For a locally compact space, replace finite singular chains by chains for which every compact subset meets the images of only finitely many simplices with nonzero coefficients. The boundary operator preserves this local finiteness, giving the Borel-Moore chain complex and its homology. On a compact space it reduces to ordinary singular homology. A noncompact manifold has a locally finite fundamental class, with its orientation local system when necessary. The chain groups and homology are zero in negative degrees.

Isotropic turbulence has statistics invariant under rotations as well as translations. With no mean velocity, its two-point velocity correlation tensor depends on separation length and on longitudinal/transverse directions, and its one-component velocity variances are equal.

For differentiable homogeneous incompressible flow, $⟨\mathrm{tr}{A}^3⟩=0$. One divergence representation is $\mathrm{tr}{A}^3={\partial }_i[u_j({\partial }_j{u}_k)({\partial }_k{u}_i)-u_i\mathrm{tr}{A}^2/2]$. Splitting the velocity gradient tensor into strain and rotation gives $⟨{\omega }_i{\omega }_j{S}_{ij}⟩=-4⟨abc⟩$. It connects mean vortex stretching to the cubic strain statistic, without requiring isotropy.

The integral scale of turbulence characterizes energy-containing velocity correlations; in isotropic turbulence one definition is $\ell ={\int }_0^{\infty }f(r)dr$. Its turnover time is of order $\ell /u$. Large-scale intensity fluctuations can change the energy supplied to the equilibrium range.

For frequency-dependent transmission with homogeneous mixing, each member of the susceptible compartment experiences force of infection $\lambda =\beta I/N=\beta i$. Consequently the total flow into the infectious compartment is $\beta SI/N$ people per unit time. The constant $\beta$ combines the contact frequency and the chance that a relevant contact causes infection, and has units of inverse time. Holding the infectious compartment's fraction fixed holds each person's force of infection fixed when $N$ changes. An unnormalized mass-action infection convention instead writes the total flow as $bSI$, where $b=\beta /N$ has units of inverse person-time; these two coefficients must not be confused.

ConsiderThe circle $r=1$ is a periodic orbit. Writing $r=1+\rho$ gives $\dot{\rho }=-(1-2\sin \theta )\rho +O({\rho }^2)$, while $\theta =t+{\theta }_0$ on the circle. Its normal Floquet multiplier is therefore $\exp [-{\int }_0^{2\pi }(1-2\sin \theta )d\theta ]=e^{-2\pi }<1$.

Inside the circle, use the increasing angle as independent variable and set $w=(1-r{)}^{-1}$. Direct differentiation gives the linear ordinary differential equationLet $A(\theta )=\theta +2\cos \theta$. The solution with $w(\pi )=1$ isAt the endpoint, $w^{\prime }(\pi )=0$ and $w^{\prime \prime }(\pi )=2$, so $w=1+(\theta -\pi {)}^2+O((\theta -\pi {)}^3)>1$ immediately before $\pi$. A crossing of $w=1$ has slope $-2\sin \theta$. Thus there is no crossing on $(0,\pi )$ compatible with this terminal behavior. AlsoEvery crossing on $(-\pi ,0)$ has positive slope, giving a unique ${\theta }_b\in (-\pi ,0)$ where $w=1$. Hence $r=1-1/w>0$ for ${\theta }_b<\theta <\pi$ and tends to zero at both ends. At the incoming endpoint $r\sim (\pi -\theta {)}^2$, so $dt=d\theta /r$ has a divergent integral. At the outgoing endpoint the zero is simple and the time integral diverges logarithmically. The curve is a homoclinic orbit to the origin.

Uniqueness for the angular scalar equation prevents positive-radius trajectories from crossing this loop. Inside it they approach the origin; outside it they approach the attracting circle. The origin therefore attracts an open set, but is not Lyapunov stable: points arbitrarily close to its outgoing branch follow trajectories to a fixed positive radius before returning or escaping. Attraction and stability are distinct properties.

A Shilnikov bifurcation involves a homoclinic orbit to a saddle-focus equilibrium in three dimensions. Write its eigenvalues asand put $\delta =-{\lambda }_{-}/{\lambda }_{+}$. The Shilnikov return map combines contraction toward the equilibrium with arbitrarily many rotations before departure.

