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The qubit density operators $\rho =(I+\mathbf{s}\cdot \mathbf{\sigma })/2$ correspond exactly to the closed unit ball: their eigenvalues are $(1\pm |\mathbf{s}|)/2$, so positivity is equivalent to $|\mathbf{s}|\le 1$. The boundary is the Bloch sphere of pure states, and the centre is $I/2$. A depolarizing channel scales this ball isotropically, possibly with a sign reversal in the larger completely positive parameter interval.

For a spatially periodic nonlinear pattern, a linear stability disturbance is written $e^{\lambda t+iq\cdot x}v(x)$ with $v$ periodic on the pattern cell. The conjugated operator $L(q)$ acts on that fixed cell and determines the sideband eigenvalues. Translation invariance supplies zero eigenfunctions at $q=0$, namely spatial derivatives of the base pattern. Their small-$q$ eigenvalues determine long-wavelength phase stability; other parts of the spectrum can give independent instabilities.

The Born rule connects a quantum state to measurement probabilities. Let $\{|j⟩\}$ be a complete orthonormal basis, finite or countably infinite, for a Hilbert space, and letAn ideal measurement distinguishing these basis vectors returns label $j$ with probabilityThis assignment is a physical measurement rule. Its compatibility with total probability one follows from orthonormality and normalization:The probabilities are the squared moduli, not the amplitudes themselves or the squares $a_j^2$. ^[ref].

If the basis is an eigenbasis of an observable, the reported numerical value is its eigenvalue. Several vectors can share one eigenvalue; the probability of that value is then the sum over the corresponding basis labels. More generally, a projective measurement with orthogonal projections $P_r$ assignsfor a pure state and a general density matrix $\rho$, respectively. This also covers mixed states and degenerate eigenspaces. The Born rule determines outcome statistics; the ideal conditional state change is specified separately by the Lüders rule. See projective measurement for that formulation and its assumptions. ^[ref].

For a worked example, take the normalized qubit quantum stateThe squared moduli add to $3/4+1/4=1$. A computational-basis measurement, equivalently a measurement of the $Z$ Pauli matrix, hasThese probabilities do not depend on the relative quantum phase $\varphi$. Now choose the Hadamard basiswhose vectors are eigenstates of the $X$ Pauli matrix, also used by the Pauli X gate. The new probability amplitudes and probabilities areBoth probabilities are nonnegative and their sum is one. The relative quantum phase becomes visible because the two probability amplitudes add before their squared modulus is taken:

A Hadamard gate followed by a computational-basis measurement realizes these $X$ statistics, since $H|\pm ⟩=|0⟩,|1⟩$, respectively. Compare the two bases using fresh, identically prepared copies: a preceding measurement may change the state. The quantum superposition section shows why replacing the preparation by the mixed state $\mathrm{diag}(3/4,1/4)$ keeps the $Z$ statistics but gives $p_{+}=p_{-}=1/2$ for every $\varphi$.

The usefulness of the Born rule is this conversion from linear algebra to experimental predictions: calculate overlaps in the basis actually measured, then take squared moduli. Its appealing geometry is that changing basis redistributes probability while preserving total squared norm. The same rule relates an energy eigenstate expansion to an energy measurement and a position wavefunction to a probability density.

Born's 1926 paper on quantum scattering interpreted the outgoing-wave coefficients in terms of the probabilities of collision outcomes; a corrective footnote specified proportionality to the square of the amplitude. This is a historical scattering formulation, rather than the general density matrix formula above. ^[ref]; ^[ref].

