For $dy={\sum }_i{A}^{i}yd{B}^i$ with constant linear maps, let $K={\sum }_i\Vert A^i{\Vert }_{\mathrm{op}}^2$ and let $D_m$ be consecutive Picard iterate differences starting from $y^{(0)}=a$. The vector Itô isometry gives $\mathbb{E}|D_m(t){|}^2\le K{\int }_0^t\mathbb{E}|D_{m-1}(s){|}^{2}ds$, proving the displayed estimate by induction. The square roots of these factorial bounds are summable on bounded time intervals. The Doob L2 maximal inequality then makes ${\sum }_m\mathbb{E}{\sup }_{t\le T}|D_m(t)|$ finite. Tonelli's theorem proves absolute uniform almost-sure convergence of the iterations, in addition to their $L^2$ convergence.

An irreducible Markov chain on a finite or countably infinite state space has at most one stationary distribution. On a countably infinite state space it exists exactly when the chain is positive recurrent. In that case, for every state $i$,where $T_i^{+}$ is the first return time. The expression determines every component uniquely. The formula follows by dividing the one visit to $i$ in each return cycle by the mean recurrence time; the strong law of large numbers for successive cycles makes this occupation fraction precise.

Every irreducible finite Markov chain is positive recurrent and has exactly one stationary distribution. On an infinite state space, an irreducible null recurrent Markov chain or transient Markov chain has none. For example, the simple symmetric random walk on the integers is null recurrent. Aperiodicity affects convergence of the one-time laws, but is unnecessary for existence or uniqueness of a stationary distribution.

On states $0,\dots ,m$, take an interior step of $+1$ or $-1$ with probability $p$ each and hold with probability $1-2p$. At an endpoint take the inward step with probability $2p$ and hold with probability $1-2p$. For $m\ge 1$ and $0<p\le 1/2$, the chain is irreducible and its stationary distribution has endpoint weights $1/(2m)$ and interior weights $1/m$.

Detailed balance proves this: the doubled inward endpoint transition requires its stationary mass to be half that of an adjacent interior point, and all interior points have equal mass. The total unnormalized weight is $1+2(m-1)+1=2m$. For $m=1$ there are just two equal endpoint masses.

When $p<1/2$, positive self-loops make the chain aperiodic, so its transition probabilities converge to this law. At $p=1/2$ every move changes parity and two-step returns are possible, giving period two and no such convergence from a fixed starting state. At $p=0$ every state is absorbing and uniqueness of the stationary law is lost.

Place $N\ge 2$ books in a row and distinguish one of them. At each step select the distinguished book with Probability $p\in (0,1)$ and each of the other $N-1$ books with Probability $q=(1-p)/(N-1)$. Swap the selected book with its left neighbour; selecting the leftmost book leaves the ordering unchanged.

The distinguished book's position $I\in \{1,\dots ,N\}$ is a birth-death chain. From an interior position it moves left with Probability $p$, moves right with Probability $q$ when the book immediately to its right is selected, and otherwise holds. Missing outward moves at the endpoints become holding. Detailed balance therefore gives ${\pi }_{i}p={\pi }_{i-1}q$ for $i\ge 2$. With $r=q/p$,This finite Truncated geometric distribution is valid even when $r>1$: both numerator and denominator then have the same negative sign. The uniform case is $p=q=1/N$, when every book is selected equally often. Large selection Probability for the distinguished book biases it toward the left end. The strict range of $p$ ensures positive motion in both directions and avoids division by zero in the ratio.

Represent a finite sparse vector by a sorted doubly linked list of its nonzero coordinates. A list header stores the dimension; every ordinary linked list node stores a distinct valid coordinate index, its nonzero value, and predecessor and successor references. In a circular linked list, the endpoints connect to the header, and an all-zero vector has the header linked to itself in both directions. The representation invariant requires strictly increasing ordinary indices, valid bounds and mutually consistent forward and backward links. The abstraction function assigns zero to every coordinate absent from the list. The header is metadata, never an ordinary coordinate.

