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For a holomorphic function on $x>0$, $0<y<l$, define boundary spectra by counterclockwise traversal: ${\rho }_L={\int }_{il}^0{e}^{-ikw}f(w)dw$, ${\rho }_B={\int }_0^{\infty }{e}^{-ikx}f(x)dx$, and ${\rho }_T=-e^{kl}{\int }_0^{\infty }{e}^{-ikx}f(x+il)dx$. The displayed reconstruction follows from the Cauchy integral formula and the ray identity ${\int }_0^{e^{i\theta }\infty }{e}^{ik(z-w)}dk=i/(z-w)$ whenever that exponential decays. The three rays are selected by the inward normals of the three sides.

// Target: analysis.bigb

For $q_t+q_x-q_{xxx}=0$ on $x>0$, its local relation has flux $(1+k^2)q-ik{q}_x-q_{xx}$. With $Q_0(k)={\int }_0^{\infty }{e}^{-ikx}q(x,0)dx$ and $G_j(k,T)={\int }_0^T{e}^{w(k)s}{\partial }_x^{j}q(0,s)ds$, its global relation is $e^{wT}\hat{q}=Q_0+(1+k^2)G_0-ik{G}_1-G_2$. In $D_{+}=\{\mathrm{Im}k>0,\mathrm{Re}w(k)<0\}$, choose the other cubic root $\nu (k)$ in the lower half-plane. Then the known spectral density is $-Q_0(\nu )+(k^2-{\nu }^2)G_0-i(k-\nu )G_1$. The remaining final-state transform has zero integral around $\partial D_{+}$ by the Cauchy integral theorem and exponential decay. The full boundary includes the real axis and the hyperbola $\mathrm{Im}k=\sqrt{1+3(\mathrm{Re}k{)}^2}$, oriented with the domain on the left.

Differential circulation of a weak magnetic field winds its flux function into progressively smaller radial scales. In a smooth eddy with angular-velocity gradient of order $U/L^2$, its radial wavenumber grows as $k_r\sim Ut/L^2$. Resistive damping becomes substantial when $\eta {\int }_0^t{k}_r^{2}dt\sim 1$, giving $t\sim (L^4/(\eta U^2){)}^{1/3}=(L/U){\mathrm{Rm}}^{1/3}$. This estimate presumes differential rotation and negligible magnetic feedback; a rigidly rotating eddy does not create this phase mixing. Finite magnetic diffusivity then permits weak interior field and boundary concentration, unlike exact ideal flux freezing.

For identical unit-density, unit-sound-speed fluids above and below a sheet on $x<0$, use time $e^{i\omega t}$ and transform $\int e^{ikx}\varphi dx$. The outgoing acoustic square-root branch obeys ${\gamma }^2=k^2-{\omega }^2$. If $J$ is the pressure jump upper minus lower, $s={\eta }^{\prime }(0)$ and $d^{+}$ the right Half-range Fourier transform of the scattered normal derivative, the Wiener-Hopf equation isThe total displacement is pinned, so its transformed second derivative is $s-k^2{\eta }^{-}$. The bare-sheet pole $b=\omega \sqrt{m/T}$ in reconstructed displacement must cancel: $J(b)=Ts$. This is an incoming-pole cancellation at a pinned membrane edge condition, distinct from the outgoing coupled-mode pole at the zero of $K$.

The 2002 experiment trapped a $6.3\mu \mathrm{m}$ bead in water with a $980\mathrm{nm}$ laser, a $100\times$, numerical aperture $1.3$ objective and a piezo-driven microscope stage. A quadrant photodiode tracked bead position at $1\mathrm{kHz}$ with approximately $500\mathrm{Hz}$ bandwidth and reported $15\mathrm{nm}$ resolution. Stage translation drove water past the fixed optical trap at $1.25\mu m/s$. Weak trapping prolonged the transient; stationary recording before translation established the position reference. Stokes drag law calibration was checked against the bead's measured diffusion coefficient. ^[ref]