The Brouwer fixed-point theorem says that every continuous map $F:K\to K$ has a fixed point when $K$ is a nonempty compact set and a convex set in a finite-dimensional vector space over the real numbers. Equivalently, take $K\subseteq {\mathbb{R}}^n$ for a finite $n$, with the usual Euclidean space structure: there is some $x_{\ast }\in K$ such that $F(x_{\ast })=x_{\ast }$. The conclusion guarantees at least one fixed point, without asserting uniqueness or telling us how quickly to find one. ^[ref]

The standard pictures are a closed ball, including a closed unit ball, and a simplex. A filled disk qualifies; its bounding circle alone does not. A probability simplex also qualifies even though its dimension is smaller than that of its ambient Euclidean space. The hypotheses have concrete meanings: being a convex set keeps every line segment between allowed points inside $K$, and compactness prevents escape or loss of a limiting point. In Euclidean space, the Heine-Borel theorem identifies compact sets with closed sets that are bounded sets. Crucially, every output must belong to the same $K$ as the input.

For a fully computable example, let $D=\{(u,v):u^2+v^2\le 1\}$ be the filled unit disk. Apply a counterclockwise quarter-turn rotation, shrink by $0.4$, and add a translation:This affine map is a continuous map. Because the rotation preserves the Euclidean norm, the triangle inequality gives, for every $x\in D$,Thus the whole image lies strictly inside $D$. More precisely, it is a disk of radius $0.4$ centred at $(0.3,0.1)$. This containment check is what makes the Brouwer fixed-point theorem applicable; a formula being a continuous map by itself would not suffice.

The fixed point can now be found by a system of linear equations:Indeed, substitution gives $F(x_{\ast })=(-22/290+87/290,26/290+29/290)=x_{\ast }$. Its Euclidean norm is $\sqrt{290}/58\approx 0.293610$, safely inside $D$. The figure shows arrows from sample inputs to their outputs; an arrow disappears at the fixed point because the displacement $F(x)-x$ vanishes there.

Repeated iterations of a map happen to succeed for this example. Starting at $x_0=(1,0)$ and setting $x_{k+1}=F(x_k)$ gives $x_1=(0.3,0.5)$, $x_2=(0.1,0.22)$, and $x_3=(0.212,0.14)$. Subtracting the fixed point equation yields $x_{k+1}-x_{\ast }=0.4R_{90}(x_k-x_{\ast })$, so the Euclidean distance to $x_{\ast }$ shrinks by exactly $0.4$ each step. This extra fact follows from the contraction mapping property and the Banach fixed-point theorem. It is additional information beyond the Brouwer fixed-point theorem.

The distinction becomes visible with two simpler continuous maps of the same disk. The pure rotation $G(u,v)=(-v,u)$ has exactly one fixed point, $(0,0)$, since $u=-v$ and $v=u$ force both to vanish. Yet iterations of a map starting at $(1,0)$ here cycle through $(0,1)$, $(-1,0)$, and $(0,-1)$ forever. Their Euclidean distance from $(0,0)$ stays one, so there is no convergence in a metric space. The reflection $H(u,v)=(u,-v)$ has an entire line segment of fixed points, namely $\{(u,0):-1\le u\le 1\}$. Starting off that line segment produces an alternating pair. Existence, uniqueness, and convergence of iterations of a map are separate questions.

Removing hypotheses changes the answer. Restrict the same quarter-turn rotation to the circle $S^1=\{(u,v):u^2+v^2=1\}$. It remains a continuous map into that compact set, but its only possible fixed point, $(0,0)$, has been removed. The circle is not a convex set. Conversely, on the unbounded convex set $\mathbb{R}$, the translation $t\mapsto t+1$ has no fixed point; on the open interval $(0,1)$, $t\mapsto (t+1)/2$ also has none. Neither domain is a compact set. The finite-dimensional closed ball statement should likewise not be extended to arbitrary infinite-dimensional vector spaces, where a closed ball need not be a compact set.

The one-dimensional version explains the connection to the intermediate value theorem. For a continuous map $f:[a,b]\to [a,b]$, put $h(t)=f(t)-t$. Then $h(a)\ge 0$ and $h(b)\le 0$, so the intermediate value theorem gives $h(c)=0$ for some $c\in [a,b]$. In higher dimensions, the Brouwer fixed-point theorem supplies a corresponding existence principle when this scalar sign argument is unavailable. Its practical value is ensuring that a proposed equation $x=F(x)$ has a solution once a suitable domain is found; it supplies no general promise of a fast algorithm.

