If $\alpha$ is a real algebraic irrational number and $\epsilon >0$, thenhas only finitely many reduced rational solutions $p/q$.

Relative to a Schauder basis, a block sequence consists of nonzero finitely supported vectors with successive disjoint supports: $\max \mathrm{supp}{x}_i<\min \mathrm{supp}{x}_{i+1}$. It is a basic sequence, with basis constant at most that of the original basis.

Relative to a Schauder basis $(e_j)$, a block basic sequence consists of nonzero finite combinations supported on successive disjoint coordinate intervals. In the l-p sequence space, disjointly supported vectors $u_n$ of norm one satisfy $\Vert \sum a_n{u}_n{\Vert }_p^p=\sum |a_n{|}^p$. Their closed span is consequently isometric to the l-p sequence space.

Write the S-matrix as $S=1+iT$. Unitarity gives $-i(T-T^{†})=T^{†}T$, so the diagonal forward matrix element obeys $2\mathrm{Im}{T}_{ii}={\sum }_f\int d{\Phi }_f|T_{fi}{|}^2$. Thus the imaginary part of a forward scattering amplitude measures the total inclusive transition rate. For an electromagnetic source, the hadronic electromagnetic-current spectral density supplies the inclusive spectral sum.

With this half-variance convention, the unforced scalar equation gives $d⟨C^2/2⟩/dt=-{ϵ}_c$ in a homogeneous incompressible flow. Authors defining dissipation of the full variance instead use $2\alpha ⟨|\nabla C{|}^2⟩$. Stating the convention is necessary when relating the dissipation rate to a scalar structure function.

Newtonian gauge removes scalar shear and writes scalar metric perturbations in terms of gravitational potentials. The fluid Euler equation then contains an explicit gradient of the Newtonian potential.

For classical nondegenerate electrons, the free-particle Boltzmann occupancy has the form $z_e\exp [-p^2/(2mkT)]$, with chemical potential measured relative to its rest energy. Counting the two spin states in phase space gives $n_e=2z_e(2\pi mkT/h^2{)}^{3/2}$. A bound level of energy $-I_n$ has relative occupancy $(g_n/g_{+})z_e{e}^{I_n/kT}$ against its parent ion. Eliminating $z_e$ gives the level-specific Saha ionization equation.

Suppose a function near a critical point has Taylor expansion $f(p+h)=f(p)+P_m(h)+o(\Vert h{\Vert }^m)$, where $P_m$ is its first nonzero homogeneous polynomial term. If $P_m$ takes both positive and negative values on directions, then $p$ is a saddle point of a scalar function. Indeed, on rays $h=rv$ the leading term is $r^m{P}_m(v)$; for each chosen direction its sign dominates the remainder for small positive $r$. This works even when the Hessian matrix is zero. For example, $P_4(\xi ,\eta )=-{\xi }^3(\xi +\eta )$ is negative on $(1,0)$ and positive on $(1,-2)$, hence detects a saddle despite having degree four.

When a saddle-node lies on a returning invariant circle, removing the equilibria creates passage around a periodic orbit through a slow bottleneck. For $\dot{x}=x^2+\eta$ with $\eta >0$, crossing a fixed interval $[-h,h]$ takes $2\arctan (h/\sqrt{\eta })/\sqrt{\eta }\sim \pi /\sqrt{\eta }$. If the remaining orbit takes bounded time, this is also its leading period. Strong transverse contraction makes the newborn periodic orbit attracting. This global saddle-node mechanism differs from a local fold occurring away from an already existing cycle.

Let $S$ be a finite set of prime numbers and let $f_S(x)$ count S-smooth numbers $1\le n\le x$, where $x\ge 1$. Each such $n$ has a unique expression $n=a{b}^2$, where $a$ is a positive squarefree integer and $b$ is a positive integer. Split every prime exponent into its parity and an even remainder to obtain this decomposition.

Every prime factor of $a$ belongs to $S$, so $a$ is the product of a subset of $S$, giving at most $2^{|S|}$ possibilities. Also $b\le \sqrt{x/a}\le \sqrt{x}$. Ignoring further restrictions on $b$ only increases the count, provingIn particular, each fixed finite-prime family has natural density zero. This differs from allowing the smoothness bound, and hence the prime set, to grow with $x$.

For radial quantum scattering, the incoming coefficient of the regular solution defines the Jost function, with convention $S(k)=F(-k)/F(k)$. A zero at $k=i\kappa$, $\kappa >0$, removes the exponentially growing component and produces a bound state, subject to regularity and normalizability. Poles of the outgoing numerator alone need not produce bound states.

A pole of a scattering matrix arising from the numerator $F(-k)$ rather than a zero of the incoming Jost function $F(k)$ can fail to correspond to a bound state. For $F(k)=(k-i\kappa )/(k+i\alpha )$, the pole at $i\kappa$ is associated with the incoming zero; the pole at $i\alpha$ comes from the numerator. The regular Eckart realization has $\alpha >\kappa$. The distinction and this explicit example are discussed in dipot.ulb.ac.be/dspace/bitstream/2013/373712/4/2306.12216.pdf.

