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A sequence assigns a value to each position in an ordered index list. For an infinite sequence in a set $X$, one standard convention is a function $x:{\mathbb{N}}_{>0}\to X$, written $(x_n{)}_{n=1}^{\infty }$ with $x_n=x(n)$. Some applications start the index at zero, and a finite sequence has only finitely many positions. Values may repeat; both order and repetition matter. ^[ref].

For example, $a_n=(-1{)}^n$ begins $-1,1,-1,1,\dots$, whereas $b_n=(-1{)}^{n+1}$ begins $1,-1,1,-1,\dots$. Their sets of values are both $\{-1,1\}$, but these are different sequences. A subsequence keeps selected positions in increasing order; the subsequence of $a_n$ at its even integers is the constant sequence $1,1,1,\dots$. A tail of a sequence retains all positions from one index onward.

Sequences express successive approximations, repeated observations, and the iterates of a recurrence relation. A limit of a sequence describes where its terms eventually concentrate. Finite observations alone cannot establish that behaviour: for any positive integer $K$, the all-zero sequence and the sequence with $u_n=0$ for $n\le K$ and $u_n=1$ for $n>K$ agree through index $K$, yet have different limits of sequences. A formula or bound controlling the entire future, such as the tail diameter of a sequence in the Cauchy sequence condition, supplies information that a finite list cannot.

For $0<a<b$ and an integer $n\ge 1$, the harmonic function on $a<r<b$ with $u(a,\theta )=0$ and $u(b,\theta )=\cos (n\theta )$ isSeparation of variables gives the radial combination $A{r}^n+B{r}^{-n}$; the two boundary values determine $A,B$. Substitution at both radii checks the normalization. The zero mode must be handled separately: for outer value one it is $u(r)=\log (r/a)/\log (b/a)$. The maximum principle supplies uniqueness among harmonic functions continuous on the closed annulus.

A channel is entanglement breaking when applying it to one side of every bipartite state always produces a separable state. Equivalently, its Choi matrix is separable, and equivalently it has a measure-and-prepare representation.

Every separable state has positive partial transpose. The condition is also sufficient for separability on $2\otimes 2$ and $2\otimes 3$ systems.

An algebraic element $x$ over $K$ is separable when its minimal polynomial over $K$ has no repeated root in a splitting field. For an irreducible polynomial $m_x$, this is equivalent to $m_x^{\prime }\ne 0$, because then $\gcd (m_x,m_x^{\prime })=1$.

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 an absolutely continuous function $f$ on $[0,T]$ with square-integrable derivative, $T>0$, and $f(0)=f(T)=0$, the endpoint form of Wirtinger's inequality isExpanding in the sine eigenfunctions of the Dirichlet Laplacian proves the bound: every squared frequency is at least $(\pi /T{)}^2$. Equality holds precisely for a multiple of $\sin (\pi t/T)$. This identifies the optimal constant, rather than merely supplying some bound. Without the endpoint condition, a nonzero constant function would violate the inequality.

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 along a stationary path. For a smooth scalar integral $I[u]=\int F(x,u,u\prime )dx$, its fixed-endpoint second variation has coefficients $P=F_{pp}$, $B=F_{up}$ and $C=F_{uu}$ evaluated along the path:The boundary term vanishes because $\eta$ vanishes at the endpoints. Linearizing the stationarity equation givesThe same equation is the stationarity equation of this quadratic form. If $P>0$ continuously on the closed interval, it is a regular second-order equation.

For $F=(p^2-\lambda u^2)/2$, it becomes $\eta \prime \prime +\lambda \eta =0$. When $\lambda >0$, the solution vanishing at zero is a multiple of $\sin (\sqrt{\lambda }x)$; its next zero at $\pi /\sqrt{\lambda }$ gives a conjugate point. More generally, solutions describe infinitesimal families of stationary paths, and their zeros enter the Jacobi positivity criterion. This is a linearized equation around a specified extremal, not the original nonlinear boundary problem.

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]

For a local matrix-valued connection one-form $A$, putIts exterior derivative isThus the Chern-Simons 3-form is a local primitive of the trace-square curvature form. For traceless unitary connection curvature, this equals the Second Chern form in the stated convention; a general unitary principal bundle needs the additional trace-product correction. This is an instance of transgression from a characteristic four-form to a three-form.^[ref]

The expression uses a chosen local frame and is not itself a globally gauge-invariant three-form. Its precise change under gauge transformations is recorded in gauge change of the Chern-Simons three-form.

For a differentiable statistical model with probability density function $f_{\theta }$, the score function of one observation is $s_{\theta }(X)={\nabla }_{\theta }\log f_{\theta }(X)$. Its outer-product Fisher information matrix iswhen the entries are integrable. It is a positive semidefinite matrix, since $a^T{I}_{1}a={\mathbb{E}}_{\theta }(a^T{s}_{\theta }{)}^2\ge 0$ for every vector $a$.

The usual information identities require regularity. For common probability support independent of $\theta$, differentiating $\int f_{\theta }=1$ under the integral gives ${\mathbb{E}}_{\theta }{s}_{\theta }=0$. For an independent and identically distributed sample, $S_n={\sum }_i{s}_{\theta }(X_i)$ then has vanishing cross terms, soIf second derivatives may also pass under the integral, differentiation once more gives $I_1(\theta )=-{\mathbb{E}}_{\theta }[{\nabla }_{\theta }^2\log f_{\theta }(X)]$. These conditions make the outer-product and expected negative-Hessian matrix definitions agree.

Parameter-dependent probability support can break these identities. For a uniform distribution $X\sim \mathrm{Uniform}(0,\theta )$, the interior log-density derivative is $-1/\theta$, whose expectation is not zero. The formal sum of these interior scores has $\mathbb{E}{S}_n^2=n^2/{\theta }^2$, rather than $n/{\theta }^2$. The moving endpoint contributes to differentiating the normalization; discarding it does not produce a regular score function. Thus an assertion of information additivity must include its regular mean-zero score function assumptions.

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