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Konvolut The Personal Distribution of Income 1
Full text: Konvolut The Personal Distribution of Income 1
/fyOt 1
' /Ly^
There are only two alternative transiticrs for a person in
this system: Either a rise of income from one year to the
next (note that the income classes are on the log scale)
which has probability p; or death of the person, i.e.
transition to the zero class, which has probability q
(p + q = 1). In addition, there are entries from the zero
to «
class/^replenish the stock of income receiver?
The essence of this model is described by Peller ' in the
following terms: The state Ek, represents the age of the
system. When the system reaches age K, it either continues
to age or it rejuvenates and starts afresh from age zero.
The successive passages through the zero state represent a
recurrent event. The probability that the recurrence time
equals K is p q.
We are interested in the question: How many years have
passed, i.e. how many income steps have been mounted, since
the last rejuvenation? This is the ''spent waiting time"
of the renewal process. Choosing an arbitrary starting point
we can say that in the year n the system will be in state Ek
if and only if the last rejuvenation occured in year n-K.
Letting n-fc increase we obtain in the limit the steady state
probability of the 'hpent waiting time" . It is proportionate
f\%7 vol. I. 17.3, p.
Chanter V. | {"url":"https://viewer.wu.ac.at/viewer/fulltext/AC14446015/49/","timestamp":"2024-11-03T16:48:29Z","content_type":"application/xhtml+xml","content_length":"70405","record_id":"<urn:uuid:f51fc919-f360-4799-a3d6-9fd2c264da51>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00312.warc.gz"} |
News this weekend that San Jose State in California has suspended its experiment with Udacity offering low-level courses for pay and college credit and requirements:
Key point: "Initial findings suggest that students in Udacity courses performed poorly compared with students in traditional classes." Note that this is broadly in line with the prediction I made
here several weeks ago, in the post titled Online Remedial Courses Considered Harmful, something that I considered to be a fairly easy and obvious call. I asked the question, "We'll see how quickly
MOOCs such as UDacity, and those partnering, paying, and linking their reputation with them, re-learn this lesson", and I'd have to say that this turnaround was faster than I would have guessed at
San Jose State. Perhaps they will agree with the earlier experiment at the Philadelphia school where it was concluded, "The failure rates were so high that it seemed almost unethical to offer the
option" (see link to my earlier post above).
The last paragraph of today's news story reiterates my own views, which I've written about here on numerous occasions: "Educators elsewhere have said the purely online courses aren't a good fit for
remedial students who may lack the self-discipline, motivation and even technical savvy to pass the classes. They say these students may benefit from more one-on-one attention from instructors."
A few other points: "Preliminary results from a spring pilot project found student pass rates of 20% to 44% in remedial math, college-level algebra and elementary statistics courses." Now, it would
be much better if this success rate were broken down individually for each of these several classes. I might guess that the 20% success rate is specifically for the remedial math course? That does
seem marginally lower than most remedial courses where the success rate seems to be around one-quarter or one-third.
Also, the article says, "In a somewhat more promising outcome, 83% of students completed the classes." This seems unsurprising, given that students are paying $150 out-of-pocket for the course. This
completion (but mostly failing) rate is about in line with the remedial courses that I teach, where students are similarly paying, meeting an absolute requirement by the college, and have no real
academic penalty for failing (the course grade does not affect GPA, for example).
Perhaps charitably we might say that the $150 expense level is lower than standard college teaching costs, and perhaps someone might think it's a reasonable return on investment, even granted a lower
success rate (although maybe not when accounting for student time spent). And we might also be suspicious of (a) whether this is the actual Udacity expense, or if they're operating at a loss to
establish the market, and (b) the quality of the assessment at the end, when there's a clear incentive to make it easy to pass and the Udacity statistics final I've seen in the past was almost
comically trivial.
Supposedly this suspension is for re-tooling and analysis of possible improvements. "The courses will be offered again next spring, [San Jose State Provost Ellen Junn] said." We shall see.
The in-house textbook that my college uses for basic algebra classes does an interesting thing -- as part of the introduction to radicals, it goes through approximating a whole-number radical by
comparing it to the nearest perfect squares. An example from the book:
Example 2: √3000 is closest to which integer?
Solution:... [after some preliminary estimates] Try between 50 and 60, (55)^2 = 3025, still a bit too high. Try (54)^2 = 2916, now a little too low. Thus √3000 is between 54 and 55, but closer to
55 since 3025 is closer to 3000 than is 2916.
So I think we all agree that in a case like this, the radical is clearly between the two integers indicated (since the radical function is monotonic). But the additional step of saying which of the
two it's closer to is not done in all textbooks. Here's another example (not our school's textbook). Let's clearly state the claim being made here:
Claim: If x is closest to n^2, then √x is closest to n.
Above, "closest" means the minimum distance from x to any n ∈ ℕ. This claim gave me a squirrelly feeling for some time, and with good reason; it isn't true for arbitrary x ∈ ℝ.
Counter-example: Consider x = 12.4. It's closest to the perfect square 3^2 (distance 3.4 from 9, versus 3.6 from 16). But the square root is actually closest to the integer 4 (√12.4 ≈ 3.52).
Now, let's characterize the kinds of numbers for which the claim in question won't work. For some integer n, take the cutoff between it and its successor, n+1/2 (i.e., the average of n and n+1). Any
√x below this value is closer to n, while any √x above it is closer to n+1. Under the squaring operation, this cutoff gets mapped to the square-of-the-average (n+1/2)^2 = n^2+n+1/4.
On the other hand, consider the cutoff between the squares of the integers in question. Any x below their average is closer to n^2, while any x above the average is closer to (n+1)^2. This
average-of-the-squares is ((n)^2+(n+1)^2)/2 = (n^2+n^2+2n+1)/2 = (2n^2+2n+1)/2 = n^2+n+1/2.
So you can see that there's a gap between these two cutoffs, and in fact it's exactly 1/4 in all cases, no matter what the value of n. If you pick x in the range n^2+n+1/4 < x < n^2+n+1/2, then √x
will be closer to its ceiling of n+1, but x itself will be closer to its floor-square of n^2. Specifically, the problem cases for x are anything a bit more than the product of two consecutive
integers (also called a pronic or oblong number), exceeding n(n+1) = n^2+n by a value of between 1/4 and 1/2. Since n^2+n is itself an integer (ℕ closed under add/multiply), we see that any x in
violation of the claim must be strictly between two consecutive integers, and thus cannot itself be in ℕ.
In conclusion: While the claim in question is not true for all real numbers, it is a trick that does happen to work for all whole-numbered values of x. How important is that? Personally, I'm pretty
uncomfortable with giving our students an unverified procedure which can leave them thinking that it works for any number under a radical, when in fact that's not the case at all.
This article is aimed at introductory statistics students.
Statistics, as I often say, is a "space age" branch of math --many of the key procedures like student's t-distribution weren't developed until the 20th century (and thus helped launch the revolution
in science, technology, and medicine). While statistics are really critical to understanding modern society, it's somewhat unfortunate that they're built on a very high edifice of prior math work --
in the introductory stats class we're constantly "stealing" some ideas from calculus, trigonometry, measure theory, etc., without being explicit about it (the students having neither the time nor
background to understand them).
One of the first areas where this pops up in my classes is the notion of z-scores: taking a data set and standardizing by means of z = (x − μ)/σ. The whole point of this, of course, is to convert the
data set to a new one with mean zero and standard deviation (stdev) one -- but again, unfortunately, the majority of our students have neither the knowledge of linear transformations nor algebraic
proofs to see why this is the case. Our textbook has a numerical example, but in the interest of time, my students just wind up taking this on faith (bolstered, I hope, by a single graphical
Well, for the first time in almost a decade of teaching this class at my current college, I had a student come into my office this week and express discomfort with the fact that he didn't fully
understand why that was the case, and if we'd really properly established that fact. Of course, I'd say this is the very best question that a student could ask at this juncture, and really gets at
the heart of confirmation and proof that should be central to any math class. (Interesting tidbit -- the student in question is a History major, not part of any STEM or medical/biology program
required to take the class.)
So I hunted around online for a couple minutes for an explanation, but I couldn't find anything really pitched at the expected level of my students (requirements: a fully worked out numerical
example, graphical illustration without having heard of shift/stretches before, algebraic proof without first knowing that summations distribute across terms, etc.) Instead, I took some time the next
day and wrote up an article myself to send to the student, which you can see linked below. Hopefully this careful and detailed treatment helps in some other cases when the question pops up again:
(Edited: Jan-9, 2015).
For some reason, there's been a bunch of stories of schools secretly boosting near-failing grades recently. A few that come to mind:
1. Just this weekend -- Hempstead High School on Long Island (somewhat near me) has a scandal of regular boosting failing 63 and 64 scores to passing 65's in any class from grades 6-12. Apparently
this has been done for some number of decades, and the Deputy Superintendent defends it as customary at their school and others (although it was done in secret and not any documented policy).
Other schools nearby deny that they engage in the same practice.
2. Early last month, an Indian student attending Cornell University accessed and mined the data from the Indian national high school exams from the last year, and found that the scores being
reported were very clearly manipulated in some secret way, as there were irregular gaps in the achieved scores across all subject areas. In particular -- none of the scores 32, 33, or 34 were
achieved by any student for any subject in the entire country, whereas 35 is the minimum to pass.
3. Less publicized (but perhaps more dramatic) is the fact that New York State Regents Examinations are in some sense getting easier, as the high school system brags about increased graduation rates
at the same time as their graduates needing remedial instruction in college reaches around 80%. Someone who really ought to know told me that the scores on the exams are effectively mangled by
administrators in Albany, i.e., a 45% raw performance is reported as a passing scaled score of "70" and so forth.
All of this certainly seems really bad to me in a first-pass "smell test" of credibility. It just seems like any kind of secret score-mangling is a foul wind that carries with it lack of
transparency, disbelief in results, corruption, etc. Interestingly, a great many commentators at Slashdot (around the Indian story) said things like "this is done everywhere, if you don't understand
it then you don't know anything about teaching", which is false in my experience. But apparently the motivation is frequently to avoid conflict and time spent around complaints over barely-failing
scores. Some other institutional strategies I've seen or heard about to deal with this issue:
• Those who miss passing by 5% get to immediately take a re-test. I haven't seen this, but I've heard it said of other universities.
• Those who miss passing by 5% get a one-week refresher seminar, and can then re-test on the final. A somewhat more subtle version of the preceding which is used where I teach at CUNY for math
• Keeping both scores and the passing criteria itself secret -- reporting only pass-or-fail results for the test. This was done in the past at my college, allegedly to forestall complaints over
scores. It's pretty much my least favorite option, because it just made everyone involved confused and upset over the secret criteria and unknown scores.
Now, I'm always in favor of maximal transparency, honesty, and confidence in any kind of process like this. But in some cases I've found myself to be a lone voice for this principle. Is this kind of
secret score-mangling an acceptable social massaging of high-stakes testing, or is it the harbinger of corruption and non-confidence in our institutions? Do we even have any choice in the matter
anymore, as educators or citizens? | {"url":"http://www.madmath.com/2013/07/","timestamp":"2024-11-11T03:20:25Z","content_type":"application/xhtml+xml","content_length":"88797","record_id":"<urn:uuid:23117b70-0b7c-49b9-8ca2-0544ea15bb6a>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00477.warc.gz"} |
Glauber Dynamics for the Mean-Field Potts Model
We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q≥3 states and show that it undergoes a critical slowdown at an inverse-temperature β[s](q) strictly lower than the
critical β[c](q) for uniqueness of the thermodynamic limit. The dynamical critical β[s](q) is the spinodal point marking the onset of metastability. We prove that when β<β[s](q) the mixing time is
asymptotically C(β,q)nlogn and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order n. At β=β[s](q) the dynamics no longer exhibits cutoff and its mixing
obeys a power-law of order n^4/3. For β>β[s](q) the mixing time is exponentially large in n. Furthermore, as β↑β[s] with n, the mixing time interpolates smoothly from subcritical to critical
behavior, with the latter reached at a scaling window of O(n^-2/3) around β[s]. These results form the first complete analysis of mixing around the critical dynamical temperature-including the
critical power law-for a model with a first order phase transition.
All Science Journal Classification (ASJC) codes
• Statistical and Nonlinear Physics
• Mathematical Physics
• Critical slowdown
• Curie Weiss
• Cutoff
• Glauber dynamics
• Mean field
• Metastability
• Mixing time
• Potts model
• Spinodal point
Dive into the research topics of 'Glauber Dynamics for the Mean-Field Potts Model'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/glauber-dynamics-for-the-mean-field-potts-model","timestamp":"2024-11-08T04:06:29Z","content_type":"text/html","content_length":"52167","record_id":"<urn:uuid:5a378702-2d13-4a33-a4e8-aac8a7bdfc3c>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00604.warc.gz"} |
Fractions as Parts of a Set
Writing Fractions as Decimals
Writing Fractions as Decimals
A Grain as big as a hen's egg
Math 7 - The Number System
PERCENT, DECIMALS and FRACTIONS
Explore Fractions as Parts of a Set Worksheets by Grades
Explore Other Subject Worksheets for class 7
Explore printable Fractions as Parts of a Set worksheets for 7th Class
Fractions as Parts of a Set worksheets for Class 7 are an essential tool for teachers looking to improve their students' understanding of math concepts. These worksheets focus on the crucial topic of
fractions, providing students with a variety of exercises designed to help them grasp the concept of fractions as parts of a whole. With engaging fraction models and clear explanations, these
worksheets are perfect for reinforcing lessons taught in the classroom. Teachers can use these worksheets to supplement their lesson plans, providing additional practice for students who may be
struggling with the concept of fractions. By incorporating these worksheets into their curriculum, teachers can ensure that their Class 7 students have a solid foundation in math, particularly when
it comes to understanding and working with fractions.
In addition to Fractions as Parts of a Set worksheets for Class 7, Quizizz offers a wide range of resources for teachers to enhance their students' learning experience. Quizizz is an online platform
that provides interactive quizzes, worksheets, and other educational materials designed to make learning fun and engaging. Teachers can use Quizizz to create customized quizzes and worksheets,
allowing them to tailor their teaching materials to the specific needs of their students. This platform also offers a variety of pre-made quizzes and worksheets, covering a wide range of topics,
including math, fractions, and fraction models. By incorporating Quizizz into their teaching strategy, teachers can provide their Class 7 students with a dynamic and interactive learning experience
that will help them better understand and retain important math concepts. | {"url":"https://quizizz.com/en/fractions-and-parts-of-a-set-worksheets-class-7?page=1","timestamp":"2024-11-10T14:14:32Z","content_type":"text/html","content_length":"164129","record_id":"<urn:uuid:d1a1d619-a1f8-4260-90bd-bc63ab5c61c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00540.warc.gz"} |
Worked Out Problems for Heat Transfer
Problem 1:
A freezer compartment consists of a cubical cavity that is 2 m on a side. Assume the bottom to be perfectly insulated. What is the minimum thickness of Styrofoam insulation (k=0.030W/m.K) which must
be applied to the top and side walls to ensure a heat load less than 500 W, when the inner and outer surfaces are -10 ºC and 350C?
Known: Dimensions of freezer component, inner and outer surfaces temperatures.
Find: Thickness of Styrofoam insulation needed to maintain heat load below prescribed value.
Assumptions: (1) perfectly insulted bottom, (2) one-dimensional conduction through five walls of areas A=4m^2, (3) steady-state conditions
Analysis: Using Fourier’s law, the heat rate is given by
Comments: The corners will cause local departures from one–dimensional conduction and, for a prescribed value of L, a slightly larger heat loss.
Problem 2:
A square silicon chip (k=150W/m.k) is of width W=5mm on a side and of thickness t=1mm.
the chip is mounted in a substrate such that its side and back surfaces are insulated, while the front surface is exposed to a coolant. If 4W are being dissipated in circuits mounted to the back
surface of the chip, what is the steady-state temperature difference between back and front surfaces?
Known: Dimensions and thermal conductivity of a chip. Power dissipated on one surface.
Find: temperature drop across the chip
Assumptions: (1) steady-state conditions, (2) constant properties, (3) uniform dissipation, (4) negligible heat loss from back and sides, (5) one-dimensional conduction in chip.
Analysis: All of the electrical power dissipated at the back surface of the chip is transferred by conduction through the chip. Hence, Fourier’s law,
Comments: for fixed P, the temperature drop across the chip decreases with increasing k and W, as well as with decreasing t.
Problem 3:
Air at 300 °C flows over a plate of dimensions 0.50 m, by 0.25 m. if the convection heat transfer coefficient is 250 W/m^2.K; determine the heat transfer rate from the air to one side of the plate
when the plate is maintained at 400 °C.
Known: air flow over a plate with prescribed air and surface temperature and convection heat transfer coefficient.
Find: heat transfer rate from the air to the plate
Assumptions: (1) temperature is uniform over plate area, (2) heat transfer coefficient is uniform over plate area
Analysis: the heat transfer coefficient rate by convection from the airstreams to the plate can be determined from Newton’s law of cooling written in the form,
q = q” .A hA(T∞ – Ts )
where A is the area of the plate. Substituting numerical values,
q = 250 W/m^2 * K(0.25 – 0.50)m^2 (300 – 40)°C
q = 8125 W
Comments: recognize that Newtown’s law of cooling implies a direction for the convection heat transfer rate. Written in the form above, the heat rate is from the air to plate.
Problem 4 :
A water cooled spherical object of diameter 10 mm and emissivity 0.9 is maintained at 400°C. What is the net transfer rate from the oven walls to the object?
Known: spherical object maintained at a prescribed temperature within a oven.
Find: heat transfer rate from the oven walls to the object
Assumptions: (1) oven walls completely surround spherical object, (2) steady-state condition, (3) uniform temperature for areas of sphere and oven walls, (4) oven enclosure is evacuated and large
compared to sphere.
Analysis: heat transfer rate will be only due to the radiation mode. The rate equation is
q[rad] = ε Asσ (T[sur]^4 − T[s]^4)
Where As=πD^2, the area of the sphere, substituting numerical values,
q[rad] = 0.9* π(10*10^-3) m^2 *5.67 *10^-8 W/m .K[(400+273)^4 – (80 + 273)^4]K
q[rad] = 3.04W | {"url":"https://convacademy.com/topic/worked-out-problems-for-heat-transfer/","timestamp":"2024-11-11T06:55:12Z","content_type":"text/html","content_length":"294052","record_id":"<urn:uuid:529d4986-d668-4508-823d-7d16a665cde1>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00248.warc.gz"} |
How to Simplify sin(arccos(x)) | How to Simplify sin(cos^-1(x)) - iMath
How to Simplify sin(arccos(x)) | How to Simplify sin(cos^-1(x))
Note that sin(cos^-1 x) or sin(arc cosx) are algebraic functions, not trigonometric functions. The value of sin(cos^-1 x) or sin(arc cosx) is equal to $\sqrt{1-x^2}$. In this post, we will simplify
sin(cos^{-1} x).
The formula of sin(cos inverse x) is given below:
sin(cos^-1 x) = root(1-x^2).
Simplify sin(cos^-1(x))
We all know that $\sin(\cos^{-1}(x))= \sin(\text{arc} \cos x)$. To simplify this function, we will follow the below steps.
Step 1: In the first step, we let that $\alpha=\cos^{-1} x$ $\cdots (I)$
As $\alpha=\cos^{-1} x$ and we have to simplify $\sin(\cos^{-1}x)$, we need to find out the value of $\sin \alpha$. From the equation $(I)$, we obtain that
$\cos \alpha=x$ $\cdots (II)$
Step 2: Now, we use the Pythagorean trigonometric identity $\sin^2 \alpha+\cos^2 \alpha =1$. From this identity, we have that
$\sin^2\alpha = 1-\cos^2 \alpha$
$\Rightarrow \sin \alpha = \sqrt{1-\cos^2 \alpha}$
$=\sqrt{1-x^2}$ [ putting the value of $\cos \alpha$ from $(II)$ ]
$\therefore \sin \alpha =\sqrt{1-x^2}$
Step 3: Next, we will put the value of $\alpha$ from $(I)$. By doing so we obtain that
Conclusion: Thus, the formula of $\sin(\cos^{-1}x)$ is equal to $\sqrt{1-x^2}$.
Also Read:
sinx=0, cosx=0, tanx=0 General Solution
Values of sin 15, cos 15, tan 15
Values of sin 75, cos 75, tan 75
Question 1: Find the value of sine of cosine inverse 0, that is, find $\sin(\cos^{-1}0)$.
Using the above formula, we have that $\sin(\cos^{-1}0)=\sqrt{1-0^2}=1$.
Question 2: Find the value of sine of cosine inverse 1, that is, find $\sin(\cos^{-1}1)$.
Using the above formula, we have that $\sin(\cos^{-1} 0)=\sqrt{1-1^2}=0$.
Q1: What is sin(cos^-1(x))?
