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Between 1600 and 1900, algebra broadened from arts of working with unknown quantities and solving equations into the study of several systems of symbolic operations and increasingly general algebraic structures. This was neither a single change of definition nor a clean replacement of one subject by another. Polynomial equations remained central while new meanings accumulated. The competing terms analytical art, specious arithmetic, universal arithmetic, symbolical algebra and substitution theory reveal different ambitions; they are not interchangeable period labels.

Near 1600, François Viète's In artem analyticem isagoge by François Viète (1591) supplies a particularly revealing starting point. His analytical art uses letters for known quantities as well as unknown ones. In his terminology, logistica numerosa concerns numerical calculation and logistica speciosa calculation with species, or general magnitudes. Vowels represent unknowns and consonants known magnitudes. This allows a single relation to represent many particular problems, but it is still tied to geometrical homogeneity: a square and a length do not simply become quantities of the same kind because they are written with letters.^[ref][ref]

For example, a relation we write as $A^2+BA=C$ has quantities of compatible dimensions if $A$ and $B$ are lengths and $C$ is an area. Letters do more than abbreviate a numerical question: they express relations between whole families of magnitudes. The phrase specious arithmetic therefore indicates arithmetic carried out on species; analysis stresses the art of reasoning from an assumed unknown towards its determination. Neither term should be reduced to the modern subject real analysis.

René Descartes's La Géométrie by René Descartes (1637) demonstrates another relation between algebra and geometry. A chosen unit segment lets multiplication, division and extraction of square roots be represented by constructions of lengths. The product of two measured lengths can itself be represented by a length relative to that unit. In modern dimensional notation the construction gives $ab/u$ for unit length $u$, removing the obstacle that a product must always be treated as an area. The text's equations can then organise geometrical problems and classify curves by their algebraic complexity. Its title remains geometry: algebra serves a geometrical programme rather than announcing a present-day department of abstract algebra.^[ref]

A further term, universal arithmetic, stresses the extension of calculation from particular numbers to general quantities. Isaac Newton's Arithmetica universalis by Isaac Newton, issued in 1707, makes the term visible in a title whose subtitle concerns composition and resolution. A title is evidence for the vocabulary, not a sufficient definition of all contemporary usage.^[ref] In Leonhard Euler's Elements of Algebra by Leonhard Euler, published in German in 1770, general numerical calculation and the solution of equations remain a substantial conception of the subject. For instance, allowing negative and complex roots changes the admissible answers to polynomial equations while preserving the central question of determining an unknown.^[ref]

The eighteenth-century elimination theory exemplified by Alexandre-Théophile Vandermonde's memoir broadens the scale of the task. Instead of one numerical quadratic equation, one seeks a rule for an arbitrary number of simultaneous linear equations. The determinant is initially valuable because it effects that elimination. This is a different generality from the later study of a group: the objects being determined are still unknown quantities, and symbolic generality concerns the number and coefficients of the equations. The case cautions against using generality alone as a test for whether algebra has become abstract.^[ref]

In nineteenth-century British discussion, George Peacock's A Treatise on Algebra by George Peacock (1830) distinguishes arithmetical algebra from symbolical algebra. If the symbols mean positive numbers, $a-(a+b)$ has no positive-number result. Yet algebraists manipulate this expression without restriction. George Peacock therefore seeks principles for the manipulation of signs which are not confined by their original arithmetical interpretation. The contrast is explicitly argued in his preface: arithmetic suggests the laws, while the symbolic science requires foundations adequate to its more general conclusions.^[ref]

The familiar extension $(-1)(-1)=1$ illustrates the stakes. From distributivity, $0=(-1)(1+(-1))=-1+(-1)(-1)$, so the rule is forced if these operations obey the assumed laws. This explains how a law can survive a change in interpretation. It does not establish that every familiar law must survive every extension: once the allowable systems themselves change, an argument must state which laws it assumes. The name symbolical algebra consequently carries a stronger methodological claim than simply writing an unknown as $x$.

