A genuinely closed steady channel with impermeable sidewalls and ends must have zero integrated axial cell flux through every section, not only zero fluid flux. In the bulk phototactic ansatz $n^{\prime }=\beta n$, ${\omega }_y=-w^{\prime }$, and $⟨p_z⟩=-K_1{w}^{\prime }/(2D_r)$, the axial flux is $[1+V_s\beta K_1/(2D_r)]\int nwdx$. Meanwhile the momentum equation, zero fluid flux and integration by parts give $g^{\prime }\int nwdx=-\mu \int (w^{\prime }{)}^{2}dx$. Thus a nontrivial bulk profile is generally incompatible with exact closed-end cell conservation unless the prefactor vanishes. Additional axial concentration variation is needed; neglecting end regions does not itself remove this global constraint.

For $n\ge 2$, a Prüfer sequence is the encoding of a labelled tree on $\{1,\dots ,n\}$ by a sequenceRepeated labels are allowed, and their order matters. To encode the tree, repeatedly remove its smallest-labelled leaf of a graph together with its incident edge, recording the label of its unique neighbour. Stop when two vertices remain; do not record their final edge. Removing a leaf of a graph leaves a tree, so the procedure makes exactly $n-2$ deletions. The choice of the smallest label makes the encoding deterministic. ^[ref]

For a concrete example, take six vertices and the set of edgesInitially $2,4,6$ label the vertices of vertex degree one. Remove $2$ and record its neighbour $1$. Now $1$ has become a leaf of a graph, so remove $1$ and record $3$. Next remove $4$ and record $3$ again; then remove $3$ and record $5$. The final edge joins $5$ and $6$. Thus the Prüfer sequence isThe deletion order is $2,1,4,3$: removing a leaf of a graph can make its neighbour a leaf of a graph, so the eligible labels change as the tree shrinks.

To decode, keep a set $L$ of remaining labels, initially $\{1,\dots ,n\}$, and the remaining sequence, initially $p$. While the sequence is nonempty, let $v$ be the smallest label in $L$ absent from the entire remaining sequence. Add an edge between $v$ and its first entry, then remove $v$ from $L$ and delete that first entry. Finally join the two labels left in $L$. For the example:

The decoder always has a choice: if the remaining sequence has length $m$, then $L$ has $m+2$ labels, so at least two of them are absent. Its first entry is still in $L$, and differs from the chosen absent label. Every removed vertex attaches to a vertex that survives longer; following these attachments eventually reaches the final pair. Consequently the decoded graph is connected, and its $n-1$ edges make it a tree. In this construction, each occurrence of $v$ contributes an incident edge, and $v$ gets one more edge at its final attachment. Thus a remaining label is absent from the remaining sequence exactly when it labels a leaf of a graph. Encoding and decoding choose the same smallest label at each step and undo each other. This bijection proves Cayley's formula and explains why every freely chosen sequence is valid. ^[ref]

For $n=2$ there is exactly one empty sequence, and decoding adds the sole edge. For $n=1$ there is one tree, with vertex degree zero, but no length-$n-2$ Prüfer sequence. This is the exception both to the encoding convention and to the Prüfer degree formula.

Heinz Prüfer's 1918 paper "Neuer Beweis eines Satzes über Permutationen" appeared in Archiv der Mathematik und Physik, third series, volume 27, pages 142–144. It gives an encoding proof of the labelled tree count. The historical title means "New proof of a theorem on permutations"; the modern Prüfer sequence permits repeated labels, so it is not a permutation. ^[ref]

For a fixed set of $k<n$ roots among $n$ labelled vertices, the number of forests with exactly one root in each graph component is $k{n}^{n-k-1}$. Orient edges toward their roots. Repeatedly remove the smallest nonroot leaf and record its parent. This gives a code of length $n-k$ whose last entry is a root. Conversely, for any such code choose the smallest remaining nonroot absent from the remaining code, join it to the first entry, and delete it. There is always a choice because the final entry is a root. These inverse operations give $n^{n-k-1}$ choices for the earlier entries and $k$ for the last. For $k=n$, the empty forest is unique.

