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When a second-order equation has a repeated indicial root and only one independent Frobenius series, a second solution has the local form

Under $\zeta =1/z$, the point $z=\infty$ is regular singular forexactly when $zP(z)$ and $z^{2}Q(z)$ are holomorphic functions of $1/z$ near $1/z=0$.

The disjoint union of $r$ triangles is $2$-regular on $3r$ vertices and has matching number $r$, attaining the regular-graph lower bound $n/3$.

For a regular curve, $s(t)={\int }_{t_0}^t|{\alpha }^{\prime }(r)|dr$ has positive derivative and hence a smooth inverse. Reparametrization by $s$ gives unit speed.

For a regular category $\mathcal{C}$, the category $\mathbf{Rel}(\mathcal{C})$ has the objects of $\mathcal{C}$ and subobjects of $A\times B$ as relations $A⇝B$. Composition takes the image of the pullback expressing existential quantification over the middle object.

For $\mathcal{C}=\mathbf{Set}$ this is the category of relations (sets); the general construction also applies to other regular categories.

At a perfectly rigid plane, every component of the displacement field vanishes.

For incident SV angle $\theta$, tangential phase matching gives $\sin \varphi =(c_P/c_S)\sin \theta$ for the reflected P-wave. It is evanescent when $\sin \theta >c_S/c_P$.

Under the standard binary inner product,The product of representing polynomials has degree at most $d-1$, so its evaluation word has even weight; equality follows by comparing dimensions.

For length-$n$ binary linear codes $C_2\subseteq C_1$, their bar product isIt has length $2n$ and dimension $\dim C_1+\dim C_2$.

For nearest-neighbour random walk on ${\mathbb{Z}}_{+}$ with upward probability $p>q$, the probability of ever hitting zero from $i$ is $(q/p{)}^i$.

A state is recurrent exactly when the sum over time of its return probabilities diverges.

A recurrent state is null recurrent when its expected return time is infinite.

A recurrent state is positive recurrent when its expected return time is finite.

A global maximum of a real-valued function is a value $f(x_0)$ satisfying $f(x)\le f(x_0)$ throughout its domain.

A global maximum is one type of extremum of a real-valued function; global minima reverse the inequality.

A bounded closed interval includes both of its endpoints:If $a=b$, it consists of one point. Its length is $b-a$. In contrast, the open interval $(a,b)$ excludes both endpoints; half-open real intervals include just one.

Including the endpoints has a useful consequence for limits: a convergent sequence whose terms lie in $[a,b]$ has its limit of a sequence in $[a,b]$ too. Finite endpoints also make this a bounded set, so it is a compact set by the Heine-Borel theorem. These two properties support both the nested interval theorem and the extreme value theorem.

The Brusselator is a two-species autocatalytic reaction model. In the parametrizationits positive homogeneous equilibrium is $(u_{\ast },v_{\ast })=(\alpha ,\beta /\alpha )$.

A Turing, or diffusion-driven, instability occurs when a spatially homogeneous equilibrium is stable to homogeneous perturbations but unstable to a nonzero spatial Fourier mode after unequal diffusion is introduced.

If a fast inhibitor satisfies $0=v_{xx}+u-v$, then its spatial Fourier transform obeysSubstitution produces a nonlocal scalar equation for the activator.

A morphogen reaction-diffusion equation models a spatial concentration by diffusion and local production or decay. Linearization at a homogeneous equilibrium $C=0$ replaces $f(C)$ by $f^{\prime }(0)C$.

It is a transport-and-reaction model for a morphogen, rather than the identity of the signaling molecule concept.

The extremal Rayleigh quotients of a self-adjoint operator are the extremal spectral values. In particular, a Hamiltonian with a discrete spectrum satisfies $R[\psi ]\ge E_0$, with equality precisely on its ground-state eigenspace. More generally, minimizing over states orthogonal to the first $j$ eigenspaces gives an upper bound on $E_j$.