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For integer $k<0$, the operator $D=(2\pi i{)}^{-1}d/dz=qd/dq$ sends a weight-$k$ level-one weak modular form to a weight-$2-k$ one after $1-k$ iterations. In the derivative transformation of a weak modular form, every lower derivative coefficient then contains a zero factor. This also holds on the subspace $M_k^{!}$ of forms meromorphic at the cusp.

With pure solid crystals, negligible solute diffusion, and the density-change flow in a pulled mushy layer, steady solute balance gives $(1-\phi )C=C_0$. If $\theta =(T-T_0)/\Delta T$, $\mathcal{C}=m{C}_0/\Delta T$, and the liquidus is $T-T_0=-m(C-C_0)$, then

If $p\in G$ and a ground-model set $D$ is dense below a forcing condition $p$, then the generic filter $G$ meets $D$. Add all incompatible forcing conditions with $p$ to $D$. The enlarged set is globally dense: a condition compatible with $p$ first has a common strengthening, then a further strengthening in $D$. Genericity meets the enlargement, and directedness excludes its incompatible part.

Relative to the meridian of a knot $\mu$ and Seifert longitude $\lambda$, an integer surgery coefficient $n$ means that the filling meridian of a solid torus is identified with $n\mu +\lambda$.

For $n\ge 2$, a self-map of $S^n$ factoring through Real projective space ${\mathbb{RP}}^n$ has degree zero when $n$ is even, and can have exactly the even degrees when $n$ is odd. In the odd case the first map lifts through the double covering and the covering has degree two. The dimension-one exception allows every integer, since ${\mathbb{RP}}^1\cong S^1$.

If $S^n\to N$ has prime degree $p$ and $N$ is a closed connected oriented $n$-manifold, every integral homology group of $N$ in degrees strictly between zero and $n$ is a finite $p$-primary abelian group. Consequently one power of $p$ annihilates all these groups. The proof combines Poincare duality over each field ${\mathbb{F}}_q$, $q\ne p$, with the universal coefficient theorem for homology and finite generation.

For a $p$-subgroup $D$ of a finite group $G$, set $I_D={\mathrm{Tr}}_D^G((kG{)}^D)$, a transfer ideal of conjugation-fixed elements in $Z(kG)$. A block of a group algebra idempotent $b$ lies in $I_D$ if and only if a defect group of a block of $b$ is conjugate to a subgroup of $D$. In particular, defect groups can equivalently be defined as minimal subgroups $D$ with $b\in I_D$.

Here is a proof using the Brauer morphism. A subgroup of smallest order with $b\in I_D$ exists, since a Sylow subgroup $P$ gives $b={\mathrm{Tr}}_P^G(b/[G:P])$. Write $b={\mathrm{Tr}}_D^G(a)$ and replace $a$ by $ba$. If ${\mathrm{Br}}_D(b)=0$, then $ba$ belongs to the kernel of the Brauer morphism, which is the sum of proper-subgroup relative traces. Hence $b\in {\sum }_{Q<D}{I}_Q$. The algebra $bZ(kG)$ is a commutative Artinian ring and a local ring, so multiplying this equality by $b$ forces some ideal $b{I}_Q$ to contain its identity $b$. This contradicts minimality. On the other hand, ${\mathrm{Br}}_E(I_D)=0$ unless $E$ is conjugate into $D$: the $E$-action on $G/D$ has no fixed coset otherwise, and nonfixed orbit sizes vanish in characteristic $p$. Thus these minimal subgroups are precisely the maximal subgroups with nonzero Brauer image. They are all conjugate. Trace transitivity proves the stated criterion for larger $D$.

In characteristic $2$, a Sylow subgroup $P$ of $A_5$ is a Klein four-group, and its normalizer is $A_4$. Since $k{A}_4$ has one block, Brauer first main theorem gives a unique block of $k{A}_5$ with defect $P$. No order-two subgroup $D$ can be a defect group of a block: both $C_{A_5}(D)$ and $C_{A_5}(P)$ equal $P$, so the Brauer morphisms of a central block idempotent at $D$ and at $P$ are equal. A nonzero image is $1$, since a group algebra of a p-group in characteristic p is local, and consequently $D$ is not maximal among subgroups with nonzero Brauer image.

For a one-sided draining gravity current with injected volume per unit width $V_0(t/\tau {)}^3$, defineThe similarity solution satisfieswhere $K=k[g^3{\tau }^3/({\nu }^3{V}_0){]}^{2/5}$. The inlet condition is $-H(0{)}^3{H}^{\prime }(0)=9$, the integral of $H+\varphi L$ is one, and both depths vanish at the advancing front. The endpoint ${\xi }_N$ gives $x_N={\xi }_{N}D{t}^2$ and depends on $K$ and $\varphi$.

For a fixed bounded measurable uniformly elliptic operator in divergence form, solutions bounded by one have uniform local Caccioppoli inequalities and Hölder continuity estimates. A diagonal subsequence argument yields a local weak $W^{1,2}$ limit, while the Arzelà-Ascoli theorem gives uniform convergence on a fixed smaller closed Euclidean ball. The weak equation passes to the limit because the coefficients are fixed and bounded. A global supremum normalization does not force the interior limit to be nonzero.

