Row additions reduce an invertible matrix to diagonal form without changing determinant. A determinant-one diagonal matrix is a product of diagonal two-coordinate blocks $\mathrm{diag}(a,a^{-1})$, each expressible as a product of six elementary transvection matrices. Therefore these matrices generate the special linear group over a field.

// Target: algebra.bigb

A group of exponent two is abelian. Give such a finite group the additive notation of a vector space over ${\mathbb{F}}_2$, with zero acting as zero and one as the identity scalar. The group axioms and $g+g=0$ verify the vector-space laws. A basis with $d$ vectors then gives exactly $2^d$ elements. Thus a finite group whose elements all have order at most two must have power-of-two order.

The electroweak scale is the characteristic energy associated with electroweak symmetry breaking. With the Standard Model Higgs doublet normalized by $⟨H⟩=(0,v/\sqrt{2}{)}^T$, its vacuum expectation value satisfies $v=(\sqrt{2}{G}_F{)}^{-1/2}\simeq 246\mathrm{GeV}$ and $m_W=gv/2$. Its smallness compared with possible ultraviolet scales motivates the hierarchy problem.

Renormalization-group evolution can change a Higgs squared-mass parameter from positive at a high input scale to negative at lower scales, making a symmetry-breaking scalar expectation energetically favorable. In the Minimal supersymmetric Standard Model, the large top Yukawa coupling can drive the up-type Higgs soft squared mass downwards as the scale is lowered. Actual breaking also requires the other Higgs-potential parameters to give a stable suitable minimum; the running of one mass alone is not sufficient.

For completely degenerate, nonrelativistic electrons in fully ionized hydrogen and helium with hydrogen mass fraction $X$, $n_e=(1+X)\rho /(2m_p)$. Filling the spin-degenerate momentum states to $p_F$ gives $n_e=8\pi p_F^3/(3h^3)$ and $P_e=(1/3){\int }_0^{p_F}(p^2/m_e)(8\pi p^2/h^3)dp=8\pi p_F^5/(15m_e{h}^3)$. Eliminating $p_F$ gives $P_e=K{\rho }^{5/3}$, whereIt is a polytropic index $3/2$ relation. Complete degeneracy suppresses thermal dependence, while the nonrelativistic assumption requires $p_F\ll m_{e}c$.

In a symmetry-plane cross-section, write $\mathbf{p}=r(\theta )\mathbf{n}(\theta )$, $v=1/r$, and $\mathbf{t}=\mathbf{n}\prime$. The elastic wave surface is $\mathbf{g}=v\mathbf{n}+v\prime \mathbf{t}$. Its derivative vanishes when $v+v\prime \prime =(r^2+2r{\prime }^2-rr\prime \prime )/r^3=0$, exactly an inflection point of the regular slowness surface. At a simple zero the second and third derivatives of $\mathbf{g}$ are linearly independent, giving an ordinary cusp. Smooth nonconvex slowness surface sheets can therefore generate cusped wavefronts.

A solid-to-fluid interface can turn spherical source P waves into parallel vertical rays if its height $Z(R)$ satisfies the displayed travel-time identity, with $r=\sqrt{R^2+Z^2}$. The positive branch through $Z(0)=z_0$ is a hyperboloid when $\alpha >{\alpha }_f$. Differentiating the travel-time identity enforces Snell law for elastic and acoustic waves. Travel time above the interface is then $z_0/\alpha +(z-z_0)/{\alpha }_f$, independent of cylindrical radius. The leading wavefront is planar although its wave amplitude need not be uniform across that plane.

For an isotropic radial wave speed $\alpha (r)$, the elastic ray equations imply conservation of $\mathbf{x}\times \mathbf{y}$, where $\mathbf{y}$ is the slowness vector. Indeed, its derivative is $\alpha \mathbf{y}\times \mathbf{y}+\mathbf{x}\times \nabla (1/\alpha )=0$. Thus each nonradial elastic ray lies in a fixed plane through the centre and $p=|\mathbf{x}\times \mathbf{y}|$ is constant. The angle $i$ is measured from the radial direction; tangential slowness continuity preserves $p$ across spherical interfaces.

For a mode $u=a{e}^{i\omega (p_{\alpha }{x}_{\alpha }+q{x}_3-t)}$ with $x_3$ positive downwards, the reduced traction is $(Tq+R)a$. Its mean vertical acoustic energy flux is the displayed expression. Propagating modes are downgoing for positive flux and upgoing for negative flux; phase-slowness sign need not agree in anisotropy. Depth-decaying evanescent modes have $\mathrm{Im}q>0$. At grazing or repeated roots use limiting radiation modes.