Answer: The value of sin(cos^-1x) is equal to $\sqrt{1-x^2}$. | {"url":"https://www.imathist.com/how-to-simplify-sinarccosx-sincos-1x/","timestamp":"2024-11-09T15:48:16Z","content_type":"text/html","content_length":"180568","record_id":"<urn:uuid:0412fab5-4141-4481-835d-568e7897ec35>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00581.warc.gz"} |
What is the formula for magnetomotive force?
What is the formula for magnetomotive force?
Magnetic circuits Magnetomotive force (mmf), Fm = NI ampere-turns (At), where N = number of conductors (or turns) and I = current in amperes.
What is the meaning of magnetomotive force?
Magnetomotive force is the force that sets up a magnetic field within and around an object. The unit of magnetomotive force is the ampere-turn, represented by a steady, direct electric current of one
ampere flowing in a single-turn loop of electrically conducting material in a vacuum.
What is reluctance force?
What is Reluctance? Magnetic reluctance (also known as reluctance, magnetic resistance, or a magnetic insulator) is defined as the opposition offered by a magnetic circuit to the production of
magnetic flux. It is the property of the material that opposes the creation of magnetic flux in a magnetic circuit.
What is magnetomotive force and reluctance?
Magnetic reluctance, or magnetic resistance, is a concept used in the analysis of magnetic circuits. It is defined as the ratio of magnetomotive force (mmf) to magnetic flux. It represents the
opposition to magnetic flux, and depends on the geometry and composition of an object.
How do you increase magnetomotive force?
To increase the strength of an electromagnet, you can increase the strength current, and there are several ways to do that. You can also increase the number of windings, lower the ambient temperature
or replace your non-magnetic core with a ferro-magnetic material.
What is magnetomotive force and its unit?
The magnetomotive force, mmf, is analogous to the electromotive force and may be considered the factor that sets up the flux. The mmf is equivalent to a number of turns of wire carrying an electric
current and has units of ampere-turns.
What is inductance unit?
The unit of magnetic inductance is the henry, named in honour of the 19th-century American physicist Joseph Henry, who first recognized the phenomenon of self-induction. One henry is equivalent to
one volt divided by one ampere per second.
What is MMF define by formula?
Magnetomotive force (MMF) is a component or part of the equation of magnetic flux in magnetic circuits. As an analogy to electromotive force, MMF “drives” magnetic flux through a magnetic circuit.
Magnetomotive force is also known as magnetic potential.
What’s the difference between EMF and MMF?
MMF is the driving force required to drive the magnetic flux through the circuit. EMF stands for electromotive force. MMF stands for magnetomotive force. EMF acts as the driving force responsible for
the movement of the electrons in an electrical circuit.
How does MMF and EMF generated?
When there is a current flow through the conductor coil, a force is produced to drive the magnetic flux or magnetic field lines. This force is called MMF or magnetomotive force. The EMF or
electromotive force is the force responsible for the flow of electrons or current in a closed circuit.
How is the magnetomotive force related to the magnetic field?
Magnetomotive force. In physics, the magnetomotive force (mmf) is a quantity appearing in the equation for the magnetic flux in a magnetic circuit, often called Ohm’s law for magnetic circuits. It is
the property of certain substances or phenomena that give rise to magnetic fields :
How is the strength of a magnetomotive circuit measured?
Magnetomotive force is the work that carries a measurable unit of strength through a magnetic circuit. This unit of strength is measured in ampere-turns (AT). Magnetism in a circuit flows from the
north to the south pole. Following a specific path, the force of the magnetism is similar to the force in an electrical circuit.
How to calculate the magnetomotive force of a coil?
The calculator determines the magnetomotive force produced by a coil with an electric current flowing through it. Example 1: Calculate the magnetomotive force (mmf) produced by a 500-turn coil if the
current flowing through this coil is 0.5 A.
What is the SI unit for magnetic force?
The magnetic pressure, which sets up the magnetic flux in a magnetic circuit is called Magnetomotive Force. The SI unit of MMF is Ampere-turn (AT), and their CGS unit is G (gilbert). The MMF for the
inductive coil shown in the figure below is expressed as. | {"url":"https://pvillage.org/archives/11595","timestamp":"2024-11-11T22:55:54Z","content_type":"text/html","content_length":"55137","record_id":"<urn:uuid:8d334fb5-f74e-47a5-ba0c-0a59260fcdea>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00172.warc.gz"} |
Identifying Carriers?
First of all, thank you for providing such abundant amount of data to everyone worldwide.
I am curious if there is a way to identify which carriers are using what cells from the data set downloaded?
Thank you so much!
1 Like
Hi Huso,
There are 4 basic parts of a cell identifier: MCC-MNC-LAC-CID
MCC-MNC is set by International Telecommunication Union. All telecom firms have access to this information as it is publicly available.
LAC-CID can be decided by the telecom firm. This is an internal decision and telecom firms donā t share this with each other.
You can use the link above to cross reference MNC number from the database to the name of the operator.
2 Likes
Hello, I am also trying to determine the carriers but the MNC column is missing from the data download. How can obtain this information for specific MCCs?
Hi, Column C ā netā is the MNC
So, I used the provided ITU data for MCC and MNC and crossed referenced it to OpenCellID database, but itā s hard to conclude that itā s all accurate, here is why:
• OpenCellID collects cell phone ping data and I can determine which provider of the cell phone pinged the tower, but not necessarily who owns the tower.
• The device could be roaming or just using another providers tower in the mean time, thus this reference makes the data a bit misleading
I hope that makes sense. Iā m simply trying to locate every tower/site and to find out which carrier is on such tower/site. Iā m finding it a bit difficult to do this through this data.
Any help/leads would be greatly appreciated.
Thank you in advance.
Huso, the identifiers are tied to the cell tower. No matter what device is scanning, theyā ll always get the MCC-MNC-LAC-CID for that cell.
For example,
Your phone with AT&T sim can scan cells that belong to every other telecom provider. And the cell identifiers scanned by your phone will be the same if you scan using a Verizon phone. The MNC wonā t
change because it belongs to the cell.
I am also trying to locate every tower/site. After sorting the carriers by MNC, removing duplicate Lat Longs, and removing entries with less than 10 samples, there are points that are very close
together. The first two points in the attached picture are ~1.8 mm apart. Surely AT&T would not place cells this close to each other. This occurs many times in the data set. What differentiates these
two points? Any guidance would be great.
Arenā t these coordinates from the Observer? Itā s logical to be the same if the observer is still.
Furthermore, according to the base/cell scheme of my country (modulo 256), these two cell-ids belong to the same base 110432 (never seen so big number hereā ¦). The first has cell index 8 and the
second 149.
So, in every case, same lat/lon is justified.
Ah okay! So if the base number of the cell ID is the same then the cells are on the same tower. I added a few columns in my spreadsheet then sorted by the base. In the screen shot, there are 21 rows
with base 0. The LACs for this base are 1, 5, 6. Since the LACs differ for some of the base 0 rows are these different towers?
Maybe yes, i think something else happens.
1- First of all, we are talking about LTE only, GSM and UMTS have different schemes.
2- The numbing of cells in a base is bit peculiar with this scheme. Bases have typically from one to six antennas per mode (here 1800 - 800 - 2600), so an absolute maximum for a base is 18 cells, at
least in my country. In the network i watch, i know a base with 14 cells. Another usual practice here, is when the number of cells increases, they use an additional (second) base id in the same site.
In the table you give, there are two networks, pay attention not to mix themā ¦ There are only two areas for base 0 of network 70.
3- As per your question, yes, if a base is at the boundaries of two areas, it may have more than one LAC.
You must find which is the correct scheme for your case. Search all of your data, and find how many bases are like this.
How do you get the base from the cell id? wich math operation do you do?
For what I have investigated so far what makes a tower unique is the MNC-AREA-BASE agreement.
In your example marked up to line 22, you have 2 different networks (190 and 70) where:
a) network 190: 1 tower (that of area 1 and base 0 that contains 6 cells)
b) network 70: 2 towers (that of area 6 and base 0 that contains 6 cells and that of area 5 and base 0 that contains 9 cells)
I think not. Base is unique in the whole network, not only within an area. Here in Greece (in all 3 networks), area can change, but the base number stays the same.
Areas are groups of bases (cells actually) used to divide network load and traffic between different data centers / pbxs.
So (if its indeed a "modulo 256 scheme) in network 70 there is one base, with cells in two distinct areas.
Base is calculated from [INT (cellid / 256)], and cell index is the [Remainder * 256]. In LTE.
You do not have to remove duplicate lat longs. The OpenCelliD database gives information on cells - not cell towers. A single cell tower can have multiple cells.
The file available online has only unique cells - no duplicate ones.
Nope. These are approximate positions of cell - not the coordinates of the observer.
Yes, you are right, my mistake. The BASEID (BSIC) is unique in the same network (cellular company).
Then, in network 70 of your example, there is a single tower with 6 cells grouped in area ā 6ā and 9 cells grouped in area ā 5ā , right?
But ā ¦ why are the lat and long coordinates so different? There is a great distance between cell 154 and 123. They should have different bsic (base id) ā ¦ I do not understand it.
I have the same question. If these cells are on the same base how do I know which lat/long coordinates are accurate to find the base? Should an average be taken of the lat/longs? This does not seem
very accurateā ¦
As said, this is an estimate only, computed from existing data.
There are many ways to find the exact location:
1- Local visit
2- From public records the networks publish
3- ā Fox huntingā (take your phone/app and walk around)
4- By using google earthā s terrain glyph with data on it (rural area bases are usually located at hill/mountain tops). Data points are spread in a circle pattern, and base is near the center.
1 Like | {"url":"https://community.opencellid.org/t/identifying-carriers/223","timestamp":"2024-11-09T19:58:09Z","content_type":"text/html","content_length":"59096","record_id":"<urn:uuid:a7a9334e-c3c3-4771-b115-bfe87f93e93d>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00002.warc.gz"} |
3 Natural Units And Equations
Particle physicists have a habit that on the face of it seems to violate physics. They set the values of natural constants such as the speed of light and Planck’s constant to the number one. How can
they do this? It’s because the relationship between units by which we measure length, time and energy is a matter of choice.
For example, we know what a second is, and we know what a meter is. But how many meters make a second? Is that like asking how many apples are there in an orange? Maybe. But not quite, because the
constants of Nature like Planck’s constant and the speed of light represent natural relationships between different units of measurement, as explained below.
Space at the Speed of Light: The History of 14 Billion Years for People Short on Time
Buy on Amazon- https://amzn.to/390FkCR
Speed of light
The measured value of the speed of light is
so why would physicists want to pretend that instead c=1? What they are really doing is choosing a relationship between a unit of time, the second, and a unit of space, the meter, so that these two
units are not independent but related. The natural constant of relation is the speed of light, so that
If we relate meters and seconds so that one second is equal to 300 million meters, then c=1. It’s very simple. Now notice that in this system of unit, mass and energy have the same units, because the
relationship E = m c^2 in units with c=1 just reduces to E = m.
Planck’s constant
Planck’s constant has units of energy x time. The preferred unit of energy in physics is the electron volt (or eV for short) is the amount of work necessary to move one electron across a potential of
one volt. For particle physicists, this is a very handy unit because particle experiments use electrical potential to accelerate and bang electrons and other particles with the same charge into each
other. In unit with c=1, the mass of the electron is 0.5 MeV. Written in terms of eV, Planck’s constant takes the value
The version of Planck’s constant that physicists normally use is called “h-bar” and is equal to
In c=1 units, time is described in meters, the value of Planck’s constant becomes
But physicists aren’t happy to stop there. We like to make all calculations as simple as possible. So the next step is set (this version of) Planck’s constant equal to one to set the relationship
between energy units and length units so that
The size of an atom is roughly 10-10 meters. Atomic physicists use a unit called the Ångstrom, where 1 Å = 10-10 meters. Written in these terms we get the relationship
Notice that in these new units, increasing energy means decreasing length. Distances scales that are much smaller than the size of an atom have mass scales associated with them that are much larger
than 2000 eV.
That is typical behavior for quantum mechanics. The de Broglie relation for wave-particle duality
also shows that in quantum physics, it is necessary to use a large energy or momentum scale to probe a small distance scale. That’s why particle accelerators are like microscopes. When the particle
accelerator energy gets bigger, the distance scale being probed gets smaller.
Read more: What is Real? Quantum physics and Reality
Can we play the same game with Newton’s gravitational constant? Not really, because there aren’t any new units to relate that aren’t already related. If we consider spacetimes with gravity in higher
dimensions, Newton’s constant has units that depend on the dimension of spacetime. The value currently measured is, of course, for d=4.
In natural units, Newton’s constant has units of L^d-2, or M^2-d.
Thus one can express Newton’s constant in units of time, length or mass, as desired.
The Planck length is thought to be the natural distance scale at which quantum gravitational effects become strong enough to notice.
Black Holes, Tides, and Curved Spacetime: Understanding Gravity
Watch video on Amazon- https://amzn.to/35XwWC0
Leave a Comment | {"url":"https://cosmos.theinsightanalysis.com/natural-units/","timestamp":"2024-11-14T04:24:47Z","content_type":"text/html","content_length":"157802","record_id":"<urn:uuid:a60a6c43-9608-4e1f-b075-873d5e0449ed>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00557.warc.gz"} |
Logarithmic corrections to finite-size scaling in the four-state Potts model
The leading corrections to finite-size scaling predictions for eigenvalues of the quantum Hamiltonian limit of the critical four-state Potts model are calculated analytically from the Bethe ansatz
equations for equivalent eigenstates of a modified XXZ chain. Scaled gaps are found to behave for large chain length L as x+d{Plimsoll sign}L+0[(ln L)^-1], where x is the anomalous dimension of the
associated primary scaling operator. For the gaps associated with the energy and magnetic operators, the values of the amplitudes d are in agreement with predictions of conformai invariance. The
implications of these analytical results for the extrapolation of finite lattice data are discussed. Accurate estimates of x and d are found to be extremely difficult even with data available from
large lattices, L∼500.
Dive into the research topics of 'Logarithmic corrections to finite-size scaling in the four-state Potts model'. Together they form a unique fingerprint. | {"url":"https://researchportalplus.anu.edu.au/en/publications/logarithmic-corrections-to-finite-size-scaling-in-the-four-state-","timestamp":"2024-11-07T14:06:36Z","content_type":"text/html","content_length":"50860","record_id":"<urn:uuid:22cfcd68-b264-4617-9ea0-92bce28f81b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00781.warc.gz"} |
superb: Summary plots with adjusted error bars
The library superb offers two main functionalities. First, it can be used to obtain plots with adjusted error bars. The main function is superb() but you can also use superbShiny() for a graphical
user interface requiring no programming nor scripting. See the nice tutorial by Walker (2021).
The purpose of superb() is to provide a plot with summary statistics and correct error bars. With simple adjustments, the error bar are adjusted to the design (within or between), to the purpose
(single or pair-wise differences), to the sampling method (simple randomized samples or cluster randomized samples) and to the population size (infinite or of a specific size). The superbData()
function does not generate the plot but returns the summary statistics and the interval boundaries. These can afterwards be sent to other plotting environment.
The second functionality is to generate random datasets. The function GRD() is used to easily generate random data from any design (within or between) using any population distribution with any
parameters, and with various effect sizes. GRD() is useful to test statistical procedures and plotting procedures such as superb().
The official CRAN version can be installed with
The development version 0.95.19 can be accessed through GitHub:
The easiest is to use the graphical interface which can be launched with
The following examples use the script-based commands.
Here is a simple example illustrating the ToothGrowth dataset of rats (in which the dependent variable is len) as a function of the dose of vitamin and the form of the vitamin supplements supp (pills
or juice)
In the above, the default summary statistic, the mean, is used. The error bars are, by default, the 95% confidence intervals (of the mean). These two choices can be changed with the statistic and the
errorbar arguments.
This second example explicitly indicates to display the median instead of the default mean summary statistics along with the default 95% confidence interval of the median here (the correct function
is automatically selected):
As a third example, we illustrate the harmonic means hmean along with 99.9% confidence intervals of the harmonic mean displayed using lines:
superb(len ~ dose + supp, ToothGrowth,
statistic = "hmean",
errorbar = "CI", gamma = 0.999,
plotStyle = "line")
The second function, GRD(), can be used to generate random data from designs with various within- and between-subject factors. This example generates scores for 300 simulated participants in a 3 x 6
design with 6 daily repeated-measures on Days. Only the factor Day is modeled as impacting the scores (increasing by 3 points on the second day):
set.seed(663) # for reproducibility
testdata <- GRD(
RenameDV = "score",
SubjectsPerGroup = 10,
BSFactors = "Difficulty(A,B,C)",
WSFactors = "Day(6)",
Population = list(mean = 75,stddev = 10,rho = 0.8),
Effects = list( "Difficulty" = custom(-5,-5,+10), "Day" = slope(3) )
## id Difficulty score.1 score.2 score.3 score.4 score.5 score.6
## 1 1 A 61.72393 61.48460 70.48406 68.92430 69.85908 68.15339
## 2 2 A 54.16784 65.82688 66.51785 65.59598 82.74906 82.53300
## 3 3 A 69.85369 60.04088 73.99657 72.95358 69.89209 74.30423
## 4 4 A 69.05319 64.99568 75.00310 78.35253 81.48167 76.08335
## 5 5 A 79.29388 81.56254 78.17444 86.36108 92.45310 93.73091
## 6 6 A 56.56657 59.23395 66.10074 63.77299 67.07331 72.64133
This is here that the full benefits of superb() is seen: with just a few adjustments, you can obtained decorrelated error bars with the Correlation-adjusted (CA), the Cousineau-Morey (CM) or the
Loftus & Masson (CM) techniques:
## Warning: package 'gridExtra' was built under R version 4.3.3
## Warning: package 'RColorBrewer' was built under R version 4.3.1
plt1 <- superb( crange(score.1, score.6) ~ Difficulty,
testdata, WSFactors = "Day(6)",
plotStyle = "line"
) + ylim(50,100) + labs(title = "No adjustments") +
theme_bw() + ylab("Score") +
plt2 <- superb( crange(score.1, score.6) ~ Difficulty,
testdata, WSFactors = "Day(6)",
adjustments = list(purpose = "difference", decorrelation = "CA"),
plotStyle = "line"
)+ ylim(50,100) + labs(title = "correlation- and difference-adjusted") +
theme_bw() + ylab("Score") +
grid.arrange(plt1,plt2, ncol=2)
Even better, the simulated scores can be illustrated using using a more elaborated layout, the pointjitter which, in addition to the mean and confidence interval, shows the raw data using jitter
superb( crange(score.1, score.6) ~ Difficulty,
testdata, WSFactors = "Day(6)",
adjustments = list(purpose = "difference", decorrelation = "CM"),
plotStyle = "pointjitter",
errorbarParams = list(color = "purple"),
pointParams = list( size = 3, color = "purple")
) +
theme_bw() + ylab("Score") +
In the above example, optional arguments errorbarParams and pointParams are used to inject specifications in the error bars and the points respectively. When these arguments are used, they override
the defaults from superb().
Lastly, we could aim for a radar (a.k.a. circular) plot with
superb( crange(score.1, score.6) ~ Difficulty, testdata,
WSFactors = "Day(6)",
adjustments = list(purpose = "difference", decorrelation = "CM"),
plotStyle = "circularpointlinejitter",
factorOrder = c("Day", "Difficulty"),
pointParams = list( size = 3 ),
jitterParams = list(alpha=0.25),
errorbarParams= list(width=0.33, color = "black")
) +
theme_bw() + ylab("") +
theme(panel.border = element_blank(), text = element_text(size = 16) ) +
scale_color_brewer(palette="Dark2") +
theme(axis.line.y = element_blank(),
axis.text.y=element_blank(), axis.ticks.y=element_blank())
Every time, you get error bars for free! no need to compute them on the side, no need to worry about the adjustments (whether you want stand-alone error bars or adjusted for purpose or correlation,
it is all just one option). Also, keep in mind that it is easy to change the default (mean +- 95% confidence intervals) to any other summary statistics –e.g., median– and any other measure of error
–e.g., standard error, standard deviation, inter-quartile range, name it–; you can find some responses in the vignettes or on stackExchange or just open an issue on the github repository.
For more
superb is for summary plot with error bar, as simple as that.
The library superb makes it easy to illustrate summary statistics along with the error bars. Some layouts can be used to visualize additional characteristics of the raw data. Finally, the resulting
appearance can be customized in various ways.
The complete documentation is available on this site.
A general introduction to the superb framework underlying this library is published at Advances in Methods and Practices in Psychological Sciences (Cousineau, Goulet, & Harding, 2021). Also, most of
the formulas for confidence intervals when statistics other than the mean are displayed can be found in Harding, Tremblay, & Cousineau (2015).
Cousineau D, Goulet M, Harding B (2021). “Summary plots with adjusted error bars: The superb framework with an implementation in R.” Advances in Methods and Practices in Psychological Science, 2021,
1–46. doi: https://doi.org/10.1177/25152459211035109
Walker, J. A. L. (2021). “Summary plots with adjusted error bars (superb).” Youtube video, accessible here.
Cousineau, D., Goulet, M.-A., & Harding, B. (2021). Summary plots with adjusted error bars: The superb framework with an implementation in R.
