The $k$ order-$k$ B-splines active on a simple-knot cell form a basis of its degree-$k-1$ polynomials. To see this from Marsden's identity, use the dual polynomials ${\psi }_i(y)={\prod }_{\ell =1}^{k-1}(y-t_{i+\ell })$. On $t_j<x<t_{j+1}$ the identity is $(y-x{)}^{k-1}={\sum }_{i=j-k+1}^j{N}_i(x){\psi }_i(y)$. Evaluation of the dual polynomials at $t_j,t_{j-1},\dots ,t_{j-k+1}$ gives a triangular coefficient matrix with nonzero diagonal. They are independent, and comparison of coefficients of $y$ in the identity gives independence of the active splines. Consequently a spline vanishes on a whole open cell exactly when all its active coefficients vanish.

For the partition-of-unity normalization, the derivative of a degree-$d$ B-spline is the displayed difference of two overlapping degree-$(d-1)$ B-splines. A zero denominator denotes a term with collapsed support, assigned zero. For simple knots the divided-difference representation $N_{i,d}(t)=(t_{i+d+1}-t_i)[t_i,\dots ,t_{i+d+1}](\tau -t{)}_{+}^d$ has the required support, smoothness and integral normalization. Differentiate the truncated power and apply the last-step divided difference recursion to obtain the formula. Repeated-knot versions follow by confluent limits, with one-sided interpretation where necessary.

Assume zero toroidal loop voltage in smooth steady ideal magnetohydrodynamics, so the poloidal flow aligns with the poloidal magnetic field. Write $\rho {\mathbf{u}}_p=\eta {\mathbf{B}}_p$. The continuity equation makes $\eta$ constant along a magnetic line. The induction equation gives the conserved field-line angular velocity ${\Omega }_F=\Omega -\eta B_{\varphi }/(\rho R)$. The azimuthal momentum equation is $\rho {\mathbf{u}}_p\cdot \nabla (R^2\Omega )={\mathbf{B}}_p\cdot \nabla (R{B}_{\varphi })/{\mu }_0$; hence $L=R^2\Omega -R{B}_{\varphi }/({\mu }_0\eta )$ is also constant along the line.

With $M_{Ap}^2={\mu }_0{\eta }^2/\rho =|{\mathbf{u}}_p{|}^2/v_{Ap}^2$, eliminating $B_{\varphi }$ givesFor $M_{Ap}^2\ll 1$ and nonsingular invariants, $\Omega ={\Omega }_F+O(M_{Ap}^2)$ and matter approximately corotates along a field line. With exactly zero poloidal flow, induction directly gives Ferraro's law of isorotation. The approximation requires the correction $M_{Ap}^2({\Omega }_F-L/R^2)$ to be small on the segment in question.

For a cold equatorial axisymmetric magnetohydrodynamic wind with $B_p=C{R}^{-2}$ in the gravity of a point mass $M$, smooth crossing of the Alfvén surface gives ${\rho }_a={\mu }_0{k}^2$ and $\ell =\omega R_a^2$. Put $x=R/R_a$, $y=\rho /{\rho }_a$, $\alpha ={\omega }^2{R}_a^3/(GM)$ and $\beta =C^2/({\mu }_0{\rho }_{a}GM{R}_a^3)$. The corotating energy, scaled by $GM/R_a$, isThe azimuthal velocity relative to field-line rotation is $\omega R_a(x-x^{-1})/(y-1)$ and the poloidal speed is $C/(\sqrt{{\mu }_0{\rho }_a}{R}_a^2{x}^{2}y)$. The apparent singularity at $x=y=1$ is interpreted through a smooth limiting slope. A nondegenerate magnetosonic crossing on a constant-$f$ curve requires $f_x=f_y=0$ and a real admissible tangent slope.

For the spatial group combined with the temporal phase circle in an equivariant Hopf problem, a complex axial isotropy subgroup is an actual stabilizer whose fixed-point space is one complex line. Its real dimension is two. One must include the kernel of the combined action when writing the full stabilizer, not just a generator having the right fixed space.