The sign of the saddle value ${\lambda }_{+}+{\lambda }_{-}={\lambda }_{+}(1-\delta )$ separates two regimes. At a nondegenerate homoclinic connection with $0<\delta <1$, the Shilnikov theorem gives countably many nearby saddle periodic orbits. When $\delta >1$, the usual generic unfolding instead creates a locally attracting periodic orbit on the appropriate side. The boundary $\delta =1$ requires additional terms.^[ref]

The spectral and nondegeneracy assumptions matter: a planar saddle loop has no rotating stable plane and does not supply this mechanism. The result is associated with Leonid Pavlovich Shilnikov; Andrey Shilnikov coauthored the cited account.

A symmetric homoclinic gluing bifurcation joins two symmetry-related periodic orbits into one orbit that traverses both lobes of a figure-eight separatrix. At the transition, the two loops are homoclinic orbits to the same saddle equilibrium. The two original cycles spend increasingly long near that saddle before the connection is reached.

A signed transverse coordinate distinguishes the lobes. Reflection can make the corresponding return map odd, so fixed points $x_{\ast }$ and $-x_{\ast }$ represent the separate cycles, while a two-cycle passing between positive and negative coordinates can represent the glued orbit. The signed gluing-map reduction describes the local power-law passage and the global reinjection; its coefficients and saddle eigenvalue ratio determine the existence and stability of these branches.

The geometry alone does not guarantee attracting cycles. For example, the exact Hamiltonian double-well homoclinic orbits form a figure-eight but are surrounded by conserved-energy families, rather than isolated attracting cycles. Dissipation, a suitable parameter unfolding and the return-map conditions must supply the gluing scenario. A damped double-well model with symmetry-related Hopf cycles is one setting in which those conditions can be investigated.

Near two homoclinic orbits of a planar saddle equilibrium with contracting ratio $\delta >1$, the signed Poincaré return map has separate offsets $\mu ,\nu$. Two single-lobe periodic orbits exist for $\mu <0$ and $\nu <0$, respectively. A two-lobe periodic orbit is represented by positive $p,q$ solving the displayed equations. Its boundary consists of the compound homoclinic orbit curves $\mu =-A{\nu }^{\delta }$ for $\nu >0$ and $\nu =-B{\mu }^{\delta }$ for $\mu >0$, with $A,B>0$. In a sufficiently small return domain the derivatives tend to zero, so each admissible symbolic branch has at most one attracting periodic orbit. Restricting to $\mu =\nu$, $A=B$ gives the usual symmetric homoclinic gluing bifurcation.

For a hyperbolic equilibrium point with one unstable eigenvalue ${\lambda }_{+}>0$ and weakest stable decay rate ${\lambda }_{-}<0$, the saddle index is $\delta =-{\lambda }_{-}/{\lambda }_{+}$. For a stable complex pair, use its real part. An incoming distance $x$ from the stable manifold takes time $T={\lambda }_{+}^{-1}\log (h/|x|)$ to reach an outgoing section at unstable coordinate magnitude $h$. A stable coordinate is multiplied by $e^{{\lambda }_{-}T}=(|x|/h{)}^{\delta }$, giving a power-law local passage. A smooth nondegenerate reinjection consequently has derivative tending to zero when $\delta >1$ and an unbounded leading derivative when $\delta <1$. This determines local contraction or expansion near a homoclinic orbit; existence of a particular attractor also depends on the global return geometry.

The circle $S^1$ and the closed interval $[0,1]$ are both compact and connected, so those two properties alone do not distinguish them. Instead examine connectedness after removing a point. Every punctured circle is connected: after rotating the missing point to $(1,0)$, the continuous map $t\mapsto (\cos t,\sin t)$ from the connected open interval $(0,2\pi )$ maps onto it. Continuous maps preserve connectedness. ^[ref]

If a homeomorphism $h:S^1\to [0,1]$ existed, choose $p=h^{-1}(1/2)$. Its restriction would give a homeomorphism between $S^1\setminus \{p\}$ and $[0,1]\setminus \{1/2\}$. But the latter splits into two nonempty disjoint open sets in its subspace topology, $[0,1/2)$ and $(1/2,1]$, contradicting connectedness of the punctured circle. Removing an endpoint of $[0,1]$ leaves a connected space; the proof deliberately chooses the preimage of an interior point. This illustrates how a simple connectedness test can rule out every possible homeomorphism.