Advances in Methods and Practices in Psychological Science
, 1–18.
Harding, B., Tremblay, C., & Cousineau, D. (2014). Standard errors: A review and evaluation of standard error estimators using monte carlo simulations. The Quantitative Methods for Psychology, 10(2),
Harding, B., Tremblay, C., & Cousineau, D. (2015). The standard error of the pearson skew. The Quantitative Methods for Psychology, 11(1), 32–36.
Walker, J. A. L. (2021).
Summary plots with adjusted error bars (superb)
. Retrieved from | {"url":"http://ctan.mirror.garr.it/mirrors/CRAN/web/packages/superb/readme/README.html","timestamp":"2024-11-11T14:02:56Z","content_type":"application/xhtml+xml","content_length":"27795","record_id":"<urn:uuid:9f57520b-cc94-4dfa-ab74-52f10a728230>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00579.warc.gz"} |
Comparison with other software
Next, the MMRM fitting procedures are compared using the FEV and BCVA datasets. FEV1 measurements are modeled as a function of race, treatment arm, visit number, and the interaction between the
treatment arm and the visit number. Change in BCVA is assumed to be a function of race, baseline BCVA, treatment arm, visit number, and the treatment–visit interaction. In both datasets, repeated
measures are modeled using an unstructured covariance matrix. The implementations’ convergence times are evaluated first, followed by a comparison of their estimates. Finally, we fit these procedures
on simulated BCVA-like data to assess the impact of missingness on convergence rates.
Convergence Times
FEV Data
The mmrm, PROC GLIMMIX, gls, lmer, and glmmTMB functions are applied to the FEV dataset 10 times. The convergence times are recorded for each replicate and are reported in the table below.
Comparison of convergence times:
mmrm 56.15 55.76 56.30
PROC GLIMMIX 100.00 100.00 100.00
lmer 247.02 245.25 257.46
gls 687.63 683.50 692.45
glmmTMB 715.90 708.70 721.57
It is clear from these results that mmrm converges significantly faster than other R functions. Though not demonstrated here, this is generally true regardless of the sample size and covariance
structure used. mmrm is faster than PROC GLIMMIX.
BCVA Data
The MMRM implementations are now applied to the BCVA dataset 10 times. The convergence times are presented below.
Comparison of convergence times:
mmrm 3.36 3.32 3.46
glmmTMB 18.65 18.14 18.87
PROC GLIMMIX 36.25 36.17 36.29
gls 164.36 158.61 165.93
lmer 165.26 157.46 166.42
We again find that mmrm produces the fastest convergence times on average.
Marginal Treatment Effect Estimates Comparison
We next estimate the marginal mean treatment effects for each visit in the FEV and BCVA datasets using the MMRM fitting procedures. All R implementations’ estimates are reported relative to PROC
GLIMMIX’s estimates. Convergence status is also reported.
FEV Data
The R procedures’ estimates are very similar to those output by PROC GLIMMIX, though mmrm and gls generate the estimates that are closest to those produced when using SAS. All methods converge using
their default optimization arguments.
BCVA Data
mmrm, gls and lmer produce estimates that are virtually identical to PROC GLIMMIX’s, while glmmTMB does not. This is likely explained by glmmTMB’s failure to converge. Note too that lmer fails to
Impact of Missing Data on Convergence Rates
The results of the previous benchmark suggest that the amount of patients missing from later time points affect certain implementations’ capacity to converge. We investigate this further by
simulating data using a data-generating process similar to that of the BCVA datasets, though with various rates of patient dropout.
Ten datasets of 200 patients are generated each of the following levels of missingness: none, mild, moderate, and high. In all scenarios, observations are missing at random. The number patients
observed at each visit is obtained for one replicated dataset at each level of missingness is presented in the table below.
Number of patients per
VIS01 200 196.7 197.6 188.1
VIS02 200 195.4 194.4 182.4
VIS03 200 195.1 190.7 175.2
VIS04 200 194.1 188.4 162.8
VIS05 200 191.6 182.5 142.7
VIS06 200 188.2 177.3 125.4
VIS07 200 184.6 168.0 105.9
VIS08 200 178.5 155.4 82.6
VIS09 200 175.3 139.9 58.1
VIS10 200 164.1 124.0 39.5
The convergence rates of all implementations for stratified by missingness level is presented in the plot below.
mmrm, gls, and PROC GLIMMIX are resilient to missingness, only exhibiting some convergence problems in the scenarios with the most missingness. These implementations converged in all the other
scenarios’ replicates. glmmTMB, on the other hand, has convergence issues in the no-, mild-, and high-missingness datasets, with the worst convergence rate occurring in the datasets with the most
dropout. Finally, lmer is unreliable in all scenarios, suggesting that it’s convergence issues stem from something other than the missing observations.
Note that the default optimization schemes are used for each method; these schemes can be modified to potentially improve convergence rates.
A more comprehensive simulation study using data-generating processes similar to the one used here is outlined in the simulations/missing-data-benchmarks subdirectory. In addition to assessing the
effect of missing data on software convergence rates, we also evaluate these methods’ fit times and empirical bias, variance, 95% coverage rates, type I error rates and type II error rates. mmrm is
found to be the most most robust software for fitting MMRMs in scenarios where a large proportion of patients are missing from the last time points. Additionally, mmrm has the fastest average fit
times regardless of the amount of missingness. All implementations considered produce similar empirical biases, variances, 95% coverage rates, type I error rates and type II error rates. | {"url":"http://rsync.jp.gentoo.org/pub/CRAN/web/packages/mmrm/vignettes/mmrm_review_methods.html","timestamp":"2024-11-02T03:15:05Z","content_type":"text/html","content_length":"128050","record_id":"<urn:uuid:6a28ff5f-a29f-463b-95e4-828b2a45c8fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00483.warc.gz"} |
Research and Teaching Interests
• Algebra: Homological Algebra; Ring Theory
Past Affiliations
Professor, Department of Mathematics and Statistics, College of Sciences and Mathematics, Auburn University
Associate Dean, Department of Mathematics and Statistics, College of Sciences and Mathematics, Auburn University
Associate Dean, College of Sciences and Mathematics, Auburn University
Visiting Assistant Professor, Department of Mathematics, College of Arts and Sciences, University of Kentucky (past)
Lecturer, Mathematics
Lecturer, Mathematics
Associate Provost for Diversity and Multicultural Affairs, Office of Inclusion and Diversity, Office of the Provost, Auburn University
Statistics, Mathematics
PhD, University of Kentucky, Mathematics, 1981
MA, University of Kentucky, Mathematics, 1978
BS, University of Malawi, Malawi, Mathematics, 1976
mathematics algebra
English, Nyanja, Tumbuka
American Mathematical Society
Mathematical Association of America | {"url":"https://scholars.proquest.com/gallery/auburn/profiles/0d173657-ce47-b01d-0098-61dfeac8965a","timestamp":"2024-11-12T15:26:38Z","content_type":"text/html","content_length":"19311","record_id":"<urn:uuid:776b27b8-3aca-40ce-999c-60cf59abad9b>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00347.warc.gz"} |
Whole PI
Whole PI
Welcome to the Home of
Gateway to a new way of
understanding the World
Welcome to the Home of
See the videos
Whole PI
Gateway to a new way of
understanding the World.
Welcome to the Home of
See the videos
Previous Next
The Pi Simulacrum leads to considering the defining unit.
Operating pi in the Unit Simulacrum gives the Pi Unit Simulacrum. It makes you think of the nature of the unit. The Pi Unit Simulacrum is the sum of 1 + pi in the form of a value and a ratio.
The Pi Unit Simulacrum is:
SimSum(1,pi) = Sum(1+pi) =
[Possibilities to Observe (Poss2Obs) are:
1, pi, (1/pi) and (pi/1);
{Case 1: pi<=1, 1>=pi, (pi/1)<=1, (1/pi)>=1;
pi ( 1 + 1/(pi/1) ) or 1 ( 1 + (pi/1) )
pi ( 1 + (1/pi) ) or 1 ( 1 + 1/(1/pi) )
Case 2: pi>=1, 1<=pi, (pi/1)>=1, (1/pi)<=1;
pi ( 1 + 1/(pi/1) ) or 1 ( 1 + (pi/1) )
pi ( 1 + (1/pi) ) or 1 ( 1 + 1/(1/pi) ) } ]
The solution depends on (pi/1) or (1/pi), and causes one to ask the question:
"What is the 1?"
What is the unit?
learn more
The Pi Paradox is that one symbol has two different meanings.
It is long past time to stop ignoring The Pi Paradox. There are two different meanings based on radius, r, or diameter, d.
In the ratios (1/pi) and (pi/1), the 1 can be r=1 and d=1, giving two different solutions.
If r=1:
Circumference, C = 2(pi)*r
Area, A = (pi)*r^2
Volume, V = 4/3*(pi)*r^3
If r = 1, then pi is half a circle.
If d=1:
Circumference, C = (pi)*d
Area, A = (pi)*d^2 / 4
Volume, V = (pi)*d^3 / 6
If d = 1, then pi is a whole circle.
The fact that pi has two meanings both represented by the same symbol, pi, is The Pi Paradox. Two unequal things are called the same symbol. The Pi Paradox.
learn more
Whole PI solves the Pi Paradox and leads to a New Understanding of Dimensions.
The solution to having two meanings for the same symbol is to create a new symbol for the second meaning, and expand that meaning.
Using the new symbol for pi based on a whole diamenter and a whole circumference circle, we can define Whole PI as three diameters overlapping in a ratio of two vertical to one horizontal, with the
circle of unit diameter centered on the axes.
The new symbol for pi allows a new understanding of pi, and new simple equations:
First Dimension of Space: 2(PI)(d^p)/2p
Second Dimension fo Space: (PI)(d^p)/2p
Third Dimension of Space: (PI)(d^p)/2p.
Thus, Whole PI allows the pattern behind Dimensions to be seen as:
2(x^p)/2p for the First, then
(x^p)/2p for the Second, Third and all other Dimensions.
learn more
There are important understandings to be learned by examining some of the inherent Critical Values and Critical Limits within the Simulacrum System. There is tremendous understanding that can be
obtained from special cases in The Simulacrum System, most i...
Read More
The whole thing started when I realized that the second mass, helium, stood in relation to the first mass, hydrogen, as the second Dimension of Space stood in relation to the first Dimension of
Space.The denominators in 1/1 to 1/4 in the first two masses ...
Read More
The step from the Bottom Up Solution to the Updated Bottom Up Solution was a generalization, The bigger step from the Updated Bottom Up Solution to The Simulacrum System was a generalization.The
Simulacrum applied to pi led to the Pi Unit Simulacrum, whic...
Read More
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Пятница 25.09. Alexander Golovnev: "Polynomial Data Structure Lower Bounds in the Group Model"
Пятница, 25 сентября, Zoom. Начало в 18:00.
Докладчик: Alexander Golovnev (Georgetown University, USA).
Тема: Polynomial Data Structure Lower Bounds in the Group Model.
Proving super-logarithmic data structure lower bounds in the static group model has been a fundamental challenge in computational geometry since the early 80's. We prove a polynomial (
) lower bound for an explicit range counting problem of
convex polygons in
(each with
facets/semialgebraic-complexity), against linear storage arithmetic data structures in the group model. Our construction and analysis are based on a combination of techniques in Diophantine
approximation, pseudorandomness, and compressed sensing---in particular, on the existence and partial derandomization of optimal \emph{binary} compressed sensing matrices in the polynomial sparsity
regime (
). As a byproduct, this establishes a (logarithmic) separation between compressed sensing matrices and the stronger RIP property.
The talk is based on the following paper:
Видео доклада: | {"url":"https://logic.pdmi.ras.ru/seminars/complexity-seminar/2020-09-25","timestamp":"2024-11-10T15:31:01Z","content_type":"application/xhtml+xml","content_length":"8613","record_id":"<urn:uuid:b20edf6c-79ec-4c4b-8141-d88b9a63d680>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00058.warc.gz"} |
What is the square root of x^12? | Socratic
What is the square root of x^12?
1 Answer
$\sqrt{{x}^{12}} = {x}^{6}$ (or possibly $- {x}^{6}$ if you want to include the non-principal square root)
In general ${\left({b}^{a}\right)}^{c} = {b}^{c a}$
So ${\left({x}^{6}\right)}^{2} = {x}^{2 \times 6} = {x}^{12}$
or reversed
${x}^{12} = {\left({x}^{6}\right)}^{2}$
$\sqrt{{x}^{12}} = \sqrt{{\left({x}^{6}\right)}^{2}} = {x}^{6}$
Impact of this question
20685 views around the world | {"url":"https://socratic.org/questions/what-is-the-square-root-of-x-12","timestamp":"2024-11-06T04:52:53Z","content_type":"text/html","content_length":"32425","record_id":"<urn:uuid:889c33f4-65b1-4479-95f1-1cd40267e51e>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00472.warc.gz"} |
/matrix -
/matrix - articles
• Matrices and vectors math for AI with Python examples
Article provides an introduction to vectors and matrices, two fundamental concepts in linear algebra, which are widely used in artificial intelligence. It explains what vectors and matrices are
and how they are defined in math. Basic operations with vectors and matrices using Python, including adding, multiplying, and transposing matrices.
Published 2 years ago in #machinelearning about #math, #matrix and #vector | {"url":"https://datachild.net/matrix","timestamp":"2024-11-07T14:00:03Z","content_type":"text/html","content_length":"2412","record_id":"<urn:uuid:9de5653b-4e91-4b85-966f-bfd5c2c8f43d>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00883.warc.gz"} |
PDL::MATLABp - Online in the Cloud
This is the command PDL::MATLABp that can be run in the OnWorks free hosting provider using one of our multiple free online workstations such as Ubuntu Online, Fedora Online, Windows online emulator
or MAC OS online emulator
PDL::MATLAB - A guide for MATLAB users.
If you are a MATLAB user, this page is for you. It explains the key differences between
MATLAB and PDL to help you get going as quickly as possible.
This document is not a tutorial
. For that, go to PDL::QuickStart. This document
the Quick Start guide, as it highlights the key differences between MATLAB and
The key difference between MATLAB and PDL is
Perl is a general purpose programming language with thousands of modules freely available
on the web. PDL is an extension of Perl. This gives PDL programs access to more features
than most numerical tools can dream of. At the same time, most syntax differences between
MATLAB and PDL are a result of its Perl foundation.
You do not have to learn much Perl to be effective with PDL
. But if you wish to learn
Perl, there is excellent documentation available on-line (<
>) or
through the command "perldoc perl". There is also a beginner's portal
Perl's module repository is called CPAN (<
>) and it has a vast array of
modules. Run "perldoc cpan" for more information.
MATLAB typically refers to vectors, matrices, and arrays. Perl already has arrays, and the
terms "vector" and "matrix" typically refer to one- and two-dimensional collections of
data. Having no good term to describe their object, PDL developers coined the term
" to give a name to their data type.
consists of a series of numbers organized as an N-dimensional data set. Piddles
provide efficient storage and fast computation of large N-dimensional matrices. They are
highly optimized for numerical work.
For more information, see "
Piddles vs Perl Arrays
" later in this document.
Unlike MATLAB, PDL does not come with a dedicated IDE. It does however come with an
interactive shell and you can use a Perl IDE to develop PDL programs.
PDL interactive shell
To start the interactive shell, open a terminal and run "perldl" or "pdl2". As in MATLAB,
the interactive shell is the best way to learn the language. To exit the shell, type
"exit", just like MATLAB.
Writing PDL programs
One popular IDE for Perl is called Padre (<
>). It is cross
platform and easy to use.
Whenever you write a stand-alone PDL program (i.e. outside the "perldl" or "pdl2" shell)
you must start the program with "use PDL;". This command imports the PDL module into
Perl. Here is a sample PDL program:
use PDL; # Import main PDL module.
use PDL::NiceSlice; # Import additional PDL module.
use PDL::AutoLoader; # Import additional PDL module.
$b = pdl [2,3,4]; # Statements end in semicolon.
$A = pdl [ [1,2,3],[4,5,6] ]; # 2-dimensional matrix.
print $A x $b->transpose;
Save this file as "myprogram.pl" and run it with:
perl myprogram.pl
New: Flexible syntax
In current versions of PDL (version 2.4.7 or later) there is a flexible matrix syntax that
can look extremely similar to MATLAB:
1) Use a ';' to delimit rows:
$b = pdl q[ 2,3,4 ];
$A = pdl q[ 1,2,3 ; 4,5,6 ];
2) Use spaces to separate elements:
$b = pdl q[ 2 3 4 ];
$A = pdl q[ 1 2 3 ; 4 5 6 ];
Basically, as long as you put a "q" in front of the opening bracket, PDL should "do what
you mean". So you can write in a syntax that is more comfortable for you.
There are two modules that MATLAB users will want to use:
Gives PDL a syntax for slices (sub-matrices) that is shorter and more familiar to
MATLAB users.
% MATLAB
b(1:5) --> Selects the first 5 elements from b.
# PDL without NiceSlice
$b->slice("0:4") --> Selects the first 5 elements from $b.
# PDL with NiceSlice
$b(0:4) --> Selects the first 5 elements from $b.
Provides a MATLAB-style autoloader for PDL. If an unknown function "foo()" is called,
PDL looks for a file called "foo.pdl". If it finds one, it reads it.
This section explains how PDL's syntax differs from MATLAB. Most MATLAB users will want to
start here.
General "gotchas"
In PDL, indices start at '0' (like C and Java), not 1 (like MATLAB or FORTRAN). For
example, if $b is an array with 5 elements, the elements would be numbered from 0 to
Displaying an object
MATLAB normally displays object contents automatically. In the PDL shells you display
objects explicitly with the "print" command or the shortcut "p":
>> a = 12
a = 12
>> b = 23; % Suppress output.
PDL Shell (perldl or pdl2):
pdl> $a = 12 # No output.
pdl> print $a # Print object.
pdl> p $a # "p" is a shorthand for "print" in the shell.
Creating Piddles
Variables in PDL
Variables always start with the '$' sign.
MATLAB: value = 42
PerlDL: $value = 42
Basic syntax
Use the "pdl" constructor to create a new
MATLAB: v = [1,2,3,4]
PerlDL: $v = pdl [1,2,3,4]
MATLAB: A = [ 1,2,3 ; 3,4,5 ]
PerlDL: $A = pdl [ [1,2,3] , [3,4,5] ]
Simple matrices
------ ------
Matrix of ones
(5) ones 5,5
Matrix of zeros
(5) zeros 5,5
Random matrix
(5) random 5,5
Linear vector 1:5 sequence 5
Notice that in PDL the parenthesis in a function call are often optional. It is
important to keep an eye out for possible ambiguities. For example:
pdl> p zeros 2, 2 + 2
Should this be interpreted as "zeros(2,2) + 2" or as "zeros 2, (2+2)"? Both are
valid statements:
pdl> p zeros(2,2) + 2
[2 2]
[2 2]
pdl> p zeros 2, (2+2)
[0 0]
[0 0]
[0 0]
[0 0]
Rather than trying to memorize Perl's order of precedence, it is best to use
parentheses to make your code unambiguous.
Linearly spaced sequences
MATLAB: >> linspace(2,10,5)
ans = 2 4 6 8 10
PerlDL: pdl> p
[2 4 6 8 10]
: Start with a 1-dimensional piddle of 5 elements and give it equally
spaced values from 2 to 10.
MATLAB has a single function call for this. On the other hand, PDL's method is more
pdl> p zeros(5,5)->xlinvals(2,10)
[ 2 4 6 8 10]
[ 2 4 6 8 10]
[ 2 4 6 8 10]
[ 2 4 6 8 10]
[ 2 4 6 8 10]
pdl> p zeros(5,5)->ylinvals(2,10)
[ 2 2 2 2 2]
[ 4 4 4 4 4]
[ 6 6 6 6 6]
[ 8 8 8 8 8]
[10 10 10 10 10]
pdl> p zeros(3,3,3)->zlinvals(2,6)
[2 2 2]
[2 2 2]
[2 2 2]
[4 4 4]
[4 4 4]
[4 4 4]
[6 6 6]
[6 6 6]
[6 6 6]
Slicing and indices
Extracting a subset from a collection of data is known as
. PDL and MATLAB
have a similar syntax for slicing, but there are two important differences:
1) PDL indices start at 0, as in C and Java. MATLAB starts indices at 1.
2) In MATLAB you think "rows and columns". In PDL, think "x and y".
MATLAB PerlDL
------ ------
>> A pdl> p $A
A = [
1 2 3 [1 2 3]
4 5 6 [4 5 6]
7 8 9 [7 8 9]
(row = 2, col = 1) (x = 0, y = 1)
>> A(2,1) pdl> p $A(0,1)
ans = [
4 [4]
(row = 2 to 3, col = 1 to 2) (x = 0 to 1, y = 1 to 2)
>> A(2:3,1:2) pdl> p $A(0:1,1:2)
ans = [
4 5 [4 5]
7 8 [7 8]
When you write a stand-alone PDL program you have to include the PDL::NiceSlice
module. See the previous section "
" for more
use PDL; # Import main PDL module.
use PDL::NiceSlice; # Nice syntax for slicing.
use PDL::AutoLoader; # MATLAB-like autoloader.
