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# 16.4 The simple pendulum  (Page 2/3)

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## Measuring acceleration due to gravity: the period of a pendulum

What is the acceleration due to gravity in a region where a simple pendulum having a length 75.000 cm has a period of 1.7357 s?

Strategy

We are asked to find $g$ given the period $T$ and the length $L$ of a pendulum. We can solve $T=2\pi \sqrt{\frac{L}{g}}$ for $g$ , assuming only that the angle of deflection is less than $\text{15º}$ .

Solution

1. Square $T=2\pi \sqrt{\frac{L}{g}}$ and solve for $g$ :
$g={4\pi }^{2}\frac{L}{{T}^{2}}.$
2. Substitute known values into the new equation:
$g={4\pi }^{2}\frac{0\text{.}\text{75000}\phantom{\rule{0.25em}{0ex}}\text{m}}{{\left(1\text{.}\text{7357 s}\right)}^{2}}.$
3. Calculate to find $g$ :
$g=9\text{.}\text{8281}\phantom{\rule{0.25em}{0ex}}\text{m}/{\text{s}}^{2}.$

Discussion

This method for determining $g$ can be very accurate. This is why length and period are given to five digits in this example. For the precision of the approximation $\text{sin θ}\approx \theta$ to be better than the precision of the pendulum length and period, the maximum displacement angle should be kept below about $\text{0.5º}$ .

## Making career connections

Knowing $g$ can be important in geological exploration; for example, a map of $g$ over large geographical regions aids the study of plate tectonics and helps in the search for oil fields and large mineral deposits.

## Take home experiment: determining $g$

Use a simple pendulum to determine the acceleration due to gravity $g$ in your own locale. Cut a piece of a string or dental floss so that it is about 1 m long. Attach a small object of high density to the end of the string (for example, a metal nut or a car key). Starting at an angle of less than $\text{10º}$ , allow the pendulum to swing and measure the pendulum’s period for 10 oscillations using a stopwatch. Calculate $g$ . How accurate is this measurement? How might it be improved?

An engineer builds two simple pendula. Both are suspended from small wires secured to the ceiling of a room. Each pendulum hovers 2 cm above the floor. Pendulum 1 has a bob with a mass of $\text{10}\phantom{\rule{0.25em}{0ex}}\text{kg}$ . Pendulum 2 has a bob with a mass of $\text{100 kg}$ . Describe how the motion of the pendula will differ if the bobs are both displaced by $\text{12º}$ .

The movement of the pendula will not differ at all because the mass of the bob has no effect on the motion of a simple pendulum. The pendula are only affected by the period (which is related to the pendulum’s length) and by the acceleration due to gravity.

## Phet explorations: pendulum lab

Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, and the amplitude of the swing. It’s easy to measure the period using the photogate timer. You can vary friction and the strength of gravity. Use the pendulum to find the value of $g$ on planet X. Notice the anharmonic behavior at large amplitude.

## Section summary

• A mass $m$ suspended by a wire of length $L$ is a simple pendulum and undergoes simple harmonic motion for amplitudes less than about $\text{15º}.$

The period of a simple pendulum is

$T=2\pi \sqrt{\frac{L}{g}},$

where $L$ is the length of the string and $g$ is the acceleration due to gravity.

## Conceptual questions

Pendulum clocks are made to run at the correct rate by adjusting the pendulum’s length. Suppose you move from one city to another where the acceleration due to gravity is slightly greater, taking your pendulum clock with you, will you have to lengthen or shorten the pendulum to keep the correct time, other factors remaining constant? Explain your answer.

## Problems&Exercises

As usual, the acceleration due to gravity in these problems is taken to be $g=9.80\phantom{\rule{0.25em}{0ex}}\text{m}/{\text{s}}^{2}$ , unless otherwise specified.

What is the length of a pendulum that has a period of 0.500 s?

6.21 cm

Some people think a pendulum with a period of 1.00 s can be driven with “mental energy” or psycho kinetically, because its period is the same as an average heartbeat. True or not, what is the length of such a pendulum?

What is the period of a 1.00-m-long pendulum?

2.01 s

How long does it take a child on a swing to complete one swing if her center of gravity is 4.00 m below the pivot?

The pendulum on a cuckoo clock is 5.00 cm long. What is its frequency?

2.23 Hz

Two parakeets sit on a swing with their combined center of mass 10.0 cm below the pivot. At what frequency do they swing?

(a) A pendulum that has a period of 3.00000 s and that is located where the acceleration due to gravity is $9\text{.}\text{79}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ is moved to a location where it the acceleration due to gravity is $9\text{.}\text{82}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ . What is its new period? (b) Explain why so many digits are needed in the value for the period, based on the relation between the period and the acceleration due to gravity.

(a) 2.99541 s

(b) Since the period is related to the square root of the acceleration of gravity, when the acceleration changes by 1% the period changes by $\left(0\text{.}\text{01}{\right)}^{2}=0\text{.}\text{01%}\text{}$ so it is necessary to have at least 4 digits after the decimal to see the changes.

A pendulum with a period of 2.00000 s in one location $\left(g=9\text{.}\text{80}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\right)$ is moved to a new location where the period is now 1.99796 s. What is the acceleration due to gravity at its new location?

(a) What is the effect on the period of a pendulum if you double its length?

(b) What is the effect on the period of a pendulum if you decrease its length by 5.00%?

(a) Period increases by a factor of 1.41 ( $\sqrt{2}$ )

(b) Period decreases to 97.5% of old period

Find the ratio of the new/old periods of a pendulum if the pendulum were transported from Earth to the Moon, where the acceleration due to gravity is $1\text{.}\text{63}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ .

At what rate will a pendulum clock run on the Moon, where the acceleration due to gravity is $1\text{.}\text{63}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ , if it keeps time accurately on Earth? That is, find the time (in hours) it takes the clock’s hour hand to make one revolution on the Moon.

Slow by a factor of 2.45

Suppose the length of a clock’s pendulum is changed by 1.000%, exactly at noon one day. What time will it read 24.00 hours later, assuming it the pendulum has kept perfect time before the change? Note that there are two answers, and perform the calculation to four-digit precision.

If a pendulum-driven clock gains 5.00 s/day, what fractional change in pendulum length must be made for it to keep perfect time?

length must increase by 0.0116%.

tree physical properties of heat
tree is a type of organism that grows very tall and have a wood trunk and branches with leaves... how is that related to heat? what did you smoke man?
what are the uses of dimensional analysis
Dimensional Analysis. The study of relationships between physical quantities with the help of their dimensions and units of measurements is called dimensional analysis. We use dimensional analysis in order to convert a unit from one form to another.
Emmanuel
meaning of OE and making of the subscript nc
Negash
kinetic functional force
what is a principal wave?
A wave the movement of particles on rest position transferring energy from one place to another
Gabche
not wave. i need to know principal wave or waves.
Haider
principle wave is a superposition of wave when two or more waves meet at a point , whose amplitude is the algebraic sum of the amplitude of the waves
kindly define principal wave not principle wave (principle of super position) if u can understand my question
Haider
what is a model?
hi
Muhanned
why are electros emitted only when the frequency of the incident radiation is greater than a certain value
b/c u have to know that for emission of electron need specific amount of energy which are gain by electron for emission . if incident rays have that amount of energy electron can be emitted, otherwise no way.
Nazir
Nazir
what is ohm's law
states that electric current in a given metallic conductor is directly proportional to the potential difference applied between its end, provided that the temperature of the conductor and other physical factors such as length and cross-sectional area remains constant. mathematically V=IR
ANIEFIOK
hi
Gundala
A body travelling at a velocity of 30ms^-1 in a straight line is brought to rest by application of brakes. if it covers a distance of 100m during this period, find the retardation.
just use v^2-u^2=2as
Gundala
how often does electrolyte emits?
alhassan
just use +€^3.7°√π%-4¢•∆¥%
v^2-u^2=2as v=0,u=30,s=100 -30^2=2a*100 -900=200a a=-900/200 a=-4.5m/s^2
akinyemi
what's acceleration
The change in position of an object with respect to time
Mfizi
Acceleration is velocity all over time
Pamilerin
hi
Stephen
It's not It's the change of velocity relative to time
Laura
Velocity is the change of position relative to time
Laura
acceleration it is the rate of change in velocity with time
Stephen
acceleration is change in velocity per rate of time
Noara
what is ohm's law
Stephen
Ohm's law is related to resistance by which volatge is the multiplication of current and resistance ( U=RI)
Laura
acceleration is the rate of change. of displacement with time.
the rate of change of velocity is called acceleration
Asma
how i don understand
how do I access the Multiple Choice Questions? the button never works and the essay one doesn't either
How do you determine the magnitude of force
mass × acceleration OR Work done ÷ distance
Seema
Which eye defect is corrected by a lens having different curvatures in two perpendicular directions?
acute astigmatism?


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# Sequences with integral variances

This is a companion to my earlier question, Sequences with integral means. This new question is, frankly, not as interesting, but it feels necessary to complete the thought.

Let $V(n)$ be the sequence whose first element is $n$, and from then onward, the next element is the smallest natural number ${\ge}1$ that ensures that the variance of all the numbers in the sequence is an integer. If there is no such number to extend $V(n)$, then it has finite length.

For example, for $n=20$, $V(20) = 20,2,2,4$. When the last element $4$ is added, the mean is $28/4=7$, and the variance sum is $$13^2 + (-5)^2 + (-5)^2 + (-3)^2 = 169+25+25+9 = 228$$ which is divisible by $n=4$ yielding a variance of $57$. (NB: An earlier version of this question divided by $n{-}1$ rather than by $n$; see the comments.) None of the smaller alternatives $1,2,3$ lead to integral variance. The $V(20)$ sequence cannot be extended beyond that terminating $4$, for the attempt to extend leads to a quadratic Diophantine equation with no solution.