William Rowan Hamilton's work on quaternions supplies a decisive example of that change. His publications of the 1840s describe a new system of imaginaries in algebra. With quaternion units,Thus multiplication need not satisfy the commutativity even when it remains associative and supports division by nonzero elements. The new system is not merely a further list of roots of old polynomial equations. It has its own operations, interpretation and laws. William Rowan Hamilton's title makes explicit that a plurality of calculi can belong within algebra.^[ref]

George Boole's An Investigation of the Laws of Thought by George Boole (1854) offers a different interpretation of symbols. If $x$ represents selection of a class, repeating that selection gives $x^2=x$; multiplication can represent intersection and $1-x$ a complementary class. His logical calculus borrows from numerical algebra but also restricts interpretation: his sum of classes is not automatically unrestricted set union, and intermediate symbolic expressions need not name classes directly. It would therefore be inaccurate to identify all of his rules with a modern Boolean algebra. The example nevertheless shows why algebraic reasoning could extend beyond magnitudes.^[ref]

The shift can also take a concrete linear form. Arthur Cayley's A Memoir on the Theory of Matrices by Arthur Cayley (1858) treats matrices with operations of their own. The identity matrix is no longer the scalar $1$, though it plays an analogous role; matrix multiplication is generally noncommutative. For example,produce $AB=\mathrm{diag}(1,0)$ and $BA=\mathrm{diag}(0,1)$. This is a checkable reason why universal arithmetic can no longer mean that every rule of ordinary numbers applies unchanged.^[ref]

Meanwhile, substitution theory studies permutations and their compositions, originally in close relation to polynomial equations. Évariste Galois's work connects the solvability by radicals of an equation with its associated group. Augustin-Louis Cauchy's work on substitutions and Heinrich Weber's 1882 definition illustrate successive forms of generalisation: from particular permutations towards arbitrary elements satisfying operational conditions. This is not a claim that one man supplied the complete modern hierarchy of groups, rings and fields at once. It identifies the change in the question: one can now investigate what follows from the laws of an operation, independently of whether the elements are numbers, substitutions or classes.^[ref]

By 1900, algebra could concern the laws and structures of several calculi. Earlier meanings persisted in teaching and in the theory of equations. The strongest account of the change therefore combines continuity with enlargement: letters for general quantities, symbolic operations with explicit laws, nonnumerical interpretations, and structural questions about those laws. Analytical art foregrounded a method of discovery, universal arithmetic foregrounded general calculation, symbolical algebra foregrounded formal operations, and substitution theory foregrounded a particular class of transformations. Their differing uses help explain the expansion more precisely than a single date for the birth of abstraction.

Use ${\omega }_0^2=k/m$ and the power formula from solution. Its normalized form isIt is zero at $\omega =0$, rises to one at ${\omega }_0$, and decays to zero as $\omega \to \infty$. The two positive half-power angular frequencies satisfyOn the low-frequency side, ${\omega }_0^2-{\omega }_1^2=\gamma {\omega }_1$; on the high-frequency side, ${\omega }_2^2-{\omega }_0^2=\gamma {\omega }_2$. Solving these quadratic equations givesConsequently the frequency bandwidth of viscous oscillator power absorption obeys the stronger exact resultFor a sharp resonance, $\gamma \ll {\omega }_0$, expansion gives ${\omega }_{1,2}={\omega }_0\mp \gamma /2+O({\gamma }^2/{\omega }_0)$, which recovers the requested approximation. The exact width above assumes the constant viscous damping coefficient in this model. Although the curve is approximately symmetric near a sharp peak, the exact half-power points satisfy ${\omega }_1{\omega }_2={\omega }_0^2$ rather than exact symmetry about ${\omega }_0$.

