Every holomorphic map $V_1/{\Lambda }_1\to V_2/{\Lambda }_2$ lifts to $z\mapsto Az+x$, where $A$ is a complex-linear map satisfying $A{\Lambda }_1\subseteq {\Lambda }_2$. Period differences of a lift lie in the discrete target Euclidean lattice, so its derivative is periodic. Boundedness on a compact fundamental domain and the Liouville theorem make this derivative constant. The linear map is unique, and the translation vector is determined modulo ${\Lambda }_2$.

The particular, forced part of a complex inclination solution. For a fixed, circular, inclined inner planet and an exterior test ring, linear Laplace-Lagrange secular theory gives $\dot{w}=-i\nu (w-w_p)$, so $w_f=w_p$ and $w=w_p+(w(0)-w_p)e^{-i\nu t}$. The ring describes a circle about the planetary tilt in an Argand diagram; relative to the planet, its mutual inclination stays constant while its ascending node precesses.

A commutative monoid is cancellative when $a+c=b+c$ implies $a=b$. Cancellation makes the natural map into its Grothendieck group injective.

Let $\mathcal{C}$ be the ordered category of all ordinals with an added greatest object $\infty$. The categorical presheaf with $X(\alpha )=\{\ast \}$ at ordinals and $X(\infty )=\varnothing$ preserves every small categorical limit: a small supremum is an ordinal unless its diagram contains $\infty$. Its category of elements consists of all ordinals, and their projection has the large colimit $\infty$. Nevertheless $X$ is not representable: an ordinal representer has bounded support, while the representer $\infty$ has a nonempty value at $\infty$. Thus preservation must include the possibly large opposite elements diagram. The dual functor $X^{\mathrm{op}}:\mathcal{C}\to {\mathbf{Set}}^{\mathrm{op}}$ also disproves the comma-projection criterion for a right adjoint under small-colimit preservation alone.

The category ${\mathcal{E}}^G$ consists of coalgebras for a comonad and their structure-preserving morphisms. Its forgetful functor has the cofree coalgebra as right adjoint. If $\mathcal{E}$ is a topos and $G$ preserves finite limits, those limits are created by the forgetful functor, while exponentials in a coalgebra topos and the subobject classifier of a coalgebra topos can be constructed as equalizers inside cofree objects.

For any real design matrix $X$ and $\lambda >0$, $(X^{T}X+\lambda I{)}^{-1}{X}^T=X^T(X{X}^T+\lambda I{)}^{-1}$. Multiplication by $X$ identifies the primal ridge regression fitted values with $K(K+\lambda I{)}^{-1}Y$, $K=X{X}^T$, the kernel-ridge hat matrix for the linear kernel. No rank assumption is needed.

For every $\eta >0$, some $\rho >0$ and $N_0$ have this property: a graph on $N\ge N_0$ vertices with fewer than $\rho N^3$ unordered triangles in a graph can be made triangle-free by deleting at most $\eta N^2$ edges. A Szemerédi regularity lemma partition deletes exceptional, intraclass, irregular and sparse-pair edges cheaply. A surviving triangle in a graph would force a positive cubic count by the regular triangle counting lemma.

An idealized classical nova that slowly ejects mass $\delta m$ isotropically from the accretor, compared with the orbital period, but quickly compared with the mass-transfer timescale, expands a circular binary star's Roche lobe. The ejecta carry accretor specific angular momentum $j_1/J=M_2/(M_{1}M)=q/M$. Taking $\delta J/J=-q\delta m/M$, $\delta M_1=-\delta m$ and $\delta M_2=0$ in $J=M_1{M}_2\sqrt{Ga/M}$ proves $\delta a/a=\delta m/M$. The lobe law $R_L\propto a(M_2/M{)}^{1/3}$ then gives $\delta R_L/R_L=4\delta m/(3M)$. A donor whose radius is unchanged moves inside its lobe.

For constant external loss $\Gamma =-\dot{J}>0$ and a donor sequence $R_2\propto M_2^{\zeta }$, the fractional gap closes at rate $2\Gamma /J$, giving $t_d=2J\delta m/(3M\Gamma )$. The binary mass-transfer contact equation gives accumulation time $t_s=J\delta m(\zeta +5/3-2q)/(2M_2\Gamma )$. ThereforeThis first-order result requires $\delta m/M\ll 1$, a positive contact denominator and a fixed donor radius during the eruption. At $\zeta =1$ it reduces to $2q/[(1+q)(4-3q)]$.

For a real point $b$ away from the initial driver, this is the maximal lifetime of the real solution ${\partial }_t{g}_t(b)=2/(g_t(b)-{\xi }_t)$. At finite lifetime its centred image collides with the Loewner driving function. A point may be visited or separated from infinity by the hull, so swallowing is not the same as an individual trace visit.

