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Inclusion-exclusion applied to the events that a permutation fixes a specified point gives the displayed formula. There are $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ choices of $k$ fixed points and $(n-k)!$ permutations fixing them. The probability that a uniformly chosen permutation is a derangement tends to $e^{-1}$.

// Target: analysis.bigb

In a slender steady porous gravity current, the dominant balance is $(h^{\beta }{)}_X=(h^{\beta }{h}_Y{)}_Y$. If its width and height scales are $W,H$, fixed downslope volume flux gives $H^{\beta }W=\text{constant}$, and transverse spreading gives $W^2\sim HX$. Eliminating $H$ yields the displayed width exponent. The porosity exponent affects transient storage but not this steady balance.

Color the vertices of a periodic graph independently red or blue with equal probabilities, and declare a vertex of its line graph open exactly when the corresponding edge has equal-colored endpoints. The resulting site percolation has marginal $1/2$ and is a finite-range dependent random field: sets at line-graph distance greater than one use disjoint color inputs. Red and blue open clusters cannot join. If the original graph has an infinite red component and an infinite blue component, its line graph consequently has at least two infinite percolation clusters. One explicit amenable graph with that property is obtained by replacing each square-lattice vertex by a complete graph on ten vertices, joining neighboring blocks completely. Blocks containing both colors have independent probability $511/512$; a Peierls argument proves that they percolate, and their infinite cluster carries both a red and a blue component. Thus finite-range dependence, even with all site marginals nondegenerate, does not substitute for finite energy in percolation uniqueness.

A bond percolation law is one-independent if the bond states in any two sets with disjoint endpoint sets are independent as collections. In particular, states of bonds in a matching in a graph are mutually independent. The law need not be invariant under translations or positively associated.

For events $A_i$, a directed graph is an independence digraph if $A_i$ is independent of the sigma-algebra generated by the events indexed by its non-outneighbors, excluding $i$. The direction records which joint independence condition is available. A cyclically oriented triangle imposes pairwise independence; it does not impose independence of all three events together.

A Riemannian metric on a smooth manifold gives a positive section of the density bundle, locally $d{\mu }_g=\sqrt{\det (g_{ij})}|d{x}^1\cdots d{x}^n|$. The absolute Jacobian determinant makes this independent of coordinates without requiring orientability. With an orientation of a smooth manifold, it corresponds to the Riemannian volume form.

The mapping class group of a compact connected oriented topological surface, for orientation-preserving maps fixing its boundary pointwise and isotopies relative to that boundary, is generated by Dehn twists about simple closed curves. Boundary-parallel twists are allowed. For a closed surface of positive genus a finite generating collection exists. Permuting boundary components requires additional generators if that larger group is used.

A nondecreasing function bounding the area of every null word of length at most $n$ is an isoperimetric function for a finite group presentation. The Dehn function is the minimal such bound. A subrecursive bound means one dominated by a total computable function. Arbitrarily enlarged bounds are not invariants: even a group presentation with zero area function admits both linear and exponential upper bounds, which are not equivalent under the usual coarse comparison.

Choose bounded-length substitutions $T$ from the first generating set to the second, and $U$ back. Each substituted relator of the second group presentation has bounded area in the first, and every generator correction $a(UT(a){)}^{-1}$ has bounded area. Substitution into a minimal second-presentation diagram and filling the correction strips givesThe reverse construction gives the opposite inequality. Thus the optimal Dehn functions are coarsely equivalent, and the existence of a recursive isoperimetric bound, hence solvability of the word problem for a group, is independent of the finite group presentation.

For a nonconstant rational function of degree $n$, its derivative has degree between $n-1$ and $2n$. A finite pole of order $m$ becomes one of order $m+1$. If infinity is a pole of order $p-q>0$, it contributes $p-q-1$ to the derivative's pole count, where $p,q$ are the degrees of coprime numerator and denominator. This proves the lower bound; the quotient rule denominator $Q^2$ proves the upper bound. The functions $z^n$ and $1/(z^n-1)$ attain the bounds.

