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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.

The canonical vorticity two-form is ${\omega }_{ab}={\nabla }_a(ϵu_b)-{\nabla }_b(ϵu_a)$, with specific enthalpy per particle $ϵ$. Potential flow $ϵu_a={\nabla }_a\chi$ has $\omega =0$ for a torsion-free connection. Contracting this identity with $u^a$ yields $ϵu^a{\nabla }_a{u}_b={\nabla }_bϵ-u_b{u}^a{\nabla }_aϵ$, the isentropic relativistic Euler equation in signature $(+---)$. This canonical condition does not mean the unweighted four-velocity one-form is closed.

For a canonical scalar with timelike gradient, its stress-energy tensor has perfect-fluid form. In a homogeneous background, $\rho ={\dot{\varphi }}^2/2+V$ and $p={\dot{\varphi }}^2/2-V$. Potential domination gives negative pressure, while the scalar's physical rest-frame sound speed is one. A constant negative background $w$ therefore does not force its perturbations to obey the adiabatic barotropic closure $\delta p=w\delta \rho$.

// Target: cosmology.bigb

A percolation measure has the finite-energy property if, conditional on all exterior states, every prescribed state of any finite collection of sites or edges has positive probability almost surely. This permits finite local modifications within events of positive probability. It is a hypothesis on the measure and does not follow merely from invariance under graph symmetries.

In bond percolation on a directed graph, independently open each directed edge with probability $p$. The out-cluster of $x$ consists of vertices reachable by open directed paths. In site percolation, independently open vertices and require every vertex of the directed path, including $x$, to be open. The same definitions allow a multigraph. The orientation matters: an infinite underlying connected graph need not contain an infinite out-cluster from a specified root.

For the Kruskal–Szekeres coordinates of a positive-mass Schwarzschild spacetime, $UV=(1-r/(2M))e^{r/(2M)}$. The displayed conformal compactification makes radial null geodesics lines of slope $\pm 1$. The Killing horizons are $T=\pm X$, and the spacelike curvature singularities $r=0$ lie at $T=\pm \pi /4$, with $|X|<\pi /4$. Two asymptotically flat exterior regions each have past null infinity, null infinity to the future, timelike infinity and spacelike infinity. The two exterior regions cannot communicate by any causal curve; future-directed curves that cross a future horizon instead end at the black-hole singularity.

In a real quadratic integer ring, multiplying by a positive norm-one fundamental unit preserves a prescribed norm. Choose an exponent so that a positive representative lies in a half-open interval $[A,A\epsilon )$. Its conjugate is the prescribed norm divided by the representative, so both integer coefficients are bounded on that interval. Enumerating these reduced representatives gives a complete finite list of unit orbits of solutions.

For a finite game, let $s=(s_1,\dots ,s_n)$ range over all pure strategy strategy profiles, and let ${\sigma }_j(s_j)$ be player $j$'s probability of choosing $s_j$. Under independent mixed strategies, the probability of $s$ is ${\prod }_j{\sigma }_j(s_j)$. The displayed sum is the expected value of player $i$'s payoff. Holding everyone else's mixed strategy fixed makes $U_i$ a linear function of ${\sigma }_i$; this is why checking pure strategy deviations suffices to check all mixed strategy deviations.

For tensor products of Pauli factors from $\{I,X,Y,Z\}$, let $k$ count the sites where both factors are nonidentity and different. Distinct nonidentity Pauli matrices anticommute, and equal or identity factors commute. Reversing the product site by site therefore gives $PQ=(-1{)}^{k}QP$. Thus two Pauli strings commute exactly when $k$ is even. The criterion applies to the perfect-correlation observables in the GHZ theorem.

Two light quarks in a symmetric spatial state and the antisymmetric colour pair inside a three-quark colour singlet require symmetric spin-flavour symmetry. Their isospin singlet and spin singlet are both antisymmetric, while their respective triplets are symmetric. Thus the allowed pairs have $(I,S_{ud})=(0,0)$ or $(1,1)$; opposite exchange parities are excluded. Coupling an isospin-zero spin-one-half quark gives an $I=0,J=1/2$ baryon or $I=1,J=1/2,3/2$ baryons.

The Swift–Hohenberg equation is a partial differential equation for pattern formation. In one spatial dimension, a cubic-quintic version isLinearizing at $w=0$ and substituting a Fourier mode $e^{\sigma t+ikx}$ gives $\sigma (k)=r-(1-k^2{)}^2$. The first unstable modes therefore have wavenumber near one when $r$ becomes positive. Replacing one by $q_0^2$ inside the differential operator selects wavenumbers near $q_0$.

There is also a useful Lyapunov functional. On a periodic interval, or with sufficient decay at infinity, defineTwice using integration by parts gives the functional derivative $\delta E/\delta w=(1+{\partial }_x^2{)}^{2}w-rw-s{w}^3+w^5$. The equation is therefore a gradient flow, andThis dissipative structure constrains the evolution without requiring the final spatial pattern to be uniform. For $s>0$, the cubic term enhances finite amplitudes while the quintic term limits growth. Near onset, a slowly varying envelope can satisfy a quintic real Ginzburg-Landau equation.

A phase modulation changes the local position of a periodic pattern. If $w_0$ has period $2\pi$ in its argument, a transverse modulation can be writtenA constant $\varphi$ translates the entire pattern; a varying $\varphi$ bends its constant-phase lines. For the basic profile $w_0(\xi )=\cos \xi$, the local wavevector is $\nabla (x+\varphi )=(1,{\varphi }_y)$. The orientation changes at first order in the slope, while the local wavenumber is $\sqrt{1+{\varphi }_y^2}=1+{\varphi }_y^2/2+\cdots$.

In the method of multiple scales, a correction $C(\xi ,Y)$ depends both on its fast coordinate $\xi =x+\varphi (Y)$ and on its slow coordinate $Y$. Differentiating at fixed physical $x$ therefore uses the chain ruleFor instance, $C(\xi ){\varphi }_{YY}$ differentiates to $C^{\prime }(\xi ){\varphi }_Y{\varphi }_{YY}+C(\xi ){\varphi }_{YYY}$, not just the second term. The translation-covariant phase expansion calculates the second derivative needed in solvability conditions. Omitting the fast-coordinate dependence can change the nonlinear coefficient of the reduced amplitude equation while leaving its linear coefficients unchanged.