From calibrated displacement and calibrated harmonic optical trap force, the experimenters evaluated the dimensionless driven-bead dissipation functionalAmong 540 trajectories, negative values became less frequent with duration. Figure 2 compared the negative/positive count ratio with $⟨e^{-{\Sigma }_t^{\mathrm{drive}}}{⟩}_{{\Sigma }_t^{\mathrm{drive}}>0}$; its agreement tested the paper's integrated transient relation. The detailed log-ratio plot in Figure 4 used simulated particle sample trajectories, not bead measurements. Near-zero sign discrimination and stage-start transients limited the earliest points. ^[ref]

For any continuous probability distribution satisfying the same-law relation, integration over $s>0$ yieldsThis conditional relation explains the comparison without equating it to the unconditional integral fluctuation theorem. The experiment supports a specified driven-system relation; it does not establish a universal same-law symmetry for any definition of heat or entropy production. In particular, a transient work-like functional should not silently acquire the actual-distribution endpoint term in finite trajectory entropy production. Negative realizations are compatible with positive mean dissipation and the Second law of thermodynamics; they do not provide a cyclic machine with reliable work output from a single thermal reservoir.

For a system coupled to a thermal reservoir at fixed temperature $T$, prepare the forward process in the initial canonical ensemble and the reverse process in the final control parameter's canonical ensemble. Reverse the protocol, assume microscopic reversibility and matching path probability support, and count $W$ as work done on the system. With Helmholtz free energy difference $\Delta F=F_{\mathrm{final}}-F_{\mathrm{initial}}$ and $\beta =1/(k_{B}T)$,The final forward distribution need not already be in thermal equilibrium. Its reverse preparation is prescribed equilibrium preparation, rather than the actual forward final distribution used in finite trajectory entropy production. Thus the paired path logarithm $\beta (W-\Delta F)$ is dissipated work in thermal units, not automatically the instantaneous total stochastic entropy production at the end of an arbitrary driven protocol. ^{[ref] [ref]}

A steady state does not by itself give every fluctuating observable an entropy production symmetry. If the dimensionless total entropy production ${\Sigma }_t$ has the same-law finite relation and a large deviation principle with rate function $I$ for ${\Sigma }_t/t$, then, on its common domain,The first expression denotes leading exponential scaling; the second follows by taking the scaled logarithm of the probability ratio. A finite-state, autonomous Markov process with time-even states, positive bidirectional rates and a positive stationary distribution supplies a standard setting for a finite total-entropy production relation. The stochastic system entropy is then bounded, so its endpoint contribution does not change the long-time rate function of bath entropy production. ^[ref]

Unbounded state spaces can have endpoint fluctuations on the scale relevant to a large deviation principle. Dropping system entropy, changing the measured observable to heat, or changing the reverse preparation therefore requires a fresh argument. The exact finite integral fluctuation theorem, a same-law finite relation, and an asymptotic rate function symmetry are distinct claims.

Take $n$ independent steps, each $+1$ with probability $p>1/2$ and $-1$ with probability $q=1-p>0$. Reverse a step sequence by reading it backward and changing each sign; use the same biased law for both experiments. With $K$ positive steps, the forward sequence weight is $p^K{q}^{n-K}$ and its reversed weight is $p^{n-K}{q}^K$. Their logarithm ratio isHere $S_n$ is dimensionless entropy production, not an entropy in joules per kelvin. A Markov chain on a three-state ring gives a state-space realization: start uniformly, take a clockwise step with probability $p$ and a counterclockwise step with probability $q$. The uniform stationary distribution makes endpoint weights cancel. A physical interpretation as exchanged heat requires local detailed balance with bath entropy $\pm k_{B}a$ per directed step; the abstract model alone specifies path probabilities.

