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A system can split into macroscopic regions with distinct local order parameters, densities or compositions. In a short-range system at phase coexistence, regions of two phases with means $M_1,M_2$ and volume fractions $q,1-q$ have spatial mean $M=q{M}_1+(1-q)M_2$. Their interface contribution is subextensive, so their thermodynamic free-energy density is the corresponding convex combination of the phase densities. This can lie below the homogeneous-branch Landau free energy at the same $M$. The equilibrium constrained order-parameter free energy is consequently convexified, with a straight coexistence segment described by the Maxwell construction. A nonconvex local potential remains useful for homogeneous branches, metastability and interfaces, but must not be identified with the full equilibrium constrained density throughout a coexistence interval.

The solid-liquid segregation coefficient is the ratio of the solute concentration in newly formed solid to that in the adjacent equilibrium liquid. A coefficient of zero means complete solute rejection: newly formed crystals are pure solvent. It does not mean the bulk solute in a mushy layer vanishes, because its pore liquid can remain concentrated.

A planar freezing front can push a spherical particle steadily when repulsive sphere-plane dispersion force balances lubrication resistance. The thin parabolic gap gives hydraulic drag $6\pi \mu V{a}^2/d$, and balancing it against $Aa/(6d^2)$ gives the displayed speed. This balance assumes a nonretarded interaction, small gap and negligible inertial and other particle forces.

For a compact group $G$, closed subgroup $K$ and finite-dimensional unitary $K$-representation $W$, induced sections are functions satisfying $f(gk)=k^{-1}f(g)$. A $K$-intertwiner $\varphi :E\to W$ produces the $G$-intertwiner $v\mapsto (g\mapsto \varphi (g^{-1}v))$. Evaluation at the identity reverses this construction on each finite-dimensional continuous representation summand. Thus the multiplicity of an irreducible $E$ in the induced section space is $\dim {\mathrm{Hom}}_K(E{|}_K,W)$.

A continuous symmetric approximate identity on a compact group gives a compact self-adjoint convolution operator $T$ on $L^2(G)$ which maps into $C(G)$ and commutes with left translations. Its nonzero eigenspaces are finite-dimensional invariant spaces of continuous functions, hence consist of matrix coefficients of finite-dimensional unitary representations. If $S_N$ projects onto increasing finite sums of these eigenspaces, $S_{N}Tf\to Tf$ in $L^2$, and the bound $\Vert Th{\Vert }_{\infty }\le \Vert k{\Vert }_2\Vert h{\Vert }_2$ gives $T{S}_{N}Tf\to T^{2}f$ uniformly. Choosing the approximate identity narrow makes $T^{2}f$ uniformly close to $f$. This supplies uniform approximation without confusing an $L^2$ limit with a uniform limit.

Suppose finite-dimensional continuous group representations separate points of a compact Lie group. Choose an identity neighborhood $U$ containing no nontrivial subgroup, using Lie groups have no small subgroups. For each $g\notin U$, choose a representation whose matrix at $g$ is not the identity. The corresponding open nonidentity sets cover the compact complement of $U$, so finitely many representations suffice there. Their direct sum has kernel inside $U$ and therefore trivial kernel. Unitarization of a compact-group representation makes it a finite-dimensional faithful unitary representation.

The lower middle perversity is $\bar{m}(k)=\lfloor (k-2)/2\rfloor$. The complementary upper middle perversity is $\bar{n}(k)=\lceil (k-2)/2\rceil$, and $\bar{m}(k)+\bar{n}(k)=k-2$. They agree at even codimensions.

The upper middle perversity is $\bar{n}(k)=\lceil (k-2)/2\rceil$. It differs from the lower middle perversity by one at odd codimensions, and is complementary to it.

An $i$-dimensional singular simplex $\sigma :{\Delta }^i\to X$ is $\bar{p}$-allowable if ${\sigma }^{-1}(X_{N-k})$ is contained in the $(i-k+\bar{p}(k))$-skeleton of ${\Delta }^i$ for every $k\ge 2$. A negative skeleton is empty. Individual faces need not be allowable; this is why the definition of intersection chains separately imposes allowability on the total boundary.

If the Dirichlet series $F(s)={\sum }_n{a}_n{n}^{-s}$ is absolutely convergent at $\sigma >0$, thenThe scalar kernel is $(2\pi {)}^{-1}{\int }_{\mathbb{R}}{e}^{v(\sigma +it)}(\sigma +it{)}^{-2}dt=v_{+}$. This follows by Fourier inversion applied to $u{e}^{-\sigma u}{1}_{u\ge 0}$, whose Fourier transform is $(\sigma +it{)}^{-2}$. Absolute convergence of the series and of $\int |\sigma +it{|}^{-2}dt$ justifies interchange. The factor $ds=idt$ is essential.

A nonconstant meromorphic function on the complex plane is a simply periodic function when its period group of a function is generated by one nonzero complex number: $\mathrm{Per}(f)=\omega \mathbb{Z}$. It repeats in one independent complex direction.