To invert the complex cosine, let $z=\cos w$ and put $u=e^{iw}$. Since the exponential function is nonzero, $u\ne 0$, and Euler's formula gives $2z=u+u^{-1}$. Multiplying by $u$ yields the quadratic equation $u^2-2zu+1=0$. ThusChoosing either square root of $1-z^2$ merely exchanges the two signs. The two values have product one, so neither vanishes; taking a complex logarithm is legitimate.

For each choice of sign, all solutions of $e^{iw}=u$ satisfy $iw=\mathrm{Log}u+2\pi in$. Therefore the full inverse relation isConversely, exponentiating any displayed value recovers one root of the quadratic equation, and hence $(e^{iw}+e^{-iw})/2=z$, proving completeness. With a multivalued complex logarithm, the $2\pi n$ term is already included and this is the source's compact formula for inverse cosine.

The principal inverse cosine requires compatible choices of logarithm branch, square root and sign. Both signs cannot represent that same principal function indiscriminately: at $z=0$, the two principal-log values are $\pi /2$ and $-\pi /2$, whereas the principal $\arccos 0$ is $\pi /2$. Thus the printed $\pm$ is correct for the inverse relation, not a pair of interchangeable principal-value formulas.

A normal mode of a stable undamped linear coupled oscillator has a single angular frequency and a fixed ratio of component displacements. All components share a sinusoidal time factor, with relative oscillation phases of zero or $\pi$ for real mode shapes. A general free motion is a superposition of the normal modes. For this system the symmetric stiffness matrix gives real eigenvalues and orthogonal eigenvectors; positive spring constants make the squared angular frequencies positive.

Write $m=m_1=m_2>0$ and $\kappa =k_{12}$. Assume ideal massless linear springs, negligible damping and positive $k_1,k_2,\kappa$. By Hooke's law and Newton's second law,The coupling spring extends by $x_2-x_1$, giving equal and opposite coupling forces.

For a normal mode, take $\mathbf{x}=\mathrm{Re}(\mathbf{a}{e}^{i\omega t})$. Nonzero eigenvectors requireSolving this quadratic equation in $m{\omega }^2$ givesA corresponding mode shape has ratioand the general real solution is $\mathbf{x}=C_{-}{\mathbf{a}}_{-}\cos ({\omega }_{-}t+{\varphi }_{-})+C_{+}{\mathbf{a}}_{+}\cos ({\omega }_{+}t+{\varphi }_{+})$. The arbitrary amplitudes and oscillation phases account for four independent initial data.

Weak coupling alone does not guarantee large beats. A useful representative sketch assumes nearly equal uncoupled angular frequencies. In the resonant case $k_1=k_2=k$ with $\kappa \ll k$, the mode shapes are $(1,1)$ and $(1,-1)$, andFor $x_1(0)=A$, $x_2(0)={\dot{x}}_1(0)={\dot{x}}_2(0)=0$, both normal modes have equal coefficients. With $\bar{\omega }=({\omega }_{+}+{\omega }_{-})/2$ and $\Delta \omega ={\omega }_{+}-{\omega }_{-}$,These rapid oscillations have slowly varying complementary wave envelopes. The initial displacement shifts almost entirely to the other mass at $t=\pi /\Delta \omega$, and returns after $2\pi /\Delta \omega$. Arbitrary initial data give a different superposition; if $|k_1-k_2|\gg \kappa$, the normal modes are localized mainly on individual masses, and strong exchange is suppressed.

Holding one mass fixed leaves the coupling spring attached and stretched by the other mass. Therefore the two constrained angular frequencies areThese differ from the angular frequencies obtained by physically removing the coupling spring. Ordinary frequencies in hertz are each corresponding $\omega /(2\pi )$.

Finally, for $k_1>k_2$ and $\kappa >0$, put $D=\sqrt{(k_1-k_2{)}^2+4{\kappa }^2}>k_1-k_2$. ThenSince ${\omega }_1>{\omega }_2$, taking positive square roots provesAt $\kappa =0$, the outer inequalities become equalities.

