For a finitely presented base group $G$ and subgroup $H$, the extension $⟨G,t\mid [t,h]=1(h\in H)⟩$ is finitely presented exactly when $H$ is finitely generated. Sufficiency follows by imposing commutation with finitely many subgroup generators. For necessity, any finite set normally generating the defining kernel uses only finitely many centralizing relators. If their subgroup $H_0$ is proper in $H$, Britton's lemma makes $t^{-1}ht{h}^{-1}$ nonidentity in the partial extension for $h\in H\setminus H_0$, contradicting the claimed kernel equality.

// Target: algebra.bigb

For a length-$r$ central series $1=G_0\le \cdots \le G_r=G$, induction from the top gives ${\Gamma }_{r-i+1}(G)\le G_i$, and induction from the bottom gives $G_i\le Z_i(G)$. Consequently the lower central series terminates at ${\Gamma }_{c+1}=1$ exactly when the upper central series reaches $Z_c=G$.

// Target: algebra.bigb

Forward time with centered advection has amplification magnitude greater than one at nonzero modes. Under a fixed nonzero Courant number, its powers blow up as the time step vanishes. If instead $\Delta t/h^2$ stays bounded, then $\log |G|\le c^2\Delta t^2/(2h^2)$, so $|G{|}^n\le \exp (c^{2}T\Delta t/(2h^2))$ is uniformly bounded on fixed time intervals. This severe parabolic refinement supplies stability without contraction; it does not validate the usual hyperbolic-step choice.

For $f\in C^3[x-h,x+h]$, the Peano kernel for the error of the central secant slope has one sign. Integrating its absolute value gives the displayed constant $h^2/6$, and a cubic polynomial attains it. At $x=1$, $h=1$, the unnormalized order-two kernel is $-{\theta }^2/2$ on $[0,1]$ and $-(2-\theta {)}^2/2$ on $[1,2]$; the integral representation includes the factor $1/2!$.

This symmetric finite difference method approximates a second derivative with $D_{4,h}u=u^{\prime \prime }-h^4{u}^{(6)}/90+O(h^6)$ for sufficiently smooth functions. Matching the constant, second-derivative and fourth-derivative coefficients of a symmetric five-node Taylor expansion determines the displayed weights uniquely; its sixth-derivative coefficient is nonzero. The Fourier symbol is $-4h^{-2}{\sin }^2(\theta /2)[1+{\sin }^2(\theta /2)/3]\le 0$. Thus the semidiscrete heat equation $\dot{U}=D_{4,h}U$ is contractive in the discrete L2 norm by the discrete Parseval identity. This says nothing by itself about stability of a subsequently chosen time update.

Collapse two distinct parallel circles of a torus to two separate points. Cutting between the circles gives two annuli, each of which becomes a two-sphere with the two collapsed points as its poles. The resulting CW complex has two vertices, two edges connecting them, and two two-cells with zero cellular boundary. Its homology is as displayed, with all higher groups zero; it is homotopy equivalent to $S^1\vee S^2\vee S^2$.

The product cells of finite CW complexes give the tensor product of their cellular chain complexes. The boundary formula is the displayed signed product rule. The sign comes from moving the outward normal of the second factor past the $p$ oriented tangent directions of the first factor. This is the cellular counterpart of the Eilenberg–Zilber theorem.

A cellular chain whose cellular boundary is zero.

A finite linear combination, with coefficients in $R$, of the oriented $n$-cells of a CW complex.

For a CW complex, the qth cellular chain group is the relative homology group of successive skeleta, a free abelian group with one generator for each q-cell. The attaching maps determine the cellular boundary, and the resulting cellular chain complex computes singular homology.

A map from the $n$-sphere to a CW complex can be homotoped into its $n$-skeleton. By compact subsets of CW complexes lie in finite subcomplexes, its image lies in a finite subcomplex. For a top-dimensional cell of dimension $d>n$, use smooth approximation of maps into Euclidean space in its interior, with a cutoff supported away from the cell boundary. The Sard theorem supplies a point in a smaller interior ball omitted by the smoothed map; the cutoff is chosen so the transition region also avoids that point. Radial retraction from the omitted point pushes the image into the cell boundary and fixes all other cells. Repeating over the finitely many higher-dimensional cells proves the assertion, without assuming the initial continuous image has measure zero.

A finite spectrum is a topological spectrum equivalent to a cell spectrum with finitely many cells. Such an object is compact in the stable homotopy category and has a Spanier-Whitehead dual. Compactness means maps out of it commute with filtered homotopy colimits of spectra.

