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The product Riemannian metric of the round unit two-sphere and a circle has Ricci curvature equal to $g_{S^2}$ on the sphere directions and zero on the circle direction. Its scalar curvature is constantly two, but no single scalar multiple of the whole metric equals that Ricci tensor. In two dimensions the identity $\mathrm{Ric}=\mathrm{Scal}g/2$ does make constant scalar curvature equivalent to an Einstein metric.

For a symmetric nonnegative integer matrix, the two neighboring prefix shapes along the diagonal are equal. In the RSK growth-diagram local rule, set $\mu =\nu$ and take alternating sums of row lengths. The two occurrences of the alternating sum of $\mu$ cancel, leaving $o(\lambda )=o(\rho )+a_{ii}$. Induction along the diagonal proves the formula. Each column contributes one to this alternating sum exactly when its length is odd.

A root system spanning a real inner product space is irreducible when it cannot be partitioned into two nonempty mutually orthogonal root systems. Its Dynkin diagram is connected. This condition concerns the roots themselves, rather than an arbitrary ambient space larger than their span.

For nonproportional roots in a reduced root system which is a crystallographic root system, the negative Cartan integers $m=2(\alpha ,\beta )/(\alpha ,\alpha )$ and $n=2(\alpha ,\beta )/(\beta ,\beta )$ satisfy $mn=4{\cos }^2\theta \in \{1,2,3\}$. At least one of $m,n$ equals $-1$, so reflecting one root in the other produces their sum. The exclusion of opposite roots is essential: their inner product is negative but their sum is zero, which is not a root.

After choosing positive roots in a root system, a negative root is the negative of a positive root. Equivalently, for a regular linear functional defining the positive roots, its value on a negative root is negative. Relative to the associated simple roots, the coefficients of a negative root are all nonpositive integers.

A root subsystem of a root system $\Phi$ is a subset that is itself a root system in its real span. In particular, if a subspace $F$ is spanned by roots, $\Phi \cap F$ is a full root subsystem: root reflections in its roots preserve both $\Phi$ and $F$. Reflection-generated subsystems need not contain every ambient root in their span. A pair is a simple root basis only when every subsystem root has integral coordinates of one sign in that pair; merely choosing two independent roots is insufficient.

The hyperplane perpendicular to a root of a root system is its root hyperplane, interpreted as the kernel of the associated linear functional on the Cartan space. The corresponding reflection is $s_{\alpha }(v)=v-2⟨v,\alpha ⟩\alpha /⟨\alpha ,\alpha ⟩$. For the B2 root system, the four distinct hyperplanes are the two coordinate axes and the two diagonals; their reflections generate the signed-permutation Weyl group of order eight.

For root spaces ${\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta }$, the Jacobi identity gives $[{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta }]\subset {\mathfrak{g}}_{\alpha +\beta }$ when $\alpha +\beta \ne 0$. If the bracket is nonzero, that nonzero weight must be a root. When $\beta =-\alpha$, the bracket instead lies in the Cartan subalgebra and can be nonzero; zero is not a root. Thus the common nonzero-bracket implication requires the opposite-root exception.

Let $q=M_1/M_2$, hold total binary mass fixed, use $R_L\propto a{q}^{3/10}$, and impose $d\log J=fd\log M_1$. The circular-binary orbital angular momentum then gives $d\log a/d\log M_1=2(q-1+f)$, whereas mass conservation gives $d\log q/d\log M_1=1+q$. Thus ${\zeta }_L=(23q-17+20f)/10$. If the donor obeys $R_1=K{M}_1^{\beta }$ with slowly varying $K$, loss of donor mass increases its overfill when ${\zeta }_L>\beta$, or $q>(17-20f+10\beta )/23$. This is a feedback criterion on the time scale of the assumed donor response, not automatically a dynamical instability for every choice of $\beta$. For $\beta =1/2$, the threshold is $(22-20f)/23$; if $f\ge 11/10$, every positive mass ratio satisfies it.

A Riemannian cut point along a geodesic in a complete Riemannian manifold is either its first conjugate point or the endpoint of another minimizing geodesic from the initial point. To prove this, approach the cut time from above and choose minimizing geodesics to the later endpoints using the Hopf-Rinow theorem. A subsequence of their initial unit vectors converges. If its limit is the original initial vector and the cut endpoint is not conjugate, the inverse function theorem for the Riemannian exponential map identifies the two geodesics locally and contradicts the shorter length of the chosen minimizers. A conjugate point earlier than the cut time is excluded by the negative Riemannian index form direction obtained by extending an endpoint-vanishing Jacobi field by zero and perturbing its endpoint corner.

In unit-curvature Real projective space of dimension at least two, Riemannian distance is $d([v],[u])=\arccos |⟨v,u⟩|$ for unit lifts. A unit-speed geodesic has lift $\cos tv+\sin tw$, with $w$ unit and perpendicular to $v$, and therefore ceases to minimize after $\pi /2$. Its normal Jacobi fields with zero initial value are $\sin t$ times parallel vector fields along a curve, giving first conjugate point at time $\pi$. The two times differ in every direction.