Use signed velocities in the shock frame; the Mach number depends on their squares. For a genuine nonradiative perfect gas shock wave, the Rankine-Hugoniot conditions for a perfect gas give a common mass flux $j={\rho }_L{u}_L={\rho }_R{u}_R$, a common momentum flux $p+\rho u^2$, and a common specific stagnation enthalpy $\gamma p/[(\gamma -1)\rho ]+u^2/2$. Here $e$ in the given equation of state is specific internal energy, despite the source's informal “energy density” wording.

Put $r={\rho }_R/{\rho }_L$, $P=p_R/p_L$ and $M^2={\rho }_L{u}_L^2/(\gamma p_L)$. Mass and momentum give $u_R=u_L/r$ and $P=1+\gamma M^2(1-1/r)$. Substituting these into the energy relation and clearing denominators givesThe $r=1$ factor is the no-jump solution. On the nontrivial shock wave branch,In particular $r-1=2(M^2-1)/[(\gamma -1)M^2+2]$. With $\gamma >1$, compression is therefore equivalent to $M_L^2>1$. The other Mach number isPositive pressure makes its denominator positive, and $M_R^2-1$ has the opposite sign to $M_L^2-1$. Thus a nontrivial physical shock wave has a supersonic side and a subsonic side; the entropy-admissible flow goes from the former to the latter.

The shock compression ratio also yields $u_L{u}_R=(\gamma -1)u_L^2/(\gamma +1)+2\gamma p_L/[(\gamma +1){\rho }_L]$. Rearranging produces the first velocity identity in the question. The pressure relation gives the second identity in the equivalent formThe numerator here contains mass density, as the original PDF shows; the converted TeX incorrectly replaces that mass density by pressure.

For reflection, the once-shocked gas is the common left state in both applications. Before rebound, its velocity in the shock frame is $a=u_s-U_{+}<0$ and the gas ahead has velocity $-U_{+}$; after rebound, it is $b=u_s+U_{-}>0$ and the gas next to the wall has velocity $U_{-}$. In both cases the right velocity minus the left velocity equals $-u_s$. Applying the first identity therefore gives the same quadratic equation,with roots $a$ and $b$. Their opposite signs distinguish the two roots, so $\boxed{(u_s-U_{+})(u_s+U_{-})=-\gamma p_s/{\rho }_s}$.

Write $q_0=p_0/p_s$, $q_1=p_1/p_s$. The pressure identity applied to each shock wave givesSquaring the root-product relation and eliminating the velocities yieldsand hence the normal shock reflection at a rigid wall relationTaking $p_0/p_s\to 0$ gives $\boxed{p_1/p_s\to (3\gamma -1)/(\gamma -1)}$; for a monatomic gas this limiting amplification is six.

For positive $\lambda ,\alpha$ and $0<s<1$, the constant term of the quadratic equation is negative. Its roots are therefore real with opposite signs, so one has positive growth. At $s>1$, both the coefficient of $\beta$ and the constant term are positive. If the roots are real they are both negative; if complex they have negative real parts. Thus the growing band remains $\boxed{0<k^2{a}^2<1}$. At $s=0$ or $1$ there is a neutral root. If $\alpha =0$, the result reduces continuously to the clean-film capillary branch with a passive zero root.

The quarter-rate limit is the strong-elasticity limit $\lambda /\alpha \ll 1$, not the weak-elasticity limit. The original PDF repeats $\alpha /\lambda \ll 1$ here; that is inconsistent with the preceding small-$\alpha$ result. In fact at $\alpha =0$ the maximum is $\lambda /4$, directly contradicting a quarter of that value in the same limit. The converted TeX repairs this particular ratio.