On the unit two-sphere this formula maps onto the torus: every torus point has arguments corresponding to $x,y\in [-1/2,1/2]$, and $x^2+y^2\le 1/2$ leaves a real $z$ on the sphere. Scaling both arguments continuously to zero is a null-homotopy, so the degree is zero. Surjectivity alone does not force nonzero degree.

Choose an oriented embedded closed disc in a closed oriented positive-dimensional manifold and collapse its complement to a point. A target point away from the collapsed value has one preimage with local degree $+1$, proving global degree one by the degree as a sum of local degrees.

For $k>0$, an epimorphism ${\pi }_1({\Sigma }_2)\to \mathbb{Z}/k$ gives a connected $k$-sheeted cover. Its Euler characteristic is $-2k$, hence its genus is $k+1$, and the lifted orientation gives degree $k$. For negative $k$ compose the $|k|$-sheeted cover with an orientation-reversing homeomorphism of the target. For $k=0$ take a constant map from the torus.

Let $M,N$ be closed connected oriented manifolds of the same dimension, and suppose $a^r$ generates the top integral cohomology of $N$. If every class of degree $|a|$ on $M$ has zero rth cup power, every continuous map $f:M\to N$ has mapping degree zero. Indeed, $f^{\ast }(a^r)=(f^{\ast }a{)}^r=0$, and evaluation on the fundamental class gives $\mathrm{deg}f=0$. For $S^2\times S^4\to {\mathbb{CP}}^3$, the degree-two class is a multiple of the sphere class $u$, whose square vanishes.

A negligible-mass radiative envelope surrounding a core of mass $M_c$, carrying constant luminosity and having constant opacity, obeys $dP/dT=16\pi acG{M}_c{T}^3/(3{\kappa }_{0}L)$. Neglecting the outer pressure integration constant gives $\Lambda =4\pi acG{M}_c/(3{\kappa }_{0}L)$. The gas pressure fraction $\beta$ is constant because $P_{\mathrm{rad}}=a{T}^4/3$, and $\rho =\mu \beta \Lambda T^3/\mathcal{R}$. Hydrostatic balance gives $T=T_s+\mu \beta G{M}_c(1/r-1/R_s)/(4\mathcal{R})$; dropping this outer-boundary constant gives the displayed deep-envelope approximation. A positive gas-pressure fraction requires $L<4\pi cG{M}_c/{\kappa }_0$.

For a one-dimensional motion with velocity $v(t)$, total distance traveled is ${\int }_a^b|v(t)|dt$, while net displacement is ${\int }_a^{b}v(t)dt$. In several dimensions, replace absolute value by the Euclidean norm of the velocity vector; this is the arc length of the trajectory.

If $v(t)=t-1$ for $0\le t\le 2$, the body reverses direction at $t=1$. Its displacement is zero, but its total distance is ${\int }_0^1(1-t)dt+{\int }_1^2(t-1)dt=1$. Splitting at reversals makes the physical meaning of the integral visible.

The lower and upper integration limits $a,b$ specify the ordered endpoints of a definite integral ${\int }_a^{b}f(x)dx$. Interchanging the integration limits reverses its sign: ${\int }_b^{a}f(x)dx=-{\int }_a^{b}f(x)dx$.

A block has finite representation type exactly when its defect group is cyclic. In the forward implication, the block bimodule restricted to its defect group on both sides contains the regular defect-group bimodule as a summand; tensoring with arbitrary defect-group modules transfers finiteness to the defect group. Conversely, vertices are contained in the defect group and a cyclic $p$-group has only finitely many indecomposable modules, so there are only finitely many possible induced sources and their summands.

For a block idempotent $e$, an $e$-Brauer pair consists of a $p$-subgroup $P$ and a block idempotent $b_P$ of $k{C}_G(P)$ for which $b_P{\mathrm{Br}}_P(e)\ne 0$. The subgroup of a maximal pair is a defect group. A maximal pair's stabilizer defines the inertial quotient of the block.

The defect of a block of a group algebra is the integer $d$ for which a defect group of a block has order $p^d$. Conjugacy of defect groups of a block makes this independent of the choice. A block of a group algebra has defect zero when its defect group of a block is trivial.