For nonnegative strictly increasing link delays and nonempty routes, let $M=B(0)$ and $L={\lim }_{\lambda \to \infty }B(\lambda )$. The inverse demand tends to infinity as flow decreases to $L$, so increasing flow slightly there gains more utility than the bounded link-cost increment. At $M$ the inverse demand is zero, whereas a used route has positive delay, so reducing flow improves the objective. A minimizer consequently has $L<f<M$ and satisfies the ordinary interior Wardrop equilibrium conditions. Lower-semicontinuous endpoint extensions permit minimization on the compact demand interval even when utility diverges at $L$.

The weight-sixteen cusp form space is spanned by $E_4\Delta$. Its leading coefficient and the Fourier expansion of a normalized Eisenstein series give $3617E_{16}=3617E_4^4-3456000E_4\Delta$. The coefficient multiplier $-3456000$ is congruent to $16320$ modulo $3617$, and is invertible there. Coefficient comparison gives the displayed modular congruence. Integrality of $E_4\Delta$ follows from the modular discriminant product.

Every eigenvalue of this nonnormal matrix is $-1$, but $e^{t{L}_h}=e^{-t}\left(\begin{array}{cc}1& t/h\\ 0& 1\end{array}\right)$. Applied to the second coordinate unit vector, its norm at $t=1$ is at least $1/(eh)$. Thus negative eigenvalues give no mesh-uniform semigroup bound by themselves. Normal matrices, uniformly conditioned eigenvector bases, or a direct dissipative energy estimate supply the missing control.

Stability of the partial differential equation requires a bound uniform over refining meshes and time steps on every fixed physical time interval. This is distinct from demanding an all-time bound for one fixed matrix. A fixed matrix is power bounded exactly when all eigenvalues are in the closed unit disk and its unit-modulus Jordan blocks have size one. A time-step family can nevertheless have harmless polynomial physical-time growth: $G_{\tau }=I+\tau N$ with $N^2=0$ gives $G_{\tau }^n=I+n\tau N$, uniformly bounded for $n\tau \le T$ despite its nontrivial unit-modulus Jordan block. Mesh dependence and the scaling of the step must therefore be kept in an eigenvalue argument.

Suppose the Bayesian deviance is locally $D(\theta )=D(\hat{\theta })+(\theta -\hat{\theta }{)}^{T}J(\theta -\hat{\theta })$ with positive-definite $J$, and the posterior is normal with mean $m$ and covariance $\Sigma$. Writing $b=m-\hat{\theta }$, expectation and variance of the quadratic form giveConsequently the usual effective parameter count in DIC is $p_D=\mathrm{tr}(J\Sigma )$, whereas half the deviance variance is $\mathrm{tr}\{(J\Sigma {)}^2\}+2b^{T}J\Sigma Jb$. A locally flat prior gives $m=\hat{\theta }$ and $\Sigma =J^{-1}$, so both counts equal the parameter dimension. Informative priors generally destroy their equality. The expectation formula requires only the first two moments; normality is used for the variance formula.

If the galactic distribution function is not restricted to positive relative energy, isotropic velocity integration gives $\rho (\psi )=4\pi \sqrt{2}{\int }_{-\infty }^{\psi }f(E)\sqrt{\psi -E}dE$. The same Abel transform argument as Eddington inversion yields the displayed formula, with lower limit $-\infty$. For $\rho ={\rho }_0{e}^{\psi /{\sigma }_0^2}$, substitution $\psi =E-{\sigma }_0^2{s}^2$ gives $f(E)={\rho }_0(2\pi {\sigma }_0^2{)}^{-3/2}{e}^{E/{\sigma }_0^2}$. An escape-energy truncation changes both the density law and the velocity tails.

Algebraic alpha quenching prescribes a field-dependent alpha effect as a closure for nonlinear mean-field dynamo feedback. The polynomial law $\alpha ={\alpha }_0-p|B{|}^2$ with $p>0$ decreases alpha, so it can saturate positive alpha but makes negative alpha more negative. It therefore differs importantly from the sign-preserving rational law ${\alpha }_0/(1+|B{|}^2/B_{\ast }^2)$. Its solutions must satisfy nonnegative squared-amplitude conditions; an assertion based only on large $|{\alpha }_0|$ need not hold for both signs.

Define a finite colouring of the positive integers by $c(n)=\lfloor {\log }_{2}n\rfloor (\mathrm{mod}2)$. No increasing infinite sequence has all the numbers $x_i+2x_j$, $i\ne j$, in one colour class.