$A = random 4,4;
print $A(0,1);
Matrix Operations
Matrix multiplication
MATLAB: A * B
PerlDL: $A x $B
Element-wise multiplication
MATLAB: A .* B
PerlDL: $A * $B
MATLAB: A'
PerlDL: $A->transpose
Functions that aggregate data
Some functions (like "sum", "max" and "min") aggregate data for an N-dimensional data set.
This is a place where MATLAB and PDL take a different approach:
In MATLAB, these functions all work along one dimension.
>> A = [ 1,5,4 ; 4,2,1 ]
A = 1 5 4
>> max(A)
ans = 4 5 4
>> max(A')
ans = 5 4
If you want the maximum for the entire data set, you can use the special A(:)
notation which basically turns the entire data set into a single 1-dimensional
>> max(A(:))
ans = 5
>> A = ones(2,2,2,2)
>> max(A(:))
ans = 1
PDL offers two functions for each feature.
sum vs sumover
avg vs average
max vs maximum
min vs minimum
long name
works over a dimension, while the
short name
works over the entire
pdl> p $A = pdl [ [1,5,4] , [4,2,1] ]
[1 5 4]
[4 2 1]
pdl> p $A->maximum
[5 4]
pdl> p $A->transpose->maximum
[4 5 4]
pdl> p $A->max
pdl> p ones(2,2,2)->max
pdl> p ones(2,2,2,2)->max
Notice that PDL aggregates horizontally while MATLAB aggregates vertically. In other
MATLAB PerlDL
max(A) == $A->transpose->maximum
max(A') == $A->maximum
: In MATLAB you think "rows and columns". In PDL, think "x and y".
Higher dimensional data sets
A related issue is how MATLAB and PDL understand data sets of higher dimension. MATLAB was
designed for 1D vectors and 2D matrices. Higher dimensional objects ("N-D arrays") were
added on top. In contrast, PDL was designed for N-dimensional piddles from the start. This
leads to a few surprises in MATLAB that don't occur in PDL:
MATLAB sees a vector as a 2D matrix.
MATLAB PerlDL
------ ------
>> vector = [1,2,3,4]; pdl> $vector = pdl [1,2,3,4]
>> size(vector) pdl> p $vector->dims
ans = 1 4 4
MATLAB sees "[1,2,3,4]" as a 2D matrix (1x4 matrix). PDL sees it as a 1D vector: A
single dimension of size 4.
But MATLAB ignores the last dimension of a 4x1x1 matrix.
MATLAB PerlDL
------ ------
>> A = ones(4,1,1); pdl> $A = ones 4,1,1
>> size(A) pdl> p $A->dims
ans = 4 1 4 1 1
And MATLAB treats a 4x1x1 matrix differently from a 1x1x4 matrix.
MATLAB PerlDL
------ ------
>> A = ones(1,1,4); pdl> $A = ones 1,1,4
>> size(A) pdl> p $A->dims
ans = 1 1 4 1 1 4
MATLAB has no direct syntax for N-D arrays.
pdl> $A = pdl [ [[1,2,3],[4,5,6]], [[2,3,4],[5,6,7]] ]
pdl> p $A->dims
Feature support.
In MATLAB, several features such as sparse matrix support are not available for N-D
arrays. In PDL, just about any feature supported by 1D and 2D piddles, is equally
supported by N-dimensional piddles. There is usually no distinction.
Loop Structures
Perl has many loop structures, but we will only show the one that is most familiar to
MATLAB users:
MATLAB PerlDL
------ ------
for i = 1:10 for $i (1..10) {
disp(i) print $i
endfor }
Never use for-loops for numerical work. Perl's for-loops are faster than MATLAB's,
but they both pale against a "vectorized" operation. PDL has many tools that
facilitate writing vectorized programs. These are beyond the scope of this guide. To
learn more, see: PDL::Indexing, PDL::Threading, and PDL::PP.
Likewise, never use 1..10 for numerical work, even outside a for-loop. 1..10 is a
Perl array. Perl arrays are designed for flexibility, not speed. Use
To learn more, see the next section.
Piddles vs Perl Arrays
It is important to note the difference between a
and a Perl array. Perl has a
general-purpose array object that can hold any type of element:
@perl_array = 1..10;
@perl_array = ( 12, "Hello" );
@perl_array = ( 1, 2, 3, \@another_perl_array,
(5) );
Perl arrays allow you to create powerful data structures (see
Data structures
but they are not designed for numerical work
. For that, use
$pdl = pdl [ 1, 2, 3, 4 ];
$pdl = sequence 10_000_000;
$pdl = ones 600, 600;
For example:
$points = pdl 1..10_000_000 # 4.7 seconds
$points = sequence 10_000_000 # milliseconds
: You can use underscores in numbers ("10_000_000" reads better than 10000000).
Perl has many conditionals, but we will only show the one that is most familiar to MATLAB
MATLAB PerlDL
------ ------
if value > MAX if ($value > $MAX) {
disp("Too large") print "Too large\n";
elseif value < MIN } elsif ($value < $MIN) {
disp("Too small") print "Too small\n";
else } else {
disp("Perfect!") print "Perfect!\n";
end }
Here is a "gotcha":
MATLAB: elseif
PerlDL: elsif
If your conditional gives a syntax error, check that you wrote your "elsif"'s
TIMTOWDI (There Is More Than One Way To Do It)
One of the most interesting differences between PDL and other tools is the expressiveness
of the Perl language. TIMTOWDI, or "There Is More Than One Way To Do It", is Perl's motto.
Perl was written by a linguist, and one of its defining properties is that statements can
be formulated in different ways to give the language a more natural feel. For example, you
are unlikely to say to a friend:
"While I am not finished, I will keep working."
Human language is more flexible than that. Instead, you are more likely to say:
"I will keep working until I am finished."
Owing to its linguistic roots, Perl is the only programming language with this sort of
flexibility. For example, Perl has traditional while-loops and if-statements:
while ( ! finished() ) {
if ( ! wife_angry() ) {
But it also offers the alternative
until ( finished() ) {
unless ( wife_angry() ) {
And Perl allows you to write loops and conditionals in "postfix" form:
keep_working() until finished();
kiss_wife() unless wife_angry();
In this way, Perl often allows you to write more natural, easy to understand code than is
possible in more restrictive programming languages.
PDL's syntax for declaring functions differs significantly from MATLAB's.
MATLAB PerlDL
------ ------
function retval = foo(x,y) sub foo {
retval = x.**2 + x.*y my ($x, $y) = @_;
endfunction return $x**2 + $x*$y;
Don't be intimidated by all the new syntax. Here is a quick run through a function
declaration in PDL:
1) "
" stands for "subroutine".
2) "
" declares variables to be local to the function.
3) "
" is a special Perl array that holds all the function parameters. This might seem
like a strange way to do functions, but it allows you to make functions that take a
variable number of parameters. For example, the following function takes any number of
parameters and adds them together:
sub mysum {
my ($i, $total) = (0, 0);
for $i (@_) {
$total += $i;
return $total;
4) You can assign values to several variables at once using the syntax:
($a, $b, $c) = (1, 2, 3);
So, in the previous examples:
# This declares two local variables and initializes them to 0.
my ($i, $total) = (0, 0);
# This takes the first two elements of @_ and puts them in $x and $y.
my ($x, $y) = @_;
5) The "
" statement gives the return value of the function, if any.
ASCII File IO
To read data files containing whitespace separated columns of numbers (as would be read
using the MATLAB
command) one uses the PDL
in PDL::IO::Misc. For a general
review of the IO functionality available in PDL, see the documentation for PDL::IO, e.g.,
"help PDL::IO" in the
shell or " pdldoc PDL::IO " from the shell command line.
Data structures
To create complex data structures, MATLAB uses "
cell arrays
" and "
structure arrays
Perl's arrays and hashes offer similar functionality but are more powerful and flexible.
This section is only a quick overview of what Perl has to offer. To learn more about this,
please go to <
> or run the command "perldoc
Perl arrays are similar to MATLAB's cell arrays, but more flexible. For example, in
MATLAB, a cell array is still fundamentally a matrix. It is made of rows, and rows
must have the same length.
array = {1, 12, 'hello'; rand(3, 2),
(3), 'junk'}
=> OK
array = {1, 12, 'hello'; rand(3, 2),
(3) }
=> ERROR
A Perl array is a general purpose, sequential data structure. It can contain any data
@array = ( [1, 12, 'hello'] , [ random(3,2), ones(3,3), 'junk' ] )
=> OK
@array = ( [1, 12, 'hello'] , [ random(3,2), ones(3,3) ] )
=> OK
@array = ( 5 , {'name' => 'Mike'} , [1, 12, 'hello'] )
=> OK
Notice that Perl array's start with the "@" prefix instead of the "$" used by
To learn about Perl arrays, please go to <http://perldoc.perl.org/perldata.html> or run the command "perldoc perldata".
Perl hashes are similar to MATLAB's structure arrays:
>> drink = struct('type', 'coke', 'size', 'large', 'myarray', {1,2,3})
>> drink.type = 'sprite'
>> drink.price = 12 % Add new field to structure array.
pdl> %drink = ( type => 'coke' , size => 'large', mypiddle => ones(3,3,3) )
pdl> $drink{type} = 'sprite'
pdl> $drink{price} = 12 # Add new field to hash.
Notice that Perl hashes start with the "%" prefix instead of the "@" for arrays and
"$" used by piddles.
To learn about Perl hashes, please go to <http://perldoc.perl.org/perldata.html> or run the command "perldoc perldata". Performance
PDL has powerful performance features, some of which are not normally available in
numerical computation tools. The following pages will guide you through these features:
: Beginner
This beginner tutorial covers the standard "vectorization" feature that you already
know from MATLAB. Use this page to learn how to avoid for-loops to make your program
more efficient.
: Intermediate
PDL's "vectorization" feature goes beyond what most numerical software can do. In
this tutorial you'll learn how to "thread" over higher dimensions, allowing you to
vectorize your program further than is possible in MATLAB.
: Intermediate
Perl comes with an easy to use benchmarks module to help you find how long it takes
to execute different parts of your code. It is a great tool to help you focus your
optimization efforts. You can read about it online
>) or through the command "perldoc
: Advanced
PDL's Pre-Processor is one of PDL's most powerful features. You write a function
definition in special markup and the pre-processor generates real C code which can be
compiled. With PDL:PP you get the full speed of native C code without having to deal
with the full complexity of the C language.
PDL has full-featured plotting abilities. Unlike MATLAB, PDL relies more on third-party
libraries (pgplot and PLplot) for its 2D plotting features. Its 3D plotting and graphics
uses OpenGL for performance and portability. PDL has three main plotting modules:
Best for
: Plotting 2D functions and data sets.
This is an interface to the venerable PGPLOT library. PGPLOT has been widely used in
the academic and scientific communities for many years. In part because of its age,
PGPLOT has some limitations compared to newer packages such as PLplot (e.g. no RGB
graphics). But it has many features that still make it popular in the scientific
Best for
: Plotting 2D functions as well as 2D and 3D data sets.
This is an interface to the PLplot plotting library. PLplot is a modern, open source
library for making scientific plots. It supports plots of both 2D and 3D data sets.
PLplot is best supported for unix/linux/macosx platforms. It has an active developers
community and support for win32 platforms is improving.
Best for
: Plotting 3D functions.
The native PDL 3D graphics library using OpenGL as a backend for 3D plots and data
visualization. With OpenGL, it is easy to manipulate the resulting 3D objects with
the mouse in real time.
Writing GUIs
Through Perl, PDL has access to all the major toolkits for creating a cross platform
graphical user interface. One popular option is wxPerl (<
These are the Perl bindings for wxWidgets, a powerful GUI toolkit for writing cross-
platform applications.
wxWidgets is designed to make your application look and feel like a native application in
every platform. For example, the Perl IDE
is written with wxPerl.
Simulink is a graphical dynamical system modeler and simulator. It can be purchased
separately as an add-on to MATLAB. PDL and Perl do not have a direct equivalent to
MATLAB's Simulink. If this feature is important to you, then take a look at
Scilab is another numerical analysis software. Like PDL, it is free and open source. It
doesn't have PDL's unique features, but it is very similar to MATLAB. Scilab comes with
(previously Scicos), a graphical system modeler and simulator similar to Simulink.
Copyright 2010 Daniel Carrera (
[email protected]
). You can distribute and/or modify this
document under the same terms as the current Perl license.
http://dev.perl.org/licenses/ Acknowledgements
I'd like to thank David Mertens, Chris Marshall and Sigrid Carrera for their immense
help reviewing earlier drafts of this guide. Without their hours of work, this
document would not be remotely as useful to MATLAB users as it is today.
Use PDL::MATLABp online using onworks.net services | {"url":"https://www.onworks.net/os-distributions/programs/pdl-matlabp-online","timestamp":"2024-11-08T06:12:16Z","content_type":"text/html","content_length":"240642","record_id":"<urn:uuid:0a399f78-bb44-406d-b92f-6ef10338773b>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00109.warc.gz"} |
av H Karlsson · 2011 — Median. PANEL A: DOMESTIC TRANSACTIONS. Enterprise Value/EBIT. 11.76 8.58. 16.39 12.37 28.26*** 30.62***. Enterprise Value/EBITDA.
EV/EBITDA is a ratio that compares a company’s Enterprise Value Enterprise Value (EV) Enterprise Value, or Firm Value, is the entire value of a firm equal to its equity value, plus net debt, plus any
minority interest, used in (EV) to its Earnings Before Interest, Taxes, Depreciation & Amortization (EBITDA EBITDA EBITDA or Earnings Before Interest, Tax, Depreciation, Amortization is a company's
profits before any of these net deductions are made. EBITDA focuses on the operating decisions
The average ev / ebitda of the companies is 5.1x with a standard deviation of 8.4x. Honda Motor Co., Ltd.'s EV / EBITDA of 8.9x ranks in the 73.1% percentile for the sector. The following table
provides additional summary stats: A company's earnings before interest, taxes, depreciation, and amortization (commonly abbreviated EBITDA, pronounced / iː b ɪ t ˈ d ɑː /, / ə ˈ b ɪ t d ɑː /, or / ˈ
ɛ b ɪ t d ɑː /) is an accounting measure calculated using a company's earnings, before interest expenses, taxes, depreciation, and amortization are subtracted, as a proxy for a company's current
operating The average ev / ebitda of the companies is 9.5x with a standard deviation of 15.4x. Bunge Limited's EV / EBITDA of 10.3x ranks in the 48.3% percentile for the sector. The following table
provides additional summary stats: Tesla EV-to-EBITDA as of today (April 26, 2021) is 165.99.
Thus as shown in the above example, the Industry average is 10, while the multiple for Company A is 6.67 and that for Company B is 11.42. EV-to-EBITDA is essentially the enterprise value (EV) of a
stock divided by its earnings before interest, taxes, depreciation and amortization (EBITDA). Average 20-day Volume greater than or Calculate the Enterprise Value (Market Cap plus Debt minus Cash) =
$69.3 + $1.4 – $ 0.3 = $70.4B; Divide the EV by 2017A EBITDA = $70.4 / $5.04 = 14.0x; Divide the EV by 2017A EBITDA = $70.4 / $5.50 = 12.8x . Download the Free Template. Enter your name and email in
the form below and download the free template now!
2013–2017. Volati acquires well-managed companies av H Karlsson · 2011 — Median.
The average 2020 EV/EBITDA for all companies is 9.3x, so PCOR has a higher EV/EBITDA and is relatively more expensive than peers. We hope this has helped you learn more about EV/EBITDA. Thanks for
reading, and good luck investing this week. Other articles …
TGT EV-to-EBITDA as of today (April 11, 2021) is 12.10. In depth view into Target EV-to-EBITDA explanation, calculation, historical data and more Over 1,010 companies were considered in this
analysis, and 472 had meaningful values.
För att beräkna EV/EBITDA räknar man på följande sätt: Steg 1. Börsvärde + nettoskuld = EV. Steg 2.
Yahoo Finance uses Capital IQ data to provide it, but there was no mention of its calculation basis. Acquisition valuation Step 2: Calculation or estimation of a target company’s EBITDA Next, we need
to calculate Dell Computer’s EBITDA. EBITDA measures a company’s overall financial performance (outcome + income; how it is progressing and improving), while EV (equivalent variation) presents the
cost of the business itself. In June 2018, the average EV/EBITDA for the company M&S was 7.4. Valuation multiples by industry, including EV/Revenue and EV/EBITDA multiples. Data includes enterprise
value multiples for 2018, 2019 and 2020.
EV/EBITDA 2020 trading multiples have on average almost reached the Mar 26, 2019 Where FCF: Free Cash Flow, WACC: Weighted Average Cost of Capital, g: annual growth rate. The estimation of the
future cash flows is usually the industry-average multiple, it must match the 2 “EV/EBITDA” refers to the ratio of enterprise value to earnings before interest, taxes, depreciation, and Jun 18,
2020 Similar to free cash flow yield and price-to-cash-flow, EV/EBITDA is a Given changes in the space, are historical average EBITDA multiples Nov 7, 2013 The “classic” EV/EBITDA ratio is much
better in capturing debt and net Statoil is subject to special taxes, which on average amount to 75% of May 13, 2019 In this video we are showcasing the use of EV to EBITDA ratio, used as a
valuation tool to compare the value of a company, including debt, Jun 24, 2019 The WACC, or Weighted Average Cost of Capital, is an enterprise level discount rate used in capitalizing debt-free
income measures and in Feb 17, 2021 For example, let's assume Company A and Company B, are identical businesses that operate with no debt.
P stockholm
7.5. P/E I jämförelse med median-bolaget har ZetaDisplay. with a Buy recommendation and NOK 65 target price. Our NOK 65 target price equals 15x EV/EBITDA for 2022, in line with the peer average. Show
this thread.
m. Book Value ps Average acquisition multiple. (EV/EBITDA) since 2004. Average organic EBITA growth,.
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The higher the EBITDA margin, the higher the EV/EBITDA multiple valuation. There isn’t a linear relationship in the size of the company and the EV/EBITDA multiple, but the small set of micro cap
companies have a EV/EBITDA multiples below the average. Average EV/EBITDA multiple is 13.9x and the median EV/EBITDA multiple is 13.8x.
I filtered out ADRs, non-US companies, companies in the miscellaneous financial services industry category (to mainly filter out closed-end funds), stocks trading below $2, market caps less than $433
million (approximately matching the average cut-off Tortoriello used), and companies that did not have a EBITDA/EV ratio EV / EBITDA Definition. The EV to EBITDA measures the ratio between enterprise
value and earnings before interest, taxes, depreciation, and amortization. This metric is important to analyze when looking at a company's true value based on how much they are valued at compared to
how money they are earning. Read full definition.
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At the end of 2019, the company traded at price to earnings of 25.3x and EV/EBITDA of 9.86x. The firm generated an average return on equity
21.3. 16.7. 17.6. 14.8. 11.9. EV/EBITDA. 9.3. | {"url":"https://valutaklhi.web.app/81125/31513.html","timestamp":"2024-11-05T23:29:18Z","content_type":"text/html","content_length":"12390","record_id":"<urn:uuid:cf91f9da-7d84-4032-a78b-f79163fed08a>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00887.warc.gz"} |
How to Find Vertical Asymptotes [Updated Guide] - The Education
One of the most common problems you’ll come across in geometry is finding vertical asymptotes. These are important because they allow us to solve for the slopes of a line on a plane, and in doing so,
we can calculate the values of the tangents to that line. Using these asymptotes is an important part of a graph’s construction.
What is Horizontal asymptote?
Horizontal asymptotes are lines in the graph of a function that indicate the y-values approach a finite number as the x-values approach infinity. This is an easy way to visualize the behavior of a
function as a function reaches infinity. It can be useful in certain situations. For instance, you may want to know how to find horizontal asymptotes for a given trig function. In this article, we
will look at a few examples to help you understand what horizontal asymptotes are and how they can be determined.
How to Find horizontal asymptotes
In order to find horizontal asymptotes, you should check the degree of the corresponding numerator and denominator of a given polynomial. In the case of a rational function, the numerator must be
larger than the denominator. The most significant digit in a polynomial must be bigger than the smallest digit in the numerator.
A horizontal asymptote is the upper bound to the behavior of a function as a x-value approaches infinity. There are two types of asymptotes: positive and negative. When a function approaches
infinity, it moves towards the constant value c. Similarly, when the function reaches -infinity, it moves toward the zero point.
Read Also: Linear Interpolation Formula
You can see these asymptotes in the following figure. As x approaches infinity, the distance between the points on the x-axis and the asymptote (which is a straight line passing by the center of the
hyperbola) becomes increasingly smaller. When x approaches -infinity, it’s distance from the asymptote to the x-axis is zero, and when x reaches +infinity, the distance is exactly one.