Q1. Is it the case that the only infinite sequences are $V(1)=1,1,1,\ldots$ and $V(2)=2,2,2,\ldots$ ?

Q2. Is there nevertheless no upperbound on the length of the longest finite sequence?

Here are some "long" sequences encountered within $n \le 1000$, of lengths $6$, $6$, $8$, and $9$ respectively: $V(61)=61,1,1,1,1,1$; $V(62)=62,2,2,2,2,2$; $V(422)=422,2,2,2,2,2,2,6$; $V(842)=842,2,2,2,2,2,2,2,8$.

-
Dividing by $n-1$, not by $n$? –  Gerry Myerson Nov 3 '13 at 22:18
@GerryMyerson: Yes, dividing by $n-1$. Of course one could define the variance differently, I was just following the typical unbiased estimate of the variance... Perhaps for number-theoretic interest, dividing by $n$ would be more natural? –  Joseph O'Rourke Nov 3 '13 at 22:27
Another unimpeachable source: the Kahn Academy! :-) Review and intuition why we divide by n-1 for the unbiased sample variance. –  Joseph O'Rourke Nov 3 '13 at 22:40
Since this is not a situation of sampling from a distribution, I don't see what an "unbiased estimate" has to do with it. You're free to use whatever formula you want, but you should at least specify it. –  Robert Israel Nov 3 '13 at 23:05
@JosephO'Rourke How about the sequences arising from the elements in oeis.org/A174554? I believe this will answer your second question (there is no upper-bound as one can produce a long string of $2$s). –  Benjamin Dickman Nov 5 '13 at 3:46

The variance of the list $n,1,1,\dots,1$ of length $k$ equals $(k-1)(n-1)^2/k^2$. In particular, if $n\equiv1\pmod k$ then this variance is an integer. It follows that if $n\equiv1\pmod{k!}$, then the sequence $V(n)$ has length at least $k$. This answers Q2: there is no upper bound on the length of such sequences.

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Article| Volume 2, ISSUE 4, 100088, December 14, 2022
• Top

# Single-photon smFRET. III. Application to pulsed illumination

Open AccessPublished:November 24, 2022

## Abstract

Förster resonance energy transfer (FRET) using pulsed illumination has been pivotal in leveraging lifetime information in FRET analysis. However, there remain major challenges in quantitative single-photon, single-molecule FRET (smFRET) data analysis under pulsed illumination including 1) simultaneously deducing kinetics and number of system states; 2) providing uncertainties over estimates, particularly uncertainty over the number of system states; and 3) taking into account detector noise sources such as cross talk and the instrument response function contributing to uncertainty; in addition to 4) other experimental noise sources such as background. Here, we implement the Bayesian nonparametric framework described in the first companion article that addresses all aforementioned issues in smFRET data analysis specialized for the case of pulsed illumination. Furthermore, we apply our method to both synthetic as well as experimental data acquired using Holliday junctions.

## Why it matters?

In the first companion article of this series, we developed new methods to analyze noisy single-molecule Förster resonance energy transfer data. These methods eliminate the requirement of a priori specifying the dimensionality of the physical model describing a molecular complex’s kinetics. Here, we apply these methods to experimentally obtained datasets with samples illuminated by laser pulses at regular time intervals. In particular, we study conformational dynamics of Holliday junctions.

## Introduction

Among the many fluorescence methods available (
• Weiss S.
Fluorescence spectroscopy of single biomolecules.
,
• Lippincott-Schwartz J.
• Snapp E.
• Kenworthy A.
Studying protein dynamics in living cells.
,
• Huang B.
• Bates M.
• Zhuang X.
Super-resolution fluorescence microscopy.
,
• Lelek M.
• Gyparaki M.T.
• Zimmer C.
• et al.
Single-molecule localization microscopy.
,
• Fazel M.
• Wester M.J.
Analysis of super-resolution single molecule localization microscopy data: A tutorial.
,
• Datta R.
• Heaster T.M.
• Skala M.C.
• et al.
Fluorescence lifetime imaging microscopy: fundamentals and advances in instrumentation, analysis, and applications.
,
• Garini Y.
• Young I.T.
• McNamara G.
Spectral imaging: principles and applications.
), single-molecule Förster resonance energy transfer (smFRET) has been useful in probing interactions and conformational changes on nanometer scales (
• Roy R.
• Hohng S.
• Ha T.
A practical guide to single-molecule FRET.
,
• Mazal H.
• Haran G.
Single-molecule FRET methods to study the dynamics of proteins at work.
,
• Schuler B.
Single-molecule FRET of protein structure and dynamics - a primer.
,
• Lu M.
• Ma X.
• Mothes W.
• et al.
Associating HIV-1 envelope glycoprotein structures with states on the virus observed by smFRET.
,
• Mooney S.M.
• Qiu R.
• Weninger K.R.
• et al.
Cancer/testis antigen PAGE4, a regulator of c-Jun transactivation, is phosphorylated by homeodomain-interacting protein kinase 1, a component of the stress-response pathway.
). This is typically achieved by estimating FRET efficiencies (and system states) at all instants of an smFRET trace and subsequently estimating transition rates. Furthermore, among different FRET modalities, FRET efficiencies are most accurately determined under pulsed illumination (
• Wozniak A.K.
• Schröder G.F.
• Oesterhelt F.
• et al.
Single-molecule FRET measures bends and kinks in DNA.
,
• Chung H.S.
• Louis J.M.
• Gopich I.V.
Analysis of fluorescence lifetime and energy transfer efficiency in single-molecule photon trajectories of fast-folding proteins.
,
• Heo W.
• Hasegawa K.
• Tahara T.
• et al.
Scanning two-dimensional fluorescence lifetime correlation spectroscopy: Conformational dynamics of DNA Holliday junction from microsecond to subsecond.
), where the FRET dyes are illuminated by short laser bursts at known times.
Under this illumination procedure, photon arrival times are recorded with respect to the immediately preceding pulse, thereby facilitating an accurate estimation of fluorescence lifetimes as well as FRET rates. As such, in this article, we will focus on single-photon smFRET analysis under pulsed illumination.
Under pulsed illumination, information on kinetic parameters present in smFRET data is traditionally learned by binned photon methods, thereby eliminating lifetime information altogether (
• Zeug A.
• Woehler A.
• et al.
Quantitative intensity-based fret approaches—a comparative snapshot.
,
• Kuppa S.
• Deveryshetty J.
• Antony E.
• et al.
Rtt105 regulates rpa function by configurationally stapling the flexible domains.
,
• Kilic Z.
• Sgouralis I.
• Pressé S.
Generalizing hmms to continuous time for fast kinetics: hidden markov jump processes.
); bulk correlative methods (
• Kapusta P.
• Wahl M.
• Enderlein J.
• et al.
,
• Ishii K.
• Tahara T.
Two-dimensional fluorescence lifetime correlation spectroscopy. 1. principle.
,
• Otosu T.
• Ishii K.
• Tahara T.
Microsecond protein dynamics observed at the single-molecule level.
); and single-photon methods (
• Chung H.S.
• Louis J.M.
• Gopich I.V.
Analysis of fluorescence lifetime and energy transfer efficiency in single-molecule photon trajectories of fast-folding proteins.
,
• Yoo J.
• Kim J.-Y.
• Chung H.S.
• et al.
Fast three-color single-molecule FRET using statistical inference.
,
• Lerner E.
• Ingargiola A.
• Weiss S.
Characterizing highly dynamic conformational states: The transcription bubble in RNAP-promoter open complex as an example.
). However, these methods are parametric, i.e., require fixing the number of system states a priori, and necessarily only learn system kinetics even though information on the number of system states is encoded in the data.
In this article, we implement a general smFRET analysis framework that was presented in Sec. 2.5.1 of the first companion manuscript (
• Saurabh A.
• Safar M.
• Pressé S.
• et al.
Single photon smFRET. I. theory and conceptual basis.
) for the case of pulsed illumination to learn full distributions. In other words, probability distributions over parameters take into account uncertainties from all existing sources such as cross talk and background. These parameters include the system transition probabilities and photophysical rates, that is, donor and acceptor relaxation and FRET rates, with special attention paid to uncertainty arising from sources such as inherent stochasticity in photon arrival times and detectors. As our main concern is deducing the number of system states using single-photon arrivals while incorporating detector effects, we leverage the formalism of infinite hidden Markov models (iHMMs) (
• Sgouralis I.
• Pressé S.
An introduction to infinite HMMs for single-molecule data analysis.
,
• Sgouralis I.
• Pressé S.
• et al.
A Bayesian nonparametric approach to single molecule Förster resonance energy transfer.
,
• Fox E.B.
• Sudderth E.B.
• Willsky A.S.
• et al.
A sticky HDP-HMM with application to speaker diarization.
,
• Teh Y.W.
• Jordan M.I.
• Blei D.M.
• et al.
Hierarchical Dirichlet processes.
,
• Jayaram S.
A constructive definition of Dirichlet priors.
,
• Pitman J.
Poisson–Dirichlet and GEM invariant distributions for split-and-merge transformations of an interval partition.
) within the Bayesian nonparametric (BNP) paradigm (
• Sgouralis I.
• Pressé S.
An introduction to infinite HMMs for single-molecule data analysis.
,
• Sgouralis I.
• Pressé S.
• et al.
A Bayesian nonparametric approach to single molecule Förster resonance energy transfer.
,
• Ferguson T.S.
A Bayesian analysis of some nonparametric problems.
,
• Gershman S.J.
• Blei D.M.
A tutorial on Bayesian nonparametric models.
,
• Sgouralis I.
• Whitmore M.
• Pressé S.
• et al.
Single molecule force spectroscopy at high data acquisition: A Bayesian nonparametric analysis.
,
• Tavakoli M.
• Taylor J.N.
• Pressé S.
• et al.
Single molecule data analysis: An introduction.
,
• Tavakoli M.
• Jazani S.
• Pressé S.
• et al.
Pitching single-focus confocal data analysis one photon at a time with Bayesian nonparametrics.
,
• Tavakoli M.
• Jazani S.
• Pressé S.
• et al.
Direct photon-by-photon analysis of time-resolved pulsed excitation data using Bayesian nonparametrics.
,
• Bryan 4th, J.S.
• Sgouralis I.
• Pressé S.
Diffraction-limited molecular cluster quantification with Bayesian nonparametrics.
,
• Fazel M.
• Jazani S.
• Pressé S.
• et al.
High resolution fluorescence lifetime maps from minimal photon counts.
). The iHMM framework assumes an a priori infinite number of system states with associated transition probabilities, where the number of system states warranted by input data is enumerated by those states most visited over the course of the system state trajectory.
Next, to benchmark our BNP-FRET sampler, we analyzed synthetic and experimental smFRET data acquired using a single confocal microscope with pulsed illumination optimized to excite donor dyes.
In particular, we employ a broad range of experimental data acquired from Holliday junctions (HJs) with an array of different kinetic rates due to varying buffer concentration of $MgCl2$ (
• Okamoto K.
• Sako Y.
State transition analysis of spontaneous branch migration of the Holliday junction by photon-based single-molecule fluorescence resonance energy transfer.
,
• Hohng S.
• Joo C.
• Ha T.
Single-molecule three-color FRET.
,
• McKinney S.A.
• Déclais A.C.
• Ha T.
• et al.
Structural dynamics of individual Holliday junctions.
,
• McKinney S.A.
• Freeman A.D.J.
• Ha T.
• et al.
Observing spontaneous branch migration of Holliday junctions one step at a time.
).