Write $\delta (y)=2^{\lfloor {\log }_{2}y\rfloor +1}-y$. If $\delta$ is unbounded on the sequence, fix a term $x$ and choose a later term $y>2x$ with $\delta (y)>2x$. With $L=2^{\lfloor {\log }_{2}y\rfloor }$, the numbers $y+2x$ and $2y+x$ lie respectively in the adjacent dyadic intervals $[L,2L)$ and $[2L,4L)$, so their colours differ. If $\delta$ is bounded by $D$ on the sequence, choose $x>2D$ and then $y>2x$. Now $y+2x=2L-\delta (y)+2x$ lies in $[2L,4L)$, whereas $2y+x=4L-2\delta (y)+x$ lies in $[4L,8L)$. Their colours again differ. The finite colouring therefore excludes the symmetric weighted-pair configuration.

The Conditional Fatou lemma gives $X_n\ge \mathbb{E}[X_{\infty }\mid {\mathcal{F}}_n]$ for the dyadic density martingale. Integration first on dyadic sets, then extension by the Monotone class theorem, makes $\nu =\mu -X_{\infty }\lambda$ a nonnegative measure. Its finite-level density is $Y_n=X_n-\mathbb{E}[X_{\infty }\mid {\mathcal{F}}_n]$, which tends to zero almost everywhere for Lebesgue measure. The density level sets separate a singular measure argument shows that this zero-limit set has $\nu$-measure zero. Thus $\nu$ and $\lambda$ are mutually singular measures, obtaining the decomposition without assuming it to identify the martingale limit.

The Einstein model of a solid represents the vibrations of $N$ chemical atoms by $3N$ independent one-dimensional quantum harmonic oscillators, all with the same positive angular frequency $\omega$. Equivalently, each chemical atom has three independent oscillator directions. The common angular frequency is a simplifying model assumption, not a claim about every vibration of a real solid. Hold the volume, $N$ and $\omega$ fixed. Define the characteristic temperature ${\theta }_E=\hslash \omega /k_B$ and the dimensionless ratio $x={\theta }_E/T$. The vibrational internal energy isThe first term is the zero-point energy $3N\hslash \omega /2$. It is constant at fixed $\omega$, so it contributes no heat capacity at constant volume. The temperature-dependent term givesThis is the vibrational heat capacity at constant volume of the model, with no expansion or other contributions included. ^[ref]

For a worked comparison, choose the model parameter ${\theta }_E=300\mathrm{K}$. The normalized heat capacity at constant volume depends only on $T/{\theta }_E$:

At $T={\theta }_E$, one mole of chemical atoms therefore has vibrational heat capacity at constant volume $3R(0.92067)\simeq 22.965\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, using the unrounded model value. Even at twice the characteristic temperature, the model remains about $2.06\%$ below the Dulong-Petit law limit. These are calculations for the ideal Einstein solid, not material measurements.

A short derivation uses the thermal partition function of a quantum harmonic oscillator. The energy eigenvalues $E_n=\hslash \omega (n+1/2)$ and their Boltzmann factors form a geometric series, giving the one-oscillator canonical partition function $z=e^{-x/2}/(1-e^{-x})$. The mean occupation is $1/(e^x-1)$, hence the displayed $U$ after summing over $3N$ independent quantum harmonic oscillators. Taking its derivative with $dx/dT=-x/T$ gives the displayed heat capacity at constant volume. ^[ref]

The two limits explain the shape. At high temperature, a Taylor series in small $x$ givesAt low temperature, $1-e^{-x}\to 1$, soThe zero-point energy remains even when the thermal response is tiny: freeze-out means that changing temperature scarcely changes the occupations, not that the chemical atoms have zero energy. This is the useful lesson of the Einstein model of a solid: the classical counting rule reappears when many levels are accessible, while discrete excitation spacings explain its failure upon cooling.

The exponential low-temperature decay is also the model's limitation. Real three-dimensional crystalline solids have acoustic phonon branches with arbitrarily small angular frequencies in the bulk limit. The Debye model includes these collective normal modes and predicts a low-temperature vibrational contribution $C_V\propto T^3$, which the single-frequency Einstein solid cannot reproduce. This qualification concerns the vibrational contribution; additional excitations can change a material's total heat capacity. ^[ref]

Einstein introduced the quantum treatment of solid vibrations in his 1907 paper Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme. Its historical importance is that quantum excitation energies explained why the classical Dulong-Petit law could fail; the modern quantum harmonic oscillator formulation above includes the zero-point energy explicitly. ^[ref]

For a closed polyhedral cone $C$, the maps $\tau \mapsto C^{\vee }\cap {\tau }^{\perp }$ and $\nu \mapsto C\cap {\nu }^{\perp }$ give inverse order-reversing bijections on faces. Choose a exposing functional in the relative interior of the dual face. The bipolar identity and an exposing functional for the original face show that applying the maps twice returns that face.

// Target: geometry-and-topology.bigb