For the additive infinite cyclic group $\mathbb{Z}$, the generating set of a group $S=\{-1,1\}$ produces the infinite line: each integer $n$ is adjacent to $n-1$ and $n+1$. Adding the moves $\pm 2$ gives $T=\{-2,-1,1,2\}$ and additional edges skipping one vertex. Each skip still costs one step in the word metric.

For integers $a,b$, these word metrics areThe absolute value counts unit moves, and the ceiling function counts pairs plus a possible leftover unit. Every $T$-move changes the integer by at most two, while moves of size two followed by at most one move of size one attain this bound. For example, the graph path $0,1,2,3,4,5$ has five steps for $S$, while $0,2,4,5$ has three for $T$. This calculation gives the concrete comparison

The picture shows only a finite window of each infinite Cayley graph. By contrast, $U=\{-2,2\}$ alone fails to be a generating set of a group for $\mathbb{Z}$: it generates the subgroup $2\mathbb{Z}$. Its move graph has two connected components of a graph, the even integers and the odd integers. Within either component the graph distance is $|a-b|/2$; between different components no finite graph path exists. Adding a shortcut is different from deleting the moves needed for connectivity.

For $a{u}_{x_1}+b{u}_{x_2}=c$ with data along $(\xi (s),\eta (s))$, the initial curve is noncharacteristic when its tangent is linearly independent of the projected characteristic vector $(a,b)$ evaluated at the initial value of $u$. The determinant is $a{\eta }^{\prime }-b{\xi }^{\prime }$, up to sign. For $C^1$ coefficients and data this permits the inverse function theorem to invert the local characteristic parametrization and gives a unique local classical solution.

Choose distinct ${\kappa }_j>0$ and weights $c_j>0$ at spectral points $i{\kappa }_j$. With $E_j=e^{-{\kappa }_{j}x+{\kappa }_j^{3}t}$ and $A_{ij}=c_j{E}_i/({\kappa }_i+{\kappa }_j)$, solve $(I+A)\Phi =E$. The reconstructed multisoliton solution is $q=-{\partial }_x(c^T\Phi )$. Since $\Lambda A+A\Lambda =E{c}^T$, cyclicity of the trace gives $c^T\Phi =-2{\partial }_x\log \det (I+A)$. The Cauchy matrix $1/({\kappa }_i+{\kappa }_j)$ is positive definite, because it is the Gram matrix of $e^{-{\kappa }_{i}s}$ on $s>0$; diagonal similarity then proves $\det (I+A)>0$. For one spectral point this yields $q=({\kappa }^2/2){\mathrm{sech}}^2[\kappa (x-{\kappa }^{2}t-x_0)/2]$.

Apply the symmetric-function involution in one alphabet to the ordinary Cauchy identity. It changes complete generators to elementary ones and, by the dual Jacobi–Trudi identity, sends $s_{\lambda }$ to $s_{{\lambda }^{\prime }}$. This proves the identity as a formal series and relates pairings with transposed Young shapes to squarefree matrix weights.

Given $F(t)={\sum }_{r\ge 0}{f}_r{t}^r$ with $f_0=1$, put $f_r=0$ for $r<0$. The Cauchy identity for symmetric functions and Jacobi–Trudi identity give ${\prod }_{i}F(t_i)={\sum }_{\lambda }{s}_{\lambda }^F{s}_{\lambda }(t)$, degree by degree in the completion. For $F(t)={\prod }_j(1-x_{j}t{)}^{-1}$ the coefficients are $s_{\lambda }(x)$; for $F(t)={\prod }_j(1+x_{j}t)$ they are $s_{{\lambda }^{\prime }}(x)$.

For a holomorphic function on a neighbourhood of a closed disc, parametrize the Cauchy derivative formula on its boundary. The conjugate Cauchy integral theorem gives $\int \overline{f(r{e}^{i\theta })}{e}^{-i\theta }d\theta =0$. Replacing $f$ in the derivative integral by $f+\bar{f}=2\mathrm{Re}f$ therefore gives the displayed identity. The boundary real part determines every nonconstant analytic coefficient; only an imaginary additive constant is lost.