For an allowed value $s=a(2k-n)$, the binomial distribution givesThis is an exact same-law finite fluctuation theorem, with no long-time approximation. For $p=3/4$, $n=9$ and $k=6$, $s=3\ln 3$ and $p_n(s)/p_n(-s)=27$. The positive mean is $⟨S_n⟩=na(p-q)$, although the all-negative path has nonzero probability $q^n$. Directly, $⟨e^{-S_n}⟩=(p{e}^{-a}+q{e}^a{)}^n=(q+p{)}^n=1$.

The left panel shows exact probability masses for $p=3/4$, with negative values highlighted; the right panel tests the logarithm ratio against $s$. These are model values, without experimental measurements or simulated measurement noise.

Consider a finite-duration Markov process with discrete, time-even states coupled to one thermal reservoir at positive temperature $T$. Let $P_F[\gamma ]$ include the initial probability ${\rho }_0(x_0)$ of the sample trajectory $\gamma$. Define $P_R$ by reversing the control protocol and initializing it with the actual forward final probability distribution ${\rho }_{\tau }$. Write $\Theta \gamma$ for the same states visited in reverse time order. Require normalized path laws with matching probability support: a path has positive forward weight exactly when its reverse has positive reverse weight. This excludes irreversible forbidden transitions and missing initial states.

Under local detailed balance, with positive heat $Q_{\mathrm{bath}}$ delivered to the thermal reservoir, the full path-weight logarithm isThe endpoint term is the change in stochastic system entropy; the conditional path ratio alone accounts for the thermal reservoir. Here $k_B$ is the Boltzmann constant. Both terms matter even for a short observation. ^[ref]

Define the conjugate reverse functional by ${\Sigma }_R[\eta ]=\ln (P_R[\eta ]/P_F[\Theta \eta ])$, so ${\Sigma }_R[\Theta \gamma ]=-{\Sigma }_F[\gamma ]$. Summing the path identity over paths with a given value gives the exact finite fluctuation theoremfor nonzero denominator, using probability masses or corresponding probability density functions. This ${\Sigma }_R$ is defined by the paired laws: it need not be the total entropy production obtained by independently restarting the thermodynamic construction for the reverse experiment. A generic driven experiment therefore does not imply $p_F(s)/p_F(-s)=e^s$.

For a Floquet multiplier $m\ne 0$ of a period-$T$ linear variational equation, a Floquet exponent is any logarithmic rate $\rho$ satisfying the displayed relation. Its real part is uniquely $T^{-1}\log |m|$, while its imaginary part is defined modulo $2\pi /T$. Negative transverse real parts give local attraction to a periodic orbit; the autonomous phase direction has multiplier one and exponent zero. Constant-coefficient amplitude equations supply exponents directly as the eigenvalues of their Jacobian matrix.

Some formulas write $Q=I_3+Y$ rather than $Q=I_3+Y_{\mathrm{usual}}/2$. Then $Y=(B+S)/2$, where $B$ is baryon number and $S$ is strangeness. The light-quark weights are $u:(1/2,1/6)$, $d:(-1/2,1/6)$, $s:(0,-1/3)$. Baryon weight rows therefore have half the usual vertical coordinates. This is a flavour convention, distinct from weak hypercharge.

For a small category with finite limits, a set-valued functor preserving them has a category of elements with finite limits, so its opposite is a filtered category. The covariant density presentation is therefore a filtered colimit of representable functors. Conversely, representable functors preserve categorical limits, and filtered colimits commute with finite limits in sets, proving the criterion.

Extend this expression by zero at zero. Gaussian decay dominates every power, so the resulting function and its derivative difference quotients vanish to every order at zero. Away from zero, differentiation replaces $p(t)$ by $2t^{3}p(t)-t^2{p}^{\prime }(t)$, another polynomial. Induction therefore gives a smooth function flat at zero. If $p$ is nonzero, its Taylor series is zero while the function is nonzero at every sufficiently small nonzero argument, so it is not real analytic there.