// Target: dynamical-systems.bigb

For $t\ge 1$, iterating an AR(1) from a fixed value $X_0=x_0$ gives the displayed finite sum. With centered white noise of variance ${\sigma }^2$ and $|\varphi |<1$,For $h\ge 0$, its covariance is $\mathrm{Cov}(X_{t+h},X_t)={\varphi }^h\mathrm{Var}(X_t)$, which depends on $t$. Even setting $x_0=\mu$ leaves a changing variance. At $\varphi =0.7$, ${\sigma }^2=1$, and $x_0=\mu$, the variances at times zero through three are $0,1,1.49,1.7301$, approaching $100/51$; this finite initialized record is not weakly stationary.

To initialize in weak stationarity, give $X_0$ mean $\mu$, variance ${\sigma }^2/(1-{\varphi }^2)$, and zero covariance with every future driving noise value. The full stationary construction sets $X_0-\mu ={\sum }_{j\ge 0}{\varphi }^j{\epsilon }_{-j}$. For discrete Gaussian white noise, equivalently draw $X_0$ from $N(\mu ,{\sigma }^2/(1-{\varphi }^2))$ independently of future noise. This gives exact strict stationarity from the first plotted point; it is the initialization used in the figure in time series. Matching these first two moments alone does not guarantee strict stationarity for non-Gaussian inputs. See also stationary initialization of a scalar linear state-space model.

For the stationary causal AR(1) with $|\varphi |<1$, matching terms in its infinite moving-average representation gives, for $h\ge 0$,Symmetry of the autocovariance extends this to negative lags. Dividing by $\gamma (0)$ gives the autocorrelation $\rho (h)={\varphi }^{|h|}$. The geometric series explains both the enlarged stationary variance and the diminishing memory: all past disturbances contribute to the present, but their squared weights have a finite sum. At $\varphi =0$, the process reduces to shifted white noise and all nonzero-lag autocorrelations vanish. See Bartlett, lecture 4: AR(1) representation and mean-square convergence.

The autoregressive operator acts as $\Phi (B)X_t=X_t-{\sum }_{j=1}^p{a}_j{X}_{t-j}$. Its coefficients are encoded by the autoregressive polynomial. An ARMA equation equates this filtered observation to the noise filtered by the moving-average operator.

A permutation of order two is a product of $k$ disjoint transpositions. Its conjugacy class has the displayed size. For $k\ge 2$, equality with the transposition class requires $(n-2)!/(n-2k)!=2^{k-1}k!$. The left side grows strictly with $n$. At $n=2k$, its ratio to the right side is one only for $k=3$; at $k=2$ it jumps from below to above one between $n=4$ and $n=5$, and for $k\ge 4$ it is already greater than one. Thus degree six is the only possible exceptional degree for an group automorphism not preserving transpositions.

In characteristic $p$, the augmentation ideal $I$ of the group algebra of a finite $p$-group is nilpotent. Choose a central element $z$ of order $p$ and put $t=z-1$, so $t^p=0$. The quotient by $t$ is the group algebra of the smaller quotient group. Induction gives $I^m\subseteq tkP$ for some $m$, hence $I^{mp}=0$. For a normal $p$-subgroup $P$ of $G$, conjugation stability gives $(IkG{)}^m=I^{m}kG$, proving nilpotence of the kernel of $kG\to k(G/P)$.

In the ordinary group algebra of the additive p-adic integers, let $I$ be the augmentation ideal. The kernel of the map to the group algebra of ${\mathbb{Z}}_p/p^r{\mathbb{Z}}_p$ is generated by $[p^{r}h]-1=([h]-1{)}^{p^r}$, so it is contained in $I^{p^r}$. The quotient is the cyclic-group algebra ${\mathbb{F}}_p[T]/(T^{p^r})$, so its augmentation ideal has zero $p^r$th power and the reverse inclusion holds. Every finite degree of the associated graded ring can be read from such a quotient, giving ${\mathrm{gr}}_I{\mathbb{F}}_p[{\mathbb{Z}}_p]\cong {\mathbb{F}}_p[X]$. The completion is ${\mathbb{F}}_p[[T]]$, but completion is not needed for the graded calculation.

The skeletal filtration of a CW complex and the cofiber sequences of spectra of its cell attachments produceFor a connective spectrum it is first quadrant and converges in each fixed total degree. Its reduced version has no contribution from the basepoint summand.

