A perturbation is a small change in a model, operator, state or parameter. It can be organized as a family $F_{\epsilon }=F_0+\epsilon G+o(\epsilon )$ in a specified norm. If $F_{\epsilon }(z_{\epsilon })=0$ and $D{F}_0(z_0)$ is invertible, differentiating at zero gives $z_0^{\prime }=-[D{F}_0(z_0){]}^{-1}G(z_0)$. The implicit function theorem justifies this continued solution locally. Singular derivatives at a bifurcation instead require an unfolding or reduced equation; smallness of the change alone does not preserve a degenerate bifurcation.

A nonempty collection of real continuous subharmonic functions on a connected Riemann surface is a Perron family if it is closed under pairwise maxima and harmonic lifting of a continuous subharmonic function on every relatively compact coordinate disk. Its pointwise supremum is either everywhere infinite or a harmonic function on a Riemann surface. At a point where the supremum is finite, take increasing disk lifts approaching that value. The Harnack inequality for harmonic functions gives a finite harmonic limit on the disk. Lifting their maxima with any other member gives a larger harmonic limit with the same value at the chosen point, hence the same limit by the strong maximum principle for harmonic functions. The supremum is therefore harmonic on that disk; connectedness propagates finiteness to the whole surface.

For even degree the augmentation hyperplane has an alternating dot product whose radical is the all-ones line. Quotienting by that line gives a nondegenerate symplectic space of dimension two less than the permutation degree. Coordinate permutations preserve the form. In degree at least six their quotient action is faithful, since two distinct weight-two coordinate vectors cannot differ by the all-ones vector. In degree six this gives $S_6\cong S{p}_4(2)$; in degree four the quotient action instead has Klein-four kernel.

For a finite group action on a finite set $X$, $R[X]$ is the free $R$-module with basis $X$, with the group permuting that basis. For a transitive set $G/H$ it is ${\mathrm{Ind}}_H^{G}R$. Disjoint unions of sets give direct sums of permutation modules.

The rank is the number of orbits on ordered pairs under the diagonal group action. Its fixed-pair count at $g$ is $\chi (g{)}^2$, since the permutation character counts fixed points. Burnside's lemma identifies its average with the orbit count, while real-valuedness makes that average the character inner product of $\chi$ with itself. For a transitive action on at least two points, rank two is equivalent to transitivity on distinct ordered pairs.

If every row and column of an $n$-by-$n$ zero-one matrix has more than $n/2$ ones, its permanent of a matrix has a fully polynomial randomized approximation scheme. The short augmenting paths in a dense bipartite graph bound makes the leading coefficient dominate the matching generating function at polynomially large activity. The weighted matching Markov chain samples polynomially bounded activities in polynomial mixing time. Estimate successive annealing ratios for a matching generating function and divide by the final activity to the power $n$.

Let $X_1,\dots ,X_n$ be independent random variables, each choosing a label from $\{1,\dots ,n\}$. Put their probabilities in the matrix $P=(p_{ij})$, where $p_{ij}=\mathbb{P}(X_i=j)$. Each row sums to one. All choices are distinct precisely when their ordered list is one of the permutations of the $n$ labels. These are mutually exclusive events, and independence makes the probability of one list $\sigma$ equal to ${\prod }_i{p}_{i,\sigma (i)}$. Adding the list probabilities gives the permanent of a matrix:In particular, this permanent lies between zero and one.

For a two-person example,The distinct outcomes $(1,2)$ and $(2,1)$ have probabilities $3/5$ and $1/20$. ThereforeThe colliding outcomes $(1,1)$ and $(2,2)$ have probabilities $3/20$ and $1/5$, totaling $0.35$. All four outcomes are shown below; their probabilities sum to one.

If every choice has the uniform distribution on a finite set of $n$ labels, then $p_{ij}=1/n$ and the distinct-choice probability is $n!/n^n$. For three choices and three labels it is $6/27=2/9$. This counts a subset of all $n^n$ ordered outcomes, not independent pairwise collision events. Independence of the original choices is essential: without it, the individual rows of $P$ do not determine the joint probabilities.