One of the most common problems high school students face with functions is determining the degree of the corresponding numerator. This is a simple task that can be accomplished with a bit of effort.
To do this, you can try a range of values for x, and then graph the function. If you don’t have access to a graph, you can also use an equation to determine the horizontal asymptotes for a particular
What is vertical asymptote
The vertical asymptote is the point at which a function is closest to an x-value. For example, a 1/x-function will have a vertical asymptote. Another example is a function which is composed of
several polynomial functions. Using this approach, the asymptote will be found by dividing the function.
Vertical asymptotes are the most common, and they can be easily determined. However, there are also some complicated types of vertical asymptotes. It’s usually best to avoid drawing them on a graph.
If you do happen to draw one, then it’s likely that the result will be a bit odd. Usually, the asymptote will look very similar to a line with a vertical slope, such as a dotted line going down.
Basically, the asymptote is the smallest distance your graph can go to the asymptote without touching it. In fact, the asymptote has a very thin barrier that will keep your graph from touching it.
Similarly, the oblique asymptote is the slanted line on the graph. The oblique asymptote can have an infinite number of vertical asymptotes. One example is a f(x) x=2 when x is 4. While this is not a
true horizontal asymptote, it is still a very good example.
The horizontal asymptote is the opposite of the oblique asymptote. It is the asymptote with a smaller degree on the top. In other words, the higher the degree on the top, the fewer the horizontal
asymptotes it has. A horizontal asymptote can be found by long division or by zooming out.
What is Oblique asymptote
Oblique asymptotes are lines on a graph, which are not straight across, nor parallel to the x-axis. This type of line is also called a slant asymptote, because of its slant. These are typically found
in functions that have a rational backbone. In order to find oblique asymptotes, you need to know how to divide a polynomial by a denominator. You can do this by long division or synthetic division.
Fortunately, long division is easy to do. If you are using a TI-89, you can use the propFrac( command.
An oblique asymptote can occur when the function gets close to infinity. For this to happen, the numerator must be one higher than the denominator. Then the function has a quotient, or asymptote, at
the oblique point. As a result, the quotient is a horizontal line, not a vertical line.
When a function gets to infinity, the remainder will tend to go to zero. This means that the graph of the function will approach the asymptote from below. It is important to note that oblique
asymptotes cannot occur for functions with a slant asymptote, like the function y=5×2. They can, however, have vertical asymptotes. These can occur for functions like the function f(x)=3×6+4x-3x+1.
To figure out which asymptotes a function has, you can divide the numerator by the denominator and look for a line with a slope. When you do, you’ll see a slanted asymptote, which looks like a
parabola with a remainder. Often, the slant asymptote is used as the equation for an oblique asymptote. | {"url":"https://theeducationlife.com/how-to-find-vertical-asymptotes/","timestamp":"2024-11-05T21:33:15Z","content_type":"text/html","content_length":"109237","record_id":"<urn:uuid:d9196c80-d275-4e08-878f-2ae084e1923c>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00023.warc.gz"} |
Factor Levels In R: Using Categorical & Ordinal Variables
This tutorial will go through factors and factor levels in R. You’ll learn how to create a factor and how to adjust factor levels.
Factors are used to store and work with variables in R.
In this tutorial, you’ll be dealing with categorical and ordinal variables. Categorical variables are variables that involve one or more categories that aren’t ordered in any specific way. An
example would be colors. Ordinal variables, on the other hand, are similar to categorical variables with the difference that ordinal variables have clear ordering of the categories. This could be
like low, medium, and high.
This is an introduction to more statistical terms. You are now slowly exploring R’s capabilities for data and statistical analysis.
Categorical Factor Levels In R
If you recall in another lesson about data frames, you used the dollar sign ($) to print out the Species column from the iris dataset. Do this again in RStudio. At the bottom-most part, there’s a
line containing Levels composed of setosa, versicolor, and virginica.
This is R’s way of handling categories in data.
If you use the unique ( ) function, R will list out the unique values in the specified column. For example, if you Run unique (iris$Species), the Console displays the three Species level of iris.
There’s no inherent ordering for these levels. You can’t say that setosa is greater than the other two color categories. R, by default, arranges them into alphabetical order.
Ordinal Factor Levels In R
Now let’s try and explore factors with inherent ordering of the category.
Create a vector and name it orders. For this example, assign that vector with data using Starbucks’ cup size names: tall, venti, and grande. Then, print it out.
These should be arranged from smallest to the biggest; it should be tall, venti, and grande. But when you Run the unique ( ) function for orders, they aren’t arranged in that order.
Here’s how to turn them into ordinal variables. First, you need to create a new vector. In this case, the vector is called new_orders_factor. Assign this vector with the factor ( ) function. Inside
this function, input the vector you want to set levels with. Then, indicate levels in the order you want them to appear.
Highlight this entire line of code and then Run it. A new Value is then added in Environment.
To check if a vector has been properly assigned as a factor, use the is.factor ( ) function. If you check the two vectors, orders and new_orders_factor, you can see that the former returns FALSE
while the new vector is indeed a factor.
A factor is a special way to store a series of texts. And though it’s a character vector, it can be stored in a way that allows it to have a given number of categories that have a specific ordering
of values or levels.
If you check using the levels ( ) function, you can see that the levels are now in the correct order.
***** Related Links *****
Objects And Object Classes In R: The Basics
Create Vectors In R: A Step-by-step Tutorial
Data Frames In R: Learning The Basics
Though this lesson may seem esoteric, you’ll see how this makes a difference when dealing with more advanced R coding. It’s important to learn about factors and levels since they often come up in
many R coding and statistical analysis.
Learn R by working on practical, real-world projects.
Unlock the full potential of R in your data analysis tasks and elevate your skills from proficient to expert.
Learn to master the art of data visualization using R. This guide covers everything from basic plots to complex, interactive visualizations.
This thread explores advanced topics in data analytics, focusing on building data pipelines, comparing SQL and R for data transformation, and applying predictive modeling techniques such as customer
churn analysis and time series forecasting in R.
A hands-on guided project to discover hidden patterns and relationships in retail transaction data using the Apriori algorithm in R.
An in-depth, hands-on course designed to teach the practical application of hierarchical clustering in R, complete with real-world examples, to enhance advanced analytical skills.
This project aims to teach the principles of prescriptive analytics and optimization through hands-on examples using the R programming language.
A comprehensive guide to effectively manipulate and transform data using the dplyr package in R.
Learn how to harness the power of Random Forest models to tackle real-world business challenges.
A comprehensive guide to writing efficient, reusable code and performing analysis using the R language.
A comprehensive guide to predicting stock price trends using Random Forest models in R.
A project aimed at optimizing inventory levels for a manufacturing company through predictive modeling using Random Forests in R.
Mastering R with Practical Projects
Advanced Data Analysis with R: From Proficiency to Mastery
The Ultimate Guide to Visualization in R Programming
Comprehensive Guide to Data Transformation and Prediction with R
Market Basket Insights Using Association Rule Learning in R
Mastering Hierarchical Clustering with R: Dendrograms and Cluster Trees in Action
Mastering Prescriptive Analytics with R: A Practical Guide
Mastering Data Manipulation in R with dplyr
Mastering Random Forest Models for Business Applications
Mastering Reusable Code and Analysis in R
Forecasting Stock Price Movements Using Random Forest in R
Supply Chain Optimization Using Random Forests and R | {"url":"https://blog.enterprisedna.co/factor-levels-in-r-using-categorical-ordinal-variables/","timestamp":"2024-11-02T07:45:25Z","content_type":"text/html","content_length":"463562","record_id":"<urn:uuid:4f8f351a-6340-48c2-9687-4e79c4732688>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00462.warc.gz"} |
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Comparing ground truth with predictions using image similarity measures
Image similarity measures play an important role in image fusion algorithms and applications, such as duplicate product detection, image clustering, visual search, change detection, quality
evaluation, and recommendation tasks. These measures essentially quantify the degree of visual and semantic similarity of a pair of images. In other words, they compare two images and return a value
that tells you how visually similar they are.
The input images may be a set of images obtained from the same scene or object taken from different environmental conditions, angles, lighting conditions, or edited transformations of the same image.
In most use cases, the degree of similarity between the images is very important. In other use cases, the aim is to consider whether two images belong to the same category.
To perform image similarity, we need to consider two elements. First, we find the set of features that can be used to describe the image content and, second, we apply a suitable metric to assess the
similarity of two images based on the feature space. An image's feature space can be for the entire image or just for a small group of pixels within the image, such as regions or objects.
The resulting measure can be different depending on the types of features. Typically, the feature space is assumed to be Euclidean. Euclidean is a space in any finite number of dimensions (originally
three-dimensional), in which points are defined by coordinates (one for each dimension) and the distance between two points is calculated by a distance formula. Generally, algorithms that are used to
assess the similarity between two images aim to reduce the semantic gap between low-level features and high-level semantics as much as possible.
Depending on the use case, there may be algorithms that can be used for measuring similarity. However, in most cases, evaluating the math for a metric and ensuring the correct implementation for your
use case is a challenge. This post introduces a Python package, developed by UP42, that has several ready-to-use algorithms for applying similarity measures.
We used this package to evaluate a final image's quality in the analysis outlined in our recent publication. After doing an inference on the image, we used image similarity measures to compare it
with the ground truth image and check whether the predicted image kept the ground truth's essential features.
How To Use Our Python Package
Similarity measure algorithms can usually be implemented by checking the formula and implementing it in your preferred coding language. However, having several options available can help remove the
effort and frustration behind understanding every formula in metric. To help with this, we've developed a Python package with eight image similarity metrics that can be for either 8-bit (visual) or
12-bit (non-visual) images. You can use this package either via the command line (CLI) or by importing it directly into your Python code, as long as you are running Python version 3.6, 3.7, or 3.8.
Here is an example of using the package via the CLI:
image-similarity-measures --org_img_path=path_to_first_img
--pred_img_path=path_to_second_img --mode=tif
--org_img_path_ indicates the path to the original image, and pred_img_path indicates a path to the predicted or disordered image, which is created from the original image. --mode is the image format
with the default set to tif. Other options are png or jpg. --write_to_file can be used to write the final result to a file. You can set it to false if you don't want a final file. Finally, --metric
is the name of the evaluation metric that, by default, is set to psnr. However, it can also be on the following:
• rmse
• ssim
• fsim
• issm
• sre
• sam
• uiq
If you want to use it in your Python code, you can do so as follows:
import cv2
import image_similarity_measures
from image_similarity_measures.quality_metrics import rmse, psnr
in_img1 = cv2.imread('img1.png')
in_img2 = cv2.imread('img2.png')
out_rmse = rmse(in_img1, in_img2)
out_psnr = psnr(in_img1, in_img2)
You can check out the repository in Github to get more information, review the code, or even contribute to the Open Source project yourself!
The Eight Similarity Measures, Explained
We have implemented eight different metrics in our python package. Each of these metrics has distinct benefits and tells you something slightly different about the similarity of different images. It
is essential to choose the metric or combination of metrics that suit your use case. For instance, PSNR, RMSE, or SRE simply measure how different the two images are. This is good to make sure that a
predicted or restored image is similar to its "target" image, but the metics don't consider the quality of the image itself. Other metrics attempt to solve this problem by considering image structure
(SSIM) or displayed features (FSIM).
Metrics such as ISSM or UIQ combine a number of different measures in order to express a more "holistic" image similarity while a metric like SAM estimates spectral signatures to measure how
faithfully the relative spectral distribution of a pixel is reconstructed, while ignoring absolute brightness.
We’ll now go through each of these eight metrics and briefly cover the theory behind each of them.
Root Mean Square Error (RMSE) measures the amount of change per pixel due to the processing. RMSE values are non-negative and a value of $0$ means the image or videos being compared are identical.
The RMSE between a reference or original image, $image1 - K$ and the enhanced or predicted image, $image2 - I(i, j)$ is given by:
$RMSE = \sqrt{\frac{1}{M*N}\sum_{i=0,j=0}^{M-1,N-1} [I(i,j) - K(i,j)]^{2}}$
Peak Signal-to-Noise Ratio (PSNR) measures the ratio between the maximum possible power of a signal and the power of corrupting noise that affects the fidelity of its representation. PSNR is usually
expressed in terms of the logarithmic decibel scale. Typical values for the PSNR in lossy image and video compression in 8-bits are between 30 and 50 dB, where higher values are more desirable. For
16-bit data, typical values for the PSNR are between 60 and 80 dB.
To compute the PSNR, the package first computes the mean-squared error (MSE) using the following equation:
$MSE = \frac{\sum_{M,N} [I_{1}(m,n) - I_{2}(m,n)]^{2}}{M*N}$
In the previous equation, $M$ and $N$ are the number of rows and columns in the input images. The PSNR is subsequently calculated using the following equation:
$PSNR = 10log_{10}(\frac{R^{2}}{MSE})$
In the PSNR calculation, $R$ is the maximum possible pixel value of the image. Please note that log transform is applied to have the value in decibels.
Structural Similar Index Measure (SSIM) quantifies image quality degradation caused by processing, such as data compression, or by losses in data transmission. SSIM is based on visible structures in
the image. In order words SSIM actually measures the perceptual difference between two similar images. The algorithm does not judge which of the two is better. However, that can be inferred from
knowing which is the original image and which has been subjected to additional processing, such as data compression. The SSIM value is between $-1$ and $1$ with $1$ indicating perfect structural
The measure between two windows $x$ and $y$:
$SSIM(x,y) = \frac{(2\mu_{x}\mu_{y} + c_{1})(2\sigma_{xy}+c_{2})}{(\mu_{x}^{2}+\mu{y}^{2} + c_{1})(\sigma^{2}_{x} + \sigma^{2}_{y} + c_{2})}$
In the above equation, $\mu x$ is the average of $x$; $\mu y$ is the average of $y$; $\sigma x$ is the variance of $x$; $\sigma y$ is the variance of $y$, $\sigma xy$ is the covariance of $x$; and
$y$, $c_{1} = (k_{1}L)^{2}$ and $c_{2} = (k_{2}L)^{2}$ are two variables that stabilize the division with a weak denominator, $L$ is the dynamic range of the pixel values (typically this is $2^{no.
of bits per pixel} - 1)$, and $k_{1} = 0.01$, $kk_{2} = 0.03$ by default.
Feature Similarity Indexing Method (FSIM) are developed with a view to compare the structural and feature similarity measures between restored and original objects. It is based on phase congruency
and gradient magnitude. The FSIM value is between $0$ and $1$, where $1$ is perfect feature similarity. For more information on this measure, you can review the original paper.
Information theoretic-based Statistic Similarity Measure (ISSM) interpolates the information theory with the statistic because the information theory has a high capability to predict the relationship
among image intensity values. This hybrid approach incorporates information theory (Shannon entropy) with a statistic (SSIM) as well as a distinct structural feature provided by edge detection
(Canny). For more information on this measure, you can review the original paper.
Signal to Reconstruction Error ratio (SRE) was originally implemented in this paper and it measures the error relative to the power of the signal. It’s been stated in the paper that using SRE is
better suited to make errors comparable between images of varying brightness. Whereas the popular PSNR would not achieve the same effect, since the peak intensity is constant. SRE is computed as:
$SRE = 10log_{10}\frac{\mu_{x}^{2}}{|\hat{x} - x|^{2} / n}$
In the SRE equation $\sigma x$ is the average value of $x$. The values of SRE are given in decibels (dBs).
Spectral Angle Mapper (SAM) is a physically-based spectral classification. The algorithm determines the spectral similarity between two spectra by calculating the angle between the spectra and
treating them as vectors in a space with dimensionality equal to the number of bands. Smaller angles represent closer matches to the reference spectrum. This technique, when used on calibrated
reflectance data, is relatively insensitive to illumination and albedo effects. SAM determines the similarity by applying the following equation:
$\alpha = \cos^{-1}\frac{\sum_{i=1}^{nb} t_{i}r_{i}}{\sqrt{\sum_{i=1}^{nb} t_{i}^{2}}\sqrt{\sum_{i=1}^{nb} r_{i}^{2}}}$
In this calculation, $t$ and $r$ are an image pixel spectrum and a reference spectrum, respectively, in a n-dimensional feature space, where $n$ equals the number of available spectral bands.
Universal Image Quality index (UIQ) is designed by modeling any image distortion as a combination of three factors:
• Loss of correlation
• Luminance distortion
• Contrast distortion
UIQ can be obtained by using the following equation:
$Q = \frac{\sigma_{xy}}{\sigma_{x}\sigma_{y}} * \frac{2\overline{x}\overline{y}}{(\overline{x})^{2} + (\overline{y})^{2}} * \frac{2\sigma_{x}\sigma_{y}}{\sigma_{x}^{2} + \sigma_{y}^{2}}$
The first component is the correlation coefficient between $x$ and $y$ images, which measures the degree of linear correlation. The second component with a value range of $[0.1]$ measures how close
the mean luminance is between the $x$ and $y$. The third component measures how similar the contrasts of the images are. The range of values for the index $Q$ is $[-1,1]$. The optimum value, $1$, is
achieved if and only if the images are identical.
Image Similarity Measures In Action
We used our Python package to evaluate the accuracy of our Super-Resolution processing block available on the UP42 marketplace. Below, you can review the pan sharpened image (ground truth) and model
(the same model we used in the block) output image which super resolves the multispectral image to have the same resolution of pan sharpened image. Please note that we did the evaluation on the
multispectral image, but the actual block will do superresolution on a pan sharpened image (For more information about this block, please have a look at this blog post and this paper).
left: ground truth image; right: predicted image
We used the command to evaluate the two images as follows:
image-similarity-measures --org_img_path=path_to_first_img --pred_img_path=path_to_second_img --mode=tif --metric=ssim
The above command provides the SSIM value, but below you can find the values for all the metrics mentioned in this blog post for the above images.
• PSNR - 30.029
• SSIM - 0.77
• FSIM - 0.54
• ISSM - 0.17
• UIQ - 0.43
• SAM - 88.87
• SRE - 64.66
• RMSE - 0.03
In Closing
Image similarity detection is a hot topic in computer vision as it’s an essential component of many applications. There are various algorithms available to perform image similarity for different use
cases. In this blog post, we introduced our new Python package that includes some of the common algorithms used for image similarity. We hope you find the Open Source package useful and welcome your
feedback, ideas, and contributions directly to the project! | {"url":"https://up42.com/blog/image-similarity-measures","timestamp":"2024-11-02T09:09:06Z","content_type":"text/html","content_length":"949945","record_id":"<urn:uuid:809e2787-3afa-4a47-80e0-78ab3cbd2594>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00035.warc.gz"} |
Wolfram|Alpha Examples: Differential Equations
Examples for
Differential Equations
A differential equation is an equation involving a function and its derivatives. It can be referred to as an ordinary differential equation (ODE) or a partial differential equation (PDE) depending on
whether or not partial derivatives are involved. Wolfram|Alpha can solve many problems under this important branch of mathematics, including solving ODEs, finding an ODE a function satisfies and
solving an ODE using a slew of numerical methods.
Ordinary Differential Equations
Solve an ODE or find an ODE a function satisfies.
Solve a linear ordinary differential equation:
Solve an inhomogeneous equation:
Solve an equation involving a parameter:
Solve a nonlinear equation:
Find differential equations satisfied by a given function: | {"url":"https://m.wolframalpha.com/examples/mathematics/differential-equations","timestamp":"2024-11-10T19:24:25Z","content_type":"text/html","content_length":"76862","record_id":"<urn:uuid:9972abbd-c54c-49e2-aa23-3549bb8ae48c>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00819.warc.gz"} |
Water Sprinkler Used For Grass Lawns Begins To Rotate As Soon As As The Water Is Supplied. Explain The Principle On Which It Works.
Water sprinkler used for grass lawns begins to rotate as soon as as the water is supplied. Explain the principle on which it works.
The working of the rotation of sprinkler is based on third law of motion. As the water comes out of the nozzle of the sprinkler, an equal and opposite reaction force comes into play. So the sprinkler
starts rotating.
Was this answer helpful?
Didn't liked the above answer ?
💡 Some Related Questions | {"url":"https://flasheducation.online/question/water-sprinkler-used-for-grass-lawns-begins-to-rotate-as-soon-as-as-the-water-is-supplied-explain-the-principle-on-which-it-works/","timestamp":"2024-11-03T09:51:08Z","content_type":"text/html","content_length":"202382","record_id":"<urn:uuid:f5050fb0-272a-40db-952d-3b83a5a123fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00225.warc.gz"} |
What does DESeq2 rlog() return exactly?
Last seen 7.1 years ago
In the documentation or the rlog() function we can find that
rlog(K_ij) = log2(q_ij) = beta_i0 + beta_ij,
Which means the function would return the log2 transformed data after normalization by a size factor, estimating dispersion, shrinking dispersion and then the the beta parameters.
Following the description of the paper accompanying DESeq2 package, it seems like the model for q_ij is:
q_ij = exp(x^T * beta)
where x is the vector of covariates and beta the vector of coefficients in glm negative binomial model.
It seems like if we only have 1 factor covariate with 2 possible levels, then x is in {0,1} and we only have two possible values for beta_1j (depending on whether x_j = 1 or 0).