For example, $(0,3,0,0,-2,0,0,5,0)$ has dimension nine and only three stored coordinates. The header's index field can store nine even though nine is not a valid coordinate index. Keeping the header distinct is therefore essential when searching for or removing entries.

An open fireplace heats nearby surroundings through thermal radiation from the flame and hot surfaces, while convection carries hot combustion gases upward. A rising gas column can create a pressure difference that draws replacement air from the room and sends combustion products through a chimney. The visible fire and the warm radiant field therefore represent only part of the complete energy balance.

For a chosen room boundary, useful net heating depends on chemical energy released, outward enthalpy flow in the exhaust, warming of the structure and the energy needed to bring replacement air to room temperature. These terms explain why the rate of fuel consumption alone does not determine the room's net heating rate. An open fireplace is a useful thermal-system example because radiative heating of a nearby surface can coexist with substantial energy transport out of the same room through moving gas.

Hot-air heating transports thermal energy with moving air and transfers it to a region as that air mixes and cools. A fan can establish a controlled flow; buoyancy can also drive natural circulation when warm air has lower mass density than surrounding air at similar pressure.

For a steady stream with approximately constant composition and specific heat capacity, its sensible-heating contribution is $\dot{Q}\simeq \dot{m}{c}_p(T_{\mathrm{supply}}-T_{\mathrm{return}})$. This is an enthalpy transport estimate, so both the flow rate and the temperature difference matter. Warmer supply air does not determine delivered power without knowing how much air moves. Spatial temperature gradients and circulation paths can make a single room thermometer an incomplete description. Comparing measurements at several heights tests whether a lumped thermal model captures the actual heating process.

Hot-water heating carries thermal energy in circulating liquid water. The water gives up sensible heat in a radiator or pipe network and returns at a lower temperature. Under approximately constant pressure and specific heat capacity, the transferred power iswhere $\dot{m}$ is the mass flow rate.

The radiator's thermal conduction and its exchange with the room through convection and thermal radiation determine how large a temperature difference is needed. A circulation pump supplies mechanical work against fluid resistance, while the main heating source replenishes the lost enthalpy. Measuring inlet and outlet temperatures together with mass flow estimates the delivered thermal power. The circulating water can remain entirely within the liquid phase, making sensible cooling rather than a phase transition the principal transport budget.

Steam heating transports thermal energy using water vapor, commonly allowing it to condense in a heat-emitting component. At an approximately fixed local pressure, condensation releases latent heat while the phase-change temperature remains nearly fixed. The condensate can then return to the source for reheating.

For mass flow rate $\dot{m}$ through a steady emitter, the ideal transferred power is $\dot{Q}=\dot{m}(h_{\mathrm{in}}-h_{\mathrm{out}})$, where $h$ is specific enthalpy. That difference can include both latent heat from condensation and sensible heat from cooling the condensate. The emitter transfers this energy to its surroundings through convection and thermal radiation. Measuring only the condensate temperature change would miss the latent contribution. The mechanism illustrates why a large heat transfer need not be accompanied by a large change in the working fluid's temperature.

An arithmetic theory is one-consistent if it does not prove $\exists x\delta (x)$ while proving every $\neg \delta (\overline{n})$, for a bounded arithmetic formula $\delta$ and all standard natural numbers $n$. If the theory proves all true bounded sentences, this is equivalent to Sigma-1 soundness: a false existential theorem has every standard instance refuted by bounded completeness; conversely a true witness and its provable negation contradict the assumption of a consistent first-order theory.

A theory has Sigma-1 soundness if every Sigma-1 formula that is a closed theorem is true in the standard natural-number structure. It is weaker than full arithmetic soundness. Under bounded completeness it is equivalent to one-consistency. The standard interpretation matters: an existential statement can have a witness in a nonstandard model while having no standard witness.

Pi-1 soundness means that every closed Pi-1 theorem is true in the standard natural numbers. Consistency plus bounded completeness implies it, since a false universal bounded statement has a standard counterexample whose arithmetic verification is provable. It is weaker than full soundness and does not prevent a consistent arithmetic theory from proving a false Sigma-1 sentence.