A useful application is positive matrix eigenvector from simplex normalization: rescale the output of a strictly positive matrix to make its entries sum to one. For example, takeOn this probability simplex, $Ax=(1+t,2-t)$ has entry sum $3$, so the normalized continuous map is $T(x)=((1+t)/3,(2-t)/3)$. Both entries are positive and sum to one, keeping the image inside the probability simplex. Its fixed point equation is $t=(1+t)/3$, hence $t=1/2$. Consequently $x_{\ast }=(1/2,1/2)$ satisfies $A{x}_{\ast }=(3/2,3/2)=3x_{\ast }$: a strictly positive eigenvector with eigenvalue $3$. In larger dimensions, the Brouwer fixed-point theorem guarantees the corresponding positive eigenvector without requiring us to solve a system of linear equations in advance. The fixed point expresses preservation of relative weights, rather than preservation of their unnormalized size.

Proofs can be approached through the no-retraction theorem or Sperner's lemma. In the latter approach, increasingly fine triangulations locate approximate fixed points; compactness supplies a convergent subsequence, and continuous maps turn its limit into an exact fixed point. ^[ref]

A primary historical reference is L. E. J. Brouwer's 1911 paper Über Abbildung von Mannigfaltigkeiten, whose digitized original is hosted by Zenodo under DOI 10.1007/BF01456812. This gives a direct route to the early literature behind the Brouwer fixed-point theorem. ^[ref]

Writing $D_r=\mathbf{x}\cdot \nabla =r{\partial }_r$, the spherical-coordinate Laplacian gives $r^2\Delta =D_r^2+D_r+{\Delta }_{S^2}$. Thus the positive angular Laplace-Beltrami operator has the displayed Cartesian expression and eigenvalues $l(l+1)$ on spherical harmonics. Omitting $D_r$ is incorrect when the square means composition: on $x_1$, the truncated expression gives $x_1$, whereas the degree-one angular eigenvalue is two.

// Target: fluid-mechanics.bigb

An eigenvector of $A$ with eigenvalue $\lambda$ gives a homogeneous solution $t^{\lambda }\mathbf{v}$ for $t>0$. If $A$ is diagonalizable, these powers form a fundamental system. A constant vector forcing has a particular solution $t(I-A{)}^{-1}\mathbf{b}$ when $1$ is not an eigenvalue. At a resonant eigenvalue or a nontrivial Jordan block, logarithmic factors in $t$ can occur, equivalently by changing time to $\log t$ and solving a constant-coefficient linear system.

For a sufficiently smooth local map fixing the origin, a local invariant centre manifold is tangent to the generalized eigenspaces with eigenvalues of modulus one. Writing the map as $(u,v)\mapsto (F(u,v),G(u,v))$ and the manifold as $v=h(u)$ gives the invariance equation $h(F(u,h(u)))=G(u,h(u))$. Finite Taylor series coefficients can be found from this equation. If all transverse eigenvalues have modulus less than one, local asymptotic stability reduces to that of the map on the centre manifold.

For eigenvalues ${\lambda }_1,\dots ,{\lambda }_d$ of a finite-order linear action, its symmetric power characters satisfy ${\sum }_{n\ge 0}{\chi }_{S^{n}V}{t}^n={\prod }_j(1-{\lambda }_{j}t{)}^{-1}$. The denominator is the alternating generating polynomial of its exterior power characters.

For a complex $n\times n$ matrix $A$, $\mathrm{tr}(A^k)$ is the sum of the $k$th powers of its eigenvalues, counted with algebraic multiplicity. This does not require diagonalizability: a similar upper triangular matrix has the eigenvalues on its diagonal, and taking powers raises those diagonal entries to their corresponding powers. Newton's identities therefore recover its characteristic polynomial from the first $n$ such traces.