For a posterior distribution with finite first moment, the posterior risk derivative at continuity points is $(\gamma +\delta )F(a)-\gamma$. Convexity therefore makes a posterior $\gamma /(\gamma +\delta )$-quantile a Bayes estimator. With atoms, the condition is $F(a^{-})\le \gamma /(\gamma +\delta )\le F(a)$. Increasing the cost of underestimation moves the optimal estimate upward.

For a commutative Noetherian ring, an annihilator $Q={\mathrm{Ann}}_R(m)$ contained in $P$ localizes to the annihilator of the nonzero $m/1$. Conversely, contract an associated prime of $M_P$ represented by $m/1$ to an ideal $Q$. Choose finitely many generators of $Q$ and one denominator outside $P$ clearing all their annihilation equations. For this denominator $s$, the element $sm$ is nonzero and has annihilator exactly $Q$. This proves the formula; finite generation of the contracted ideal is the key use of Noetherianity.

// Destination: noncommutative-algebra.bigb

For a partial assignment problem, add its fixed cost to the sum, over remaining tasks, of each task's cheapest available machine cost. This is a lower bound because it relaxes the requirement that different tasks use distinct machines. It is an admissible bound for branch and bound, even though the minimizing choices may assign the same machine repeatedly.

// Target: mathematical-optimization.bigb

For an abelian extension $L/K$ of number fields, restriction to an intermediate field takes each Artin symbol to that of the smaller extension. For a finite base extension $E/K$, restriction identifies $\mathrm{Gal}(LE/E)$ with $\mathrm{Gal}(L/(L\cap E))$. At an unramified prime $\mathfrak{q}$ of $E$ above $\mathfrak{p}$, its residue-field Frobenius restricts to the $f(\mathfrak{q}/\mathfrak{p})$th power of the Frobenius automorphism at $\mathfrak{p}$. Since $N_{E/K}\mathfrak{q}={\mathfrak{p}}^{f(\mathfrak{q}/\mathfrak{p})}$, the group homomorphism property proves the displayed identity for fractional ideals supported over unramified primes.

For a finite abelian extension of number fields, local reciprocity maps into the local decomposition groups combine to a continuous map on the idele group: outside finitely many places they kill units, and an idele has nonunit components only finitely often. The Artin reciprocity law is equivalent to the compatibility that this product kills every principal idele. Conversely, restrict a global continuous Artin homomorphism to each single-place embedding of $K_v^{\ast }$. These maps reconstruct the global map, by density of finite-component ideles and triviality on a basic open subgroup of units. Merely having unrelated local maps omits the essential principal-idele compatibility.

For an integral projective curve on a smooth projective surface, blowing up a point of multiplicity $m$ gives $C^{\prime }={\pi }^{\ast }C-mE$. The intersection formula for blowing up a surface and canonical divisor formula for a surface blowup yield $C^{\prime 2}=C^2-m^2$ and $K_{X^{\prime }}\cdot C^{\prime }=K_X\cdot C+m$. Substitute both in the arithmetic adjunction formula on a smooth surface. This does not require the transformed curve to be smooth.

For area $A(R)=R^2$ and smooth data $q(\theta ,R_0)=f(\theta )$ at $R_0>0$, the transformed distance is $\zeta =R_0\log (R/R_0)$. If $M=\sup f^{\prime }>0$, the first shock wave occurs at the displayed radius; if $M\le 0$, there is no forward characteristic crossing. Nontrivial bounded data cannot instead be imposed at $R=0$ in this outgoing simple-wave model: along every characteristic, $Rq$ is constant, so a bounded limit at the origin forces that constant to vanish. With distance measured from a finite emitting surface, use $A(Z)=(R_0+Z{)}^2$ and subtract $R_0$ from the shock radius.

Restrict the affine parameter to $|\lambda |<1$, before either transverse Jacobi factor vanishes at a caustic.

In flat spacetime take a twist-free beam's two screen Jacobi factors to be $1+\lambda$ and $1-\lambda$. They solve the zero-curvature geodesic deviation equation. Their product is the displayed area, and subtracting the trace from their optical matrix gives the displayed nonnegative shear. Direct differentiation yields $(A^{1/2}{)}^{\prime \prime }=-(1-{\lambda }^2{)}^{-3/2}=-{\sigma }^2{A}^{1/2}$, an explicit check of the focusing sign.