This factor converts substellar incident flux into the representative flux used in an atmospheric model. Uniform global averaging gives $f_r=1/4$; dayside averaging gives $1/2$ if albedo is treated separately. The convention must be stated when defining an irradiation temperature.

With no axial electric current, cylindrical magnetostatic pressure balance reduces to constant $p+B_z^2/(8\pi )$. For $p=p_0{e}^{-R^2/a^2}$ and $B_z(0)=0$, its radial solutions areAs $R\downarrow 0$, $B_z\sim \sigma \sqrt{8\pi p_0}R/a$ and $j_{\varphi }\to -\sigma c\sqrt{8\pi p_0}/(4\pi a)$. These finite scalar limits do not give a smooth function of Cartesian position: $R=\sqrt{x^2+y^2}$ has a cusp at the axis, and the azimuthal unit vector has no unique limit there. Thus these data describe radial equilibrium away from the axis, but cannot satisfy smooth vector-field regularity on the axis without modifying the data.

For constant axial magnetic field and $p=p_0(1+R^2/a^2{)}^{-k}$ with $p_0>0$ and $k>0$, cylindrical magnetostatic pressure balance gives, with $x=R^2/a^2$,The nontrivial electric current is finite at infinity exactly when $k>1$, in which case $I^2(\infty )=2\pi c^2{p}_0{a}^2/(k-1)$. Either sign of $I$ is possible because the pressure balance fixes its square.

A wheel graph is the join of graphs of a cycle graph and a single vertex. Every wheel with a rim of at least three vertices has a $K_4$ graph minor: partition the rim into three nonempty consecutive connected sets and use its centre as the fourth branch set of a graph minor.

The standard three-cut Chacon transformation is an invertible measure-preserving transformation built by cutting and stacking with spacer vector $(0,1,0)$ at every stage. Cut the current tower into three equal subcolumns, place one spacer above the middle, and stack left to right. With initial width $2/3$ and height one on a unit interval, level counts and widths are $h_j=(3^{j+1}-1)/2$ and $w_j=2/3^{j+1}$. Tower measure tends to one, using total spacer measure $1/3$. Tower words obey $W_{j+1}=W_j{W}_{j}1W_j$.

For a level-one cusp form and any Dirichlet character modulo $N>1$, this translation sum is a cusp form on ${\Gamma }_1(N)\cap {\Gamma }_0(N^2)$. Conjugating that subgroup by $\left(\begin{array}{cc}1& j/N\\ 0& 1\end{array}\right)$ yields integral determinant-one matrices; cusp holomorphy under rational slash operators supplies all cusp conditions. Its exact Fourier coefficients at positive indices are $a_n(f){\sum }_{j\in (\mathbb{Z}/N\mathbb{Z}{)}^{\times }}\chi (j{)}^{-1}{e}^{2\pi inj/N}$. For a primitive Dirichlet character they equal $g(\overline{\chi })\chi (n)a_n(f)$, by the finite Fourier transform of a primitive Dirichlet character. For imprimitive characters the sum can be nonzero at nonunits, so the simplified twist formula need not hold.

For absolutely continuous first-event time $T$ with event type $J$, the event-free survivor function satisfies $S(t)=\exp (-{\int }_0^t{\sum }_j{h}_j(u)du)$. The cause-specific hazard definition gives unconditional event-type density $S(t)h_j(t)$, so $F_j(t)={\int }_0^{t}S(u)h_j(u)du$. Consequently ${\sum }_j{F}_j(t)=1-S(t)$ for exhaustive disjoint causes. Independent latent cause times are unnecessary. Substituting $1-e^{-\int h_j}$ instead estimates a hypothetical net risk and generally overstates observed incidence.

The $n$th cumulant is the coefficient of $t^n/n!$ in the cumulant-generating function. The first two cumulants are the expected value and variance; the third is the third central moment.

Normalize three distinct points of the Riemann sphere by a Möbius transformation $f$ sending them to $\infty ,0,1$. A fourth point lies on their generalized circle exactly when $f$ takes a real value there: the image of that generalized circle is the extended real line. Other standard cross-ratio ordering conventions differ by real fractional-linear transformations and preserve this reality criterion.

For cross-intersecting families $\mathcal{A},\mathcal{B}$ with member sizes at most $r,s$ respectively, there is a finite set $X$ with $|X|\le r{\sum }_{j=0}^s{r}^j$ such that $A\cap B\cap X\ne \varnothing$ for every $A\in \mathcal{A},B\in \mathcal{B}$. Take a bounded-size transversal kernel ${\mathcal{A}}_0$ preserving hitting sets of size at most $s$ and set $X={\bigcup }_{A\in {\mathcal{A}}_0}A$. For each $B$, the set $B\cap X$ is a hitting set for ${\mathcal{A}}_0$, hence for all of $\mathcal{A}$. This proves the assertion even for infinite set families; if either family is empty, take $X=\varnothing$.