The vector of Riemann constants is the parameter-independent class obtained from the zeros of $p\mapsto \theta (u(p)-z)$ using the displayed formula. Under this sign convention the theta divisor is $W_{g-1}-\kappa$. It depends on the base point and normalized symplectic marking; some accounts use its negative. Changing the marking or sign convention must also change the translation formula consistently.

If $p\mapsto \theta (u(p)-z)$ is not identically zero, its zero divisor $D_z$ has these properties. The logarithmic derivative changes by $-2\pi i{\omega }_j$ across the paired $b_j$ translation, giving zero count $g$. For a variation of $z$, the logarithm changes there by $2\pi i\delta z_j$; integrating by parts in the first-moment contour integral gives $\delta \mathrm{AJ}(D_z)=\delta z$. Thus the difference is constant. Weighted zero divisors, rather than a distinct-zero assumption, make the statement valid at collisions.

The tangent cone to a theta divisor at a degree-$g-1$ line bundle $L$ is cut out in $H^1({\mathcal{O}}_X)$ by the determinant of $\xi ⌣-:H^0(L)\to H^1(L)$. Under Serre duality its matrix entries are $⟨\xi ,s_i{t}_j⟩$ for bases of $H^0(L)$ and $H^0(K_X\otimes L^{-1})$. For $h^0(L)=1$, its tangent hyperplane is annihilated by the nonzero holomorphic differential form $s_1{t}_1$.

For a square evaluation matrix representing a degree-$g-1$ line bundle $L$, remove an invertible minor of size $N-h^0(L)$. Holomorphic row and column operations leave a square matrix $M$ of size $r=h^0(L)$ with zero constant term. Its linear part in a tangent direction $\xi \in H^1({\mathcal{O}}_X)$ is $\xi ⌣-:H^0(L)\to H^1(L)$. The invertible cup-product direction for a line bundle on a curve makes the leading degree-$r$ determinant nonzero, establishing order exactly $r$ rather than just a lower bound.

The null deviance is the unscaled deviance of the intercept-only mean model relative to the saturated model at a common dispersion parameter $\varphi$, retaining any known generalized linear model offset. It supplies a baseline against which predictor terms improve the fit. Its difference from a fitted model's residual deviance, divided by $\varphi$, is the statistic for a likelihood-ratio test of those added terms, with nuisance parameters handled consistently in both fits. In Poisson and binomial families, $\varphi =1$.

For a torsion Spin-c structure, the relative grading of Heegaard Floer homology lifts canonically to a rational absolute grading, normalized by the bottom plus generator of $S^3$ having degree zero. Multiplication by $U$ has degree $-2$, and a Heegaard Floer cobordism map has degree $(c_1^2-2\chi -3\sigma )/4$. The correction term $d(Y,\mathfrak{s})$ is the lowest grading of the image of $H{F}^{\infty }$ in $H{F}^{+}$ for a rational homology sphere. A pure tower then has bottom plus degree $d$ and top minus degree $d-2$ in the $i<0$ minus convention. Keeping that shift explicit prevents errors of two in surgery and mixed-invariant calculations.

In a strictly cold flow governed by ideal magnetohydrodynamics the sound speed vanishes, so the slow magnetosonic wave speed is zero. A nonzero-speed outflow therefore has no ordinary finite interior slow crossing. For the cold radial magnetohydrodynamic wind integral, $f_y=-\alpha (x-x^{-1}{)}^2/(y-1{)}^3-\beta /(x^4{y}^3)<0$ on the sub-Alfvénic branch $y>1$, assuming $\beta >0$. A small retained specific enthalpy is needed to locate a finite slow point; the zero-temperature limit can move it to the launch boundary. The usual simultaneous numerator and denominator regularity conditions still apply to nondegenerate critical points.

In the reduced theory the product of colors a and b decomposes with multiplicity one into colors c satisfying $|a-b|\le c\le \min (a+b,2r-4-a-b)$ and $a+b+c$ even. These rules follow by iterating $1\otimes j=(j-1)\oplus (j+1)$, with color $r-1$ set to zero. The closure traces give $d_a{d}_b={\sum }_c{N}_{ab}^c{d}_c$, and self-duality makes the multiplicities symmetric in all three colors.

The Jones-Wenzl idempotent recursion decomposes $f_j\otimes 1$ into the orthogonal $f_{j+1}$ and $f_{j-1}$ channels. The Jones-Wenzl partial trace supplies the nonzero scalar normalizing the cap-cup channel. At the highest surviving color the $f_{r-1}$ channel is zero. Induction therefore decomposes every tensor power into the surviving colors. A planar matching between different projected colors has a turnback on the larger side and vanishes; between equal colors only the identity matching remains. Their endomorphism spaces are consequently one-dimensional, and the resulting finite matrix blocks split. Taking direct sums and splitting these idempotents produces a semisimple category, supplying the complete resolutions needed in fusion of Jones-Wenzl colors and the Kirby color handle-slide identity.