In a commutative diagram consisting of two horizontally adjacent squares, if both squares are pullbacks in a category, their outside rectangle is a pullback in a category. If the right square and the outside rectangle are pullbacks in a category, the left square is a pullback in a category. To prove the first assertion, factor a compatible pair of morphisms first through the right square and then through the left square. For the second assertion, factor through the rectangle and use uniqueness in the right square to recover the prescribed middle morphism. Each factorization is unique by the same universal properties.

Given a pullback in a category $A=B{\times }_{D}C$ and a morphism $D^{\prime }\to D$, assume the indicated pullbacks in a category exist. The four base-changed objects form a pullback in a category. A compatible pair of morphisms into $B{\times }_D{D}^{\prime }$ and $C{\times }_D{D}^{\prime }$ gives morphisms into $B,C$ and a common morphism into $D^{\prime }$. The first two morphisms factor uniquely through $A$, and their common image in $D$ permits unique factorization through $A{\times }_D{D}^{\prime }$. This proves the displayed canonical isomorphism without an assumption that the whole category is complete.

Choose the illustrative local modelHere $t$ is the Celsius temperature, measured relative to $0{}^{\circ }\mathrm{C}$, so $\alpha t$ is dimensionless. For this chosen affine function, the derivative and inverse relation areCalculated values are $R(0{}^{\circ }\mathrm{C})=100\Omega$, $R(20{}^{\circ }\mathrm{C})=107.70\Omega$, $R(25{}^{\circ }\mathrm{C})=109.625\Omega$, and $R(50{}^{\circ }\mathrm{C})=119.25\Omega$. These are model outputs, not measurements or an assertion that any real Pt100 follows this line exactly. The number $0.00385$ here is the chosen slope parameter; it is not a universal local derivative for all platinum resistance thermometers. Curvature, manufacturing differences, and the fitted calibration interval matter. A wider-range calibration uses a validated response curve rather than extrapolating this line indefinitely.

For a separate synthetic thermometer calibration, stipulate $R_a=100.20\Omega$ at $t_a=0{}^{\circ }\mathrm{C}$ and $R_b=119.60\Omega$ at $t_b=50{}^{\circ }\mathrm{C}$. A two-point affine function has slope $0.388\Omega /{}^{\circ }\mathrm{C}$ and givesThus $R=109.90\Omega$ would indicate $25{}^{\circ }\mathrm{C}$. If an additional synthetic check at reference $25{}^{\circ }\mathrm{C}$ instead supplied $110.10\Omega$, the line would indicate $25.515\dots {}^{\circ }\mathrm{C}$. The $0.20\Omega$ residual exposes disagreement that two fitted points cannot reveal. This separate example changes both offset and slope; it is not a replacement specification for the nominal Pt100 local linear model.

For every scheme $X$, projection $p:X\times {\mathbb{P}}^n\to X$ is a closed map, and remains so after every base change. Hence projective space is universally closed over its base. The assertion concerns the underlying scheme points, not just rational points over a non-algebraically closed field.^[ref]

The existence part of the valuative criterion explains the role of projective space. Given a valuation ring $V$ with fraction field $K$, a $K$-point $[a_0:\cdots :a_n]$ can be rescaled so that every coordinate belongs to $V$ and one coordinate is a unit. Indeed, among the finitely many nonzero coordinates choose $a_j$ whose principal fractional ideal contains all the others; then $a_i/a_j\in V$ and $a_j/a_j=1$. These coordinates define the required morphism of schemes $\mathrm{Spec}V\to {\mathbb{P}}^n$. Since projective space is also separated and of finite type, this proves that projection is a proper morphism.

On a Noetherian scheme, one can instead use finitely many homogeneous equations and the Projective Nullstellensatz to express the image as closed rank-defect conditions on graded multiplication map for a projective fiber. An arbitrary affine base need not give a finitely generated defining ideal, so that finite-equation hypothesis must not be assumed in the general statement.

For an isolated spherical stellar system in equilibrium with constant mass-to-light ratio, the virial theorem gives $2K+W=0$. Spherical averaging gives $K=3\pi {\rm Y}\int R\mu {\sigma }_p^{2}dR$, while $W=-4\pi G{{\rm Y}}^2\int r\lambda Ldr$. Dividing yields the displayed estimator, independent of the radial velocity-anisotropy parameter. It requires the full aperture and negligible boundary stress. A radially varying mass-to-light ratio or an independent dark matter halo prevents interpreting this expression as a general reconstruction of the mass profile.

For a scale-free tracer $\nu \propto r^{-k}$ in a point-mass gravitational potential, with $k>1$ and constant velocity-anisotropy parameter, the Spherical Jeans equation gives ${\sigma }_r^2=G{M}_{\bullet }/[(k+1-2\beta )r]$ when the outer pressure term vanishes. Project the tracer density and its second stellar velocity moment along the line of sight. The substitution $r=R\sec \chi$ reduces both integrals to powers of $\cos \chi$ and hence the beta function, yielding the displayed expression. The ratio of the two numerator angular integrals is $k/(k+1)$. This makes the mass-anisotropy ambiguity explicit. At $k=2$ the velocity-anisotropy parameter dependence cancels and ${\sigma }_{\mathrm{los}}^2=2G{M}_{\bullet }/(3\pi R)$; that special slope does not remove the degeneracy in general. Physical use also requires a nonnegative tracer distribution and control of outer departures from the power law.