For $\alpha \gg \lambda$ and wavenumbers near the maximum, $S=\alpha s\gg L$. In the quadratic equation, the growing root is $O(\lambda )$, so its square and the $-L\beta$ term are small compared with $S\beta$. The leading balance is $S\beta -LS/4=0$. ConsequentlyThe second root is a rapidly damped surfactant-relaxation mode. Thus the stated quarter-rate result is correct after reversing the printed small-parameter ratio.

Substituting $\eta =D_s\beta$ and $\xi =D_d\theta$ into the previous geometric relation verifiesFor a point mass gravitational lens, the given physical deflection yieldsAn aligned source has $\beta =0$, and therefore ${\theta }^2={\theta }_E^2$. Rotational symmetry turns these signed intersections into an Einstein ring, whose Einstein radius isFor an offset source, multiplying $\beta =\theta -{\theta }_E^2/\theta$ by $\theta$ gives a quadratic equation. Its two image positions areFor $\beta >0$, ${\theta }_{+}>0$ lies on the source side of the lens and ${\theta }_{-}<0$ lies on the opposite side.

At a fixed point $x=y=p$, the quadratic equation is $p^2+(1+b)p-\mu =0$. ConsequentlyThe Jacobian matrix is $\left(\begin{array}{cc}0& 1\\ -b& -2p\end{array}\right)$, and its Floquet multipliers solve ${\lambda }^2+2p\lambda +b=0$. The Jury stability criterion requires $|b|<1$ and $1+2p+b>0$, $1-2p+b>0$. With $b>-1$, this becomes $-1<b<1$, $-(1+b)/2<p<(1+b)/2$. The lower branch never satisfies it; the upper branch is attracting precisely whenAt the lower boundary the two fixed points merge with Floquet multipliers $1,b$, a saddle-node bifurcation for $b\ne 1$. At the upper boundary the upper point has Floquet multipliers $-1,-b$, giving a generic period-doubling bifurcation for $-1<b<1$.

Its direction can be established explicitly. A distinct two-cycle alternates values $y,z$ satisfying $(1+b)z=\mu -y^2$, $(1+b)y=\mu -z^2$. Subtracting gives $y+z=1+b$, and substitution givesIt exists beyond the fixed-point period-doubling bifurcation. The second-iterate Jacobian matrix has determinant $b^2$ and trace $4yz-2b=4[(1+b{)}^2-\mu ]-2b$. Applying the same stability inequalities shows attraction for $-1<b<1$ andThus the first period-doubling bifurcation is supercritical; the next upper boundary has a Floquet multiplier $-1$ for the two-cycle. These are the fixed-point and two-cycle thresholds of the Hénon map.

At $b=1$ the map preserves phase-space area, so attraction is impossible. The fixed-point fold is at $\mu =-1$, with double Floquet multiplier $+1$, and is a conservative saddle-centre bifurcation. The upper branch $p_{+}=-1+\sqrt{1+\mu }$ is linearly elliptic for $-1<\mu <3$; the lower branch is a saddle fixed point of a map. At $\mu =3$, the upper branch has double Floquet multiplier $-1$ and becomes a saddle fixed point of a map for larger $\mu$. The two-cycle $y,z=1\pm \sqrt{\mu -3}$ is elliptic for $3<\mu <4$, then reaches double Floquet multiplier $-1$ for the second iterate at $\mu =4$. The partial sketch distinguishes elliptic, neutrally stable Floquet multipliers from attracting states. Resonances may require additional nonlinear analysis; unit Floquet multiplier moduli alone do not prove nonlinear Lyapunov stability.

The $b=1$ segment is a conservative Hénon stability boundary, not an ordinary nondegenerate dissipative Neimark–Sacker bifurcation. Indeed for $b>0$ constant phase-space area scaling is $\mathrm{area}(F(D))=b\mathrm{area}(D)$. A simple invariant closed curve would enclose a bounded invariant set $D$ of positive phase-space area, forcing $b=1$. Thus no such closed invariant curve can bifurcate into $b\ne 1$ within this constant-determinant family.