When I run rlog on the raw count data, the transformed counts are still different (even though similar) for each column even when belonging to the same class (with the same covariate).
It would be great if one of the developers could answer this question. I would greatly appreciate it.
Entering edit mode
1) Yes.
2) We calculate one prior variance for the whole matrix: "The prior variance is found by matching the 97.5% quantile of a zero-centered normal distribution to the 95% quantile of the absolute values
in the LFC matrix."
3-4) Yes, if blind=TRUE, otherwise we use the dispersion trend already calculated using the experimental design (see vignette discussion of blind=TRUE or FALSE)
5) Yes.
The idea is to shrink sample-to-sample differences when there is little information (low counts) and to preserve these differences when there is information (high counts).
ADD REPLY • link | {"url":"https://support.bioconductor.org/p/79398/","timestamp":"2024-11-02T17:51:12Z","content_type":"text/html","content_length":"23116","record_id":"<urn:uuid:a53f0089-38e6-416a-86c3-4c4e31097cf8>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00465.warc.gz"} |
If statement referencing another sheet returns invalid operation
So I am using this:
=IF({Master Bookings Sheet-V2.0 Range 1} = [Cutting Room Number]@row, "Unavailable", "Available")
Above is the sheet where I'm checking the value of "Cutting Room Number" from another sheet called "Master Bookings Sheet V2.0". I have to check: if the value in "Cutting Room Number" in the above
sheet is mentioned in the referenced sheet then "Unavailable" else "Available".
It's supposed to be a simple IF but it returns #INVALID OPERATION
Any help is appreciated.
Best Answers
• @pdv90 Your formula isn't working because you are comparing an array to a single cell. You can try the following
=IF(CONTAINS([Cutting Room Number]@row, {Master Bookings Sheet-V2.0 Range 1}), "Unavailable", "Available")
• Hey @pdv90,
I think you should try and use COUNTIF within the IF so then if the COUNTIF returns anything above 0 it will show as Unavailable.
The reason the formula is not working is that IF doesnt know how to compare a range to a cell.
You can also use Vlookup a similar way within the IF.
Let me know if you need help with writing the formula.
Itai Perez
Reporting and Project Manager
If you found my comment helpful any reaction, Insightful, Awsome etc... would be appreciated🙂
• @pdv90 Your formula isn't working because you are comparing an array to a single cell. You can try the following
=IF(CONTAINS([Cutting Room Number]@row, {Master Bookings Sheet-V2.0 Range 1}), "Unavailable", "Available")
• Hey @pdv90,
I think you should try and use COUNTIF within the IF so then if the COUNTIF returns anything above 0 it will show as Unavailable.
The reason the formula is not working is that IF doesnt know how to compare a range to a cell.
You can also use Vlookup a similar way within the IF.
Let me know if you need help with writing the formula.
Itai Perez
Reporting and Project Manager
If you found my comment helpful any reaction, Insightful, Awsome etc... would be appreciated🙂
• Thank you so much Eric & Itai. It's working now
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/109959/if-statement-referencing-another-sheet-returns-invalid-operation","timestamp":"2024-11-02T02:45:15Z","content_type":"text/html","content_length":"430611","record_id":"<urn:uuid:8d6fa0e1-3e64-48af-8b4e-55820cd78377>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00374.warc.gz"} |
Factors affecting auger length in context of auger length
08 Sep 2024
Title: An Exploration of the Factors Affecting Auger Length: A Theoretical Analysis
Abstract: Augers are a crucial component in various industries, including construction, agriculture, and waste management. The length of an auger is a critical parameter that affects its performance,
efficiency, and overall effectiveness. This article aims to investigate the factors influencing auger length, providing a comprehensive understanding of the underlying principles.
Introduction: The length of an auger (L) is a function of several variables, including the type of material being handled (M), the desired flow rate (Q), and the mechanical properties of the auger
itself. Understanding these relationships is essential for designing efficient and effective augers.
Factors Affecting Auger Length:
1. Material Type (M)
The type of material being handled has a significant impact on auger length. Different materials have varying densities, viscosities, and flow characteristics, which affect the required auger length.
The relationship between material type and auger length can be represented by:
L ∝ M^(-1/2) … (1)
where L is the auger length, and M is a dimensionless parameter representing the material’s properties.
2. Desired Flow Rate (Q)
The desired flow rate of the material also influences auger length. A higher flow rate requires a longer auger to maintain efficient conveying. The relationship between flow rate and auger length can
be represented by:
L ∝ Q^(1/3) … (2)
where L is the auger length, and Q is the desired flow rate.
3. Mechanical Properties of the Auger
The mechanical properties of the auger itself, such as its diameter (D), thickness (T), and material strength (S), also affect auger length. A stronger, thicker auger can handle more material and
maintain a longer length. The relationship between mechanical properties and auger length can be represented by:
L ∝ D^2 * T^(-1) * S^(1/3) … (3)
where L is the auger length, D is the diameter, T is the thickness, and S is the material strength.
Conclusion: The factors affecting auger length are complex and interdependent. Understanding these relationships is essential for designing efficient and effective augers. The formulas presented in
this article provide a theoretical framework for analyzing the impact of various variables on auger length. Further research is needed to validate these findings and explore their practical
• [1] Auger Design Manual, Industry Standard (2019)
• [2] Material Handling Systems, Engineering Handbook (2020)
Related articles for ‘auger length’ :
• Reading: Factors affecting auger length in context of auger length
Calculators for ‘auger length’ | {"url":"https://blog.truegeometry.com/tutorials/education/b63bd388dc9ca557429349d3bc8e675a/JSON_TO_ARTCL_Factors_affecting_auger_length_in_context_of_auger_length.html","timestamp":"2024-11-04T20:44:31Z","content_type":"text/html","content_length":"16404","record_id":"<urn:uuid:a9b53242-25fe-4019-ae75-2f2f7a97eac0>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00747.warc.gz"} |
Programming Framework
Programming Framework#
OpenDP is based on a conceptual model that defines the characteristics of privacy-preserving operations and provides a way for components to be assembled into programs with desired behavior. This
model, known as the OpenDP Programming Framework, is described in the paper A Programming Framework for OpenDP. The framework is designed with a precise and verifiable means of capturing the
privacy-relevant aspects of an algorithm, while remaining highly flexible and extensible. OpenDP (the software library) is intended to be a faithful implementation of that approach. Because OpenDP is
based on a well-defined model, users can create applications with rigorous privacy properties.
The OpenDP Programming Framework consists of a set of high-level conceptual elements. We’ll cover the highlights here, which should be enough for you to get acquainted with OpenDP programming. If
you’re interested in more of the details and motivations behind the framework, you’re encouraged to read the paper. There is also an illustrative notebook A Framework to Understand DP.
• Measurements are randomized mappings from a private, potentially sensitive dataset or value to an arbitrary output value that is safe to release. They are a controlled means of introducing
privacy protection (e.g. noise) to a computation. An example of a measurement is one that adds Laplace noise to a value.
• Transformations are deterministic mappings from a private dataset to another private dataset or value. They are used to summarize or transform values in some way. An example of a transformation
is one which calculates the mean of a set of values.
• Domains are sets which identify the possible values that some object can take. They are used to constrain the input or output of measurements and transformations. Examples of domains are the
integers between 1 and 10, or vectors of length 5 containing floating point numbers.
• Measures and metrics are things that specify distances between two mathematical objects.
□ Measures characterize a distance between two probability distributions. An example measure is the “max-divergence” of pure differential privacy.
□ Metrics capture a distance between two private datasets or values. An example metric is “symmetric distance” (counting the number of additions or removals).
• Privacy maps and stability maps are functions that characterize the relationship between “closeness” of operation inputs and operation outputs. They are the glue that binds everything together.
□ A privacy map is a statement about a measurement. It’s a function that takes an input distance (in a specific metric) and emits the smallest upper bound on the output distance (in a specific
measure). A privacy map lets you make assertions about a measurement when the measurement is evaluated on any pair of neighboring datasets. It’s guaranteed that any pair of measurement inputs
within the input distance will always produce a pair of measurement outputs within the output distance.
□ A stability relation is a statement about a transformation. It’s also a function that takes an input distance (in a specific metric) and emits the smallest upper bound on the output distance
(in a specific metric, possibly different from the input metric). A stability map lets you make assertions about the behavior of a transformation when that transformation is evaluated on any
pair of neighboring datasets. It’s guaranteed that any pair of transformation inputs within the input distance will always produce transformation outputs within the output distance.
Maps capture the notion of closeness in a very general way, allowing the extension of OpenDP to different definitions of privacy.
As you can see, these elements are interdependent and support each other. The interaction of these elements is what gives the OpenDP Programming Framework its flexibility and expressiveness.
Key Points#
You don’t need to know all the details of the Programming Framework to write OpenDP applications, but it helps understand some of the key points:
• OpenDP calculations are built by assembling a measurement from a number of constituent transformations and measurements, typically through chaining or composition.
• Measurements don’t have a static privacy loss specified when constructing the measurement. Instead, measurements are typically constructed by specifying the scale of noise, and the loss is
bounded by the resulting privacy relation. This requires some extra work compared to specifying the loss directly, but OpenDP provides some utilities to make this easier on the programmer, and
the benefit is greatly increased flexibility of the framework as a whole. | {"url":"https://docs.opendp.org/en/stable/api/user-guide/programming-framework/index.html","timestamp":"2024-11-14T10:36:35Z","content_type":"text/html","content_length":"34872","record_id":"<urn:uuid:7107cc9a-0446-4440-9614-e70cd76c4f75>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00142.warc.gz"} |
Analysis of student beliefs in a tertiary preparatory mathematics course
Carmichael, Colin S. and Taylor, Janet A.. 2005. "Analysis of student beliefs in a tertiary preparatory mathematics course." Kingfisher Delta '05: 5th Southern Hemisphere Conference on Undergraduate
Mathematics and Statistics Teaching and Learning. Fraser Island, Australia 22 - 26 Nov 2005 London, United Kingdom. https://doi.org/10.1080/00207390500271065
Paper/ Analysis of student beliefs in a tertiary preparatory mathematics course
Presentation Paper
Authors Carmichael, Colin S. (Author) and Taylor, Janet A. (Author)
Journal or
Proceedings International Journal of Mathematical Education in Science and Technology
Journal 36 (7), pp. 713-719
Number of 7
Year 2005
Place of London, United Kingdom
ISSN 0020-739X
Object https://doi.org/10.1080/00207390500271065
Web Address
(URL) of http://www.tandf.co.uk/journals/titles/0020739X.asp
Conference/ Kingfisher Delta '05: 5th Southern Hemisphere Conference on Undergraduate Mathematics and Statistics Teaching and Learning
Kingfisher Delta '05: 5th Southern Hemisphere Conference on Undergraduate Mathematics and Statistics Teaching and Learning
Event Date
Event 22 to end of 26 Nov 2005
Event Location
Fraser Island, Australia
Every year approximately 800 students enrol in the tertiary preparatory course TPP7181 at the University of Southern Queensland. Successful completion of this course will allow students
to enrol in either further preparatory level mathematics courses or undergraduate study. For many of the students enrolled in this course, the study of mathematics was undertaken quite
Abstract some time ago and usually in a school setting. Drop out rates for this course are quite high and it is hypothesized that motivation may be a key factor in determining student success or
otherwise. In this study scales assessing self-efficacy were utilized in an attempt to gauge aspects of the motivation of students enrolled in the course. Initial results suggest that
only specific measures of student confidence predict their performance and that both gender and age mediate the strength of this prediction.
Keywords tertiary preparatory mathematics; course; student; TPP7181
ANZSRC Field
of Research 520102. Educational psychology
390303. Higher education
390109. Mathematics and numeracy curriculum and pedagogy
Public Notes Files associated with this item cannot be displayed due to copyright restrictions.
Byline Learning and Teaching Support Unit
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[EM] Simulations with social welfare functions
David Cary dcarysysb at yahoo.com
Thu May 25 23:45:21 PDT 2006
--- bql at bolson.org wrote:
> To answer my own question, I think the attached perl script nicely
> shows
> the difference between std-dev and gini by this output:
The Gini Coefficient is invariant under scaling, but not under
translation. Standard deviation is invariant under translation, but
not under scaling. If you want a better comparison between the two,
you might try comparing Gini to stdev/mean. Also, Gini, like
standard deviation, can be calculated for a population or a sample.
The Perl code is inconsistently using the formulas for the Gini of a
population, but the standard deviation of a sample.
The Wikipedia article is confusing to the point of being erroneous.
To calculate the Gini coefficient for a population, use the Brown
formula with:
X_k = k / n
Y_k = S_k / S_n
S_k = sum of w_i for i = 1 to k
(w_i) i = 1 to n, is the sequence of values (e.g. income) for each
member of the population, sorted in increasing order.
With this setup, the Brown formula can be restated as:
T = sum of w_i * (n+1-i) for i = 1 to n
G = 1 - 2*(T/S_n - 0.5) / n
This is algebraically equivalent to the formula given earlier on this
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Decimal to Fractional Odds Converter | Today's Betting Tips
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Q: Are there any other types of odds formats?
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Convert annual to monthly interest rate formula
22 Oct 2018 To convert an annual interest rate to monthly, use the formula "i" divided by "n," or interest divided by payment periods. For example, to determine Calculating interest month-by-month
is an essential skill. To calculate a monthly interest rate, divide the annual rate by 12 to account for the 12 months in the year. Convert the annual rate from a percent to a decimal by dividing by
The annual percentage rate (APR) that you are charged on a loan may not be the The amount of interest you effectively pay is greater the more frequently the interest is compounded. the formula FV=pv
(r/n)^nt that would equalize the APR and effective rate. However, one compounds daily and the other one monthly. 24 Aug 2010 i have to convert pthly effective rates to an annual rate, can somebody
help. calculating interest per annum, when you are given a quarterly rate. you could have worked in months using an effective monthly interest rate. Convert Flat Interest Rate (a.k.a simple interest)
to Effective Interest Rate here. Use Loanstreet's online interest rate calculator to calculate Personal Loans, Car Loans & Hire Purchase interest rates. Monthly Installment Amount. RM 2,250.00 It is
used to compare the annual interest between loans with different compounding terms (daily, monthly, quarterly, semi-annually, annually, or other). It is also APY Calculator to Calculate Annual
Percentage Yield from a Stated Nominal Interest What APY is, how to calculate it, how to convert it back to APR, and how it or annual interest rate of 4.875% compounded monthly, would translate to
an Solving Problems with Non-Annual Periods on the TI BAII Plus the length of a period is one month, and you must convert the variables to a monthly basis in order For example, when calculating the
monthly interest rate, you should do the However, you make your interest payments monthly, so your mortgage lender need to find the rate that compounded monthly, results in an effective annual rate
of If you are comfortable using the formula to calculate the present value of an
Annual Percentage Rate (APR) Calculator. Loan Amount. $. Interest Rate. %. Term. Yr. Finance Charges (Added to loan amount). $. Prepaid Finance Charges
The annual percentage rate (APR) of a loan is the interest you pay each year represented as a percentage of the loan balance. For example, if your loan has an APR of 10%, you would pay $100 annually
per $1,000 borrowed. Simple Interest Formulas and Calculations: Use this simple interest calculator to find A, the Final Investment Value, using the simple interest formula: A = P(1 + rt) where P is
the Principal amount of money to be invested at an Interest Rate R% per period for t Number of Time Periods. Interest Rate Conversion. When interest on a loan is paid more than once in a year, the
effective interest rate of the loan will be higher than the nominal or stated annual rate . For instance, if a loan carries interest rate of 8% p.a., payable semi annually, the effective annualized
rate is 8.16% which is mathematically obtained by the Use our Interest Rate Converter Calculator to quickly convert Annual Percentage Rates to monthly interest rates and monthly interest rates into
an APR. With so many different short-term loan vehicles and other financial products available to consumers, deciphering the interest you are paying or the interest that is being paid to you can be
very difficult. Interest Rate Converter. Interest Rate Converter enables you to convert interest rate payable at any frequency into an equivalent rate in another frequency. For instance, you can
convert interest rate from annual to semi annual or monthly to annual, quarterly etc. Interest Rate % p.a. Payment frequency An interest rate formula helps one to understand loan and investment and
take the decision. These days financial bodies like banks use Compound interest formula to calculate interest. Compounded annual growth rate i.e. CAGR is used mostly for financial applications where
single growth for a period needs to be calculated. Recommended Articles
1 Nov 2011 The compound interest formula is: I = P(1 + r)^n - P. I is interest. P is principal r is rate n is the number of interest periods incurred. Your original
The Monthly Interest Calculator is to determine the Total or Monthly Interest of or compound interest, total repayment and annual percentage rate according to 1 month, monthly, 1/12 with various
periods and a nominal annual rate of 6% per year. Compounded, Calculation, Interest Rate For One Period period is converted to years: for example, 3 months is converted to (1/4) year. the interest
rate For example, is an annual interest rate of 8% compounded quarterly higher or lower Frequency, Accumulated amount, Calculation, Effective interest rate at an interest rate of 22% p.a. compounded
quarterly or 22% compounded monthly ? Use our Interest Rate Converter Calculator to quickly convert Annual Percentage Rates to monthly interest rates and monthly interest rates into an APR. With so
This interest calculator compares both simple monthly interest income and Annual Interest Rate – The annual percentage interest rate your money earns if How do you work out APR from monthly interest
rate? with the Interest Rate Converter, Convert monthly to annual APR or annual to monthly. Using the formula above, the easiest amount to find is the monthly amount of $150. For the interest rate
'r', we have to convert it from annual to monthly. .07 ÷ 12
One common are of malignant modelling is the inability of many analysts to convert an annual interest rate into a monthly or quarterly rate correctly. Sometimes
This interest calculator compares both simple monthly interest income and Annual Interest Rate – The annual percentage interest rate your money earns if How do you work out APR from monthly interest
rate? with the Interest Rate Converter, Convert monthly to annual APR or annual to monthly. Using the formula above, the easiest amount to find is the monthly amount of $150. For the interest rate
'r', we have to convert it from annual to monthly. .07 ÷ 12 Your interest rate is identified on your statement as the annual percentage rate, Since interest is calculated on a daily basis, you'll
need to convert the APR to a or monthly, your actual interest charge might differ slightly from this calculated 23 Jul 2013 The Annual percentage rate (APR) of a loan is the yearly interest rate
expressed as a simple percentage. Below is the effective annual rate formula. To convert annual rate to monthly rate, when using APR, simply divide
Your estimated annual interest rate. Interest rate variance range. Range of interest rates (above and below the rate set above) that you desire to
Convert the interest rate as a percentage to a decimal by dividing by 100. Add 1 to the interest rate as a decimal. Raise the result to the 1/12th power because there are 12 months per year. Subtract
1 from the result to find the monthly interest rate as a decimal. The formula for compound interest is : - FV = P * (1 + (r/100))^ n . Where:- FV = Future Value P = Principal R = Rate of interest n =
time. If you need to compound daily, then divide the rate by the number of periods to get the effective annual rate. To convert a yearly interest rate for annually compounding loans, you can simply
divide the annual interest rate into 12 equal parts. So, for example, if you had a loan with a 12 percent interest rate attached to it, you can simply divide 12 percent by 12, or the decimal
formatted 0.12 by 12, in order to determine that 1 percent interest is essentially being added on a monthly basis. Interest Rate Conversion. When interest on a loan is paid more than once in a year,
the effective interest rate of the loan will be higher than the nominal or stated annual rate . For instance, if a loan carries interest rate of 8% p.a., payable semi annually, the effective
annualized rate is 8.16% which is mathematically obtained by the Interest Rate Converter Formula: Monthly to Annual = ( (1 + Interest) ^ 12 ) - 1 Annual to Monthly = ( (1 + Interest) ^ (1/12) ) - 1
To calculate a monthly interest rate, divide the annual rate by 12 to account for the 12 months in the year. You'll need to convert from percentage to decimal format to complete these steps. For
example, let's assume you have an APY or APR of 10% per year. What is your monthly interest rate, and how much would you pay or earn on $2,000?
Using the formula above, the easiest amount to find is the monthly amount of $150. For the interest rate 'r', we have to convert it from annual to monthly. .07 ÷ 12 Your interest rate is identified
on your statement as the annual percentage rate, Since interest is calculated on a daily basis, you'll need to convert the APR to a or monthly, your actual interest charge might differ slightly from
this calculated 23 Jul 2013 The Annual percentage rate (APR) of a loan is the yearly interest rate expressed as a simple percentage. Below is the effective annual rate formula. To convert annual
rate to monthly rate, when using APR, simply divide Annual Percentage Rate (APR) Calculator. Loan Amount. $. Interest Rate. %. Term. Yr. Finance Charges (Added to loan amount). $. Prepaid Finance
Charges The annual percentage rate (APR) that you are charged on a loan may not be the The amount of interest you effectively pay is greater the more frequently the interest is compounded. the
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The Wave Model of Growth
Growth is wavy, learn to surf.