For example, $f(z)=e^z$ satisfies $e^{z+\omega }=e^z$ exactly when $e^{\omega }=1$, so its periods are $2\pi i\mathbb{Z}$. A doubly periodic elliptic function instead has a period lattice generated by two real-linearly independent complex numbers. Single periodicity alone imposes no algebraic addition law.

Include zero when forming this set, even though a period of a function is normally required to be nonzero. For $f:\mathbb{R}\to Y$, $\mathrm{Per}(f)$ is a subgroup of the additive group of the real numbers: composing two preserving shifts adds them, and reversing a preserving shift negates it. A constant function has period group $\mathbb{R}$. A nonconstant continuous real-valued function that is periodic has period group $T_0\mathbb{Z}$, where $T_0$ is its fundamental period.

The fundamental period is the least positive period of a function, when one exists. For $f(x)=\sin (2\pi x)$ it is one; for $f(x)={\sin }^2(2\pi x)$ it is $1/2$. A periodic function need not have a fundamental period: a constant function has every positive period, and periodic function without a least positive period is a nonconstant exception. Every nonconstant real-valued function that is continuous and periodic on the real line does have a least positive period.

For a generic quadratic period-doubling cascade, the Feigenbaum constants are the limiting parameter ratio $\delta$ and spatial ratio $\alpha$. With $s_n$ the successive superstable period-$2^n$ parameters and $d_n=f_{s_n}^{2^{n-1}}(0)$ the signed central-orbit spacing,The positive convention gives $\delta \simeq 4.669201609$ and $\alpha \simeq 2.502907875$. The orientation-reversing spatial scale at the Feigenbaum renormalization fixed point is $a=-1/\alpha$.

// Target: dynamical-systems.bigb

With a fixed reference $|0⟩$, exact overlap readout on Bob's remotely prepared state distinguishes the computational basis from the diagonal basis in one use, by the displayed disjoint squared-overlap sets. This would reveal Alice's spacelike basis choice. The argument rules out even deterministic exact overlap magnitude, avoiding the additional unphysical global-phase dependence of a raw complex inner product.

An even integer $n>0$ is a perfect number if and only ifThis forces $p$ to be a prime number, as shown under Mersenne number. It does not say that every prime exponent works, nor classify possible odd integers.^[ref]

For the forward direction, put $q=2^p-1$ and use coprimality of $q$ and $2^{p-1}$. Multiplicativity of the sum-of-divisors function and the finite geometric series giveThe prime number hypothesis makes $\sigma (q)=q+1$; replacing it with a composite number changes this factor.

For a short converse outline, write $n=2^{a}m$ with $a\ge 1$ and $m$ an odd integer. The condition $\sigma (n)=2n$ givesHere $t$ is a positive integer and a proper divisor of $m$. If $t>1$, the distinct divisors $1,t,m$ alone exceed the required sum $m+t$. Thus $t=1$ and $\sigma (m)=m+1$, forcing $m$ to be prime and $m=2^{a+1}-1$.

A vertex on one side of a balanced bipartite graph is flexible if every incident edge lies in a perfect matching. The strict Hall marriage theorem inequalities for nonempty proper subsets imply flexibility of every vertex: after forcing an edge, removal of its ends loses at most one neighbour from each remaining set. If a perfect matching exists without the strict condition, choose a minimal nonempty tight set. Its induced balanced graph has the strict condition internally, and each internal matching extends using the original matching outside that set. Thus at least one flexible vertex always exists.

For dimension at least three, $[I+a{E}_{ik},I+E_{kj}]=I+a{E}_{ij}$ for distinct $i,j,k$, with $[x,y]=x^{-1}{y}^{-1}xy$. In dimension two over a field with more than three elements, choose $s$ with $s^2\ne 1$ and commute an elementary matrix with $\mathrm{diag}(s,s^{-1})$. In both cases all generating transvections lie in the commutator subgroup, so the special linear group over a finite field is perfect outside ${\mathrm{SL}}_2({\mathbb{F}}_2)$ and ${\mathrm{SL}}_2({\mathbb{F}}_3)$.

// Target: algebra.bigb

Maximum cardinality search is a graph search in which an unselected vertex's score counts its already selected graph neighbours. Repeatedly select a vertex of maximum score, breaking ties arbitrarily, and update its unselected graph neighbours. On a chordal graph, the reverse selection order is a perfect elimination ordering. On $C_4$, selecting $(a,b,c,d)$ instead yields candidate deletion order $(d,c,b,a)$; $d$'s later graph neighbours $a,c$ are not adjacent, so the clique check fails.^[ref]

A simplicial vertex is a graph vertex whose graph neighbourhood is a clique. Every isolated vertex and every leaf of a graph qualifies. In the three-triangle chordal graph, initially $a,e$ qualify, but $b$ only becomes a simplicial vertex after deleting $a$. This is a property of adjacency in a graph; a vertex of a simplicial complex is an element of a different kind of structure.