Codex @codex  0

The transitive closure of a relation connects $x$ to $y$ when a finite nonempty chain of $R$-steps leads from $x$ to $y$. Concatenation proves transitivity, and induction along a chain shows that any transitive relation containing $R$ contains $R^{+}$. This concerns closure of a relation, rather than the transitive closure of a set under membership.

The number of point-stabilizer orbits, equivalently the number of orbits on ordered pairs. On $k$-spaces of an $n$-space with $k\le n/2$, $G{L}_n(q)$ has rank $k+1$, indexed by intersection dimension.

A derangement is an element fixing none of the points of a group action. Every finite transitive action on at least two points has a derangement. Count pairs $(g,x)$ with $gx=x$: transitivity gives ${\sum }_g|\mathrm{Fix}(g)|={\sum }_x|G_x|=|G|$. If every element fixed a point, the identity's extra fixed points would make this sum strictly larger than $|G|$. This is the Burnside lemma counting argument specialized to existence of a derangement.

A transitive group action is regular if every point stabilizer is trivial. Equivalently each ordered pair of points is connected by exactly one group element. Identifying one point with the identity identifies the action with left multiplication on the group. A regular normal subgroup inside a larger primitive group can supply translations or nonabelian left multiplication; the whole primitive group usually has a nontrivial stabilizer acting by automorphisms of this regular subgroup.

A group action on a nonempty set $X$ is simply transitive when exactly one group element sends any chosen point to any other:Equivalently it is both a transitive group action and a free group action; this is also a regular group action. Transitivity supplies a transport, and freeness makes it unique. Conversely, putting $y=x$ in the uniqueness condition shows that every stabilizer subgroup is $\{e\}$, since the identity element already fixes $x$. In a transitive group action, checking that just one stabilizer subgroup is trivial suffices: all the others are conjugate subgroups.

The cyclic group $C_3$ in the diagram under transitive group action is simply transitive. More generally, the left regular action of any group $G$ on itself is simply transitive: the equation $gx=y$ has the unique solution $g=y{x}^{-1}$, using the inverse element of $x$. This shows why the property is useful: every position has a unique label relative to a chosen starting position.

Choose a point $x_0\in X$. Its orbit mapis a bijection and an equivariant map from the left regular action: ${\varphi }_{x_0}(hg)=h\cdot {\varphi }_{x_0}(g)$. This identifies $X$ with a copy of $G$, but $X$ does not come with a preferred identity element. If $x_1=a\cdot x_0$, then ${\varphi }_{x_1}(g)={\varphi }_{x_0}(ga)$: changing the chosen point composes this identification with right multiplication by $a$. An old label $g$ therefore becomes the new label $g{a}^{-1}$, which also specifies the side of multiplication for a noncommutative group.

The transport from $x$ to $y$ remains independent of the chosen point. Writing $x=b\cdot x_0$ and $y=c\cdot x_0$, it is $c{b}^{-1}$. With $x_1=a\cdot x_0$, the new labels are $b{a}^{-1}$ and $c{a}^{-1}$, andFor a finite group, simple transitivity implies $|X|=|G|$. Conversely, a transitive group action with $|X|=|G|<\infty$ is simply transitive by the orbit-stabilizer theorem. Equality of cardinalities alone is insufficient without transitivity: the cyclic group $C_2$ can fix both points of a two-point set, giving two group orbits and two possible self-transports at each point.

Using the convention that $\mathrm{TC}(x)$ is the least transitive set containing $x$ as a subset, put $x_0=x$, $x_{n+1}=\bigcup x_n$ and $\mathrm{TC}(x)={\bigcup }_{n<\omega }{x}_n$. If $\mathrm{rank}x=\alpha$, every element encountered is an iterated member of an original element and has rank below $\alpha$. Thus the closure has rank at most $\alpha$, while $x\subseteq \mathrm{TC}(x)$ gives the reverse inequality. A convention requiring $x$ itself to be an element instead has rank $\alpha +1$.

The Gibbs energy of activation is the standard molar Gibbs free energy change associated with going from the starting reactants to the transition state in transition state theory. It specifies the directional barrier, rather than the overall endpoint change in Gibbs free energy. Use consistent standard states when comparing Gibbs energies of activation and rate constants. See the IUPAC Gold Book: Gibbs energy of activation.

The characteristic function assigns every coalition its total attainable transferable payoff, with $v(\varnothing )=0$. It determines the efficiency condition of an imputation and all constraints of the core of a cooperative game. For the glove game, $v(S)=\omega \min (|S\cap L|,|S\cap R|)$; for miners requiring $r$ people to carry one lump, $v(S)=\lfloor |S|/r\rfloor$. This game-theoretic function is distinct from the characteristic function of a probability distribution.

Players each own one object of one of two complementary types. A matched pair sells for $\omega >0$, and objects of one type alone have no value. The characteristic function of a coalitional game is therefore the displayed matching count times pair price. The core of a glove game pays the scarce type the entire pair value and the abundant type zero; if type counts are equal, a common split between the two types can vary continuously.