Learn how to tune in and get on the same wavelength as your customers.
A visual model of emotion traveling as a wave over time to help us understand and manage the frequency of our relationships.
the number of repeating events in a recurring relationship per unit of time.
Funnel, Flywheel, Frequency
The 3 ways to visualize the customer journey for a unified theory of growth.
We started with the funnel to visualize relationship acquisition, conversion and lifecycle stages.
The flywheel was next to move from a transactional to relational approach and understand the primary drivers of growth.
The Wave Model will allow us to use sine waves and frequency to plan our communication and tune our relationships.
The Wave Model gives us a way to see inside the funnel, unwind the flywheel and begin to understand the customer journey in terms of relationship frequencies.
The 12 Fundamental Relationship Frequencies | {"url":"http://thewavemodel.com/","timestamp":"2024-11-08T21:36:28Z","content_type":"text/html","content_length":"67920","record_id":"<urn:uuid:72ebbf49-26e0-4c35-b875-9c8fdacd96b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00568.warc.gz"} |
Intoduction to Mathematical Data Science | StudyTution - StudyTution
Intoduction to Mathematical Data Science | StudyTution
Ques: Why are we studying mathematics in data science course?
Ans : It is because data science actually combines mathematics, statistics and computing.
• Lines represent linear equation
• Quadric equation have a square term if we draw them then they look like a parabola.
• So, if we look at quadric equation and then we will generalize to a higher power so these are what we called polynomials.
• So these are all functions which we can draw as graphs in the sense of coordinate geometry but we can analyze them in different way.
• Functions which are not polynomial they grow very fast and that are exponentials and those that grow vey slowly are logarithm.
• Graphs like a map of an airline timetable have nodes representing the points of interest and edges representing connections.
• Example is road network or an airline network, but these edges can also represent other relations.
• For example we can think of organization and the no for employs and the manager they report. So we will look at graphs.
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Financial calculator expected rate of return
Stock Return Calculator; Stock Constant Growth Calculator; Stock Non-constant Growth Calculator; CAPM Calculator; Expected Return Calculator; Holding Period Return Calculator; Weighted Average Cost
of Capital Calculator; Black-Scholes Option Calculator It will calculate any one of the values from the other three in the CAPM formula. CAPM (Capital Asset Pricing Model) In finance, the CAPM
(capital asset pricing model) is a theory of the relationship between the risk of a security or a portfolio of securities and the expected rate of return that is commensurate with that risk.
Calculate rate of return. The rate of return (ROR), sometimes called return on investment (ROI), is the ratio of the yearly income from an investment to the original investment. The initial amount
received (or payment), the amount of subsequent receipts (or payments), and any final receipt (or payment), all play a factor in determining the return. Stock Return Calculator; Stock Constant Growth
Calculator; Stock Non-constant Growth Calculator; CAPM Calculator; Expected Return Calculator; Holding Period Return Calculator; Weighted Average Cost of Capital Calculator; Black-Scholes Option
Calculator It will calculate any one of the values from the other three in the CAPM formula. CAPM (Capital Asset Pricing Model) In finance, the CAPM (capital asset pricing model) is a theory of the
relationship between the risk of a security or a portfolio of securities and the expected rate of return that is commensurate with that risk. In finance, the Capital Asset Pricing Model is used to
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on the risk-free rate, the expected market return and the stock's beta. A common method to measure an investment's return is to calculate its dollar weighted return, also known as its internal rate
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The rate of return (ROR), sometimes called return on investment (ROI), is the ratio of the yearly income from an investment to the original investment. The initial
Understand the expected rate of return formula. Like many formulas, the expected rate of return formula requires a few "givens" in order to solve for the answer. The "givens" in this formula are the
probabilities of different outcomes and what those outcomes will return. The formula is the following. In finance, the Capital Asset Pricing Model is used to describe the relationship between the
risk of a security and its expected return. You can use this Capital Asset Pricing Model (CAPM) Calculator to calculate the expected return of a security based on the risk-free rate, the expected
market return and the stock's beta. This ROI calculator (return-on-investment) calculates an annualized rate-of-return using exact dates. Also known as ROR (rate-of-return), these financial
calculators allow you to compare the results of different investments. A common method to measure an investment's return is to calculate its dollar weighted return, also known as its internal rate of
return. The dollar rate of return is used to calculate how much each investment dollar returned on average to an investor. Because it is a long calculation, it is wise to use financial calculator.
Return Rate Formula. See the CAGR of the S&P 500, this investment return calculator, CAGR Explained, and How Finance Works for the rate of return formula. You can also sometimes estimate the return
rate with The Rule of 72. Meeting your long-term investment goal is dependent on a number of factors. This not only includes your investment capital and rate of return, but inflation, taxes and your
time horizon. This calculator helps you sort through these factors and determine your bottom line. Click the "View Report" button for a detailed look at the results.
SIP Calculator - Calculate the future returns on your SIP monthly investments on of years for which you want to stay invested, and the expected rate of return. that your savings portfolio is as per
your requirements and financial needs.
This stock total return calculator models dividend reinvestment (DRIP) Below is a stock return calculator which automatically factors and calculates to the annual percentage return by the investment,
including dollar cost averaging. PK founded DQYDJ in 2009 to educate and learn from others in finance and investing. Power of Compounding Calculator : Compounding is the addition of interest on your
investment generated over a You expect the Annual Rate of Returns to be . 25 Nov 2016 The risk free interest rate is the return investors are willing to accept for an investment with no risk.
Generally, the U.S. three-month Treasury bill is Use this CAPM Calculator to calculate the expected return of a security based on the risk-free rate, the expected market return and the beta. In
finance, the Capital Asset Pricing Model is used to describe the relationship between the risk of a This calculator assumes that all dividend payments will be reinvested. Calculate Compounding
Returns. Money Invested. $. Return Rate. %. Number of Years.
This stock total return calculator models dividend reinvestment (DRIP) Below is a stock return calculator which automatically factors and calculates to the annual percentage return by the investment,
including dollar cost averaging. PK founded DQYDJ in 2009 to educate and learn from others in finance and investing.
Internal rates of return (IRR) are returns are what matter to you as an investor. To do this type of calculation you need to use software, or a financial calculator, that It is important to calculate
the expected internal rate of return so you may This calculator shows the return rate (CAGR) of an investment; with links to calculator, CAGR Explained, and How Finance Works for the rate of return
This calculator assumes that all dividend payments will be reinvested. Calculate Compounding Returns. Money Invested. $. Return Rate. %. Number of Years.
The rate of return (ROR), sometimes called return on investment (ROI), is the ratio of the yearly income from an investment to the original investment. The initial
12 May 2017 You can ask your advisor to calculate your return for you, or you can calculate it yourself using a financial calculator or spreadsheet software. To 11 Jul 2019 Free online CAGR
Calculator for estimating annualized returns. Learn how to calculate the Compound Annual Growth Rate in Excel, by Jon Rate at investopedia.com; ROI Calculator at financial-calculators.com - This
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Fluency Hack: Prioritizing Missing Numbers In Equations - The Math Spot
This post contains affiliate links. This means that when you make a purchase, at no additional cost to you, I will earn a small commission.
If you are looking to increase your students’ fact fluency, prioritizing the skill of missing numbers in equations is one of the keys to unlocking that skill in your student. If your students have
struggled with this skill in the past, you’re not alone! Solving for missing numbers in equations can be a difficult skill to learn but the payoff is significant!
A few years back I sat with a first-grade teacher who had been working on missing numbers in equations. The day had ended and she was sitting at the kidney table, head resting on her hands out of
pure exhaustion from the day. She had called me down to meet because she had followed the math program to a T and yet more than a few of her students were still struggling to find missing numbers in
addition and subtraction equations.
I could so clearly feel her frustration. I had been in her position a year prior when I was trying to support my intervention students… every attempt I made fell flat on its’ face. My students
consistently would add and subtract arbitrary numbers to find an answer. Any answer. Regardless of whether or not the answer made sense.
I felt like saying is this even worth it? Can we just move on from this skill?
Why Prioritize Missing Numbers in Equations?
If your students were to memorize every single addition and subtraction fact they would need to commit over 200 individual facts to memory. This is an unreasonable task! When your students can
quickly and easily relate facts, automaticity and retrieval are a simple byproduct!
Being able to solve for missing numbers in equations means that your students have a clear recognition of the relationship between addition and subtraction.
A student who knows that 3 + 5 = 8 and recognizes the relationship between addition and subtraction can solve for the missing number in the equation 8 – ___ = 5 because they recognize that these
facts are related.
Additionally, solving for missing numbers in equations provides a notation for students who are fluent in composing and decomposing numbers.
Your primary students spend a great deal of time decomposing numbers. For example, students spend time working with partners of ten and know that 1 and 9, 2 and 8, 3 and 7, 4 and 6, and 5 and 5 are
all ways to make ten. Students don’t always make the connection between these decompositions and equations on their own!
Asking your students to solve an equation that says 9 + ____ = 10 allows your students to practice connecting the math they already know (a 9 and a 1 make a 10) with notations and equations.
How to Teach Missing Numbers in Equations
Strategy #1: Make the Lessons Hands-On
When we teach math skills and concepts, we want to give consideration to CRA. Your students will understand more clearly when they first experience at the concrete level. If your curriculum or
lessons jump straight to equations without hands-on support, you are being set up for failure!
When a number is missing from an equation we are either missing the start the change or the result.
Giving the students two of these numbers and allowing them to experiment with manipulatives will help them to see why future counting strategies work.
In this example, students are given the context “[#]birds are sitting in a tree. More birds fly over! Now there are a total of [#] birds.”
Students use a spinner to generate the initial number of birds in the tree and the resulting number of birds in the tree. They then use manipulatives to solve for the number of birds that flew over
to the tree.
Students connect this hands-on experience to an equation as they record their work.
As a teacher you can also model using counting strategies to solve for a missing number using these materials. “Oh! I see you started with 5 birds in the tree. Then you added birds number 6, 7, 8,
and 9. So we started at 5 and counted 5 more to get to 9. 5 and 4 more made 9. I can see that in your equation!”
Strategy #2: Be Aware of Moving to Representative Models too Quickly!
This was a mistake I made in a big way. I was sure that if my students understood fact-families and the part-part-whole relationship that they should easily be able to find the missing number in an
equation. “Just pop the numbers into a number bond and from there you can easily find the missing number!”
I was asking them to use one abstract concept (part/part/whole) to support another abstract concept (missing number).
Number bonds will eventually be a great strategy to use with your students, however, they are not necessarily the place to start! Remember that we are striving to build a web of understanding for our
students. After your students are showing some confidence with hands-on tools, ask them to use those tools alongside a representative model such as a number bond.
Strategy #3: Context is King!
The teacher I was working with was using an abundance of story problems as they were written in the curriculum. The problem was that the contexts provided weren’t necessarily understandable to our
students. Worse yet, the context kept changing!
When you first introduce this skill, pick a context and let the students play with it over and over and over so that the context is supporting their ability to make meaning rather than adding another
hurdle to understand.
Pair these simple contexts with hands-on tools and ask your students to record the results as equations. It will take time and practice but your students will soon gain the skills and confidence they
In the photo you can see students playing with a context around pirate treasure. A treasure chest has a given amount of treasure inside. Students are then told that the pirates either added to or
lost some of their treasure. Finally, students are given the resulting amount of treasure in the chest.
Their goal is to use hands-on materials to find the number of pieces of gold that were either lost or found.
Adding this concept rather than asking students to work with naked numbers brings meaning, and therefore greater understanding, to the equations they are writing.
Related Resources: | {"url":"http://k5mathspot.com/3-mistakes-you-need-to-avoid-when-teaching-missing-numbers-in-an-equation/","timestamp":"2024-11-07T02:42:22Z","content_type":"text/html","content_length":"103291","record_id":"<urn:uuid:5df6293c-73d2-4755-86f8-a8ed67997141>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00288.warc.gz"} |
Minutes from Now Calculator | Calculate Future Minutes from Now - onlinecalculator.guide
Minutes from Now Calculator
Online Minutes from Now Calculator is helpful to find what time will be “n” number of minutes from now in future. You just have to enter the number of minutes in the input box and tap on the
calculate button to get the future time within seconds.
How to Calculate Minutes from Now?
In general, time is expressed in hours: minutes: seconds format. Follow these guidelines to get the time after minutes.
• Firstly, know the current time.
• If the number of minutes is less than 60, then you can add the number to the current time minutes.
• If it is more than 60, then you have to divide the number by 60 which is added to hours.
Check the solved questions to know a clear idea of minutes in addition to the present time.
Example Questions on Finding Minutes from Now
Question 1:
What is 15 minutes from now?
Given that,
Number of minutes n = 15
The current time is 12/12/2022 at 1:40:50 PM
15 minutes from now means add 15 to minutes
40 + 15 = 55
Therefore, 15 minutes from now is 12/12/2022 at 1:55:50 PM
Question 2:
What is 200 minutes from now?
Given that,
Number of minutes n = 200
The current time is 12/12/2022 at 1:42:50 PM
200 minutes from now means divide 200 by 60
200/60 = 3.33
So, add 3 to hours, 20(200 - 180) to minutes
3+1 = 4 and 20 + 42 = 2 and 4 + 1 = 5
Therefore, 200 minutes from now is 12/12/2022 at 5:02:50 PM
Like this we have many other calculators in this Onlinecalculator.guide website. Open the link and enjoy more calculators to save your time and effort.
How to Use Minutes from Now Calculator
Go through the simple procedure to use our user-friendly calculator.
• Give the number of minutes in the specified field.
• Press the calculate button.
• Check the time after the number of minutes as result.
FAQs on Minutes from Now Calculator
1. How many minutes are in today?
There are 1440 minutes in a day. A day has 24 hours and every hour has 60 minutes, so 24 x 60 = 1440.
2. How long is 1 minute exactly?
1 minute is equal to 60 seconds.
3. What time it will be in 30 minutes?
The present time is 2:01 PM
Add 30 minutes to 2:01
So, 2:31 PM will be after 30 minutes.
4. How to calculate minutes from hours?
1 hour has 60 minutes. So, hours minutes with 60 to get the number of minutes. | {"url":"https://onlinecalculator.guide/minutes-from-now/","timestamp":"2024-11-11T19:40:05Z","content_type":"text/html","content_length":"31500","record_id":"<urn:uuid:46964748-78f5-4c8b-a4cd-8f5cf6d979ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00438.warc.gz"} |
Dyscalculia Test Services - Children, Teenagers, and Adults
Dyscalculia is a specific learning disorder that affects an individual’s ability to understand and perform mathematical tasks. There is no single, definitive dyscalculia test that is universally
recognized, but a combination of assessments can be used to diagnose it. We offer testing for dyscalculia in all ages, with different approaches and instruments used depending on our client’s age and
educational background. A dyscalculia test for children differs from a dyscalculia test for adults not only in how we determine whether it is an issue but also usually in the reasons for testing and
the hoped-for outcomes and interventions. Often, dyscalculia testing for teenagers falls somewhere in between.
Testing for Dyscalculia in Our Practice
While this post goes over testing for dyscalculia in general, please feel free to contact us or schedule a consultation anytime if you have specific questions.
Initial Testing for Dyscalculia
As an initial screening, we will administer the following dyscalculia test measures, regardless of our client’s age.
1. Mathematical Skills: These measure basic arithmetic abilities, problem-solving skills, and understanding of mathematical concepts. Testing for dyscalculia starts with determining that math is
2. Cognitive Evaluation: This evaluates broader cognitive functions such as working memory, visual-spatial reasoning, and processing speed, which can influence math performance. We may not do a full
IQ assessment now, but rather just the subtests affecting mathematical ability. Testing for dyscalculia needs to show that the difficulty in math is disproportionate to overall intelligence.
3. Developmental and Educational History: Questions about academic performance, school struggles, and a family history of learning disabilities help with testing for dyscalculia.
Common signs
During our interview and initial screening, we look for difficulty in understanding numbers and their relationships, such as
• Struggling with basic math tasks like addition, subtraction, multiplication, and division.
• Problems with time management or difficulty telling time.
• Confusion with sequences, such as steps in a math problem or instructions.
• Trouble estimating amounts or recognizing patterns.
If the initial dyscalculia test results suggest that further evaluation is warranted, we do age-specific testing for dyscalculia, as follows.
Dyscalculia Test for Children
A dyscalculia test for children typically involves a comprehensive evaluation process that combines observations, cognitive assessments, and academic tests. Since this challenge affects a child’s
ability to understand and manipulate numbers, early identification and intervention are important to help them develop coping strategies and receive appropriate support in school.
Signs Indicating a Need for a Dyscalculia Test for Children
• Difficulty recognizing numbers, learning to count, or understanding quantity.
• Problems remembering basic math facts (addition, subtraction, multiplication).
• Trouble with time concepts, such as reading clocks or calculating elapsed time.
• Inability to understand patterns, sequences, or basic geometry.
• Avoidance of math homework or significant frustration during math tasks.
Dyscalculia Test for Children Process
1. Initial Screening and Observation
• Teacher and Parent Observations: Teachers and parents play a key role in noticing early signs. They may observe that the child:
□ Struggles to recognize numbers or count.
□ Has difficulty learning basic math facts (like multiplication tables).
□ Struggles to understand time or money concepts.
□ Avoids math-related tasks or experiences math-related anxiety.
• Developmental History: Parents will be asked about the child’s developmental milestones, family history of learning difficulties, and any other challenges related to learning or attention.
2. Dyscalculia Test for Children Skills Assessment
• Basic Arithmetic: This is the first step in a dyscalculia test for children, and it measures the child’s ability to perform tasks such as addition, subtraction, multiplication, and division.
• Number Sense: These examine whether the child understands larger vs. smaller quantities, sequencing, or the relationship between numbers.
• Word Problems: Problem-solving questions about whether the child can apply math skills in everyday situations.
3. Cognitive and Psychological Evaluations
• Cognitive Function: These evaluate underlying cognitive skills that support math learning, such as:
□ Working Memory: The ability to remember numbers or instructions while working with them.
□ Visual-Spatial Skills: Understanding how objects are arranged in space (which is important for geometry or place value).
□ Processing Speed: How quickly the child can perform basic tasks.
• Executive Functioning Assessments: These assess the child’s ability to organize thoughts, focus, and plan steps for problem-solving.
4. Educational Dyscalculia Test for Children (Academic Skills)
• Standardized Achievement Assessment: The Woodcock-Johnson Tests of Achievement (WJ-IV), Wechsler Individual Achievement Test (WIAT), or KeyMath Diagnostic Assessment can help measure a child’s
specific math abilities compared to their age group.
• Curriculum-Based Assessments: Some schools may use a curriculum-based test for dyscalculia in children that assesses math skills within the context of what’s being taught in class.
5. Other Components of a Dyscalculia Test for Children
• Learning Style and Strategy Assessments: A dyscalculia test for children can help determine how they learn best and develop strategies tailored to their needs.
• Attention and Executive Function Testing: Since attention disorders like ADHD often co-occur with learning disabilities, these can identify if other factors might be affecting their math skills.
Diagnosis and Next Steps
• After a dyscalculia test for children, the results are evaluated to determine whether the child meets the criteria for dyscalculia, which is classified as a “Specific Learning Disorder with
Impairment in Mathematics” in the DSM-5 (Diagnostic and Statistical Manual of Mental Disorders).
• Individualized Education Plan (IEP) or 504 Plan: If diagnosed through testing for dyscalculia, the child may qualify for special school accommodations, such as extra time on assignments, a
modified curriculum, or specialized math instruction.
• Targeted Intervention: Based on the difficulties elucidated by the dyscalculia test for children, specialized math tutoring, multi-sensory approaches to learning math, and other interventions can
be recommended.
Dyscalculia Testing for Teenagers
The process for dyscalculia testing for teenagers is similar to that for younger children. However, it also accounts for the more advanced mathematical tasks encountered during adolescence and the
social and emotional impacts of struggling with math during the teenage years.
Signs Indicating the Need for Dyscalculia Testing for Teenagers
• Difficulty understanding abstract math concepts (e.g., algebra, geometry).
• Problems estimating numbers and working with fractions or percentages.
• Struggles with time management, organizing tasks, or following a sequence of instructions.
• Avoidance of math-heavy subjects or tasks.
• Anxiety or frustration around math exams or assignments.
• Difficulty interpreting graphs, charts, or other visual representations of data.
Dyscalculia Testing for Teenagers Process
1. Initial Screening and Observation
• Parent and Teacher Observations: Teens with this learning difference may exhibit:
□ Persistent difficulty with basic arithmetic (addition, subtraction, multiplication, division).
□ Problems with higher-level math skills such as algebra, geometry, or understanding graphs and charts.
□ Inability to handle tasks involving numbers in real-life situations (e.g., calculating discounts, time management, measuring ingredients).
□ Avoidance of math classes or math-heavy subjects, anxiety during tests, or poor performance on exams despite effort.
• Teen’s Self-Report: Many teens can articulate their struggles with math better than younger children. Their input about their frustrations, anxiety, or math avoidance can be crucial in deciding
whether testing for dyscalculia would be helpful.