This one-step family interpolates between the Forward Euler method at $a=0$, the trapezoidal rule at $a=1/2$, and the Backward Euler method at $a=1$. Its stability function is $R(z)=[1+(1-a)z]/[1-az]$. It is first order except at one half, where it is second order. Applying it to spatial diffusion gives the theta diffusion stability criterion.

An $s$-stage Runge-Kutta method has stability function $R(z)=1+z{b}^T(I-zA{)}^{-1}1$, a rational function with numerator and denominator degrees at most $s$. Order $2s$ forces $R(z)-e^z=O(z^{2s+1})$, so uniqueness of the diagonal Padé approximant identifies $R$ with $[s/s{]}_{e^z}$. That approximant has no poles in the closed left half-plane and modulus at most one there. Hence every such maximal-order method is A-stable. Since the Padé approximant numerator and denominator are coprime and both have degree $s$, the original stage determinant cannot have hidden canceled poles in that half-plane.

For sufficiently regular vector fields, for example bounded fields with bounded derivatives through order three when $2<p<3$, the solution map of a rough differential equation exists uniquely and is locally continuous in the rough path metric. Ordinary differential equations driven by smooth approximations therefore converge whenever their enhanced drivers converge. The theorem makes the enhancement, rather than just the original path, the correct input for a continuous solution map.

For a finite time grid, suppose a nonnegative error is zero on neighboring points and obeys $a_{s,t}\le a_{s,u}+a_{u,t}+L\omega (s,t{)}^{1/p}{a}_{s,u}+A\omega (s,t{)}^{\theta }$. For a control function for rough paths and $\theta >1$, sufficiently small control intervals have the displayed bound with constants independent of the grid. Splitting near half the control size makes the higher-order errors contract, and handles a crossing single grid interval by its zero error. Applying this to local Euler composition errors yields compactness of approximate pathlevel solutions.

A pathlevel solution is the ordinary state path satisfying a local rough Taylor expansion with a remainder of order $\theta >1$ in a control function for rough paths. It specifies the first level of the solution, rather than requiring the full lifted solution as part of the definition. With vector fields of regularity ${\mathrm{Lip}}^{\gamma }$, $\gamma >p$, the Davie lemma bounds discrete Euler errors uniformly; compactness and passage to the limit give such a path.

Write $r_3(N)$ for the maximum cardinality of a subset of $[N]$ containing no nonconstant three-term arithmetic progression. There is an absolute constant $C$ such that every three-term-progression-free subset $A\subseteq [N]$ satisfies $|A|\le CN/\log \log N$ for $N\ge 3$. A quantitative Roth density-increment step gives, for subset density $\delta$ and $N\ge 4\cdot 10^6{\delta }^{-4}$, a arithmetic progression of length at least ${\delta }^2\sqrt{N}/2000$ and increased subset density at least $\delta +{\delta }^2/64$. The increment decreases reciprocal subset density by at least $1/65$, so fewer than $\lceil 65/\delta \rceil$ iterations are possible. If $N_j$ is the successive length, then $\log N_{j+1}\ge \frac{1}{2}\log N_j-\log (2000/{\delta }^2)$, using the initial subset density to bound all later ones. Hence $\log N_j\ge 2^{-j}\log N-2\log (2000/{\delta }^2)$. If $\delta \log \log N$ exceeds a sufficiently large absolute constant, all these intervals remain above the density increment threshold, a contradiction. This proves the displayed bound. The result is a classical quantitative version of Roth theorem on three-term arithmetic progressions, without asserting an optimal bound.

Embed $A\subseteq [N]$ in the odd cyclic group $G=\mathbb{Z}/(2N+1)\mathbb{Z}$, so additive three-term relations in $A$ cannot wrap around. Form three copies of $G$, putting edges $x\sim y$ when $y-x\in A$, $y\sim z$ when $z-y\in A$, and $x\sim z$ when $(z-x)/2\in A$. A triangle in a graph encodes $a+b=2c$ in $A$. If $A$ has no nonconstant arithmetic progression, every triangle in a graph has $a=b=c$; there are exactly $|G||A|$ triangles in a graph, and they are edge-disjoint. Thus at least $|G||A|$ edges must be removed to eliminate them. For fixed positive subset density of $A$, this is a positive fraction of the squared vertex count, whereas the triangle in a graph count is only quadratic. The triangle removal lemma permits deletion of an arbitrarily small quadratic fraction when the cubic triangle in a graph subset density is sufficiently small. This contradiction establishes Roth theorem on three-term arithmetic progressions.

If a density-$\delta$ subset of a sufficiently long integer interval has no nonconstant three-term arithmetic progression, Fourier expansion of the progression count gives a large coefficient of its balanced indicator. Dirichlet approximation supplies a step at most $\sqrt{N}$ on which that character varies slowly. Partitioning the interval into equal-length progressions of that step, and controlling the short tails, produces the displayed density increment. Iteration must terminate since density cannot exceed one. The same argument treats the three-point pattern with coefficients $2,-3,1$.