If the probability of occupation at every site in a finite set depends only on its cardinality, the law is exchangeable. The product self-duality of symmetric exclusion and conservation of the finite dual particle count imply that all these occupation-product moments are unchanged by evolution. Inclusion-exclusion principle reconstructs every finite cylinder probability from them, proving invariance. This direct argument does not require a representation theorem for exchangeable laws.

In an affine variety with coordinate ring $A$, every dense open subset $U$ contains a dense principal open subset. Write its closed complement as $V(I)$. Density says that $I$ is contained in none of the finitely many minimal prime ideals of the reduced ring $A$. Prime avoidance supplies $s\in I$ outside all these primes. Thus $D(s)\subseteq U$, and $s$ is a non-zero-divisor. Noetherian Zariski topology also makes $U$ a finite union of principal opens; one can add $D(s)$ to this cover. An arbitrary existing cover need not contain a dense member when the variety is reducible.

For an isolated solenoidal magnetic field, zero flux through every sphere removes the degree-zero spherical harmonic of $P=\mathbf{B}\cdot \mathbf{x}$. In a poloidal-toroidal decomposition, let ${\mathcal{L}}^2=-{\Delta }_{S^2}$. The spherical poloidal component obeysIf $P={\sum }_{l\ge 1,m}{p}_{lm}(r)Y_{lm}$, put $D_{lm}={\int }_0^{\infty }[r^2|p_{lm}^{\prime }{|}^2+l(l+1)|p_{lm}{|}^2]dr$. The two sides of the bound are respectively $\sum D_{lm}/[l(l+1)]$ and $\frac{1}{2}\sum D_{lm}$. Since $l(l+1)\ge 2$, the bound follows, with equality for degree-one radial scalars. Together with the radial-flow dynamo energy bound it gives $(q/\eta {)}^2\ge 2E_P/E_{\mathrm{total}}$ for a marginal or growing dynamo field.

// Target: fluid-mechanics.bigb

Twice integrating by parts gives $R(f^{\prime \prime })=\int f{f}^{(4)}$. Estimate this by averaging a fourth-derivative kernel density estimator at the observations. Its expectation contains diagonal term $K^{(4)}(0)/(n{g}^5)$ and leading smoothing bias $-g^2{\mu }_2(K)R(f^{\prime \prime \prime })/2$. Balancing them yields pilot scale $g\asymp R(f^{\prime \prime \prime }{)}^{-1/7}{n}^{-1/7}$. With suitable derivatives and tails, the estimation error is bounded by $O_p(g^2+(n{g}^5{)}^{-1}+n^{-1/2}+n^{-1}{g}^{-9/2})$, giving an $O_p(n^{-2/7})$ relative-bandwidth bound after substitution. More carefully calibrated bias cancellation can improve that bound.

// Target: statistical-inference.bigb

Three distinct points of multiplicity two use the entire arithmetic genus three of an integral projective curve of degree four. One point blowup at each gives arithmetic genus zero. The normalization of an algebraic curve then has genus zero and is the projective line over an algebraically closed field, by rationality of a smooth projective genus-zero curve. This allows double points with one branch as well as ordinary nodes.

The piecewise linear functions $h_n$, for distinct integer indices, are linearly independent over the integers. Restrict a finite integer relation to $y=1$. At $x=n$, only $h_n$ has a jump of one in its slope; hence the coefficient of that function must be zero. This proves that the underlying group of the two-generated free Abelian lattice-ordered group has infinite rank of an abelian group.

For a smooth projective curve $C$ of genus $g$ over an algebraically closed field $k$, its Picard scheme represents the sheafified Picard functor of a curve. It is a disjoint union of smooth proper schemes ${\mathrm{Pic}}_{C/k}^d$, indexed by the degree of a line bundle, each of dimension $g$. Its $k$-points are degree-$d$ line bundles up to isomorphism.^[ref]

The degree-zero component is the Jacobian variety. Its group action on every degree component, by tensor product of sheaves, is free and transitive in families: degree-$d$ components are torsors rather than canonically chosen copies of the Jacobian variety. A chosen degree-$d$ line bundle supplies such an identification by tensoring. Over a general field, the same geometric components exist for a smooth projective geometrically connected curve, but they need not have a rational point, so that identification cannot be assumed over the ground field.

The construction has concrete charts: on the locus of nonspecial degree-$g$ line bundles, Riemann-Roch theorem gives one independent global section, whose zero divisor recovers the bundle. Over the algebraically closed field, translates of this open locus cover the Picard scheme.^[ref]

Independent successive applications of the phase-flip channel $D_{ϵ}(\rho )=(1-ϵ)\rho +ϵZ\rho Z$ preserve diagonal entries and multiply off-diagonal entries by $(1-2ϵ{)}^n$. Hence $D_{ϵ}^n=D_{{ϵ}_n}$ with the displayed parameter, for nonnegative integers $n$. If $0<ϵ<1/2$, the limit is the completely dephasing channel, equivalently a nonselective projective measurement in the computational basis. A time-interval law can be iterated this way only under the memoryless composition assumption.