For the Jordan block $A=\left(\begin{array}{cc}2& 1\\ 0& 2\end{array}\right)$, $\mathrm{tr}A=4$ and $\mathrm{tr}(A^2)=8$. Hence $e_1=4$ and $e_2=(4^2-8)/2=4$, giving $\det (zI-A)=z^2-4z+4=(z-2{)}^2$. The repeated eigenvalue is recovered even though this matrix is not diagonalizable.^[ref]

A characteristic speed is an eigenvalue of the flux Jacobian. Along a corresponding characteristic curve, an associated wave variable obeys an ordinary differential equation obtained by projecting the governing system onto the matching left eigenvector.

At an element $g$ of a finite group, a complex character value is the sum of the roots of unity which are the eigenvalues of its representing matrix. It is therefore an algebraic integer, and every algebraic conjugate has modulus at most the degree.

A chemostat steady state has constant nutrient concentration and biomass concentration even while nutrient and cells flow through the vessel. It is an equilibrium of the chemostat ordinary differential equations, rather than a culture in which all growth has stopped. Assume $D>0$ and the Monod equation. Setting both derivatives to zero gives the always-existing chemostat washout equilibrium $(s_{\ast },x_{\ast })=(s_{\mathrm{in}},0)$. A positive biomass concentration instead requires growth to balance removal, $\mu (s_{\ast })=D$, whenceThe positive chemostat steady state exists exactly whenThe weaker condition $D<{\mu }_{\max }$ only makes the denominator positive; it does not ensure $s_{\ast }<s_{\mathrm{in}}$ or $x_{\ast }>0$. Because the Monod equation is increasing, the feed must permit a specific growth rate greater than $D$. At a positive chemostat steady state, increasing feed nutrient concentration at fixed $D$ increases the biomass concentration without changing $s_{\ast }$, within the one-nutrient assumptions.

For a concrete chemostat, let ${\mu }_{\max }=1{\mathrm{h}}^{-1}$, $K_s=1\mathrm{g}{\mathrm{L}}^{-1}$, $s_{\mathrm{in}}=9\mathrm{g}{\mathrm{L}}^{-1}$ and $Y=0.5\mathrm{g}\text{ biomass}/(\mathrm{g}\text{ nutrient})$. At $D=0.5{\mathrm{h}}^{-1}$,The nutrient supply minus outflow is $D(s_{\mathrm{in}}-s_{\ast })=4\mathrm{g}{\mathrm{L}}^{-1}{\mathrm{h}}^{-1}$, matching nutrient consumption $\mu (s_{\ast })x_{\ast }/Y=4\mathrm{g}{\mathrm{L}}^{-1}{\mathrm{h}}^{-1}$. Biomass production and removal both equal $2\mathrm{g}{\mathrm{L}}^{-1}{\mathrm{h}}^{-1}$. The chemostat washout threshold is $D_c=\mu (9)=0.9{\mathrm{h}}^{-1}$. For example, $D=0.95{\mathrm{h}}^{-1}<{\mu }_{\max }$ formally gives $s_{\ast }=19\mathrm{g}{\mathrm{L}}^{-1}$ and $x_{\ast }=-5\mathrm{g}{\mathrm{L}}^{-1}$, which cannot describe a physical culture.

There is a useful exact simplification: the nutrient-equivalent concentration $z=s+x/Y$ satisfies $\dot{z}=D(s_{\mathrm{in}}-z)$, so $z(t)=s_{\mathrm{in}}+[z(0)-s_{\mathrm{in}}]e^{-Dt}$. This combines dissolved nutrient and the nutrient equivalent of biomass; it does not assert that these two variables account for every chemical constituent. In coordinates $(z,x)$, the Jacobian matrix at the positive chemostat steady state has eigenvalues $-D$ and $-{\mu }^{\prime }(s_{\ast })x_{\ast }/Y$, both negative. Hence the positive chemostat steady state is locally asymptotically stable. In the worked example the eigenvalues are $-0.5{\mathrm{h}}^{-1}$ and $-2{\mathrm{h}}^{-1}$.