## Materials and methods

### Terminology convention

To be consistent throughout our three-part article, we precisely define some terms as follows.
• 1. A macromolecular complex under study is always referred to as a system.
• 2. The configurations through which a system transitions are termed system states, typically labeled using $σ$.
• 3. FRET dyes undergo quantum mechanical transitions between photophysical states, typically labeled using $ψ$.
• 4. A system-FRET combination is always referred to as a composite.
• 5. A composite undergoes transitions among its superstates, typically labeled using $φ$.
• 6. All transition rates are typically labeled using $λ$.
• 7. The symbol $N$ is generally used to represent the total number of discretized time windows, typically labeled with $n$.
• 8. The symbol $wn$ is generally used to represent the observations in the $n$-th time window.

### Forward model and inverse strategy

In this section, we first briefly illustrate the adaptation of the general formalism described in our first companion article (
• Saurabh A.
• Safar M.
• Pressé S.
• et al.
Single photon smFRET. I. theory and conceptual basis.
) to the pulsed illumination case. Next, we present a specialized inference procedure. The details of the framework not provided herein can be found in the supporting material.
As before, we consider a molecular complex labeled with a donor-acceptor FRET pair. As the molecular complex transitions through its $Mσ$ system states indexed by $σ1:Mσ$, laser pulses (optimized to excite the donor) separated by time $τ$ may excite either the donor or acceptor to drive transitions among the photophysical states, $ψ1:Mψ$, as defined in the first companion article (
• Saurabh A.
• Safar M.
• Pressé S.
• et al.
Single photon smFRET. I. theory and conceptual basis.
). Such photophysical transitions lead to photon emissions that may be detected in either donor or acceptor channels. The set of $N$ observations, e.g., photon arrival times, from $N$ pulses are recorded as
$w={w1,w2,…,wN}.$
(1)

Here, each individual measurement is a pair $wn=(μnd,μna)$, where $μnd$ and $μna$ are the recorded arrival times (also known as microtimes) after the $n$-th pulse in both donor and acceptor channels, respectively. In cases where there is no photon detection, we denote the absent microtimes with $μnd=∅$ and $μna=∅$ for donor and acceptor channels, respectively.
As is clear from Fig. 1, smFRET traces are inherently stochastic due to the nature of photon excitation, emission, and noise introduced by detector electronics. To analyze such stochastic systems, we begin with the most generic likelihood derived in Eq. 51 of the first companion article (
• Saurabh A.
• Safar M.
• Pressé S.
• et al.
Single photon smFRET. I. theory and conceptual basis.
),
$L∝ρstartQ1…Qn…QNρnormT,$
(2)

where $ρstart$ is the initial probability vector for the system-FRET composite to be in one of $Mϕ=Mσ×Mψ$ superstates, and $ρnorm$ is a vector that sums the elements of the propagated probability vector. Here, we recall that $Qn$ is the transition probability matrix between pulses at $tn$ and $tn+1$, characterizing system-FRET composite transitions among superstates.
The propagators $Qn$ above adopt different forms depending on whether a photon is detected or not during the associated period. Their most general forms are derived in the section on illumination features in the first companion article (
• Saurabh A.
• Safar M.
• Pressé S.
• et al.
Single photon smFRET. I. theory and conceptual basis.
). However, these propagators involve computationally expensive integrals, and thus we make a few approximations here as follows: 1) we assume that the system state remains the same over an interpulse period since typical system kinetic timescales (typically 1 ms or more) are much longer than interpulse periods ($≈100$ ns) (
• McKinney S.A.
• Déclais A.C.
• Ha T.
• et al.
Structural dynamics of individual Holliday junctions.
,
• Panyutin I.G.
• Biswas I.
• Hsieh P.
A pivotal role for the structure of the Holliday junction in DNA branch migration.
), and 2) the interpulse period ($≈100$ ns) is longer than the donor and acceptor lifetimes ($≈$ a few ns) (
• McKinney S.A.
• Déclais A.C.
• Ha T.
• et al.
Structural dynamics of individual Holliday junctions.
,
• Panyutin I.G.
• Biswas I.
• Hsieh P.
A pivotal role for the structure of the Holliday junction in DNA branch migration.
) such that they relax to the ground state before the next pulse. Furthermore, we will demonstrate a specialized sampling scheme under these physically motivated approximations.
The immediate implications of the first assumption are that the system transitions may now, to a good approximation, only occur at the beginning of each pulse. Consequently, the evolution of the FRET pair between two consecutive pulses is now exclusively photophysical, as the system state remains the same during interpulse times. As such, the system now evolves in equally spaced discrete time steps of size $τ$, where the system state trajectory can be written as
$s1:N={s1,s2,…,sn,…,sN−1,sN},$

where $sn$ is the system state between pulses $n$ and $n+1$. The stochastic evolution of the system states in such discrete steps is then determined by the transition probability matrix designated by $Πσ$. For example, in the simplest case of a molecular complex with two system states $σ1:2$, this matrix is computed as follows:
$Πσ=exp(τ[∗λσ1→σ2λσ2→σ2∗])=[πσ1→σ1πσ1→σ2πσ2→σ1πσ2→σ2],$
(3)

where the matrix in the exponential contains transition rates among the system states and the $∗$ represents the negative row sum.
Next, by assumption two, we can further suppose that the fluorophores always start in the ground state at the beginning of every pulse. As a result, we treat pulses independently and write the probability of observation $wn$ as
$p(wn|sn,Gψ)=ρgroundQnψ(sn)ρnormT,$
(4)

where $ρground$ denotes the probability vector when the FRET pair is in the ground state at the beginning of each pulse, $Gψ$ is the generator matrix with only photophysical transition rates, and $Qnψ(sn)$ is the photophysical propagator for the $n$-th interpulse period.
We further organize the observation probabilities of Eq. 4 into a newly defined detection matrix $Dnσ$ with its elements given by $(Dnσ)sn→σj=p(wn|sn,Gψ)$. Here, we note that the index $j$ does not appear on the right-hand side because the system state does not change during an interpulse window, resulting in the independence of observation probability from the next system state, $sn+1$. The explicit formulas for the observation probabilities are provided in the supporting material.
Now, using the matrix $Dnσ$, we define the reduced propagators for each interpulse period as
$Πnσ=Πσ⊙Dnσ,$
(5)

where $⊙$ denotes the element-by-element product.
Finally, using these simplified propagators, we can write the likelihood for an smFRET trace under pulsed illumination as
$L=p(w|ρstart,Πσ,Gψ)∝ρstartΠ1σΠ2σ…ΠNσρnormT,$
(6)

as also introduced in the section on illumination features in the first companion article (
• Saurabh A.
• Safar M.
• Pressé S.
• et al.
Single photon smFRET. I. theory and conceptual basis.
). This form of the likelihood is advantageous in that it allows empty pulses to be computed as a simple product, greatly reducing computational cost.
In the following, we first illustrate a parametric inference procedure assuming a given number of system states. We next generalize the procedure developed to the nonparametric case to deduce the number of system states along with the rest of parameters.