For a locally integrable periodic function, its Fourier series coefficients are $c_m={\int }_0^{1}f(x)e^{-2\pi imx}dx$. Window the function to one period by $w(x)=H(x)-H(x-1)$. With the stated Fourier transform convention,There is no extra factor of $2\pi$: the Fourier coefficient already integrates over a period of length one.

For a completely arbitrary periodic distribution, multiplication by a discontinuous window need not be defined, for example at a derivative of a delta supported on a period boundary. The sharp-window assertion therefore presupposes a windowing convention or enough regularity at the endpoints. The smooth-window Fourier coefficients of a periodic distribution use a smooth compactly supported partition-of-unity window $\chi$ with ${\sum }_{j\in \mathbb{Z}}\chi (x+j)=1$; then $c_m=\widehat{\chi f}(2\pi m)$ in the distribution sense. The logarithmic example below is locally integrable, so the printed sharp window is legitimate and no ambiguity affects it.

For $f(x)=\log |x-1/2|$, put $u=x-1/2$. Symmetry gives, for integer $m\ne 0$,An integration by parts has no endpoint term: $u\log u\to 0$ at zero and $\sin (\pi m)=0$ at the upper endpoint. It follows thatwhere the sine integral is $\mathrm{Si}(z)={\int }_0^z\sin t/tdt$. Its limiting value $\pi /2$ follows from the Abel-regularized calculation in part (ii); integration by parts of its tail gives $\mathrm{Si}(\pi m)=\pi /2-(-1{)}^m/(\pi m)+O(m^{-3})$ for positive integer $m$. Hence the Fourier coefficients of a periodic logarithmic singularity haveThe first term is the requested leading behaviour. Its alternating sign records the singularity at half a period, and its $m^{-1}$ decay records the logarithmic cusp. The $m^{-2}$ correction comes from the derivative mismatch across the periodically joined endpoints. Also $c_0=-1-\log 2$ and $c_{-m}=c_m$, consistent with the real symmetric profile.

On the tensor product state space, the total angular momentum is $\mathbf{J}={\mathbf{J}}^{(1)}\otimes I+I\otimes {\mathbf{J}}^{(2)}$, conventionally abbreviated ${\mathbf{J}}^{(1)}+{\mathbf{J}}^{(2)}$. Operators on different factors commute. Define $J_{\pm }^{(i)}=J_1^{(i)}\pm i{J}_2^{(i)}$. Expanding the square givesThe first three terms commute with $J_3=J_3^{(1)}+J_3^{(2)}$. In a mixed term one subsystem's magnetic quantum number increases and the other's decreases, so its commutator with total $J_3$ is zero. Hence $\boxed{[{\mathbf{J}}^2,J_3]=0}$.

Use the standard identities $[J_3,J_{\pm }]=\pm J_{\pm }$ and $J_{\pm }|j,m⟩=\sqrt{(j\mp m)(j\pm m+1)}|j,m\pm 1⟩$. The PDF has the commutator in the reversed order with unchanged sign and prints $m+1$ also for lowering; these are sign/label slips. The standard identities are consistent with the meaning of raising and lowering and are used here.

Put $s=j_1+j_2$. The unique product state of magnetic quantum number $-s$ isFor $j_1,j_2>0$, define the orthonormal product states $|a⟩=|j_1,-j_1+1⟩|j_2,-j_2⟩$ and $|b⟩=|j_1,-j_1⟩|j_2,-j_2+1⟩$. Raising the preceding state givesThe normalized orthogonal combination is annihilated by $J_{-}$, because its two lowering amplitudes cancel. It is consequently the lowest-weight state with $J=s-1$:up to an overall phase. For $j_1=3,j_2=1$, the outcome $m_1=-3,m_2=0$ is the $|b⟩$ component. The Born rule therefore gives

For the order-18 group, the index-two subgroup $P$ is normal and $G=⟨P,t⟩$. Restrict a complex irreducible representation $V$ to the finite abelian group $P$. Simultaneous diagonalization decomposes it into character spaces $V_{\chi }$. The inversion relation makes $t$ exchange $V_{\chi }$ and $V_{{\chi }^{-1}}$. The sum of spaces belonging to any such character orbit is $G$-invariant, so irreducibility leaves only one orbit.

If the character is trivial, all of $P$ is in the representation's kernel. Otherwise $\chi$ has order three and $\chi \ne {\chi }^{-1}$. The only spaces are $V_{\chi }$ and $V_{{\chi }^{-1}}$, and both have the same kernel in $P$. A nontrivial character of $P\cong C_3\times C_3$ maps onto $C_3$, so that kernel has order three. It acts trivially on all of $V$. Therefore the group has no faithful irreducible complex representation. In fact choosing nonzero $v\in V_{\chi }$ makes $\mathrm{span}(v,tv)$ invariant, so these nontrivial-character irreducibles have degree two.