Generic folds and flips away from these degeneracies persist under sufficiently small smooth perturbations: their simple unit Floquet multiplier, transverse parameter crossing and nonzero leading nonlinear coefficient persist, while the other Floquet multiplier stays away from the unit circle. The implicit function theorem then continues the bifurcation curves and their criticality. The unqualified perturbation claim needs this restriction. Arbitrary perturbations need not preserve the conservative $b=1$ behavior. For example, in local complex coordinates at a nonresonant elliptic fixed point, the area-preserving map property gives a cubic normal form $z^{\prime }=e^{i\theta }z(1+i\tau |z{|}^2)+\cdots$ with zero cubic radial damping. Composing with the arbitrarily small radial perturbation $z\mapsto (1-\eta |z{|}^2)z$ introduces radial coefficient $-\eta$. Varying the modulus through one can now give a genuine Neimark–Sacker bifurcation. Conservative degeneracies and resonances are therefore not covered by a blanket structural-stability assertion.

Begin with two points at distance one. A ruler draws the line through two known points; a compass draws a circle with a known centre and a constructible radius. New points are intersections of such lines and circles. A real number is constructible when it occurs as a coordinate of a point obtained by finitely many such operations; a complex point is constructible when both coordinates are.

If current coordinates lie in a field $K\subseteq \mathbb{R}$, a line-line intersection solves linear equations over $K$, while line-circle or circle-circle intersections reduce to a quadratic equation over $K$. Thus each step requires at most adjoining a square root. Consequently every constructible number lies in a towerConversely field sums and differences are obtained by laying off segments, products and quotients by similar triangles, and a positive square root by the geometric-mean construction in a semicircle. Hence every real number in such a tower is constructible. This proves the quadratic-tower criterion, not merely its necessity.

In particular a constructible algebraic number has degree a power of two. Degree alone is not sufficient in general; the exact criterion is containment in a quadratic tower. Equivalently the Galois group of its normal closure is a finite two-group: a tower's normal closure remains a two-extension, and a two-group admits a chain of subgroups with successive index two, yielding the converse by Galois correspondence.

The standard impossibility results follow. Doubling a unit-volume cube requires $\sqrt[3]{2}$, whose polynomial $X^3-2$ is irreducible by Eisenstein and has degree three. Trisecting an arbitrary angle would in particular trisect $60$ degrees, requiring $2\cos 20^{\circ }$, a root of $X^3-3X-1$. This cubic has no rational root and is irreducible, so this construction is impossible, although some particular angles can be trisected. Squaring the unit circle requires side length $\sqrt{\pi }$, impossible because $\pi$ is transcendental whereas constructible numbers are algebraic.

For a regular $n$-gon, constructibility is equivalent to that of $e^{2\pi i/n}$. The cyclotomic extension is Galois of degree $\phi (n)$, so it is constructible exactly when this degree is a power of two. Writing the prime factorization of $n$ shows the equivalent conditionIndeed odd prime exponents must be one, and each $p_i-1$ must be a power of two. If $2^m+1$ is prime, $m$ must itself be a power of two by factoring when $m$ has an odd divisor. Conversely this factorization makes the cyclotomic Galois group a two-group and provides a quadratic tower. This gives both positive constructions, such as the regular seventeen-gon, and obstructions, such as the regular seven-gon.

Let the equilibrium ideal gas be at rest, with uniform ${\rho }_0,p_0$ and ${\mathbf{B}}_0=B_0{\mathbf{e}}_z$. This local wave calculation neglects gravity, viscosity and magnetic diffusion. Define the adiabatic sound speed $c\equiv v_s=(\gamma p_0/{\rho }_0{)}^{1/2}$, the Alfvén speed $v_a=B_0/({\mu }_0{\rho }_0{)}^{1/2}$, and a wavevector $\mathbf{k}=k(\sin \theta ,0,\cos \theta )$, where $k=|\mathbf{k}|$ and $\theta$ is its angle to the equilibrium magnetic field. Perturbations have dependence $e^{i\mathbf{k}\cdot \mathbf{x}-i\omega t}$ and phase speed $v=\omega /k$.