Suppose $N$ is a nonnegative integer count of items and each batch holds at most $k$ items, where $k$ is a positive integer. The smallest sufficient number of batches isIndeed, $bk\ge N$ exactly when the integer $b$ is at least $N/k$. If $N=0$, no batches are needed. If $N$ is a multiple of $k$, no partly filled batch is needed.

For $102$ items with capacity $20$ per batch, $b=\lceil 5.1\rceil =6$. Five batches can be full and the sixth holds two items, leaving $6\cdot 20-102=18$ unused places. This is a different question from the five complete groups counted by $\lfloor 102/20\rfloor$.

Writing the Euclidean division $N=kq+r$, $0\le r<k$, gives the equivalent rule $b=q$ when $r=0$, and $b=q+1$ when $r>0$. This explains the ceiling function's jump: one extra item beyond a complete multiple demands a whole additional batch. For exact integer arithmetic, the identity $b=\lfloor (N+k-1)/k\rfloor$ avoids a fractional intermediate value; an implementation still needs an integer representation large enough to hold $N+k-1$.

Myt1 is a protein kinase distinct from Wee1. The cloned Xenopus laevis Myt1 phosphorylated Cdc2 at both inhibitory sites Thr14 and Tyr15; depletion experiments implicated it as the predominant Thr14-directed activity in Xenopus laevis egg extracts.^[ref]

A CDK-activating kinase is a protein kinase supplying activating phosphorylation to a cyclin-dependent kinase 1 preparation. The Xenopus laevis experiment above distinguished this activity from the catalytic activity of Cdc2 itself. CAK names that function here, without asserting that every species uses the same protein assembly.

Write $F(x)={\int }_0^1(1+x{)}^t(1-x{)}^{1-t}dt=x/\mathrm{arctanh}x$. Pairing $t$ and $1-t$ after differentiating twice givesThe power series of $(1-x^2{)}^{-3/2}$ has strictly positive even coefficients, and the other factor has nonnegative even coefficients. Thus every even coefficient of $F^{\prime \prime }$ is negative. Since $F$ is even with $F(0)=1$, the displayed sign pattern follows. This supplies a short positivity argument for the first Dahlquist barrier.

For a weak horizontal forcing proportional to $e^{imz}$ in an ideal uniformly stratified nonrotating fluid, the low-frequency hydrostatic approximation gives the displayed one-dimensional wave equation for the streamfunction amplitude. Two oppositely propagating characteristic integrals determine the response. A source confined to $|x|\le a$ and switched on at time zero produces no response beyond $|x|\le ct+a$ in this reduced equation. The bound on mixed forcing derivative $|f_{xt}|\ll |mN{f}_{\max }|$ controls the neglected vertical acceleration in characteristic magnitude.

In four dimensions a single commuting Majorana spinor is specified by a two-component complex spinor $\xi$. Its future Dirac current is proportional to $({\xi }^{†}\xi ,{\xi }^{†}\mathbf{\sigma }\xi )$, and the Pauli matrix multiplication law implies $|{\xi }^{†}\mathbf{\sigma }\xi {|}^2=({\xi }^{†}\xi {)}^2$. Thus it is null. A Dirac spinor or a sum of independent Majorana currents can be timelike. This distinction matters when applying a timelike-current classification to minimal real supersymmetry.

For a standard Cauchy process, the increment characteristic function is $\exp [-(t-s)|u|]$. Scaling space by ${\alpha }^{-1}$ and time by $\alpha$ leaves this expression unchanged and preserves independent increments, proving equality of all finite-dimensional distributions. Equivalently, $\alpha X_{t/\alpha }$ has the original process law. Scaling both space and time by $\alpha$ instead changes the exponent by ${\alpha }^2$ and is not an invariance unless $\alpha =1$.

A complete Cauchy hypersurface in Minkowski spacetime, such as $t=0$, has full domain of dependence and empty Cauchy horizons. This does not forbid horizons for a non-Cauchy initial surface in the same spacetime. The smooth edgeless spacelike hyperboloid $t=\sqrt{a^2+r^2}$, $a>0$, has domain of dependence $t>r$ and a past Cauchy horizon on the future light cone of the origin. Every causal curve starting inside that cone crosses the hyperboloid in the direction needed for dependence: $t^2-r^2$ passes through $a^2$, while a null generator of the cone never meets it. The time-reversed hyperboloid gives the analogous future horizon.