#### Inference procedure: Parametric sampler

With the likelihood at hand, we construct the posterior as follows
$p(ρstart,Πσ,Gψ|w)∝p(w|ρstart,Πσ,Gψ)p(ρstart)p(Gψ)p(Πσ),$
(7)

where we assume that the unknown parameters, including the initial probability vector, $ρstart$, the photophysical transition rates in the generator matrix $Gψ$, and the transition probabilities among system states in propagator $Πσ$ are independent, allowing us to conveniently write the prior on these parameters as a product (the last three terms on right hand side). Here, we can sample the set of unknowns using the above posterior with the Gibbs sampling procedure described in the first companion article (see the section describing inverse strategy in (
• Saurabh A.
• Safar M.
• Pressé S.
• et al.
Single photon smFRET. I. theory and conceptual basis.
)). However, a computationally more convenient inference procedure that allows direct sampling is accomplished by writing the posterior of Eq. 7 as a marginalization (sum) over state trajectories as follows
$p(ρstart,Πσ,Gψ|w)=∑s1:Np(ρstart,Πσ,Gψ,s1:N|w)∝∑s1:Np(w|Πσ,Gψ,s1:N)p(ρstart)p(Gψ)p(Πσ)p(s1:N|ρstart,Πσ),$
(8)

where $s1:N={s1,s2…,sN}$ denotes a system state trajectory. Now, we can use the nonmarginal posterior
$p(ρstart,Πσ,Gψ,s1:N|w)∝p(w|Πσ,Gψ,s1:N)p(ρstart)p(Gψ)p(Πσ)p(s1:N|ρstart,Πσ)$
(9)

to sample the trajectory $s1:N$, which, in turn, allows direct sampling of the elements of propagator $Πσ$ described shortly. For priors on $ρstart$ and rates in $Gψ$, we, respectively, use Dirichlet and Gamma distributions similar to Eqs. 65 and 66 of the first companion article (
• Saurabh A.
• Safar M.
• Pressé S.
• et al.
Single photon smFRET. I. theory and conceptual basis.
). We sample the system state trajectory $s1:N$ by recursively sampling the states using a forward filtering backward sampling algorithm described in section S4.3.
Finally, for each row in the propagator $Πσ$, we use a Dirichlet prior
$πm∼Dirichlet(αβ),m=1,2,…,Mσ,$
(10)

where $Mσ$ is the number of system states and $πm$ denotes the $m$-th row of the propagator. Here, the hyperparameters $α$ and $β$ are, respectively, the concentration parameter and a vector of length $Mσ$ described in the first companion article (see Section 3.2.2 of (
• Saurabh A.
• Safar M.
• Pressé S.
• et al.
Single photon smFRET. I. theory and conceptual basis.
)). We can now directly generate samples for the transition probability vectors $πm$ of length $Mσ$ via prior-likelihood conjugacy as (see section S4.3)
$πm∼Dirichlet(nm+αβ),m=1,2,…,Mσmax,$

where the vector $nm$ collects the number of times each transition out of system state $σm$ occurs obtained using the system state trajectory.
After constructing the posterior, we can make inferences on the parameters by drawing samples from the posterior. However, as the resulting posterior has a nonanalytical form, it cannot be directly sampled. Therefore, we develop a Markov chain Monte Carlo sampling procedure (
• Bryan 4th, J.S.
• Sgouralis I.
• Pressé S.
Diffraction-limited molecular cluster quantification with Bayesian nonparametrics.
,
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• Jazani S.
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,
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Equation of state calculations by fast computing machines.
,
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Monte Carlo sampling methods using Markov chains and their applications.
,
• Fazel M.
• Wester M.J.
• Lidke K.A.
• et al.
Bayesian multiple emitter fitting using reversible jump Markov chain Monte Carlo.
,
• Jazani S.
• Sgouralis I.
• Pressé S.
• et al.
An alternative framework for fluorescence correlation spectroscopy.
) to draw samples from the posterior.
Our Markov chain Monte Carlo sampling scheme follows a Gibbs sampling technique, sweeping through updates of the set of parameters in the following order: 1) photophysical transition rates including donor relaxation rates $λd$ (inverse of donor lifetime), acceptor relaxation rate $λa$ (inverse of acceptor lifetime), FRET rates $λσ1:MσFRET$ for each system state, and excitation rate (inverse of excitation probability $πex$) using the Metropolis-Hastings(MH) step; 2) transition probabilities between system states, $π1:Mσ$, by directly drawing samples from the posterior; 3) the system states trajectory, $S$, using a forward-backward sampling procedure (
• Rabiner L.R.
A tutorial on hidden Markov models and selected applications in speech recognition.
); and 4) the initial probabilities, $ρstart$, by taking direct samples. In the end, the chains of samples drawn can be used for subsequent numerical analysis.

#### Inference procedure: Nonparametrics sampler

The smFRET data analysis method illustrated above assumes a given number of system states, $Mσ$. However, in many applications, the number of system states is not specified a priori. Here, we describe a generalization of our parametric method to address this shortcoming and estimate the number of system states simultaneously along with other unknown parameters.
We accomplish this by modifying our previously introduced parametric posterior as follows. First, we suppose an infinite number of system states $(Mσ→∞)$ for the likelihood introduced previously and learn the transition matrix $Πσ$. The number of system states can then be interpreted as those appreciably visited over the course of the trajectory.
To incorporate this infinite system state space into our inference strategy, we leverage the iHMMs (
• Sgouralis I.
• Pressé S.
An introduction to infinite HMMs for single-molecule data analysis.
,
• Sgouralis I.
• Pressé S.
• et al.
A Bayesian nonparametric approach to single molecule Förster resonance energy transfer.
,
• Teh Y.W.
• Jordan M.I.
• Blei D.M.
• et al.
Hierarchical Dirichlet processes.
,
• Jayaram S.
A constructive definition of Dirichlet priors.
,
• Pitman J.
Poisson–Dirichlet and GEM invariant distributions for split-and-merge transformations of an interval partition.
) from the BNP repertoire, placing a hierarchical Dirichlet process prior over the infinite set of system states as described in the first companion article (the inverse strategy section in (
• Saurabh A.
• Safar M.
• Pressé S.
• et al.
Single photon smFRET. I. theory and conceptual basis.
)). However, as detailed in the first companion manuscript (the inverse strategy section in (
• Saurabh A.
• Safar M.
• Pressé S.
• et al.
Single photon smFRET. I. theory and conceptual basis.
)), dealing with an infinite number of random variables, though feasible, is not computationally efficient, and we approximate this infinite value with a large number, $Mσmax$, reducing our hierarchical Dirichlet process prior to
$β∼Dirichlet(γMσmax,…,γMσmax),πm∼Dirichlet(αβ),m=1,…,Mσmax.$

Here, $β$ denotes the base probability vector of length $Mσmax$ serving itself as a prior on the probability transition matrix $Πσ$, and $πm$ is the $m$-th row of $Πσ$. Moreover, $γ$ is a positive scalar hyperparameter of the Dirichlet process prior often chosen to be one. As such, we ascribe identical weights across the state space a priori for computational convenience (
• Teh Y.W.
• Jordan M.I.
• Blei D.M.
• et al.
Hierarchical Dirichlet processes.
,
• Jayaram S.
A constructive definition of Dirichlet priors.
,
• Fazel M.
• Alexander V.
• Pressé S.
• et al.
Fluorescence lifetime: beating the irf and interpulse window.
).
Now, equipped with the nonparametric posterior, we proceed to simultaneously make inferences on transition probabilities, excited-state escape rates, and the remaining parameters. To do so, we employ the Gibbs sampling scheme detailed in the inverse strategy section in the first companion article (
• Saurabh A.
• Safar M.
• Pressé S.
• et al.
Single photon smFRET. I. theory and conceptual basis.
), except that we must now also sample the system state trajectory $s1:N$. More details on the overall sampling scheme are found in section S4 of the supporting material.

## Results

The main objective of our method is to learn full distributions over 1) transition probabilities among $Mσmax$ system states determining, in turn, the corresponding system transition rates and the effective number of system states, and 2) photophysical transition rates, including FRET rates $λ1:MFRET$, and fluorophores’ relaxation rates (inverse of lifetimes) $λa$ and $λd$.
To sample from distributions over these parameters, the BNP-FRET sampler requires input data comprised of photon arrival time traces from both donor and acceptor channels as well as a set of precalibrated input parameters including camera effects such as cross talk matrix and detection efficiency (see Sec. 2.4 and example V of the first companion article (
• Saurabh A.
• Safar M.
• Pressé S.
• et al.
Single photon smFRET. I. theory and conceptual basis.
)); background emission (see the section on background in the first companion article and Section S2.4); and the instrument response function (IRF) (see illumination features sectoin in the first companion article (
• Saurabh A.
• Safar M.
• Pressé S.
• et al.
Single photon smFRET. I. theory and conceptual basis.
) and Section S2.3).
Here, we first show that our method samples posteriors over a set of parameters employing realistic synthetic data generated using the Gillespie algorithm (
• Gillespie D.T.
A general method for numerically simulating the stochastic time evolution of coupled chemical reactions.
) to simulate system and photophysical transitions while incorporating detector artefacts such as crosstalk (see the synthetic data generation section in the first companion article (
• Saurabh A.
• Safar M.
• Pressé S.
• et al.
Single photon smFRET. I. theory and conceptual basis.
)). The list of parameters used in data generation for all figures is provided in Section S6. Furthermore, prior hyperparameters used in the analysis of synthetic and experimental data are listed in Section S3.
We first show that our method works for the simplest case of slow transitions compared with the interpulse period (25 ns) with two system states using synthetic data (see Fig. 2). Next, we proceed to tackle more challenging synthetic data with three system states and higher transition rates (Fig. 3). We show that our nonparametric algorithm correctly infers system transition probabilities and thus the number of system states (see Fig. 3).
After demonstrating the performance of our method using synthetic data, we use experimental data to investigate the kinetics of HJs under different $MgCl2$ concentrations in buffer (see Fig. 4).