A first-order theory is Sigma-0 sound when every arithmetic bounded arithmetic sentence it proves is true in the standard natural numbers. For a first-order theory proving every true bounded arithmetic sentence, this is equivalent to first-order consistency: a false bounded theorem would have a provable bounded negation, and an inconsistent first-order theory proves the false bounded arithmetic sentence $0=1$.

A first-order theory is omega-consistent when it never proves $\exists x\varphi (x)$ while proving $\neg \varphi (\bar{n})$ for every standard natural number $n$. Restricting the matrix to arithmetic bounded arithmetic formulas gives one-consistency. This condition concerns standard instances and is stronger than first-order consistency.

Suppose $w$ is holomorphic off $[0,1]$, tends to zero at infinity, has sufficiently regular boundary values in the interval interior, and has endpoint growth $O(|z-a{|}^{-\alpha })$ for some $\alpha <1$ at each endpoint $a$. If its jump is integrable, the displayed reconstruction follows by deforming a Cauchy integral formula contour onto the two sides of the interval. An endpoint circle of radius $\epsilon$ contributes $O({\epsilon }^{1-\alpha })$, which vanishes. The two oppositely oriented banks leave precisely the jump. Without decay at infinity, an entire remainder can survive; a nonzero constant is the simplest example.

For $0<\alpha <1$ and $-1<x<1$, the endpoint powers are integrable and the interior pole is taken as a Cauchy principal value. The analytic function $w(z)=(z-1{)}^{\alpha -1}/(z+1{)}^{\alpha }$, positive for $z>1$ and decaying like $1/z$, has boundary values $-e^{\pm i\pi \alpha }(1-t{)}^{\alpha -1}/(1+t{)}^{\alpha }$. Its jump gives a Cauchy reconstruction, while the mean of its boundary values and the Sokhotski–Plemelj formula give the displayed identity. At $\alpha =1/2$ the principal value is zero.

On a positively oriented smooth simple closed contour, set $\mathcal{C}f=(\pi i{)}^{-1}\mathrm{PV}{\int }_{\Gamma }f(t)/(t-z)dt$. The Sokhotski–Plemelj theorem gives the sum and difference of the two Cauchy boundary values. Changing the sign of the exterior branch converts the equation $\mathcal{C}f=g$ into a normalized additive Riemann-Hilbert problem with jump $g$. Applying the boundary formula again yields $f=\mathcal{C}g$, proving the involution on a suitable smooth or Hölder function class.

For $u\in H_0^1(0,1)$, write $u(x)=(1-x){\int }_0^x{u}^{\prime }-x{\int }_x^1{u}^{\prime }$. The squared $L^2$ norm of this piecewise constant test kernel is $x(1-x)$, so the Cauchy-Schwarz inequality gives the displayed pointwise estimate. Integrating against a nonnegative weight $w$ gives $\int w|u{|}^2\le [\int wx(1-x)]\int |u^{\prime }{|}^2$. In particular $w=x^2$ gives the constant $1/20$, useful for controlling a negative quadratic potential by derivative energy. The estimate is sharp at each fixed evaluation point, whose equality profile is a tent function, although its integrated constant need not be sharp.

For a positive integer-valued random variable with finite mean $\mu$, the size-biased distribution has probabilities $k\mathbb{P}(Y=k)/\mu$. They sum to one by the definition of expected value. Multiplying the differentiated probability generating function by $s/\mu$ gives the displayed identity for $0\le s<1$, with the endpoint interpreted by its limit.

For the positive-support geometric distribution with success probability $p$, size bias gives $\mathbb{P}(\tilde{Y}=k)=k{p}^2(1-p{)}^{k-1}$. The sum of two independent copies has $k$ positive-integer decompositions at value $k+1$, so its shifted probability is exactly this expression. Its probability generating function is $p^{2}s/(1-(1-p)s{)}^2$, also obtained by the size-biased probability generating function identity.