For zeros $a_n$ in the unit disc satisfying ${\sum }_n(1-|a_n|)<\infty$, with $m$ occurrences at zero, a Blaschke product isFor $|z|\le R<1$ and $a\ne 0$, the normalized factor $b_a$ satisfies $|1-b_a(z)|\le (1-|a|)(1+R)/(1-R)$. The product therefore converges locally uniformly; outside its prescribed zeros the logarithms of its tail factors converge absolutely, so it has no additional zeros. Every finite product has modulus at most one in the disc, and the limit does too. Since $|b_a(w)|$ is pseudohyperbolic distance from $w$ to $a$, its modulus can be estimated geometrically.

For compact $K$ in a plane domain $\Omega$, its holomorphic convex hull isIt is $K$ together with the components of $\hat{\mathbb{C}}\setminus K$ lying entirely in $\Omega$. Inclusion of those components follows from the maximum modulus principle. For exclusion, suppose $w$ lies in a complementary component meeting $\hat{\mathbb{C}}\setminus \Omega$. Choose an allowed pole there. The open-and-closed resolvent argument in the proof of Runge theorem approximates $(z-w{)}^{-1}$ uniformly on $K$ by functions $h$ holomorphic on $\Omega$. Then $1-(z-w)h(z)$ has value one at $w$ and arbitrarily small modulus on $K$, separating $w$ from the hull. For $\Omega =\mathbb{C}$, this is the polynomial hull in one complex variable.

A complex-valued function is antiholomorphic when its complex conjugate is a holomorphic function. In one complex dimension it locally depends on $\overline{z}$ and satisfies ${\partial }_{z}f=0$. A nonconstant antiholomorphic map reverses the orientation induced by the complex coordinates. Thus complex conjugating a holomorphic sphere map reverses its topological degree while preserving its Dirichlet energy.

Every closed discrete subset of a plane domain is the zero set of a holomorphic function, with assigned finite positive multiplicities if desired. Exhaust the domain by relatively filled compact sets. For a prescribed point outside an earlier compact, choose a rational factor with that zero and its possible pole outside the domain in the same complementary component. A slit joining zero and pole allows a holomorphic logarithm near the earlier compact. Approximate that logarithm by Runge theorem and multiply the rational factor by the negative exponential of the approximant. The resulting factors tend to one with summable errors on each compact. Their infinite product has the prescribed zeros and no additional zeros.

// Destination: analysis.bigb

A map from an open subset of $\mathbb{C}$ to a complex Banach space is holomorphic when the displayed limit exists in the Banach-space norm at every point. A norm-convergent power series is holomorphic inside its radius of convergence: on every smaller disk the series of derivatives converges uniformly by the geometric coefficient bound, allowing differentiation term by term. Applying a continuous linear functional produces an ordinary scalar holomorphic function. Scalar Cauchy identities therefore imply the corresponding vector-integral identities when bounded linear functionals separate points, as they do by the Hahn-Banach theorem.

A holomorphic function with constant modulus on a connected domain is constant. If $h=u+iv$ and $u^2+v^2=c^2>0$, differentiation and the Cauchy-Riemann equations give $u{u}_x+v{v}_x=0$ and $-u{v}_x+v{u}_x=0$. Their coefficient determinant is $-c^2$, so $u_x=v_x=0$, and all first derivatives vanish. When $c=0$, the conclusion is immediate. Consequently two nonvanishing holomorphic functions with equal moduli differ by a single constant phase on a connected domain.

If a holomorphic function of $w$ is periodic with period one, then $F((\log z)/(2\pi i))$ is independent of the choice of a local logarithm: two choices differ by $2\pi ik$, hence their arguments differ by an integer. Consequently it defines a holomorphic function on the image of the original domain under $w\mapsto e^{2\pi iw}$, provided that domain is invariant under integer shifts. Local logarithms prove holomorphy across an arbitrary global branch cut. Singularities on an integer lattice all map to $z=1$.

A holomorphic function that is not identically zero has a zero of multiplicity $m$ at $z_0$ if $f(z)=(z-z_0{)}^{m}h(z)$ for a holomorphic function $h$ with $h(z_0)\ne 0$. Its Taylor series identifies $m$ as the first nonzero coefficient index. Logarithmic differentiation gives $f^{\prime }/f=m/(z-z_0)+h^{\prime }/h$, so the residue of $f^{\prime }/f$ is $m$. Counting zeros with zero multiplicity repeats that zero $m$ times.

The analytic Nevanlinna class consists of holomorphic functions on the unit disc with uniformly bounded means of their positive logarithmic modulus. Here $m$ is normalized angular measure. A bounded holomorphic function belongs to this class, but the class also permits unbounded functions. It is the analytic version of bounded characteristic.