### Simulated data analysis

To help validate BNPs on smFRET single-photon data, we start with a simple case of a two-state system and select kinetics similar to those of the experimental data sets, c.f., the HJ in 10 mm $MgCl2$, with escape rates of 40 s−1 for both system states (
• Kilic Z.
• Sgouralis I.
• Pressé S.
• et al.
Extraction of rapid kinetics from smFRET measurements using integrative detectors.
). The generated system state trajectory and photon traces over a period of 500 ms from both channels are shown in Fig. 2 a.
Fig. 2 b shows the bivariate posterior distribution over FRET efficiencies, $εFRET$, defined as $εFRET=λFRET/(λFRET+λd)$, and system escape rates, i.e., obtained by computing the logarithm of the propagator matrix, with two peaks corresponding to the two system states most visited by the sampler. Furthermore, the ground truths, designated by red dots, fall within the posterior with a relative error of less than 3% from the posterior modes. The results for the remaining parameters, including donor and acceptor transition rates, FRET transition rates, and system transition probabilities, are presented in Section S7.
To showcase the critical role played by BNPs, we also consider the more difficult case of a sample with three system states and faster system state kinetics ranging over 1,200–2,500 $s−1$. We do so by simulating photon traces in both donor and acceptor channels over a period of ∼150 ms. A 50 ms section of the synthetic photon trace is shown in Fig 3 a.
Using direct photon arrivals from the generated photon trace, we find that the most probable system state trajectories sampled by BNP-FRET visit the correct number of system states, as shown in Fig 3 b, while inferring all other parameters. Furthermore, the BNP-FRET sampler estimates the system transition rates and thus the escape rates (i.e., sum of transition rates out of a given state) where the ground-truth escape rates differ from the posterior peaks by a relative average error of less than 8%. The results for the remaining parameters are provided in Section S7.

### Experimental data analysis: HJ

In this section, we benchmark our method over a wide range of kinetic rates employing experimental data acquired using HJ under varying buffer $MgCl2$ concentrations (
• Heo W.
• Hasegawa K.
• Tahara T.
• et al.
Scanning two-dimensional fluorescence lifetime correlation spectroscopy: Conformational dynamics of DNA Holliday junction from microsecond to subsecond.
,
• Kilic Z.
• Sgouralis I.
• Pressé S.
• et al.
Extraction of rapid kinetics from smFRET measurements using integrative detectors.
).
HJs are four-way double-helical DNA junctions existing in various structural configurations (
• McKinney S.A.
• Déclais A.C.
• Ha T.
• et al.
Structural dynamics of individual Holliday junctions.
,
• Karymov M.
• Daniel D.
• Lyubchenko Y.L.
• et al.
Holliday junction dynamics and branch migration: single-molecule analysis.
,
• Ferapontova E.E.
• Mountford C.P.
• Mount A.R.
• et al.
Electrochemical control of a DNA Holliday Junction nanoswitch by Mg2+ ions.
). When not interacting with multivalent metal ions, electrostatic repulsion between negatively charged phosphate groups of the four helical arms forces HJs to assume a wide configuration where the arms lie along the two diagonals of a square. However, in the presence of ions, such as Mg2+, interaction with the phosphate groups results in electrostatic screening. This reduced repulsion induces transitions to what is believed to be primarily two compact stacked configurations/conformations. The transitions between both conformations necessitates passing through the intermediate open configuration. Since, at high ion concentrations, displacing ions away from the phosphate group becomes increasingly difficult, in this scenario, we anticipate smaller transition rates between both conformations.
The HJ kinetic rates have been studied using both fluorescence lifetime correlation spectroscopy (
• Heo W.
• Hasegawa K.
• Tahara T.
• et al.
Scanning two-dimensional fluorescence lifetime correlation spectroscopy: Conformational dynamics of DNA Holliday junction from microsecond to subsecond.
) and HMM analysis (
• Pirchi M.
• Tsukanov R.
• Nir E.
• et al.
Photon-by-photon hidden Markov model analysis for microsecond single-molecule FRET kinetics.
) on diffusing HJs assuming a priori a pair of high and low FRET system states. As expected, these previous studies show kinetic rates decreasing with increasing $MgCl2$ concentrations (
• McKinney S.A.
• Déclais A.C.
• Ha T.
• et al.
Structural dynamics of individual Holliday junctions.
,
• Panyutin I.G.
• Biswas I.
• Hsieh P.
A pivotal role for the structure of the Holliday junction in DNA branch migration.
) and correspondingly longer dwells.
Here, our method, free from averaging and binning that are otherwise common in HMM analysis, is particularly well suited to learn the rapid kinetics at low Mg2+ concentrations. We apply our BNP-FRET to data acquired from HJs at 1, 3, 5, and 10 mM $MgCl2$ concentrations and sample the photophysical transition rates and the system transition probabilities.
The acquired bivariate posterior distributions over the FRET efficiencies and escape rates (computed via the logarithm of the system transition probability matrix $Πσ$) are presented in Fig. 4. Moreover, estimates for the other parameters can be found in Section S7. We note that our results are obtained on a single-molecule basis with a photon budget of $104–105$ photons.
For all four concentrations (see Fig. 4), our BNP-FRET sampler most frequently visited only two system states, while this was given as an input to the other analysis methods (
• Heo W.
• Hasegawa K.
• Tahara T.
• et al.
Scanning two-dimensional fluorescence lifetime correlation spectroscopy: Conformational dynamics of DNA Holliday junction from microsecond to subsecond.
,
• Pirchi M.
• Tsukanov R.
• Nir E.
• et al.
Photon-by-photon hidden Markov model analysis for microsecond single-molecule FRET kinetics.
). Moreover, both escape rates are found to have similar values with an average of approximately 1,400s−1 (1 mM $MgCl2$), 140s−1 (3 mM $MgCl2$), 72s−1 (5 mM $MgCl2$), and 41 s−1 (10 mM $MgCl2$). These escape rates are in close agreement with values reported by fluorescence lifetime correlation spectroscopy and H2MM methods (
• Heo W.
• Hasegawa K.
• Tahara T.
• et al.
Scanning two-dimensional fluorescence lifetime correlation spectroscopy: Conformational dynamics of DNA Holliday junction from microsecond to subsecond.
,
• Pirchi M.
• Tsukanov R.
• Nir E.
• et al.
Photon-by-photon hidden Markov model analysis for microsecond single-molecule FRET kinetics.
) of $≈1,300$ s−1 (1 mM $MgCl2$), $≈170$ (3 mM $MgCl2$), $≈100$ (5 mM $MgCl2$), and $≈60$ s−1 (10 mM $MgCl2$), which lie well within the bounds of our posteriors shown in Fig. 4 while simultaneously, and self-consistently, learning a number of system states.

### Experimental data acquisition

In this section, we describe the protocol for preparing the surface-immobilized HJ sample labeled with a FRET pair and the procedure for recording smFRET traces from individual immobilized molecules. The sample preparation and recording of data follow previous work (
• Kilic Z.
• Sgouralis I.
• Pressé S.
Generalizing HMMs to continuous time for fast kinetics: Hidden Markov jump processes.
).

#### Sample preparation

The HJ used in this work consists of four DNA strands whose sequences are as follows:
• R-strand: 5′-CGA TGA GCA CCG CTC GGC TCA ACT GGC AGT CG-3′
• H-strand: 5′-CAT CTT AGT AGC AGC GCG AGC GGT GCT CAT CG-3′
• X-strand: 5′-biotin-TCTTT CGA CTG CCA GTT GAG CGC TTG CTA GGA GGA GC-3′
• B-strand: 5′-GCT CCT CCT AGC AAG CCG CTG CTA CTA AGA TG-3’
For surface immobilization, the X-strand was labeled with biotin at the 5′ end. For FRET measurements, the donor (ATTO-532) and acceptor (ATTO-647N) dyes were introduced into the H- and B-strands, respectively. In both cases, the dyes were labeled to thymine nucleotide at the 6th position from the 5′ ends of the respective strands (shown as T). All DNA samples (labeled or unlabeled) were purchased from JBioS (Shinjuku-ku, Japan) in the high-performance liquid chromatography purified form and were used without any further purification.
The HJ complex was prepared by mixing 1 mM solutions of R-, H-, B-, and X-strands in TN buffer (10 mM Tris-$HCl$ with 50 mm $NaCl$, pH 8) at a 3:2:3:3 molar ratio, annealing the mixture at 94°C for 4 minutes, and gradually cooling it down (2°C–3°C  min−1) to room temperature (25°C ). For smFRET measurements, we used a sample chamber (SecureSeal, GBL621502, Grace Bio-Labs, Bend, OR, USA) with a biotin-PEG-SVA (biotin-poly(ethylene glycol)-succinimidyl valerate)-coated coverslip. The chamber was first incubated with streptavidin (0.1 mg mL−1 in TN buffer) for 20 min. This was followed by washing the chamber with TN buffer (3 times) and injection of 1 nM HJ solution (with respect to its H-strand) for 3–10 s. After this incubation period, the chamber was rinsed with TN buffer (3 times) to remove unbound DNA, and it was filled with TN buffer containing 1 mM (or 5 mM) $MgCl2$ and 2 mM Trolox for smFRET measurements.

