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The one-dimensional affine group over ${\mathbb{F}}_q$ consists of maps $x\mapsto ax+b$ with $a\ne 0$.

The Sierpinski triangle is formed recursively by retaining the three corner subtriangles after subdividing an equilateral triangle into four congruent pieces.

If $L(y_0;{\lambda }_0)=0$ and $y_0,y$ satisfy homogeneous separated boundary conditions, the Lagrange identity givesThus $L(y;{\lambda }_0)=f$ can be solved only if $f$ is orthogonal to the null mode $y_0$.

A second-order scalar differential equation is in self-adjoint form when it can be writtenMultiplication by $y$ and integration by parts then gives an energy identity.

For a closed homogeneous epidemic, integrating $dS/dR$ relates the ultimately susceptible population to cumulative infections through an exponential equation.

The relation constrains the final outcome of compartmental models rather than describing their whole time-dependent solution.

The force of infection is the instantaneous infection hazard experienced by a susceptible individual. Under homogeneous frequency-dependent mixing it is $\beta I(t)/N$.

For $f(0)=f(T)=0$,with equality for a multiple of $\sin (\pi t/T)$.

A conjugate point marks a nonzero endpoint-vanishing Jacobi variation and the loss of positive definiteness of the second variation.

For the energy functional of a Riemannian manifold, this variational condition yields conjugate points along a geodesic.

The Jacobi equation is the linearization of an Euler-Lagrange equation about a stationary path. Its solutions describe infinitesimal one-parameter families of stationary paths.

For $x=(u,t)$ and $\rho =\Vert u{\Vert }_2$,

No cyclic device can have as its sole effect the extraction of heat from one thermal reservoir and its complete conversion into net work. A putative device absorbing $Q_h>0$ and returning to its starting state with $W=Q_h$ and no other transfer would conserve energy, yet violate the Second law of thermodynamics. Rejecting heat to colder surroundings is an essential part of the ordinary two-thermal reservoir heat engine budget.^[ref]

The cyclic and sole-effect qualifications matter. A single isothermal expansion of an ideal gas can absorb heat and deliver equal work while leaving its volume changed; restoring the initial state must also be counted. Likewise, a device discharging stored internal energy or consuming fuel has another effect. Converting supplied electrical work completely to heat through Joule heating reverses the proposed conversion and is allowed. The statement also allows cyclic devices that exchange heat with one thermal reservoir while consuming net work; it forbids positive net work output with that single heat source as the only external change.

Conversely, a hypothetical complete-conversion device could take $100\mathrm{J}$ from the hot thermal reservoir and supply $100\mathrm{J}$ of work to an ideal refrigerator that extracts $100\mathrm{J}$ from the cold thermal reservoir and delivers $200\mathrm{J}$ to the hot one. At $600\mathrm{K}$ and $300\mathrm{K}$, that refrigerator is itself admissible: its reservoir entropy changes cancel. The composite would consume no external work and move a net $100\mathrm{J}$ from cold to hot, violating the Clausius statement of the second law. Together with the construction there, this proves the equivalence of the two impossibility statements.^[ref]

For the two fixed positive-temperature thermal reservoir hypotheses, Carnot's theorem gives the sharper restriction $\eta \le 1-T_c/T_h<1$. A reversible thermodynamic process reaches that bound in the ideal Carnot cycle; an ordinary device's smaller thermal efficiency reflects its entropy production and measurement boundary. An inefficient observed engine is compatible with the Second law of thermodynamics, rather than evidence that the bound has been experimentally determined.

No cyclic device can have as its sole effect the transfer of heat from a colder thermal reservoir to a hotter thermal reservoir. The word “sole” includes the surroundings: the device must return to its starting state, and there must be no net supplied work, depletion of a stored resource, or accompanying change elsewhere. The statement excludes a self-running refrigerator, while allowing an ordinary refrigerator or heat pump supplied with work.^[ref]

For a proposed transfer $Q>0$ from a fixed cold thermal reservoir at $T_c$ to a fixed hot thermal reservoir at $T_h>T_c>0$, with no other effect, the total entropy change would beThe first law of thermodynamics permits this energy balance; the Second law of thermodynamics forbids the negative total entropy change. For $Q=600\mathrm{J}$, $T_c=300\mathrm{K}$ and $T_h=600\mathrm{K}$, it is $-1\mathrm{J}{\mathrm{K}}^{-1}$. Direct heat transfer in the opposite direction has positive total entropy change. A powered heat pump has a different budget: it supplies work and delivers more heat to the hot side than it extracts from the cold side.

A short equivalence argument links this restriction to the Kelvin-Planck statement of the second law. Couple a hypothetical device that moves $600\mathrm{J}$ from cold to hot with no other effect to the heat engine illustrated above, which absorbs $1000\mathrm{J}$, rejects $600\mathrm{J}$, and delivers $400\mathrm{J}$. The cold-side transfers cancel; the combined cyclic device takes $400\mathrm{J}$ from the hot thermal reservoir and delivers exactly $400\mathrm{J}$ of work, violating the Kelvin-Planck statement of the second law. Matching transfers is an ideal composition argument, not a claim that such a device exists.^[ref]

The Chern-Simons 3-formsatisfies $dY=C_2$.

The Fisher information is $I_n(\theta )=\mathbb{E}[S_n{S}_n^T]$ and equals $n{I}_1(\theta )$ for an independent identically distributed sample.

Under regularity permitting differentiation under the integral, ${\mathbb{E}}_{\theta }{S}_1(\theta )={\nabla }_{\theta }\int f(x,\theta )dx=0$.

The advanced coordinate $v=t+r_{\ast }$ puts the radial Schwarzschild metric into the formwhich is regular at the future horizon.

These are the advanced, ingoing member of the Eddington-Finkelstein coordinates.

Writing $y=b^{-1}\sin \varphi +\Delta y$ in $y^{\prime \prime }+y=3M{y}^2$ and retaining first order in $M$ givesThe displacement of the outgoing zero is $4M/b$.

A tempered distribution is a continuous linear functional on the Schwartz space. Tempered distributions include functions of at most polynomial growth and admit a Fourier transform by duality.

If a finite group has a faithful irreducible complex representation, every central element acts as a scalar by Schur lemma. Faithfulness embeds the centre into ${\mathbb{C}}^{\times }$, and every finite subgroup of ${\mathbb{C}}^{\times }$ is cyclic. Hence the group centre must be cyclic.

The kernel and image of an intertwiner are invariant subspaces. Irreducibility therefore makes a nonzero intertwiner injective and surjective. For an endomorphism $T$ of a finite-dimensional irreducible complex representation, choose an eigenvalue $\lambda$. The noninvertible intertwiner $T-\lambda I$ must be zero, so $T=\lambda I$.