For negligible inertia, periodic forcing $p_e=ϵP_e\sin kX$, resistance $R\propto {\alpha }^{-n}$ and prescribed mean area ${\alpha }_0$, write $Q=-c{\alpha }_0+{ϵ}^2{Q}_2+\cdots$. First-order area obeys ${\tilde{P}}_0^{\prime }{\alpha }_1^{\prime }+R_{0}c{\alpha }_1/{\alpha }_0=-k{P}_e\cos kX$. Its mean square is $P_e^2/[2({\tilde{P}}_0^{\prime 2}+(R_{0}c/(k{\alpha }_0){)}^2)]$. Averaging the second-order friction balance gives $Q_2=c(n+1)⟨{\alpha }_1^2⟩/{\alpha }_0$, proving the displayed forward pumping rate. Periodicity entails zero mean pressure drop; the mean-area convention fixes the expansion's otherwise free constant.

A holomorphic periodic point is attracting when its return multiplier has modulus strictly less than one. For the return map $G=R^n$, choose a local coordinate at the fixed point and a number $q$ strictly between $|G^{\prime }(p)|$ and one. Taylor expansion gives $|G(z)-p|\le q|z-p|$ on a sufficiently small disc. It maps that disc into itself, and successive returns converge uniformly to $p$ there. Composing with the finitely many intervening maps gives locally uniform convergence to the periodic orbit and places its basin in the Fatou set.

A holomorphic periodic point is repelling when its return multiplier has modulus greater than one. It belongs to the Julia set: if the iterates were normal near it, a subsequence of the return iterates, all fixing the point, would converge locally uniformly in the spherical metric. Near their fixed finite value the limit lies in a finite coordinate chart, so Cauchy's derivative estimate would bound their derivatives at the point. Those derivatives are the successive powers of the multiplier and are unbounded. The same argument in a reciprocal coordinate handles infinity.

An exact $n$-cycle consists of $n$ distinct points $w_0,\dots ,w_{n-1}$ with $R(w_j)=w_{j+1}$, indices read modulo $n$. Every point has exact period $n$. The return multiplier is ${\prod }_{j=0}^{n-1}{R}^{\prime }(w_j)$, computed in local charts if necessary; cyclically changing the starting point leaves it unchanged. Thus an attracting, superattracting or repelling return is of the same type at every point of the cycle. If a period-$n$ root coincides with a lower-period point at a parameter limit, multiplicity in the equation $R^n(z)=z$ need not represent distinct points of an exact cycle.

A generalized flip is a codimension-two period-doubling point with a simple multiplier $-1$, a vanishing cubic coefficient in its scalar centre-manifold map, and a nonzero quintic coefficient. In the odd normal form $z^{\prime }=(-1+\beta )z+c\alpha z^3+d{z}^5+\cdots$, symmetric two-cycles solve $\beta +c\alpha z^2+d{z}^4=0$. The discriminant of this quadratic in $z^2$ produces a fold of two-cycles where its positive roots collide. For $c=3$, $d=-3$, this fold is $\beta =-3{\alpha }^2/4$, $z^2=\alpha /2$, $\alpha >0$. At $\alpha =0$ the amplitude is proportional to $|\beta {|}^{1/4}$.

A period-doubling cascade is a succession of period-doubling bifurcations creating attracting periods $1,2,4,8,\dots$, with bifurcation parameters tending to an accumulation value. Superstable periodic orbits inside successive stability intervals give a second parameter sequence with the same limiting parameter-scaling ratio in a quadratic universality class.