For a strictly positive law $\mu$ satisfying the FKG lattice condition on a finite Boolean lattice, and an increasing function $g$, tilt to ${\mu }_t(\omega )\propto e^{tg(\omega )}\mu (\omega )$. For $t\ge 0$, the lattice inequality and $g(\omega \vee \eta )\ge g(\omega )$ give the Holley condition for ${\mu }_t{\ge }_{\mathrm{st}}\mu$. For any increasing $f$, differentiate ${\mathbb{E}}_{{\mu }_t}f\ge {\mathbb{E}}_{\mu }f$ at $t=0$ from the right. The derivative is ${\mathrm{Cov}}_{\mu }(f,g)$, proving positive association of random variables.

For a photon of energy $h\nu$ hitting a stationary target of mass $m_2$, the incoming invariant energy satisfies $W^2=m_2^2{c}^4+2m_2{c}^{2}h\nu$. The massive multi-product threshold is the displayed value when ${\sum }_i{\mu }_i>m_2$. If the algebraic bound is nonpositive, the lower limit of allowed positive frequencies can be zero. A single final product instead fixes the frequency by invariant-mass equality, and a final massless particle can make a nominal threshold an infimum rather than an attained value.

// Target: classical-mechanics.bigb

With these parameters the expected number of $(n-1)$-cliques is asymptotic to $\sqrt{n}$, yet the probability of any such clique tends to zero. The complement graph has about $\frac{1}{2}\log n$ edges, and with probability tending to one two of them are vertex disjoint. Deleting one vertex cannot remove both edges. Thus divergence of the expected clique count alone does not give an appearance theorem for arbitrary growing clique size.

For $f(z)=(az+b)/(cz+d)$ with distinct finite fixed points of a Möbius transformation $\alpha ,\beta$, the fixed-point equation gives $f(z)-\alpha =(a-c\alpha )(z-\alpha )/(cz+d)$. Dividing the two identities gives $k=(a-c\alpha )/(a-c\beta )$, which is nonzero because $(a-c\alpha )(a-c\beta )=ad-bc\ne 0$. The identities extend over the Riemann sphere. This conjugates the transformation to a dilation.

At design sites $i/n$, let ${\tilde{Y}}_n$ be the step function taking response $Y_i$ on $((i-1)/n,i/n]$. Project it onto an interval-adapted wavelet basis through resolution $J$. Each coefficient is ${\hat{c}}_{\lambda }={\sum }_i{Y}_i{\int }_{I_i}{b}_{\lambda }(t)dt$. Its expectation is the corresponding coefficient of the step function of sampled regression means. For a Haar wavelet basis, dyadic sample size and resolution at most the sampling scale, this is exactly $n^{-1}{\sum }_i{Y}_i{b}_{\lambda }(i/n)$ with right-closed pointwise versions. Independent errors of variance ${\sigma }^2$ give coefficient variance at most ${\sigma }^2/n$: Cauchy-Schwarz inequality on each sampling cell yields ${\sum }_i({\int }_{I_i}{b}_{\lambda }{)}^2\le \Vert b_{\lambda }{\Vert }_2^2/n$.

On a unit-square uniform mesh with $h=1/(M+1)$ and zero Dirichlet boundary conditions, the semidiscrete five-point Laplacian has eigenvalues $-4h^{-2}[{\sin }^2(j\pi h/2)+{\sin }^2(k\pi h/2)]$. Adding reaction $\kappa I$ gives a contractive continuous-time grid evolution exactly under the displayed threshold. The threshold is $2{\pi }^2-{\pi }^4{h}^2/6+O(h^4)$, slightly below the continuum threshold. This is an all-time non-growth criterion; finite-time mesh-uniform bounds also hold for fixed larger reaction rates.

For a smooth solution with bounded fourth derivatives on the unit square, the five-point local residual is $O(h^2)$. The smallest eigenvalue of the unscaled Dirichlet matrix is $8{\sin }^2(\pi h/2)\ge 8h^2$. With $M^2$ interior points and $M\asymp h^{-1}$, the residual has unweighted Euclidean norm $O(h)$. The equation $Ae=h^2\tau$ therefore gives unweighted error $O(h)$, equivalent to second-order error in the area-weighted grid norm.