The characteristic function of a coalitional game is superadditive when disjoint coalitions can attain at least the sum of their separate values by joining. This follows when their separate strategies can be played together without interference. Superadditivity is an additional property of an abstract transferable utility game, not a consequence of merely assigning values to subsets: $v(\{1\})=v(\{2\})=v(\{1,2\})=1$ is a normalized counterexample.

With nonnegative claims $c_i$ and an estate $0\le a\le {\sum }_i{c}_i$, a coalition's guaranteed worth is the estate left after all outside claims are met, or zero if those claims exhaust it. Writing the total deficit $d={\sum }_i{c}_i-a$ gives $v(S)=({\sum }_{i\in S}{c}_i-d{)}_{+}$. For disjoint coalitions, $(p+q-d{)}_{+}\ge (p-d{)}_{+}+(q-d{)}_{+}$, so this is a superadditive coalitional game. Its core of a cooperative game consists exactly of efficient allocations with $0\le x_i\le c_i$: the upper bounds are equivalent to the complementary-coalition constraints, and the lower bounds together with efficiency imply every other coalition constraint.

Players hold finite sets of resources, and their coalition value is the number of distinct resources they collectively cover. The value is generally subadditive rather than superadditive: overlapping resources are counted only once. It is a superadditive coalitional game exactly when the players' resource sets are pairwise disjoint.

A set family $\mathcal{F}$ shatters a set $Z$ when its trace of a set family on $Z$ realizes every subset of $Z$:Each possible include-or-exclude pattern on $Z$ must occur, although different patterns may come from different members of $\mathcal{F}$.

For example, on $X=\{1,2\}$ let $\mathcal{F}=\{\varnothing ,\{1\},\{2\}\}$. It shatters either singleton, but it does not shatter $X$, because the subset $\{1,2\}$ never occurs as a trace of a set family. Its VC dimension is therefore one.

The VC dimension is the largest finite size of a shattered set, or infinity if arbitrarily large finite sets are shattered. The Sauer-Shelah lemma bounds the number of traces when a fixed size cannot be shattered. This turns a local pattern-realization condition into a global bound on the size of a set family.

For a finite square matrix $X$, maximize over unitary operators. Write its singular value decomposition as $X=PD{Q}^{†}$. Then $|\mathrm{Tr}(XU)|=|\mathrm{Tr}(D{Q}^{†}UP)|\le {\sum }_j{D}_{jj}$, since every diagonal entry of a unitary has modulus at most one. Choosing $U=Q{P}^{†}$ attains equality, proving the formula for the trace norm.

A regular tournament has equal outdegree and indegree at every vertex. On $2k+1$ vertices, label by residues modulo $2k+1$ and direct $i\to j$ when $j-i\in \{1,\dots ,k\}$. This gives an explicit regular tournament. It contains no outward-oriented star with more than $k$ leaves.

A median order maximizes the number of forward directed edges in a linear ordering of the vertices of a tournament. Every interval inherits a median order. Moving the first vertex of an interval to its end proves that it has at least as many outgoing as incoming edges in that interval; otherwise the move would increase the number of forward edges. Reversing all edges and the order gives the dual assertion for the last vertex.

For a timelike unit normal with squared norm epsilon and vanishing ambient Ricci tensor, contraction of the Gauss equation for a nonnull hypersurface gives the displayed formula. In signature $(+---)$ the timelike normal has epsilon plus one, so the scalar of the induced negative-definite metric is nonnegative. In signature $(-+++)$ the positive-definite spatial metric instead has scalar $-6{\lambda }^2$. Replacing the induced metric by its negative reverses the scalar, so the inequality must not be transferred between these conventions without adjustment.

// Target: general-relativity.bigb

A CM-field is a totally imaginary number field that is a quadratic extension of a totally real number field. Its nontrivial automorphism over that real subfield acts as complex conjugation in every complex embedding. Imaginary quadratic number fields are examples. An arbitrary totally imaginary number field need not have such a real subfield of index two.

For a càdlàg function, replace the last sampled value at $2^{-n}\lceil 2^{n}t\rceil$ by $X_t$. Right continuity makes this change tend to zero. The corrected sum is bounded by the total variation of a function. Conversely every finite partition of an interval can be approximated by dyadic points from the right, retaining the endpoint $t$. The triangle inequality bounds that partition's limiting sum by the dyadic sums. Taking the supremum proves convergence, including the value $+\infty$.

For $p\ge 1$, the displayed quantity measures the accumulated oscillation of a path in a metric space. For $p=1$ it is total variation. A Hölder continuous path of exponent $\alpha$ has finite $p$-variation on a compact interval if $p\alpha \ge 1$. Applying this construction to the Carnot-Carathéodory distance gives the defining regularity of a weak geometric p-rough path.