#### smFRET measurements

The smFRET traces from individual HJs were recorded using a custom-built confocal microscope (Eclipse Ti, Nikon, Tokyo, Japan) equipped with the Perfect Focus System, a sample scanning piezo stage (Nano control B16-055), and a time-correlated single-photon counting module (SPC-130EM, Berlin, Germany).
The broadband light generated by a supercontinuum laser operating at 40 MHz (SC-400-4, Fianium, Southampton, UK) was filtered with a band-pass filter (FF01-525/30, Semrock, West Henrietta, NJ, USA) for exciting the donor dye, ATTO-532. This excitation light was introduced to the microscope using a single-mode optical fiber (P5-460B-PCAPC-1, Thorlabs, Newton, NJ, USA) and directed onto the sample using a dichroic mirror (ZT532/640rpc, Chroma, Cambridge, MA, USA) and a water immersion objective lens (Nikon Plan Apo IR 60×, numerical aperture: 1.27).
The excitation light was focused onto the top surface of the coverslip, and, during measurements, the focusing condition was maintained using the Perfect Focus System. The fluorescence signals were collected by the same objective, passed through the dichroic mirror, and guided to the detection assembly (Thorlabs DFM1/M) using a multimode fiber (Thorlabs M50L02S-A). Note that this multimode fiber (core diameter: 50 μm) also acts as the confocal pinhole. In the detection assembly, the fluorescence signals from the donor and acceptor dyes were separated using a dichroic mirror (ZT633rdc, Chroma Technology, Bellows Falls, VT, USA), filtered using band-pass filters (Chroma ET585/65m for donor and Semrock FF02-685/40 for acceptor), and detected using separate hybrid detectors (Becker and Hickl HPM-100-40-C).
For each detected photon, its macrotime (absolute arrival time from the start of the measurement) was recorded with 25.2 ns resolution and its microtime (relative delay from the excitation pulse) was recorded with 6.1 ps resolution using the time-correlated single-photon counting module operating in time-tagging mode. A router (Becker and Hickl HRT-41) was used to process the signals from the donor and acceptor detectors.
For recording smFRET traces from individual HJs, we first imaged a 10 × 3 μm area of the sample using the piezo stage by scanning it linearly at a speed of 1 μm s−1 in the x direction and with an increment of 0.1 μm in the y direction. Individual HJs appeared as isolated bright spots in the image.
Next, we fitted the obtained donor and acceptor intensity images with multiple 2D Gaussian functions to determine the precise locations of individual HJs. Note that, during this image acquisition, the laser excitation power was kept to a minimum (∼1 μW at the back aperture of the objective lens) to avoid photobleaching the dyes. In addition, we also employed an electronic shutter (Suruga Seiki, Shizuoka, Japan) in the laser excitation path to control the sample excitation as required.
Using the precise locations of individual HJs obtained, we recorded 30 s-long smFRET traces for each molecule by moving them to the center of the excitation beam using the piezo stage. For each trace, the laser excitation was blocked (using the shutter) for the first 5 s and was allowed to excite the sample for the remaining 25 s. Note that the smFRET traces were recorded using 40 μW laser excitation (at the back aperture of the objective lens) to maximize the fluorescence photons emitted from the dyes. We automated the process of acquiring smFRET traces from different molecules sequentially and executed it using a program written in house on Igor Pro (Wavemetrics, Portland, OR, USA).

## Discussion

The sensitivity of smFRET under pulsed illumination has been exploited to investigate many different molecular interactions and geometries (
• Roy R.
• Hohng S.
• Ha T.
A practical guide to single-molecule FRET.
,
• Mazal H.
• Haran G.
Single-molecule FRET methods to study the dynamics of proteins at work.
,
• Schuler B.
Single-molecule FRET of protein structure and dynamics - a primer.
,
• Lu M.
• Ma X.
• Mothes W.
• et al.
Associating HIV-1 envelope glycoprotein structures with states on the virus observed by smFRET.
,
• Sebolt-Leopold J.S.
• English J.M.
Mechanisms of drug inhibition of signalling molecules.
). However, quantitative interpretation of smFRET data faces serious challenges including an unknown number of system states and robust propagation of uncertainty from noise sources such as detectors and background. These challenges ultimately mitigate our ability to determine full distributions over all relevant unknowns and, traditionally, have resulted in data pre- or postprocessing compromising the information otherwise encoded in the rawest form of data: single-photon arrivals.
Here, we provide a general BNP framework for smFRET data analysis starting from single-photon arrivals under a pulsed illumination setting. We simultaneously learn transition probabilities among system states as well as determine photophysical rates by incorporating existing sources of uncertainty such as background and cross talk.
We benchmark our method using both experimental and simulated data. That is, we first show that our method correctly learns parameters for the simplest case with two system states and slow system transition rates. Moreover, we test our method on more challenging cases with more than two states using synthetic data and obtain correct estimations for the system state transition probabilities and thus the number of system states along with the remaining parameters of interest. To further assess our method’s performance, we analyzed experimental data from HJs suspended in solutions with a range of $MgCl2$ concentrations. These data were previously processed using other techniques assuming a fixed number of system states by binning photon arrival times (
• Heo W.
• Hasegawa K.
• Tahara T.
• et al.
Scanning two-dimensional fluorescence lifetime correlation spectroscopy: Conformational dynamics of DNA Holliday junction from microsecond to subsecond.
).
Despite multiple advantages mentioned above for BNP-FRET, BNPs always come with an added computational cost as they take full advantage of information from single-photon arrival times and all existing sources of uncertainty. For this version of our general BNP method simplified for pulsed illumination, we further reduced the computational complexity by grouping empty pulses together. Therefore, the computational complexity increased only linearly with the number of input photons as the photons are treated independently.
The method described in this paper assumes a Gaussian IRF. However, the developed framework is not limited to a specific form for the IRF and can be used for data collected using any type of IRF by modifying Eq. 4. Furthermore, the framework is flexible in accommodating different illumination techniques such as alternating color pulses, which are typically used to directly excite the acceptor fluorophores. This can be achieved by simple modification of the propagator $Qnψ$ in Eq. 4. A future extension of this method could relax the assumption of a static sample by adding spatial dependence to the excitation rate as we explored in previous works (
• Tavakoli M.
• Jazani S.
• Pressé S.
• et al.
Pitching single-focus confocal data analysis one photon at a time with Bayesian nonparametrics.
,
• Jazani S.
• Sgouralis I.
• Pressé S.
• et al.
An alternative framework for fluorescence correlation spectroscopy.
,
• Jazani S.
• Sgouralis I.
• Pressé S.
A method for single molecule tracking using a conventional single-focus confocal setup.
). This would allow our method to learn the dynamics of diffusing molecules, as well as their photophysical and system state transition rates.

### Code availability

The BNP-FRET software package is available on Github at https://github.com/LabPresse/BNP-FRET.

## Acknowledgments

We thank Dr. Zeliha Kilic, Weiqing Xu, and J. Shephard Bryan IV for their contributions and insight into the project. We also thank Dr. Douglas Shepherd for providing insight into the workings of detectors and other experimental equipment and Irina Gopich for discussions on FRET likelihoods. T.T. acknowledges support from the JSPS KAKENHI (grant no. 18H05265). S.P. acknowledges support from the NIH NIGMS (R01GM130745) for supporting early efforts in nonparametrics and NIH NIGMS (R01GM134426) for supporting single-photon efforts.

### Declaration of interests

The authors declare no competing interests.

## Supporting material

• Document S1. Supporting material, Figures S1–S7, and Tables S1–S3

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text:
# Min and Max temperature of path in 3D field

1. Jan 23, 2015

### Tuppe

1. The problem statement, all variables and given/known data
Temperature varies by function T(x,y,z)=3x + 4y + 2z

Path is given by r(t)={
$$x(t)=\frac{t^3}{30}+\frac{16t}{9}+7$$
$$y(t)=-\frac{t^3}{120}-\frac{13t^2}{30}+28$$
$$z(t)=\frac{13t^2}{60}+\frac{4t}{3}-14$$

$$t\in \left[0,10\right]$$

Question: What is the maximum and minumun temperatures of that path?

2. Relevant equations

There's a tip in the assignment to use the chain rule.

3. The attempt at a solution

$$\frac{dT}{dt}=\frac{\partial T}{\partial x}\cdot \frac{\partial x}{\partial t}+\frac{\partial T\:}{\partial \:y}\cdot \:\frac{\partial \:y}{\partial \:t}+\frac{\partial T\:}{\partial \:z}\cdot \:\frac{\partial z}{\partial t}$$

$$\frac{dT}{dt}=3\left(\frac{t^2}{10}+\frac{16}{9}\right)+4\left(-\frac{t\left(3t+104\right)}{120}\right)+2\left(\frac{13t}{30}+\frac{4}{3}\right)$$

$$\frac{dT}{dt}=\frac{1}{5}\left(t^2-13t+40\right)$$

Then I thought to separate the variables and integrate it to get T, but I end but with constant C.
If I ignore the constant, I end up with wrong answer(min -9/20, max 8)
I cannot think how to continue. I don't know the correct answer either.

Do you think the idea is correct to this point?
Maybe it's something obvious, I might be just baffled with all these derivation steps.

Thank you for reading!

Last edited: Jan 23, 2015
2. Jan 23, 2015

### HallsofIvy

Staff Emeritus
I don't understand why you would first differentiate and then "integrate it to get T". You were given T! You seem to have lost track of what you were trying to do.

The reason you differentiated, I hope, was to set the derivative equal to 0 to find the value of t that gives a max or min. Then put whatever value of t you get back into the original equations for x, y, and z.

3. Jan 23, 2015

### Tuppe

Hey, thank you for fast response!

That's exactly what I should do, but for some reason there was so much variables that I didn't realize the obvious course of action.
To my surprise, the correct answer happened to be at the limits of t=0 and t=10 for both, so the resoults from the differentation didn't have any affect.

Thank you for clear explanation!

Last edited: Jan 23, 2015


text:
## Percents

### Learning Outcomes

• Given the part and the whole, write a percent
• Evaluate changes in amounts with percent calculations
• Calculate both relative and absolute change of a quantity

### A percent is a fraction

Recall that a fraction is written $\dfrac{a}{b},$ where $a$ and $b$ are integers and $b \neq 0$. In a fraction, $a$ is called the numerator and $b$ is called the denominator.

percent can be expressed as a fraction, that is a ratio, of some part of a quantity out of the whole quantity,  $\dfrac{\text{part}}{\text{whole}}$.

Ex. Suppose you take an informal poll of your classmates to find out how many of them like pizza. You find that, out of 25 classmates, 20 of them like pizza. You can represent your findings as a ratio of how many like pizza out of how many classmates you asked.

$\dfrac{20}{25}$ represents the 20 out of 25 classmates who like pizza.

