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The unit radial vector field is $E=\nabla r$ on the punctured normal neighbourhood, where $r$ is the radial distance in a normal neighbourhood. Along ${\gamma }_u(t)={\exp }_p(tu)$ with $|u|=1$, it satisfies $E({\gamma }_u(t))={\dot{\gamma }}_u(t)$. The scaled radial field $W=rE=\nabla (r^2/2)$ is smooth at the centre and vanishes there. These conventions have different associated radial functions.

// Target: riemannian-geometry.bigb

Put $z=a_1/\sqrt{a_2}$ using the root with positive real part. The additional resolvent singularity at $s+a_3=z^2/4$ lies on the physical square-root sheet only when $\mathrm{Re}z>0$; otherwise the apparent saddle is on the other sheet. The singularity at $s=-a_3$ remains, including a quartic degeneracy when $a_1=0$. The rightmost physical singularity determines the exponential fixed-position growth rate. Compare it with the temporal maximum $-\mathrm{Re}{a}_3+\max (\mathrm{Re}{a}_1,0{)}^2/(4\mathrm{Re}{a}_2)$ to distinguish absolute wave-packet instability from convective wave-packet instability. Equality cases require their algebraic prefactors.

Reality of the boundary derivatives gives $H_1(\bar{k})=\overline{H_1(k)}$ and $H_2(-\bar{k})=\overline{H_2(k)}$, and the same identities for $J_1,J_2$. The global relation and its conjugate express both quarter-plane oblique boundary spectral functions in terms of the known $H_1,H_2$ and the single bounded holomorphic function $\Phi$ in the upper half-plane. For $\gamma ={\beta }_1+{\beta }_2\in \{0,\pi /2,\pi \}$, its coefficients on the positive real and positive imaginary reconstruction rays are opposite, because $e^{4i\gamma }=1$. Closing the first-quadrant contour removes $\Phi$ by the Cauchy integral theorem.

// Target: analysis.bigb

With only up/down valence quarks and no antiquarks, the quark number sum rule and leading parton model expression give the charge-squared number moment $\int F_2(x)dx/x={\sum }_f{Q}_f^2{N}_f$. The squared charges are $4/9$ for up and $1/9$ for down, so the proton has $8/9+1/9=1$ and the neutron has $4/9+2/9=2/3$. Restricting the flavour list to up/down does not itself remove antiquarks; the no-sea assumption is essential for these individual moments.

In units $\hslash =1$, a particle of mass $m$ with potential $V(q)=-Fq$ has classical path $q_c(t)=q_0+[(q_1-q_0)/T-FT/(2m)]t+F{t}^2/(2m)$. Its action and exact quantum-mechanical propagator areThe fluctuation operator is the free one because the potential is linear. This is a special case of classical-path factorization of a quadratic path integral.

For input dimension $D$, an input of entropy $S(\rho )$ gives output entropy $h_2(p)+(1-p)S(\rho )$ and reference-output entropy $h_2(p)+pS(\rho )$. These follow from entropy of an orthogonal quantum mixture and purity of the input purification. Thus the quantum mutual information is $2(1-p)S(\rho )$, maximized at $\rho =I_D/D$. The entanglement-assisted classical capacity is consequently $2(1-p){\log }_{2}D$ bits per use; for a binary input it is $2(1-p)$.

Three Fourier modes form a spatial resonance when their wavevectors sum to zero. Products of the conjugate second and third modes then force the first mode, with cyclic counterparts. Such terms couple both amplitudes and phases, unlike self-saturation terms proportional only to squared magnitudes. For three equal wavenumbers, the vectors are separated by $120$ degrees and support hexagonal pattern interactions.

In $L={\mathbb{Q}}_2(\sqrt{2})$, set $\pi =\sqrt{2}$ and normalize $v_L(\pi )=1$, so $v_L(2)=2$. The two terms of $a+b\pi$ have valuations of opposite parity, giving ${\mathcal{O}}_L={\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\pi$. The nonidentity automorphism sends $\pi$ to $-\pi$, and $v_L(-2\pi )=3$. The lower ramification numbering therefore gives the displayed groups. The different exponent is three.

For traceless generators, the tensor product of Lie algebra representations has generators $t_a\otimes I+I\otimes s_a$. The trace of their summed squares is the sum of the two factor Casimir traces; the cross term vanishes because $\mathrm{tr}{t}_a=\mathrm{tr}{s}_a=0$. Decomposition into irreducible representations, with multiplicities counted, gives the displayed identity. Without tracelessness an extra term $2{\sum }_a\mathrm{tr}{t}_a\mathrm{tr}{s}_a$ is present. A one-dimensional abelian representation of charge one tensored with itself has Casimir four, not the sum two; semisimple Lie algebras avoid this problem because every generator is a sum of commutators.

For generators normalized by ${\mathrm{tr}}_3(t_a{t}_b)={\delta }_{ab}/2$, the quadratic Casimir operator on the irreducible SU(3) representation of Dynkin labels $(a,b)$ has the displayed scalar value. In SU(3) highest-weight coordinates $(p,q)$ it is $p^2+q^2+p+\sqrt{3}q$.

To derive this, write the operator as $H_1^2+H_2^2+{\sum }_{i=1}^3(E_{+}^i{E}_{-}^i+E_{-}^i{E}_{+}^i)$. On a highest-weight vector, the raising operators vanish, and the three root commutators sum to $H_1+\sqrt{3}{H}_2$. Centrality and the Schur lemma extend this value to the representation. The singlet, fundamental, adjoint and decuplet values are $0,4/3,3,6$, respectively. Tensor labels and multiplet dimensions are summarized in the Particle Data Group's SU(n) multiplet guide.

