When load-gap equilibration is fast compared with chemical events, an opposing force gives an exponential gap distribution. The probability of room for an addition is $e^{-Fa/(k_{B}T)}$. With unloaded addition rate $u$ and removal rate $w$, assumed force-independent except for this steric gating, the velocity is the displayed expression. For $u>w$, stall occurs at $F_s=(k_{B}T/a)\log (u/w)$. Concentration is included in $u=k_{\mathrm{on}}c$ if the on-rate constant is bimolecular.

A pentagonal prism has two pentagonal bases and five quadrilateral side faces, with three edges meeting at every vertex. Radial projection from an interior point gives a spherical polygon decomposition with five quadrilaterals and two pentagons.

A quadrilateral is a polygon with four sides and four vertices of a polygon. For a simple quadrilateral, its interior angles sum to $2\pi$ by dividing it along an interior diagonal of a polygon into two triangles. Rectangles, squares and parallelograms are special cases. The qualifier convex quadrilateral excludes a reflex interior angle; a self-crossing four-segment chain requires a different interior convention. Four side lengths alone generally leave freedom to vary the angles and area.

A diagonal of a polygon joins two vertices of a polygon that are not endpoints of the same side of a polygon. A square has two diagonals of a polygon, forming a two-element group orbit under its dihedral group of symmetries.

A side of a polygon is a line segment joining two consecutive vertices of a polygon in its boundary.

A vertex of a polygon is an endpoint where consecutive sides of a polygon meet.

Replacing the flat target metric in the Polyakov action by a spacetime metric $g_{\mu \nu }(X)$ gives the displayed string sigma model. The embedding fields $X^{\mu }$ map the worldsheet into target spacetime, while $h_{ab}$ is an independent worldsheet metric. Classical worldsheet diffeomorphisms and Weyl transformations permit conformal gauge; quantum consistency additionally requires vanishing sigma-model beta functions.

An experimental comparison joined a two-ended diol chain, HOPCLOH, with a bifunctional isocyanate, at reported isocyanate-to-diol ratio $1.14:1$. The researchers used oven-dried $25\mathrm{mL}$ round-bottom flasks and an IKA RET magnetic hotplate with a metal carrier. They varied solvent, concentration, reaction temperature ($50$ or $61{}^{\circ }\mathrm{C}$), and duration ($1$ or $8\mathrm{h}$).^[ref]

A Bruker $500\mathrm{MHz}$ NMR spectroscopy instrument measured residual hydroxy group-adjacent signal relative to an ester signal. Independently, size-exclusion chromatography used ResiPore columns at $70{}^{\circ }\mathrm{C}$, a refractive-index detector, and polystyrene calibration. Figure 8 reports sharply increasing apparent number-average molar mass beyond the reported conversion $0.95$. A deliberately water-exposed isocyanate comparison reproduced a $1625\mathrm{c}{\mathrm{m}}^{-1}$ infrared spectroscopy band assigned to urea-containing products. The authors warn that tiny residual NMR spectroscopy signals reduce integration precision.^[ref]

The inference is increasing chain size alongside depletion of reactive groups, consistent with step-growth polymerization. It is not an exact measurement of $X_n$ in the toy model. The feed contains already extended diol chains: counting their constituent repeat units differs from counting each feed molecule as one unit. Their varying molar masses, calibration against a different polymer, residual starting material, and competing chemical reactions further obstruct a direct conversion-to-$X_n$ substitution. Report which functional groups the conversion readout measures before comparing it with stoichiometric imbalance in step growth.

Useful proposed controls, distinct from the reported water comparison, are pre-reaction spectra and matched mixtures omitting each joining partner in turn. Keep solvent, heating, and signal normalization matched. Disappearance of an isocyanate signal in the omitted-diol control would reveal competing consumption rather than establish chain joining. Agreement between residual-group and independently calibrated chain-size readouts is stronger evidence than calculating both quantities from the same Carothers equation.

Hydrogen fluoride is $\mathrm{HF}$, a strongly polar diatomic molecule. Hydrogen bonding gives its condensed phases and molecular aggregates behavior different from that of an isolated gas-phase molecule.

In water it is a weak acid by the equilibrium extent of proton transfer, despite having a very polar bond. Bond strength, solvation, and interactions among dissolved species must be considered together; electronegativity alone does not determine aqueous acid strength.

Hydrogen chloride is the polar diatomic molecule $\mathrm{HCl}$. Its bond has unequal electron sharing because chlorine attracts electron density more strongly than hydrogen.

The gas and an aqueous solution are different descriptions: dissolved hydrogen chloride transfers protons strongly to water and is called hydrochloric acid. Its rotational-vibrational spectrum also provides a simple molecular probe of bond length, force constant, and isotopic mass effects.

For $F\in L^1(\mathbb{R})$ the periodization is absolutely convergent almost everywhere in a period by integrating the sum of absolute values. Its Fourier series coefficient is $c_n=\tilde{F}(n)/(2\pi )$, where $\tilde{F}(\omega )={\int }_{\mathbb{R}}F(x)e^{-i\omega x}dx$. Absolute integrability permits integrating term by term and changing variables across the disjoint period intervals. This coefficient identity does not by itself assert pointwise convergence of the ordinary Fourier series. Even uniform convergence of the original periodization cannot repair arbitrary point values: $F=1_{\{0\}}$ has zero transform and a nonzero periodization at multiples of $2\pi$. Additional regularity, such as a continuous piecewise continuously differentiable periodization, validates the pointwise Poisson summation formula in this setting.