To find out what percent of the 25 asked said they like pizza, divide the numerator by the denominator, then multiply by 100.

$\dfrac{20}{25} = 20 \div 25 = 0.80 = 80 \%$

Percent literally means “per 100,” or “parts per hundred.” When we write 40%, this is equivalent to the fraction $\displaystyle\frac{40}{100}$ or the decimal 0.40. Notice that 80 out of 200 and 10 out of 25 are also 40%, since $\displaystyle\frac{80}{200}=\frac{10}{25}=\frac{40}{100}$.

A visual depiction of 40%

### convert a percent to a  decimal or fraction

To do mathematical calculations with a given percent, we must first write it in numerical form. A percent may be represented as a percent, a fraction, or a decimal.

Convert a percent to a fraction

1. Write the percent over a denominator of $100$ and drop the percent symbol %.
2. Reduce the resulting fraction as needed.

Ex. $80 \% =\dfrac{80}{100}=\dfrac{4\cdot 20}{5\cdot 20}=\dfrac{4}{5}$

Convert a percent to a decimal

There are three methods for writing a percent as a decimal.

• You can write the percent as a fraction, simply fully, then divide the numerator by the denominator.
• $80 \% =\dfrac{80}{100}=\dfrac{4\cdot 20}{5\cdot 20}=\dfrac{4}{5}=0.8$
• You can write the percent as a fraction, simplify to a denominator of 10, 100, 1000, etc., then rewrite as a decimal.
• $80 \% =\dfrac{80}{100}=\dfrac{8\cdot 10}{10\cdot 10}=\dfrac{8}{10}=\text{ eight tenths }=0.8$
• Write the percent without the percent symbol %, then place a decimal after the ones place and move it two places to the left.
• $80 \% =80.0=0.80=0.8$

### Percent

If we have a part that is some percent of a whole, then $\displaystyle\text{percent}=\frac{\text{part}}{\text{whole}}$, or equivalently, $\text{percent}\cdot\text{whole}=\text{part}$.

To do calculations using percents, we write the percent as a decimal or fraction.

The video and following few examples demonstrate how to convert between percent, fraction, and decimal representations.

### Example

In a survey, 243 out of 400 people state that they like dogs. What percent is this?

### Example

Write each as a percent:

1. $\displaystyle\frac{1}{4}$
2. 0.02
3. 2.35

### Try It

Throughout this text, you will be given opportunities to answer questions and know immediately whether you answered correctly. To answer the question below, do the calculation on a separate piece of paper and enter your answer in the box. Click on the submit button , and if you are correct, a green box will appear around your answer.  If you are incorrect, a red box will appear.  You can click on “Try Another Version of This Question” as many times as you like. Practice all you want!

In the news, you hear “tuition is expected to increase by 7% next year.” If tuition this year was $1200 per quarter, what will it be next year? ### Try It ### Example The value of a car dropped from$7400 to $6800 over the last year. What percent decrease is this? ### Absolute and Relative Change Given two quantities, Absolute change =$\displaystyle|\text{ending quantity}-\text{starting quantity}|$ Relative change: $\displaystyle\frac{\text{absolute change}}{\text{starting quantity}}$ • Absolute change has the same units as the original quantity. • Relative change gives a percent change. The starting quantity is called the base of the percent change. For a deeper dive on absolute and relative change, using the examples on this page, view the following video. The following examples demonstrate how different perspectives of the same information can aid or hinder the understanding of a situation. ### Example There are about 75 QFC supermarkets in the United States. Albertsons has about 215 stores. Compare the size of the two companies. ### Example Suppose a stock drops in value by 60% one week, then increases in value the next week by 75%. Is the value higher or lower than where it started? A video walk-through of this example can be seen here. ### Try It ### Example A Seattle Times article on high school graduation rates reported “The number of schools graduating 60 percent or fewer students in four years—sometimes referred to as ‘dropout factories’—decreased by 17 during that time period. The number of kids attending schools with such low graduation rates was cut in half.” 1. Is the “decreased by 17” number a useful comparison? 2. Considering the last sentence, can we conclude that the number of “dropout factories” was originally 34? ### Example Let’s return to the example from the last page. In the 2004 vice-presidential debates, Democratic candidate John Edwards claimed that US forces have suffered “90% of the coalition casualties” in Iraq. Cheney disputed this, saying that in fact Iraqi security forces and coalition allies “have taken almost 50 percent” of the casualties. Who is correct? A detailed explanation of these examples can be viewed here. ### Think About It In the 2012 presidential elections, one candidate argued that “the president’s plan will cut$716 billion from Medicare, leading to fewer services for seniors,” while the other candidate rebuts that “our plan does not cut current spending and actually expands benefits for seniors, while implementing cost saving measures.” Are these claims in conflict, in agreement, or not comparable because they’re talking about different things?

We’ll wrap up our review of percents with a couple cautions. First, when talking about a change of quantities that are already measured in percents, we have to be careful in how we describe the change.

### Example

A politician’s support increases from 40% of voters to 50% of voters. Describe the change.

Lastly, a caution against averaging percents.

### Example

A basketball player scores on 40% of 2-point field goal attempts, and on 30% of 3-point of field goal attempts. Find the player’s overall field goal percentage.


text:
# STRONG LAWS OF LARGE NUMBERS FOR RANDOM UPPER-SEMICONTINUOUS FUZZY SETS

Kim, Yun-Kyong

• Published : 2002.08.01
• 59 7

#### Abstract

In this paper, we concern with SLLN for sums Of in-dependent random upper-semicontinuous fuzzy sets. We first give a generalization of SLLN for sums of independent and level-wise identically distributed random fuzzy sets, and establish a SLLN for sums of random fuzzy sets which is independent and compactly uniformly integrable in the strong sense. As a result, a SLLN for sums of independent and strongly tight random fuzzy sets is obtained.

#### Keywords

random fuzzy sets;strong law of large numbers;compactly uniform integrability;tightness

#### References

1. Bull. Inst. Math. Academia Sinica v.10 Tightness and strong laws of large numbers in Banach spaces P. Z. Daffer;R. L. Taylor
2. Ann. Probab. v.13 Convexification in limit laws of random sets in Banach spaces Z. Artstein;J. C. Hansen https://doi.org/10.1214/aop/1176993083
3. Fuzzy Sets and Systems v.64 Marcinkiewicz-type law of large numbers for fuzzy random variables D. H. Hong;H. J. Kim https://doi.org/10.1016/0165-0114(94)90161-9
4. Fuzzy Sets and Systems v.59 A law of large numbers for random sets T. Uemura https://doi.org/10.1016/0165-0114(93)90197-P
5. Fuzzy Sets and Systems v.120 Kolmogorov/s strong law of laarge numbers for fuzzy random variables S. Y. Joo;Y. K. Kim https://doi.org/10.1016/S0165-0114(99)00140-2
6. Jour. Math. Anal. Appl. v.235 On strong laws of large numbers for random upper semicontinuous functions I. S. Molchanov https://doi.org/10.1006/jmaa.1999.6403
7. Proc. 5th Berkeley Symp. Math. Statist. Prob. v.2 Integration of correspondences G. Debreu
8. Bull. Inst. Math. Academia Sinica v.13 A strong law of large numbers for random sets in Banach spaces R. L. Taylor;H. Inoue
9. Fuzzy Sets and Systems v.31 Measurability concepts for fuzzy mappings D. Butnariu https://doi.org/10.1016/0165-0114(89)90068-7
10. Ann. Probab. v.3 A strong law of large numbers for random compact sets Z. Artstein;R. A. Vitale https://doi.org/10.1214/aop/1176996275
11. Proc. Roy. Soc. Lond. Ser. A v.407 Limit theorems for fuzzy random variables E. P. Klement;M. L. Puri;D. A. Ralescu https://doi.org/10.1098/rspa.1986.0091
12. Fuzzy Sets and Systems v.129 Measurability for fuzzy valued functions Y. K. Kim https://doi.org/10.1016/S0165-0114(01)00121-X
13. Jour. Math. Anal. Appl. v.246 Topological properties on the space of fuzzy sets S. Y. Joo;Y. K. Kim https://doi.org/10.1006/jmaa.2000.6820
14. Jour. Math. Anal. Appl. v.114 Fuzzy random variables M. L. Puri;D. A. Ralescu
15. Convergence of probability measures P. Billingsley
16. Fuzzy Sets and Systems v.3 Convex fuzzy sets R. Lowen https://doi.org/10.1016/0165-0114(80)90025-1
17. Fuzzy Sets and Systems v.41 A strong law of large numbers for fuzzy random sets H. Inoue https://doi.org/10.1016/0165-0114(91)90132-A
18. Jour. Korean Statist. Soc. v.31 Strong law of large numbers for tight fuzzy random variables S. Y. Joo

#### Cited by

1. Strong laws of large numbers for tight fuzzy random variables vol.13, pp.6, 2003, https://doi.org/10.5391/JKIIS.2003.13.6.754


text:
# Show that potential energy is conserved

Tags:
1. Apr 19, 2017

### gelfand

1. The problem statement, all variables and given/known data

potential energy function of :

$$U(x) = 4x^2 + 3$$

And have to

i) Work out the equation of motion

ii) Prove explicitly that the total energy is conserved

2. Relevant equations

$$F = \frac{dU}{dt}$$

3. The attempt at a solution

I would say that I have the force of

$$F = 8x$$

By differentiating the given potential energy function. I need to work out the
equation of motion, what I have an object with mass $m$.

So this means that I have

$$F = 8x = ma$$

Then I have that

$$a = \frac{8x}{m}$$

Is this an equation of motion? I mean, it's acceleration, or should I find for
$v(t)$ and $x(t)$ as well as this?