A compact object (mathematics) is an object whose represented Hom functor preserves filtered colimits. This preservation condition differs from the extra right-adjoint property defining a tiny object.

An adjunction is a natural bijection $\mathcal{D}(LB,A)\cong \mathcal{C}(B,RA)$.

A diagram of shape $\mathcal{J}$ in $\mathcal{C}$ is a functor $D:\mathcal{J}\to \mathcal{C}$.

An object $P$ is projective when every morphism $P\to B$ lifts through every epimorphism $A\twoheadrightarrow B$. Equivalently, the functor $\mathcal{C}(P,-)$ preserves epimorphisms.

An object with the displayed property is called small projective; the term 0-presentable here uses this all-small-colimits convention. In a presheaf category, every representable presheaf has this property by the Yoneda lemma and pointwise computation of colimits. Conversely the canonical colimit presentation of a presheaf shows that the identity on such an object factors through one representable, making it a retract. If the indexing category has equalizers, a representable retract is again representable by the representable retract criterion.

A categorical morphism is an arrow with a specified source and target in a category. An arrow $f:A\to B$ composes with $g:B\to C$ to give $gf:A\to C$. Composition is associative, and every object has an identity arrow acting as a unit. A functor preserves the sources, targets, identities and compositions of these arrows. In a locally small category, the arrows between fixed objects form a hom-set.

Reversing the arrows in a categorical statement produces its dual statement. A theorem holding for all categories therefore has a dual theorem obtained by applying it to the opposite category; for example, categorical limits are exchanged with colimits.

Transition state theory relates an elementary reaction's rate constant to a Gibbs energy of activation. The Eyring expression used in the catalyst example assumes every passage counted at the transition state succeeds. More generally, a factor accounting for unsuccessful passage multiplies it. Comparing barriers alone gives the quoted speedup only when the remaining prefactors agree. Free-energy profiles can depend on temperature, so the example's values at $300\mathrm{K}$ do not determine a temperature curve. See the IUPAC Gold Book: transition state theory.

A Hadamard space is a complete CAT(0) space. Equivalently, it is a complete metric space in which every pair $x,y$ has a midpoint $m$ satisfyingfor every $z$. A midpoint means $d(x,m)=d(m,y)=d(x,y)/2$. Substituting a second midpoint for $z$ forces their distance to be zero, so the midpoint is unique.

Iterating midpoints defines the joining geodesic first at dyadic parameters. Completeness then supplies its points at every real parameter. The midpoint inequality extends by this approximation to the Hadamard squared-distance inequality, and hence to Euclidean triangle comparison. In particular, closed balls have geodesic convexity.

The completeness condition belongs in both formulations. The rational numbers with the usual distance have all rational midpoints and satisfy the displayed inequality, but are incomplete. A full real-parameter geodesic from zero to one would require irrational points. Thus the midpoint inequality alone does not make that space a Hadamard space.

For an integral domain $A$ with fraction field $K$, realize Cartier divisors as invertible fractional ideals, with local divisor equation $f$ corresponding to the ideal $fA$. This is the $\mathcal{O}(-D)$ convention. The maps in the displayed exact sequence are unit inclusion, $f\mapsto fA$, and $I\mapsto [I]$. A principal fractional ideal is trivial as a divisor exactly when its generator is a unit. An invertible fractional ideal is trivial in the Picard group of a ring exactly when it is principal. Every rank-one projective module embeds in $K$ and produces an invertible fractional ideal, proving surjectivity. Local multiplication identifies $I{\otimes }_{A}J$ with $IJ$, so these are group homomorphisms.

For a morphism of varieties $f:X\to Y$, pull back local defining rational functions of a Cartier divisor $D$ whenever they remain nonzero on each component of $X$. Their ratios remain regular units, defining $f^{\ast }D$. Equivalently, pull back the associated line bundle and its rational section. A dominant morphism always permits this construction.