The linearized ideal magnetohydrodynamic equations for mass conservation, pressure, induction and momentum giveThe relation between pressure and mass density follows from eliminating $\mathbf{k}\cdot {\mathbf{u}}^{\prime }$ between the two material evolution equations; it applies to the nonzero-frequency wave sector. The induction result automatically has $\mathbf{k}\cdot \mathbf{b}=0$.

The $y$ MHD wave polarization is perpendicular to both the field and wavevector. It has $\delta p=\delta \rho =0$ and obeysThus the transverse Alfvén wave hasIn the $x$-$z$ plane the two remaining velocity components instead obey the symmetric matrix eigenproblemTaking its determinant gives $v^4-(c^2+v_a^2)v^2+c^2{v}_a^2{\cos }^2\theta =0$. Solving this quadratic equation in $v^2$ derives both magnetoacoustic wave phase speeds,The plus sign is the fast magnetosonic wave, the minus sign the slow magnetosonic wave. The radicand is nonnegative, since it equals $[(c^2-v_a^2{)}^2+4c^2{v}_a^2{\sin }^2\theta ]/4$. Both squared speeds are nonnegative. Each nonzero mode has forward and backward propagation; an advected entropy disturbance is a separate zero-frequency mode in this static equilibrium.

Take $k>0$; reflection in $x$ supplies the result for negative wavenumber. Since each region has uniform base velocity, the perturbation is an incompressible, irrotational flow with a velocity potential satisfying the Laplace equation. Its decaying outer forms areFor a varicose mode of a planar jet, reflection symmetry makes the inner potential proportional to $\cosh (ky)$; for a sinuous mode of a planar jet, it is proportional to $\sinh (ky)$.

At the upper interface, the linearized kinematic boundary condition isThe first equation gives ${\varphi }_{+}(h)=-\sigma \eta /k$. The second givesThe linearized Unsteady Bernoulli equation gives $p^{\prime }=-\rho ({\partial }_t+U{\partial }_x)\varphi$. Applying pressure continuity therefore produces the equal-density top-hat planar-jet dispersion relationSolving the quadratic equation givesEquivalently,The plus sign is the unstable Kelvin-Helmholtz instability branch. In particular, the two modes have exactly the same growth rate,Its limiting forms areThus neither type grows faster in this equal-density idealization, although their phase speeds differ. The unbounded large-$k$ growth is the familiar short-wave pathology of an inviscid vortex sheet.

The horizontally invariant magnetized shearing-sheet equations are linear in the horizontal fields, so perturbations about the equilibrium satisfy the same equations. The fixed surface boundary conditions require $\delta B_x=\delta B_y=0$ at $z=\pm H$. Choose a normal mode withwhere the vertical wavenumber is $k=m\pi /(2H)$, $m=1,2,\dots$. With the Alfvén frequencythe four amplitude equations becomeEliminating the velocities leavesA nonzero amplitude requires the determinant of this system to vanish, yielding the ideal magnetorotational dispersion relationEquivalently, with ${\kappa }_r^2=2(2-q){\Omega }^2$,As a quadratic equation for $s^2$, its discriminant is ${\kappa }_r^4+16{\Omega }^2{\omega }_a^2>0$. If $0<{\omega }_a^2<2q{\Omega }^2$, its constant term is negative, so one root $s^2$ is positive and there is an exponentially growing mode. If ${\omega }_a^2>2q{\Omega }^2$, both the constant term and the coefficient of $s^2$ are positive, giving two negative roots and only oscillatory modes. Equality is marginal.

