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If $f:U\to V$ is a holomorphic function between planar open sets and $\psi :V\to \mathbb{R}$ is a harmonic function, then $\psi \circ f$ is harmonic. In fact the chain rule and the Cauchy-Riemann equations, with $f=u+iv$, giveHere the two gradients are orthogonal and have equal length, while $u,v$ are harmonic. The formula holds even where $f^{\prime }=0$. A biholomorphism therefore induces inverse correspondences between harmonic functions by function composition with it and its inverse.

Production of red blood cells from dividing precursors, distinguished from division of mature mammalian red blood cells.

With inhibited bacterial growth $b_t=b_{xx}+gb(1-\eta b)$ and unchanged depletion $g_t=-gb$, a positive travelling wave leaving density $b_1$ obeys $c{b}_1=c-\eta {\int }_{\mathbb{R}}g{b}^{2}dy$. This follows by integrating both travelling-wave equations between their constant endpoint states. Thus $b_1<1$ for $\eta >0$: the yield behind the front is smaller than for uninhibited growth.

A triangular group presentation has relators of word length at most three. Every finitely presented group admits one. For a relator $s_1\cdots s_{\ell }$ of length $\ell \ge 4$, introduce generators $y_2,\dots ,y_{\ell -2}$ and replace it byThe defining relations say $y_j=s_1\cdots s_j$; eliminating those generators by Tietze transformations recovers the original relation. Every new relator has length three. Geometrically these operations triangulate each relator disk in its presentation complex by inserting diagonals, so they preserve the fundamental group.

A group antihomomorphism between groups reverses multiplication: $f(gh)=f(h)f(g)$. Inversion $g\mapsto g^{-1}$ is an example. An antihomomorphism is a group homomorphism when the multiplication in the target is replaced by its reverse. In a nonabelian target the reversal matters: the group homomorphism and antihomomorphism identities generally differ.

Let $G$ be a finite group and $\varphi :G\to H$ a surjective group homomorphism. Choose a preimage $g$ of $h$. Since $\varphi (g{)}^{o(g)}=e$, the order of a group element $m=o(h)$ divides $o(g)$; write $o(g)=mr$. Then $g^r$ has order $m$, because $(g^r{)}^s=e$ holds exactly when $mr\mid rs$, or equivalently $m\mid s$. This proves existence of an element of order $m$ in $G$. It need not be a preimage of $h$: for the quotient map ${\mathbb{Z}}_4\to {\mathbb{Z}}_2$, both preimages of the nonidentity target element have order $4$, while the order-$2$ element of the source maps to zero. Thus the assertion concerns element-order existence, not splitting of the quotient map.

For a finite group $G$, transitive finite group actions are classified up to equivalence by conjugacy classes of subgroups: the class of $H$ gives the coset action on $G/H$. Every finite action decomposes uniquely as a disjoint union of its orbits, so its equivalence type is a multiset of these transitive types. Consequently the coefficient of $t^n$ in the displayed product, with one factor for every subgroup conjugacy class, counts actions on $n$ points up to relabeling. Each factor expands as a geometric series recording any multiplicity of that orbit type. For both $C_6$ and $S_3$ there is exactly one transitive type of each size $1,2,3,6$; their coefficient at degree six is $4+2+1+1=8$.

Assuming choice, every nonempty set admits a group operation. A finite set receives a cyclic group structure. For an infinite set, a maximal partial group obtained by Zorn has infinite carrier $A$ with smaller complement; infinite-cardinal doubling implies the whole set is equipotent with $A$. Transport its operation along a bijection $j$. This final group need not extend a previously chosen partial group on its original subset; transport changes labels and only asserts existence.

Two identical stationary devices counting operations at the same rate per proper time act as clocks. If the upper one sends start and finish light signals over a fixed path to the lower one in a stationary metric, both propagation delays are identical in coordinate time and cancel from the interval comparison. The lower elapsed proper time divided by the upper one is $\sqrt{g_{tt}(r_2)/g_{tt}(r_1)}$. In the weak field the relative lag is $g\Delta h/c^2$, giving the displayed operation deficit. A rate of $10^{12}$ operations per second, a week and a height difference of ten metres give a deficit of roughly $660$ operations. This observable tests the radial clock-rate factor; it does not independently determine the radial spatial metric coefficient because the common signal delay cancels.

A face colouring of a plane graph assigns colours to its faces of a plane graph so that distinct faces sharing an edge have different colours. Equivalently it is a graph colouring of the dual graph, disregarding dual loops that arise when an original bridge is incident to only one face on both sides. Isolated contact at a vertex does not impose a colour constraint.

For the dual graph of a plane tree, the face-adjacency graph is the original tree, because taking the geometric dual twice recovers the original embedding on the sphere. Every nontrivial tree is a bipartite graph, so its dual requires exactly two face colours, even though that dual has only one vertex and all its edges are loops. A graph drawing must retain the nesting of those loops to preserve the correct face-adjacency graph.

The set of edges of a specified graph. In a simple graph, an edge is determined by its endpoints; a multigraph additionally distinguishes parallel edges with the same endpoints. A subgraph can be specified by selecting a vertex set and an edge set whose endpoints belong to it.

For this nonsymmetric matrix, consider the formal residual update $r^{+}=r-(r^{T}r)/(r^{T}Ar)Ar$, with nonzero $r=(x,y{)}^T$. Put $D=2x^2-2xy-2y^2$ and $B=5x^2+4xy+3y^2$. Algebra gives $r^{+}=(B/D)(-y,x{)}^T$ when $D\ne 0$. But $B+D=7(x+y/7{)}^2+6y^2/7>0$ and $B-D=3(x+y{)}^2+2y^2>0$. Hence $B>|D|$ and $\Vert r^{+}\Vert >\Vert r\Vert$ for every defined nonzero step. At $D=0$, the quotient is undefined. Thus the iteration cannot converge to zero linear-system residual from any nonzero initial residual. This concrete divergence proof distinguishes failure of an algorithm from mere failure to meet a sufficient theorem's hypotheses.

An analytical method using an enzyme-coupled reaction to measure glucose; the selectivity, calibration and intended use require validation.