For $e_3=[e_1,e_2]$, $f_3=-[f_1,f_2]$, the operator $C=h_1^2+h_2^2+h_1{h}_2+3(h_1+h_2)+3{\sum }_{i=1}^3{f}_i{e}_i$ commutes with all generators. In its commutator with $e_1$, the Cartan polynomial contributes $3h_1{e}_1$, while the last sum contributes $-3h_1{e}_1$; the two mixed terms involving $f_2{e}_3$ cancel. On highest weight $(p,q)$ its eigenvalue is $p^2+q^2+pq+3p+3q$. This normalization is three times the conventional SU(3) quadratic Casimir eigenvalue.

Sampling order-three Cardinal B-splines $N_j$ at $x_i=i+3/2$ gives $N_i(x_i)=3/4$, $N_{i-1}(x_i)=N_{i+1}(x_i)=1/8$, and all other entries zero. The B-spline collocation matrix therefore has row strict diagonal dominance margin at least $1/2$, giving inverse operator norm at most two. More precisely, with $q=3-2\sqrt{2}$, its absolute inverse row sums are $v_i=2[1-(q^i+q^{n+1-i})/(1+q^{n+1})]$. To see this, change signs by $D_{ii}=(-1{)}^i$: $DAD$ has negative off-diagonal entries and a nonnegative inverse given by a convergent Neumann series. Its inverse row sums solve $6v_i-v_{i-1}-v_{i+1}=8$, $v_0=v_{n+1}=0$. Thus $\Vert A^{-1}{\Vert }_{\infty }=v_{\lfloor (n+1)/2\rfloor }<2$, tending to two as the dimension grows.

If $X\stackrel{f}{\to }Y\stackrel{g}{\to }Z$ has proper composite and $g$ is separated, the graph $X\to X{\times }_{Z}Y$ is a closed immersion, being a base change of the diagonal of $g$. The projection $X{\times }_{Z}Y\to Y$ is a base change of the proper composite. Their composition is $f$, so $f$ is proper.

Suppose $gf$ is proper, $f$ is surjective, and $g$ is separated and of finite type. Surjectivity of scheme morphisms is preserved by base change: a nonempty fiber remains nonempty after a residue-field extension. After any base change to $Z$, for a closed subset $C$ of the new target of $f$, its inverse image is closed and has image $g(C)$ under the proper composite. Thus $g(C)$ is closed. This proves universal closedness of $g$, completing the three conditions for properness.

For a finite-type affine morphism of Noetherian schemes, apply its valuative lifting property to each domain quotient $B/\mathfrak{q}$ for a minimal prime and all valuation rings of its fraction field containing the image of $A$. The lift forces $B/\mathfrak{q}$ into every such valuation ring, hence makes it integral over that image. For each $b\in B$, multiply the finitely many resulting monic equations. Its value at $b$ lies in the nilradical. The nilradical of a Noetherian ring is nilpotent, so a power of this monic polynomial annihilates $b$. Thus $B$ is integral over $A$. Finitely many algebra generators, each satisfying a monic equation, have their powers reduced to finitely many monomials, proving that $B$ is a finite $A$-module. Nilpotents cannot be ignored in this proof.

A variety over a field $k$ is complete when its map to $\mathrm{Spec}k$ is universally closed: after any base change, projection sends closed subsets to closed subsets. For a separated finite-type variety, this is equivalent to being proper. In the classical algebraically closed setting one may test projections $X\times Y\to Y$ for arbitrary varieties. Images of rational points over a non-algebraically-closed field must not replace geometric images.

Over ${\mathbb{F}}_5$, putThese determinant-one matrices satisfy $i^2=j^2=-I$, $ij=-ji$, and $r^3=I$. Conjugation by $r$ cycles $i,ij,j$. Thus $⟨i,j⟩$ is a quaternion group of order eight, normalized by an element of order three. Their generated subgroup has order $24$ in $S{L}_2(5)$, hence order $12$ after quotienting by $\{\pm I\}$. Its index-five coset action, together with simplicity and perfectness, identifies $PS{L}_2(5)$ with the alternating group $A_5$.

For two projectively equivalent connections that are Levi-Civita connections, contraction of their difference gives $(n+1)V_c={\bar{\Gamma }}^a{}_{ac}-{\Gamma }^a{}_{ac}$. The Jacobi formula gives ${\Gamma }^a{}_{ac}={\partial }_c\log \sqrt{|\det g|}$. Subtraction proves the displayed expression. The two determinant densities transform by the same coordinate Jacobian factor, so their ratio is a scalar and its derivative a covector. The formula is a necessary consequence of projective equivalence, not a test asserting that arbitrary metric tensors have identical geodesics.

Let $\tau$ be an affine parameter for a geodesic of $\nabla$ and suppose ${\bar{\nabla }}_{X}Y-{\nabla }_{X}Y=V(X)Y+V(Y)X$. Then ${\bar{\nabla }}_{\dot{\gamma }}\dot{\gamma }=2V(\dot{\gamma })\dot{\gamma }$. For the new velocity $d\gamma /dt=\dot{\gamma }/t^{\prime }$, the acceleration is $(t^{\prime }{)}^{-2}[{\bar{\nabla }}_{\dot{\gamma }}\dot{\gamma }-(t^{\prime \prime }/t^{\prime })\dot{\gamma }]$, which vanishes under the displayed differential equation. Its nonzero solution $t^{\prime }=C\exp (2\int V(\dot{\gamma })d\tau )$ changes the parameter while preserving the curve.

The logarithm of the profile likelihood is the full log-likelihood evaluated at the nuisance fit constrained to the specified parameter of interest. It preserves likelihood ratios and maximizers, but is not generally a normalized sampling log-density. Conditioning and integrating out nuisance parameters are different operations.