// Target: fluid-mechanics.bigb

A counting process which, conditional on a random nonnegative locally integrable intensity $\Lambda$, is an Inhomogeneous Poisson process. For $L_t={\int }_0^t\Lambda (s)ds$, conditioning gives $\mathbb{E}N(t)=\mathbb{E}{L}_t$ and $\mathrm{Var}N(t)=\mathbb{E}{L}_t+\mathrm{Var}{L}_t$. Thus finite-variance counts are never underdispersed. Randomness of the intensity may induce dependence between increments which are independent conditionally.

For a rate-$\lambda$ Poisson process, keeping every second arrival gives a renewal process with independent interarrival times equal to sums of two exponential waiting times. Put $x=\lambda t$. Its count mean and variance are $(2x+e^{-2x}-1)/4$ and $(4x-8x{e}^{-2x}-e^{-4x}+1)/16$. The variance is strictly smaller than the mean for $x>0$: sixteen times their difference has derivative $4[-1+4x{e}^{-2x}+e^{-4x}]<0$ and is zero at zero, because $\sinh (2x)>2x$. Therefore this process is not a Cox process.

For a unit-rate Poisson process, $h(\eta )=(1+\eta )\log (1+\eta )-\eta$ gives the displayed maximal deviation bound for $L,\eta >0$. Poisson exponential martingales and the Ville inequality control upper and lower deviations; the upper-tail rate $h$ also bounds the lower-tail probability. Since $h(0)=0$ and $h^{\prime }(\eta )=\log (1+\eta )>0$, the rate is strictly positive for every positive deviation level.

For a unit-rate Poisson process and real $\theta$, the displayed process is a nonnegative martingale with initial value one. Independent increments give $\mathbb{E}{e}^{\theta (N_t-N_s)}=\exp ((t-s)(e^{\theta }-1))$, proving its conditional mean property. The Ville inequality turns this exponential moment into a maximal tail bound.

For a homogeneous Poisson process of rate $\lambda >0$, $N(t)/t\to \lambda$ almost surely: unit-time increments are independent Poisson variables and the strong law of large numbers applies, then monotonicity fills the intervals between integers. At its $n$th point $Y_n$, simplicity gives $N(Y_n)=n$. Consequently $Y_n/n\to 1/\lambda$. Rate and mean interarrival time are reciprocals.

A Poisson process observed for a fixed duration $H$ has likelihood proportional to ${\lambda }^{N_H}{e}^{-\lambda H}$. Complete independent exponential distribution gaps observed until a fixed event count $n$ have likelihood ${\lambda }^n{e}^{-\lambda T_n}$. Both give count divided by exposure, but the exposure is random in the second design. Under fixed exposure the total count is sufficient: conditional on it, the event locations are independent of the rate, so exact timestamps add no rate information to interval counts. Gap estimates and fixed-horizon estimates should not be compared without distinguishing their stopping rules and any final censored gap.

Let $T_n=\inf \{t\ge 0:N_t\ge n\}$ be the $n$th arrival time, and set $T_0=0$. In a Homogeneous Poisson process of rate $\lambda >0$, the gaps $W_n=T_n-T_{n-1}$ are independent and identically distributed random variables with an exponential distribution:Here $\mathrm{Exp}(\lambda )$ uses the rate convention. Under the scale convention $\theta =1/\lambda$, the same probability density function is ${\theta }^{-1}{e}^{-w/\theta }$; a simulation function accepting a scale must receive $1/\lambda$. Conversely, summing independent rate-$\lambda$ exponential random variables to form $T_n$ and counting those sums constructs a Homogeneous Poisson process.^[ref]

The first-gap formula follows immediately from $\{W_1>w\}=\{N_w=0\}$ and the Poisson distribution. Independence of all successive gaps is the additional interval-theorem assertion; deterministic-time independent increments alone should not simply be substituted at random arrival times without justification. The memorylessness of the exponential distribution givesAt the toy rate of two per minute, the mean gap is half a minute, the median is $(\log 2)/2\approx 0.346574$ minutes, and the probability of waiting longer than half a minute is $e^{-1}\approx 0.367879$. Here $\log$ is the natural logarithm. Already having waited a minute without an arrival leaves this same conditional probability of waiting another half minute. The exponential distribution describes a model with no aging of the remaining wait; a scheduled arrival with a fixed gap has no such property.

Let $\Pi$ be a Poisson point process with intensity measure $\mu$ and let $f$ be measurable. If the pushforwardis locally finite, then the image counting measure is Poisson with intensity $f_{\ast }\mu$. If one requires a simple point process, the pushforward must also be diffuse, so distinct source points do not produce coincident image points with positive probability.

Let $N$ be a Poisson point process with intensity measure $\mu$, and let $g$ be a real-valued measurable function. For the ordinary Poisson sum $S_g=\int gdN$, the condition $\int |g|d\mu <\infty$ givesIf also $\int g^{2}d\mu <\infty$, thenExpand the square: equal-index terms contribute $\int g^{2}d\mu$, while the Poisson factorial moment measure makes distinct-index terms contribute $(\int gd\mu {)}^2$. Subtracting the squared expected value proves the variance formula. The Campbell first-moment formula also holds for general point processes; this variance identity uses the Poisson assumption.^[ref]

Square integrability alone is insufficient for the ordinary sum on an infinite-intensity space. Take intensity $dx$ on $[1,\infty )$ and $g(x)=1/x$. Then $\int g^{2}dx=1$, but the sum is infinite almost surely: the ordered points satisfy $Z_i/i\to 1$ by the strong law for Poisson arrival times, so comparison with the harmonic series proves divergence.

For this same $g$, the compensated Poisson integral exists as a mean-zero limit under convergence in Lp with $p=2$ and has variance $1$. Its definition is the L2 construction of compensated Poisson integrals, rather than subtraction of two infinite ordinary integrals. When the total intensity is finite, Cauchy-Schwarz inequality makes L2 imply L1, explaining why the shorter variance statement works in that special case.^[ref]