// Target: dynamical-systems.bigb

Slip-line field theory solves certain two-dimensional problems of plastic deformation by constructing two orthogonal families of characteristic curves, called slip lines. Its classical assumptions include plane strain, negligible inertia and body force, and a rigid, perfectly plastic material. The curves are aligned with the directions of maximum in-plane shear stress, rather than the principal stress directions.^[ref]

The curve network encodes both the stress field and constraints on admissible velocity. Indentation of a thick workpiece by a rigid flat punch provides a motivating application: the construction predicts a plastic flow region and the load needed to sustain that flow. It is useful because geometry can turn coupled field equations into relations integrated along curves. A slip line is a continuum characteristic, however, not necessarily a microscopic crystal slip system or the actual path of a moving material particle.

In antiplane perfect plasticity, velocity $w=v_3$ and shear stresses $a={\sigma }_{13},b={\sigma }_{23}$ depend only on $x_1,x_2$. Force balance is $a_{,1}+b_{,2}=0$, the fixed yield surface is $f(a,b)=k$, and the associated flow rule is $\nabla w=2\dot{\lambda }(f_a,f_b)$ with $\dot{\lambda }\ge 0$. Differentiating the yield equation in each coordinate and combining it with equilibrium shows that $a,b,w$ are constant along characteristic curves tangent to $(-f_b,f_a)$. For example $-f_b{a}_{,1}+f_a{a}_{,2}=f_b{b}_{,2}+f_a{a}_{,2}=0$. Nonzero regular yield gradient is needed for a characteristic direction.

A perfect fluid with equation of state $P=\rho$ is called a stiff fluid. For positive energy density, its constant equation-of-state parameter is $w=1$. The cosmological continuity equation gives $\rho \propto a^{-6}$. In a flat universe dominated by this fluid, the Friedmann equation then gives $a(t)\propto (t-t_B{)}^{1/3}$. This statement assumes a nonzero fluid density and negligible other components.

For metric signature $(+---)$, a perfect fluid has $T_{ab}=(\rho +p)u_a{u}_b-p{g}_{ab}$. The Einstein field equations and static Killing field is a Ricci eigenvector imply $(\rho +p)(k\cdot u)u_{[a}{k}_{b]}=0$. Two nonzero timelike vectors cannot be orthogonal, so $k\cdot u\ne 0$. If $\rho +p\ne 0$, the four-velocity is parallel to the static Killing vector. Choosing both future-directed gives the displayed normalization. For $\rho +p=0$, the stress-energy tensor is proportional to the metric tensor and does not determine a preferred fluid velocity; the strict enthalpy hypothesis is therefore material.

For number density $n$, rest-frame energy density $\rho$ and pressure $p$, the enthalpy per particle is $ϵ=(\rho +p)/n$. In an isentropic barotropic fluid, $d\rho =ϵdn$ and $dp=ndϵ$. It differs from enthalpy per unit mass by the particle mass when that mass is fixed. Using $ϵ$ as thermodynamic variable gives $n=p^{\prime }(ϵ)$ and $\rho =ϵp^{\prime }(ϵ)-p(ϵ)$.

The number current is $J^a=n{u}^a$, where $n$ is the rest-frame number density and $u^a$ the unit four-velocity. Its covariant continuity equation is ${\nabla }_a{J}^a=0$, equivalently $u^a{\nabla }_{a}n+n{\nabla }_a{u}^a=0$. Integration over a spacetime volume and the divergence theorem equate the particle numbers crossing its boundary. This conservation law is separate from stress-energy conservation.

In signature $(+---)$, the matter action $S=\int \sqrt{-g}[p(ϵ)+n((\nabla \chi {)}^2-{ϵ}^2)/(2ϵ)]$ uses independent $n,ϵ,\chi$. Variations of $n,ϵ,\chi$ give $(\nabla \chi {)}^2={ϵ}^2$, $p^{\prime }(ϵ)=n$, and particle number conservation in a relativistic fluid for $u_a={\nabla }_a\chi /ϵ$. Its Hilbert stress-energy tensor is $T_{ab}=(\rho +p)u_a{u}_b-p{g}_{ab}$. The action describes isentropic potential flow, with vanishing canonical vorticity of a relativistic fluid, rather than every vortical perfect-fluid solution.