In which case I would have

$$v(t) = \int a(t) dt$$

Which in this case is found as (having the mass in the equation seems unusual?)

$$v(t) = v_0 + \frac{1}{2m}8x^2 = v_0 + \frac{4}{m} x^2$$

So then from this I have that

$$x(t) = x_0 + v_0t + \frac{4}{3m}x^3$$

And this would be all of the equations of motion for this 1D case?

Then I need to prove that energy is conserved here, and I've no idea how to go

I've not been given any frictional forces, so it seems like it's just a given
that I'm going to have

$$W + PE_0 + KE_0 = PE_f + KE_f + \text{Energy(Lost)}$$

Here I can remove work $W$ and the energy lost for

$$PE_0 + KE_0 = PE_f + KE_f$$

And I need to do something with these?

Potential energy - I have the potential energy function given as part of the
problem which is

$$U(x) = 4x^2 + 3$$

Then I can sub this into the energy expression as
$$4x_0^2 + 3 + KE_0 = 4x_f^2 + 3 + KE_f$$

Getting rid of the constants seems pretty harmless

$$4x_0^2 + KE_0 = 4x_f^2 + KE_f$$

Now I'm really not sure what I should do from here, sub in kinetic formulas of
$K = \frac{1}{2}mv^2$?

$$4x_0^2 + \frac{1}{2}mv_0^2 = 4x_f^2 + \frac{1}{2}mv_f^2$$

I'm not sure if I can arrange this to be 'nicer' in any way either, I'm purely
thinking in algebra at the moment though not physics :S

$$8(x_0^2 - x_f^2) = m(v_f^2 - v_0^2)$$

I'm not sure if differentiation should do anything nice here, but I really have
no idea what I'm doing with this.

Thanks

2. Apr 19, 2017

### haruspex

Dividing energy by time gives power, not force.

3. Apr 20, 2017

### gelfand

OK $F = - \frac{dU}{dx}$ sorry , i'm still unsure about the question

4. Apr 20, 2017

### haruspex

You got a=8x/m ok, but you cannot integrate that wrt t directly. The expression you got for v(t) was the integral wrt x (which just gets you back to U).

There is a useful trick for solving equations like $\ddot x=f(x)$. Multiply both sides by $\dot x$, then integrate dt.


text:
0
Research Papers: Jets, Wakes, and Impingement Cooling

# Film-Cooling Enhancement of the Mist Vertical Wall Jet on the Cylindrical Channel Surface With Heat Transfer

[+] Author and Article Information
V. I. Terekhov

Kutateladze Institute of Thermophysics, Russian Academy of Sciences, Siberian Branch, 1 Academician Lavrent'ev Avenue, Novosibirsk 630090, Russiaterekhov@itp.nsc.ru

M. A. Pakhomov

Kutateladze Institute of Thermophysics, Russian Academy of Sciences, Siberian Branch, 1 Academician Lavrent'ev Avenue, Novosibirsk 630090, Russia

J. Heat Transfer 131(6), 062201 (Mar 30, 2009) (10 pages) doi:10.1115/1.3082404 History: Received March 25, 2007; Revised October 10, 2008; Published March 30, 2009

## Abstract

Results of a numerical study of heat and mass transfer in a gas-droplet wall jet developing over a surface with supplied heat flux are reported. The calculation model is based on an Eulerian/Eulerian approach using a turbulence model that allows for the dynamic and thermal interactions between the phases. This approach is based on using the kinetic equation of probability density function for the coordinates, velocities, and temperatures of droplets in the turbulent flow (Derevich, I. V., “The Hydrodynamics and Heat Transfer and Mass Transfer of Particles Under Conditions of Turbulent Flow of Gas Suspension in a Pipe and in an Axisymmetric Jet  ,” 2002, High Temp., 40, pp. 78–91). The effects due to many factors are traced, including the dispersed-phase concentration in the wall jet, the droplet diameter, the blowing ratio $m=ρSUS/ρ1U1$, the main-flow temperature, and the wall heat flux. A considerable enhancement of heat transfer at relatively low mass concentrations of droplets $MLS$ in the jet (more than twofold at $MLS≤0.05$) has been revealed.

###### FIGURES IN THIS ARTICLE
<>
Copyright © 2009 by American Society of Mechanical Engineers
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## Figures

Figure 1

Schematic of the gas-droplet wall jet propagating along the wall in the presence of heat flux: (1) air and water droplets and (2) main air flow

Figure 2

Variation in momentum thickness at the injection of a single-phase jet; solid curve—calculation by formula 11, dashed line—calculation by this numerical model, and points—experimental data: (1) Ref. 33, (2) Ref. 34, and (3) Ref. 35

Figure 3

Heat transfer in the cooling region: (1) calculation by the numerical model and (2) as predicted by Eq. 14

Figure 4

Profiles of gaseous phase velocity at the injection of the gas-droplet wall jet: (1) x/S=0, (2) 25, (3) 50, (4) 100, (5) 150, and (6) 200 (qW=5 kW/m2, T1=373 K, TS=TLS=293 K, U1=50 m/s, dS=30 μm, m=0.8, MVS=0.014, MLS=0.05)

Figure 5

Variation in droplet velocity in the two-phase jet over the channel length: (1) x/S=0, (2) 25, and (3) 50. Conditions adopted in the calculations are the same as in Fig. 4.

Figure 6

Effect of droplet concentration MLS (a) and droplet diameter (b) on the gas-flow turbulence in the wall jet: (1) dS=10 μm, (2) 30 μm, (3) 50 μm, and (4) 100 μm(qW=5 kW/m2, T1=373 K, TS=TLS=293 K, U1=50 m/s, m=0.8, MVS=0.014, x/S=50)

Figure 7

Distribution of flow quantities in the two-phase film flow over the channel cross section: (1) ML/MLS, (2) d/dS, (3) Θ, (4) ΘH, (5) U/U1, (6) (MVW−MV)/(MVW−M0), and (7) profile of Θ=U/U0=(y/δt)1/7(qW=5 kW/m2, T1=373 K, TS=TLS=293 K,U1=50 m/s, m=0.8, MVS=0.014, MLS=0.05, dS=30 μm, x/S=50)

Figure 8

Profiles of the mass concentration of steam over the transverse coordinate: (1) x/S=25, (2) 50, (3) 100, (4) 150, and (5) 200. Conditions adopted in the calculations are the same as in Fig. 4.

Figure 9

Wall surface temperature versus the droplet concentration in the two-phase jet: (1) MLS=0, (2) 0.005, (3) 0.01, (4) 0.025, (5) 0.05, (6) TS=TLS=293 K, and (7) T1=373 K(qW=5 kW/m2, T1=373 K, TS=TLS=293 K, U1=50 m/s, m=0.8, MVS=0.014, dS=30 μm)

Figure 10

Heat transfer intensification ratio versus axial coordinate for air main-flow temperature: (1) T1=323 K, (2) 373 K, (3) 423 K, and (4) 473 K (qW=5 kW/m2, TS=TLS=293 K, U1=50 m/s, m=0.8, dS=30 μm, MVS=0.014, MLS=0.05)

Figure 11

Heat transfer in the gas-droplet flow for various values of the relative heat flux q¯×103: (1) 1.4, (2) 2, (3) 2.8, and (4) 5.5 (T1=373 K, TS=TLS=293 K, U1=50 m/s, m=0.8, MVS=0.014, MLS=0.05)

Figure 12

Effect of blowing ratio m on heat transfer augmentation: (1) x/S=25, (2) 50, (3) 100, and (4) 200 (qW=5 kW/m2, T1=373 K, TS=TLS=293 K, U1=50 m/s, MVS=0.014, MLS=0.05, x/S=50)

Figure 13

Heat transfer in gas-droplet jets with different initial droplet diameters: dS=10 μm, (2) 30 μm, (3) 50 μm, and (4) 100 μm

Figure 14

Variation in the droplet size over the cylindrical channel length: (a) for various particle sizes: (1) dS=10 μm, (2) 30 μm, (3) 50 μm, and (4) 100 μm(MLS=0.05); (b) for various dispersed-phase mass concentrations: (1) MLS=0.005, (2) 0.01, (3) 0.025, and (4) 0.05 (dS=30 μm); (c) cross-section profiles of droplet diameter: (1) x/S=0, (2) 25, (3) 50, (4) 100, and (5) 200 (qW=5 kW/m2, T1=373 K, TS=TLS=293 K, U1=50 m/s, m=0.8, MVS=0.014, MLS=0.05)

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text:
# Escher Illusions in LaTeX

Author
Julien Cretel
AbstractA drawing in TikZ. The first picture draws an impossible brick, which induces an optical illusion similar to that triggered by Escher's impossible cube. The second picture draws a Penrose triangle, another similar optical illusion.


text:
Prealgebra

Key Terms

PrealgebraKey Terms

Key Terms

binomial
A binomial is a polynomial with exactly two terms.
degree of a constant
The degree of a constant is $00$.
degree of a polynomial
The degree of a polynomial is the highest degree of all its terms.
degree of a term
The degree of a term of a polynomial is the exponent of its variable.
greatest common factor
The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.
monomial
A term of the form $axmaxm$, where $aa$ is a constant and $mm$ is a whole number, is called a monomial.
negative exponent
If $nn$ is a positive integer and $a≠0a≠0$, then $a−n=1ana−n=1an$.
polynomial
A polynomial is a monomial, or two or more monomials, combined by addition or subtraction.
scientific notation
A number expressed in scientific notation when it is of the form $a×10n,a×10n,$ where $a≥1a≥1$ and $a<10,a<10,$ and $nn$ is an integer.
trinomial
A trinomial is a polynomial with exactly three terms.
zero exponent
If $aa$ is a non-zero number, then $a0=1a0=1$. Any nonzero number raised to the zero power is 1.
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