The lowest allowed vertical wavenumber, $k=\pi /(2H)$, is the last to be stabilized as $|B_z|$ increases. Therefore the finite-thickness magnetorotational instability criterion isThere is also a vertically uniform velocity mode with zero magnetic perturbation; for the usual orbitally stable regime $q<2$, it is just stable epicyclic motion. For $q>2$, that uniform mode is already hydrodynamically unstable, independently of the magnetic criterion. At $q=2$ it is marginal. The criterion above concerns the magnetic modes.

This is a birth-death process with birth and death ratesIts transition diagram has the two arrowsThe gain into state $n$ comes from a death in state $n+1$ or a birth in state $n-1$, while the loss is the sum of both rates out of state $n$. The birth-death master equation is consequently

Multiplying by $n$, summing over the nonnegative integers, and shifting the summation indices shows that each birth contributes $+1$ and each death contributes $-1$. Thus the first-moment equation of a birth-death process givesAt a stationary state, use the variance identity $⟨n^2⟩=⟨n{⟩}^2+(\Delta n{)}^2$. Solving the resulting quadratic equation givesThe minus root is inadmissible whenever it is negative, namely when $(\Delta n{)}^2<\gamma /\alpha$. Equality would give zero mean, which is also incompatible with a positive immigration rate $\gamma$, so for $\gamma >0$ the minus branch requires $(\Delta n{)}^2>\gamma /\alpha$ even to be a possible mean.

For a continuum approximation, write $b(n)=\gamma +\beta n$ and $d(n)=\alpha n(n-1)$. Applying the Kramers-Moyal expansion to the two gain terms and retaining derivatives through second order gives the Fokker-Planck equationwhere the negative drift and infinitesimal jump variance areThis truncation requires the typical population and the scale on which $P$ varies to be much larger than the unit jump size, with the rates varying smoothly across that scale.

The positive zero of $g$ isso in particular $n_{\ast }\sim \beta /\alpha$ as $\alpha /\beta \to 0$. Put $x=n-n_{\ast }$. The linear noise approximation useswhereand, because $b(n_{\ast })=d(n_{\ast })$,In a stationary state with zero Fokker-Planck probability current,Its normalized solution is the normal distributionIt follows that

These estimates agree with the exact stationary first-moment relation on its plus branch to leading order: inserting $(\Delta n{)}^2\sim \beta /\alpha$ gives $⟨n⟩\sim \beta /\alpha$. Moreover,Thus the stationary mass lies far from the boundary $n=0$, is narrow relative to its mean, yet changes across many lattice sites. These are precisely the large-population and slow-variation conditions needed for the diffusion approximation.

A nondegenerate hyperbola centred at the origin has a quadratic equationThere are no linear terms because the origin is its centre. Its asymptotes are the zero set of the homogeneous quadratic form $A{x}^2+2Bxy+C{y}^2$. Requiring the directions $y=x$ and $y=-x$ gives $A+2B+C=A-2B+C=0$, hence $B=0$ and $C=-A$. Since the conic section is nondegenerate, $A$ and $F$ are nonzero. Thus all such hyperbolas with prescribed asymptotes are exactlyFor $c>0$ these rectangular hyperbolas open along the $x$-axis, and for $c<0$ along the $y$-axis. Conversely, each displayed hyperbola has the required centre and asymptotes. Two distinct members have different values of $x^2-y^2$, so they cannot intersect.

For the second pair of asymptotes, the same argument gives $H_2:4x^2-y^2=d$, $d\ne 0$. The intersections of concentric hyperbolas therefore satisfyIf both right sides are positive, there are four independent sign choices. If precisely one is zero and the other is positive, there are two points. If either is negative there are none; both cannot vanish because $c\ne 0$. Each possibility occurs: take $(c,d)=(1,5)$ for four, $(1,4)$ for two, and $(1,1)$ for none. Hence

The double-angle formula for the tangent function gives $1=2t/(1-t^2)$. Here $0<t<1$, because $0<\pi /8<\pi /4$, so the denominator is nonzero. Rearranging yields the quadratic equation