution, resolving the Algol paradox. Conservative binary mass transfer shrinks the orbit before equal masses and widens it after reversal, because $a\propto (M_d{M}_a{)}^{-2}$ at fixed total mass and orbital angular momentum.