For a singular projective Weierstrass cubic, its smooth locus of a variety contains the point at infinity and carries the chord-and-tangent group law. The singular point is excluded. Over an algebraic closure, a node gives the multiplicative algebraic group and a cusp gives the additive group. A node whose branches are not defined over the base field gives a nonsplit one-dimensional algebraic torus. This group is distinct from an elliptic curve, whose projective model is smooth.

For a finite family generating the unit ideal in a commutative ring, this augmented Čech cochain complex of any module is exact. One proof clears the finitely many denominators and cocycle relations by powers $f_i^N$, writes $1={\sum }_i{a}_i{f}_i^N$, and contracts an alternating cocycle by the weighted insertions ${\sum }_i{a}_i{f}_i^N{c}_{iI}$. The argument uses vanishing criterion in a module localization and does not require the module to be finitely generated.

For $x$ in the derived algebra of a complex matrix Lie algebra $L$, define $y$ by multiplication by $\overline{\lambda }$ on each generalized eigenspace $V_{\lambda }$ of $x$. Polynomial interpolation and adjoint compatibility of additive Jordan decomposition make $\mathrm{ad}y$ a polynomial in $\mathrm{ad}x$ with zero constant term, so $[y,L]\subseteq [L,L]$. Trace orthogonality of $[L,L]$ and $L$ forces the displayed sum to vanish. Thus $x$ is a nilpotent endomorphism, and the Engel theorem implies solvability. The auxiliary $y$ and the Jordan components are not required to lie in $L$.

Choose a directed path ${\eta }_{xy}$ of positive-probability transitions for each ordered pair of states in a finite reversible Markov chain. Write $Q(e)=\pi (u)P(u,v)$ for a directed edge $e=(u,v)$, and let $N_e(\eta )$ count its occurrences. DefineThe Cauchy-Schwarz inequality bounds $(f(x)-f(y){)}^2$ by $|{\eta }_{xy}|$ times the sum of squared differences along the path. Multiply by $\pi (x)\pi (y)/2$, sum and interchange finite sums. The identity ${\mathrm{Var}}_{\pi }(f)=\frac{1}{2}{\sum }_{x,y}\pi (x)\pi (y)(f(x)-f(y){)}^2$ proves the bound. This convention counts directed edges and retains the factor $1/2$ in the Dirichlet form of a Markov chain.

If a meromorphic top form has at most a simple pole along a smooth complex analytic hypersurface $Y=\{z=0\}$, write $\alpha =(dz/z)\wedge \beta$ with $\beta$ holomorphic. Its residue is $\beta {|}_Y$, a holomorphic section of the canonical bundle $K_Y$. A change of defining coordinate $t=az$, with $a$ holomorphic and nonzero, changes $dt/t$ by the holomorphic form $da/a$ and leaves this restriction invariant. Placing the normal factor first fixes the sign convention.

For $p\ge 2$ and a continuous local martingale starting at zero, the Itô formula for $|X{|}^p$, the Doob Lp maximal inequality and Hölder's inequality yield $\mathbb{E}{\sup }_{s\le t}|X_s{|}^p\le C_p\mathbb{E}⟨X{⟩}_t^{p/2}$. One possible constant is $C_p=[p(p-1)(p/(p-1){)}^p/2{]}^{p/2}$. Stop at increasing absolute-value levels to remove an initial boundedness assumption. This proves the upper half of the Burkholder-Davis-Gundy inequalities in this range.

An invariant path-integral measure and BRST symmetry action imply this identity when every separated insertion $G_i$ is BRST-closed. The bracket is the graded commutator, so it becomes an anticommutator for an odd $\mathcal{O}$. The arbitrary operator need not itself be closed. Exact insertions therefore decouple from physical correlators.

For auxiliary-field covariant gauge fixing, varying the parameter inserts a BRST-exact operator. Its correlations with separated physical BRST-closed operators vanish by the BRST Ward identity. Normalized physical correlators are therefore independent of the parameter under the invariant-measure and boundary assumptions.

When separated operators $G_i$ are BRST-closed, the graded BRST Ward identity makes an exact insertion vanish. This is the correlator form of identifying closed representatives that differ by exact terms in BRST cohomology. The statement presupposes an invariant measure and suitable boundary conditions.

For a complete orthonormal basis of $L^2({\mathbb{R}}_{+})$, the Parseval identity for a Hilbertian basis identifies the sum of the feature products with $⟨1_{[0,s]},1_{[0,t]}⟩=\min (s,t)$. The Cauchy-Schwarz inequality gives absolute convergence.

For a finite quiver over an algebraically closed field, an indecomposable finite-dimensional quiver representation that is not a brick module contains a brick with nonzero self-extensions. The Ringel form then gives $q_Q(\dim B)=1-\dim {\mathrm{Ext}}_Q^1(B,B)\le 0$, impossible for a positive definite Tits form of a quiver.