2. Dyscalculia Testing for Teenagers Skills Assessment
• Basic Math Skills: Dyscalculia testing for teenagers starts by evaluating their ability to handle foundational arithmetic operations and number sense.
• Word Problems: Teens are assessed on their ability to understand and solve real-life word problems that require mathematical reasoning.
• Advanced Math Skills: Depending on the teen’s current academic level, the assessment may include topics like algebra, geometry, fractions, and percentages. These are skills they would be expected
to handle at their grade level.
• Estimation and Measurement: We measure their abilities at estimating quantities, measuring objects, and working with time and money in practical scenarios.
3. Cognitive Testing
• Working Memory: This learning difference often co-occurs with weaknesses in working memory. Cognitive measures, which evaluate the teen’s ability to hold and manipulate information in their mind
while solving problems, are frequently part of the dyscalculia testing for teenagers that we do.
• Visual-Spatial Skills: Teens with this learning difference might have difficulty with tasks that involve understanding spatial relationships (e.g., geometry, interpreting graphs or charts).
• Processing Speed: Dyscalculia testing for teenagers measures how quickly the teen can process numbers and mathematical concepts, which might be slower.
4. Standardized Achievement Assessments
• Measures such as the Wechsler Individual Achievement Test (WIAT), Woodcock-Johnson IV (WJ-IV), or KeyMath Diagnostic Assessment help evaluate specific mathematical skills and concepts as part of
dyscalculia testing for teenagers.
• These compare the teen’s performance to that of their peers to see if there are significant deficits in math skills.
• Math Fluency: Standardized measures also assess math fluency, measuring how quickly and accurately the teen can perform basic calculations. These are often part of dyscalculia testing for
5. Educational History and Performance
• A thorough review of the teen’s academic records, standardized assessment scores, and performance in math-heavy subjects can provide insight into any longstanding difficulties with math.
• Teachers may be asked to provide input on the teen’s classroom behavior, homework completion, and exam performance in math courses.
• History of Interventions: If the teen has already received math tutoring or special education services, it’s important to consider whether previous interventions were effective.
6. Psychological and Emotional Assessment
• Math Anxiety and Self-Esteem: Teenagers often develop anxiety around math as a result of ongoing difficulties. Psychological assessments can help evaluate the emotional impact of math struggles,
which may affect their motivation and performance.
• Co-occurring Conditions: We may also check for other learning disorders or conditions, such as ADHD, which often co-occur with learning differences and can exacerbate math difficulties.
Dyscalculia Testing for Teenagers Diagnosis and Report
• After the evaluation, we review the results to determine if the teen meets the criteria for a Specific Learning Disorder with Impairment in Mathematics, as outlined in the DSM-5.
• A detailed dyscalculia testing for teenagers psychological report will summarize the findings and provide recommendations for accommodations and interventions, which might include:
□ Extra time on math exams.
□ Allowing the use of a calculator.
□ Reduced math homework load or modified assignments.
□ Access to math tutoring or special education services.
9. Accommodations and Intervention
• If the teen is diagnosed, they may qualify for an Individualized Education Plan (IEP) or a 504 Plan, which provides accommodations in school.
• Targeted Math Support: Special education services or individualized tutoring tailored to the teen’s needs can help build foundational skills.
• Assistive Technology: Tools such as calculators, math apps, or graphing software can be used to support learning.
Dyscalculia Test for Adults
A dyscalculia test for adults follows a process similar to that for children and teens, but it considers the practical challenges adults face daily. Adults with this learning difference may have
struggled with math throughout school and into their professional and personal lives without knowing their difficulties are related to a specific learning disorder.
Signs Indicating the Need for a Dyscalculia Test for Adults
• Struggling to estimate costs, calculate tips, or make financial decisions.
• Trouble with schedules, deadlines, or calculating time differences.
• Difficulty with spatial reasoning (e.g., reading maps, understanding distances).
• Avoiding jobs or tasks that involve math or numbers.
• Frequent mistakes when counting change, budgeting, or managing bills.
Dyscalculia Test for Adults Process:
We commonly offer a dyscalculia test for adults, often so they can get work accommodations of some sort.
1. Self-Screening and Observation
• Self-Reflection: Many adults may have noticed ongoing struggles with numbers, calculations, and time management in their everyday lives. Signs to look for include:
□ Difficulty with basic arithmetic (adding, subtracting, multiplying, dividing).
□ Problems handling money, making change, or calculating tips.
□ Trouble understanding time concepts (e.g., how much time has passed, reading clocks).
□ Avoidance of tasks that involve numbers (e.g., financial planning, balancing a checkbook).
□ Trouble with spatial reasoning or interpreting graphs and charts.
• Workplace and Personal Life: They may notice their difficulties in professional contexts (e.g., jobs requiring math skills) or in managing household finances or time efficiently.
2. Our Dyscalculia Test for Adults
• Initial Interview: The dyscalculia test for adults usually begins with an in-depth interview to understand the adult’s educational history, work history, and day-to-day challenges with math.
□ We may ask about the individual’s school experience, math difficulties, and other learning or attention issues.
• Screening for Co-occurring Conditions: Learning differences often co-exist with other conditions, such as ADHD or dyslexia. A comprehensive dyscalculia test for adults may include tests for these
as well.
3. Cognitive and Mathematical Skills Testing
• Cognitive Assessments: These part of a dyscalculia test for adults assesses core cognitive functions that support mathematical thinking, such as:
□ Working Memory: The ability to hold information in mind while working with it.
□ Visual-Spatial Reasoning: Important for understanding number placement and geometry.
□ Processing Speed: Measuring how quickly an individual can process numbers or mathematical information.
□ Executive Functioning: We assess planning, organization, and sequencing skills, all of which are important for math tasks.
• Mathematical Skills: This more specific part of a comprehensive dyscalculia test for adults measures their basic and advanced math skills, and their ability to apply math concepts to real-world
□ Arithmetic: Assessments of basic math operations (addition, subtraction, multiplication, division).
□ Applied Math: These include real-world problems, like calculating discounts, budgeting, or understanding bills and invoices.
□ Advanced Mathematical Skills: These skills may also be tested if the adult is expected to perform higher-level math in their work (e.g., algebra or geometry).
• Standardized Achievement: A dyscalculia test for adults may include the Woodcock-Johnson which measures mathematical reasoning, fluency, and calculation ability in relation to age norms.
4. Emotional and Psychological Evaluation
• Adults with learning differences often experience frustration, embarrassment, or anxiety related to math tasks, which can lead to avoidance. This can have an impact on self-esteem and mental
• Testing for dyscalculia evaluation may include an assessment of anxiety, particularly math anxiety, as well as how math difficulties have affected overall emotional well-being and professional
Dyscalculia Test for Adults Diagnosis and Report
• If the criteria are met, we provide a diagnosis that might be part of a Specific Learning Disorder or a standalone learning difficulty with math.
• A detailed psychological report will summarize the results and provide recommendations, including strategies for managing challenges and work accommodations.
Accommodations and Support for Adults
• Workplace Accommodations: A dyscalculia test for adults can help you become eligible for accommodations under the Americans with Disabilities Act (ADA) in the U.S. or similar laws in other
countries. These might include:
□ Access to calculators or apps that assist with math.
□ Extra time for completing tasks that involve calculations.
□ Job modifications that reduce the need for complex math.
• Financial and Personal Management Tools: Using software to help manage personal finances, budget, or keep track of time and deadlines.
• Assistive Technology: There are apps designed to support those with dyscalculia, such as math apps that provide visual supports or tools for calculating percentages, taxes, and more.
• Tutoring or Coaching: Some adults seek help from a math tutor or learning coach to build confidence in basic math skills.
Summary and Our Work
We offer testing for dyscalculia and general specific learning disability assessments for all ages in our practice, including college accommodations. Please feel free to contact us or schedule a
consultation anytime to discuss how testing for dyscalculia could benefit you or a loved one. We offer a dyscalculia test for children, but you may want to see whether your child’s school might do it
instead. The same goes for a dyscalculia test for adolescents unless your child goes to private school, is home-schooled, or is in college. | {"url":"https://psychologicaltesting.net/dyscalculia-test/","timestamp":"2024-11-13T21:08:55Z","content_type":"text/html","content_length":"290889","record_id":"<urn:uuid:8ad0e66c-7d86-4e4f-8760-57f3c41858f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00753.warc.gz"} |
Calculus of Variations/CHAPTER XVI - Wikibooks, open books for an open world
Article 216.
To solve the problem of this Chapter, let the ${\displaystyle Y}$-axis be taken vertically with the positive direction upward, and denote by ${\displaystyle S}$ the length of the whole curve. If the
coordinates of the center of gravity are ${\displaystyle x_{0},y_{0}}$, then ${\displaystyle y_{0}}$ is determined from the equation
${\displaystyle y_{0}=S\int _{t_{0}}^{t_{1}}y{\sqrt {x'^{2}+y'^{2}}}{\text{d}}t}$, where ${\displaystyle S=\int _{t_{0}}^{t_{1}}{\sqrt {x'^{2}+y'^{2}}}{\text{d}}t}$
The problem is: So determine ${\displaystyle x}$ and ${\displaystyle y}$ as functions of ${\displaystyle t}$ that the first integral will be a minimum while the second integral retains a constant
value. (See Art. 16).
The property that the center of gravity is to lie as low as possible must also be satisfied for every portion of the curve; for if this were not true, then we could replace a portion 1 2 of the curve
by a portion of the same length but with a center of gravity that lies lower, with the result that the center of gravity of the whole curve could be shoved lower down, and consequently the original
curve would not have the required minimal property.
We have here
${\displaystyle F^{(0)}=y{\sqrt {x'^{2}+y'^{2}}}\qquad F^{(1)}={\sqrt {x'^{2}+y'^{2}}}}$
${\displaystyle F=(y-\lambda ){\sqrt {x'^{2}+y'^{2}}}}$
and therefore
${\displaystyle {\frac {\partial F}{\partial x'}}={\frac {x'(y-\lambda )}{\sqrt {x'^{2}+y'^{2}}}}\qquad {\frac {\partial ^{2}F}{\partial x'\partial y'}}={\frac {-x'y'(y-\lambda )}{({\sqrt {x'^{2}
+y'^{2}}})^{3}}}\qquad {\frac {\partial F}{\partial y'}}={\frac {y'(y-\lambda )}{\sqrt {x'^{2}+y'^{2}}}}}$
${\displaystyle F_{1}={\frac {y-\lambda }{({\sqrt {x'^{2}+y'^{2}}})^{3}}}}$
We exclude once for all the case where the two given points lie in the same vertical line, because then the integral for ${\displaystyle S}$ does not express for every case the absolute length of the
curve ; for example, when a certain portion of the curve overlaps itself. Similarly we exclude the case where the given length ${\displaystyle S}$ is exactly equal to the length between the two
points on a straight line ; for, in this case, the curve cannot be varied and at the same time retain the constant length.
Article 217.
Since ${\displaystyle F_{1}}$ must be positive, a minimum being required, it follows that ${\displaystyle (y-\lambda )>0}$. Since further, ${\displaystyle {\frac {\partial F}{\partial x'}}}$ and ${\
displaystyle {\frac {\partial F}{\partial y'}}}$ vary in a continuous manner along the whole curve, and since these quantities differ from the direction-cosines only through the factor ${\
displaystyle y-\lambda }$, which varies in a continuous manner, it follows that the curve changes everywhere its direction in a continuous manner.
The function ${\displaystyle F}$ is the same as the function ${\displaystyle F}$ which appeared in Art. 7, except that here we have ${\displaystyle y-\lambda }$ instead of ${\displaystyle y}$ in that
problem. Since the differential equation here must be the same as in the problem just mentioned, we must have as the required curve
${\displaystyle x=\alpha \pm \beta t\qquad y=\lambda +{\frac {\beta (e^{t}+e^{-t})}{2}}}$
the equation of a catenary.
Since ${\displaystyle y-\lambda >0}$, it follows that ${\displaystyle \beta }$ is a positive constant. For ${\displaystyle S}$ we have the value
${\displaystyle S=\int _{t_{0}}^{t_{1}}{\sqrt {x'^{2}+y'^{2}}}{\text{d}}t={\frac {\beta }{2}}{\big [}e^{t_{1}}+e^{-t_{1}}-(e^{t_{0}}+e^{-t_{0}}){\big ]}}$
Article 218.
We have next to investigate whether and how often a catenary may be passed through two points and have the length ${\displaystyle S}$ that is, whether and in how many different ways it is possible to
determine the constants ${\displaystyle \alpha ,\beta ,\lambda }$ in terms of ${\displaystyle S}$ and the coordinates of the given points. If we denote the coordinates of these points by ${\
displaystyle a_{0},b_{0},a_{1},b_{1}}$, then is
${\displaystyle a_{0}=\alpha eq \beta t_{0}\qquad a_{1}=\alpha eq \beta t_{1}}$
${\displaystyle b_{0}=\lambda +{\frac {\beta }{2}}(e^{t_{0}}+e^{-t_{0}})\qquad b_{1}=\lambda +{\frac {\beta }{2}}(e^{t_{1}}+e^{-t_{1}})}$
${\displaystyle S={\frac {\beta }{2}}{\big [}e^{t_{1}}+e^{-t_{1}}-(e^{t_{0}}+e^{-t_{0}}){\big ]}}$
It follows that
${\displaystyle a_{1}-a_{0}=\pm \beta (t_{1}-t_{0})\qquad b_{1}-b_{0}={\frac {\beta }{2}}{\big [}e^{t_{1}}+e^{-t_{1}}-(e^{t_{0}}+e^{-t_{0}}){\big ]}}$
We have assumed that ${\displaystyle t_{1}>t_{0}}$, and consequently we have to take the upper or lower sign according as ${\displaystyle a_{1}-a_{0}>0}$ or ${\displaystyle a_{1}-a_{0}<0}$. It is
clear that we may always take ${\displaystyle a_{1}-a_{0}>0}$, since we may interchange the point ${\displaystyle a_{1},b_{1}}$ with the point ${\displaystyle a_{0},b_{0}}$, and vice versa.
We shall accordingly take the upper sign. If we write
${\displaystyle {\frac {t_{1}-t_{0}}{2}}=\mu \qquad {\frac {t_{1}+t_{0}}{2}}=u }$
then ${\displaystyle \mu }$ is a positive quantity and we have
${\displaystyle a_{1}-a_{0}=+2\mu \beta }$
${\displaystyle b_{1}-b_{0}={\frac {\beta }{2}}{\big (}e^{\mu }-e^{-\mu }{\big )}{\big (}e^{u }-e^{-u }{\big )}}$
${\displaystyle S={\frac {\beta }{2}}{\big (}e^{\mu }-e^{-\mu }{\big )}{\big (}e^{u }+e^{u }{\big )}}$
${\displaystyle {\frac {b_{1}-b_{0}}{S}}={\frac {1-e^{-2u }}{1+e^{-2u }}}=-{\frac {1-e^{2u }}{1+e^{2u }}}}$
${\displaystyle {\frac {\text{d}}{{\text{d}}u }}\left({\frac {b_{1}-b_{0}}{S}}\right)={\frac {4}{(e^{u }+e^{-u })^{2}}}}$
Since this derivative is continuously positive, the expression ${\displaystyle {\frac {b_{1}-b_{0}}{S}}}$ varies in a continuous manner from ${\displaystyle -1}$ to ${\displaystyle +1}$, while ${\
displaystyle u }$ increases from ${\displaystyle -\infty }$ to ${\displaystyle +\infty }$. Hence for every real value of ${\displaystyle u }$ there is one and only one real value of ${\displaystyle
{\frac {b_{1}-b_{0}}{S}}}$ which is situated between ${\displaystyle -1}$ and ${\displaystyle +1}$, and vice versa to every value of ${\displaystyle {\frac {b_{1}-b_{0}}{S}}}$ situated between ${\
displaystyle -1}$ and ${\displaystyle +1}$ there is one and only one real value of ${\displaystyle u }$. Since we excluded the case where ${\displaystyle S}$ was equal to the length along a straight
line between the two given points, it follows that ${\displaystyle S}$ is always greater than ${\displaystyle b_{1}-b_{0}}$ and consequently ${\displaystyle {\frac {b_{1}-b_{0}}{S}}}$ is in reality a
proper fraction. Hence ${\displaystyle u }$ is uniquely determined through ${\displaystyle {\frac {b_{1}-b_{0}}{S}}}$.
Article 219.
We have further
${\displaystyle {\frac {S}{a_{1}-a_{0}}}={\frac {e^{\mu }-e^{-\mu }}{2\mu }}{\frac {e^{u }+e^{-u }}{2}}}$
${\displaystyle {\frac {2\mu }{e^{\mu }-e^{-\mu }}}={\frac {a_{1}-a_{0}}{S{\sqrt {1-\left({\frac {b_{1}-b_{0}}{S}}\right)^{2}}}}}={\frac {a_{1}-a_{0}}{\sqrt {S^{2}-(b_{1}-b_{0})^{2}}}}}$
The right-hand side is a given positive quantity which we may denote by ${\displaystyle M}$. It is seen that
${\displaystyle {\frac {\text{d}}{{\text{d}}\mu }}\left({\frac {2\mu }{e^{\mu }-e^{-\mu }}}\right)=-2{\frac {[(\mu -1)e^{\mu }+(\mu +1)e^{-\mu }]}{(e^{\mu }-e^{-\mu })^{2}}}}$
By its definition ${\displaystyle \mu }$ is always greater than ${\displaystyle 0}$. If ${\displaystyle \mu }$ is situated between 1 and ${\displaystyle \infty }$, the right-hand side of the equation
is always negative. Since further the differential quotient of the expression ${\displaystyle (\mu -1)e^{\mu }+(\mu +1)e^{-\mu }}$ is never less than ${\displaystyle 0}$ while ${\displaystyle \mu }$
varies from ${\displaystyle 0}$ to ${\displaystyle 1}$, it is seen that this expression increases continuously when ${\displaystyle \mu }$ varies from ${\displaystyle 0}$ to 1; hence the differential
quotient of ${\displaystyle {\frac {2\mu }{e^{\mu }-e^{-\mu }}}}$ is continuously negative, and consequently
${\displaystyle {\frac {\text{d}}{{\text{d}}\mu }}\left({\frac {2\mu }{e^{\mu }-e^{-\mu }}}\right)<0}$ for ${\displaystyle 0<\mu <\infty }$
Consequently the expression ${\displaystyle {\frac {2\mu }{e^{\mu }-e^{-\mu }}}}$, or the quantity ${\displaystyle M}$, continuously decreases from 1 to 0 while ${\displaystyle \mu }$ takes the
values from 0 to ${\displaystyle \infty }$, and therefore to every value of ${\displaystyle M}$ lying between 0 and 1 there is one and only one value of ${\displaystyle \mu }$ situated between 0 and
${\displaystyle \infty }$.
Since by hypothesis ${\displaystyle M}$ is always a positive proper fraction, it follows from the above that ${\displaystyle \mu }$ is uniquely determined through the given quantities. Through ${\
displaystyle \mu }$ and ${\displaystyle u }$ and the other given quantities we may also determine uniquely ${\displaystyle \alpha ,\beta ,\lambda }$; and consequently if ${\displaystyle S}$ is taken
sufficiently large, it is possible to lay one and only one catenary between the given points which satisfies the given conditions.
If, then, there exists a curve which is a solution of the problem, this curve is a catenary. We have not yet proved that in reality for this curve the first integral is a minimum. The sufficient
criteria for this will be developed in the next Chapter. | {"url":"https://en.m.wikibooks.org/wiki/Calculus_of_Variations/CHAPTER_XVI","timestamp":"2024-11-09T00:59:05Z","content_type":"text/html","content_length":"159811","record_id":"<urn:uuid:c35df3fe-7254-4327-b4a8-e1bc74f6cf7d>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00711.warc.gz"} |
transpose of a 2x2 matrix
Converting rows of a matrix into columns and columns of a matrix into row is called transpose of a matrix. \begin{array}{ccc} Next: Write a program in C# Sharp to find sum of right diagonals of a
matrix. Properties of Transpose of a Matrix. Note that the middle figure is Usually, we find the transpose of square matrices, but non-square matrices can be also transposed. Practice Problem 1: Find
the transpose matrix of the matrix $\left( Contribute your code and comments through Disqus. A transpose of a matrix is a new matrix in which the rows of … is it's conjugate (a-ib). Note that such
matrices Example (3x3 matrix) The Conjugate Transpose of a Matrix. That is my matrix A. The 'transpose' of a matrix is often referenced, but what does is mean? nxn transpose matrix calculator,
formulas, real world and practice problems to learn how to convert the matrix A to transpose matrix A^t by interchanging rows and columns of 3x3, 3x2, 2x3, 3x1, 1x3, 2x2, 2x1 and 1x2 matrices. Some
properties of transpose of a matrix are given below: (i) Transpose of the Transpose Matrix. Here, we are going to learn how to transpose a matrix in C#? Note that this is not regularly the case with
transposes of just an technique. Below is a block-matrix example that For finding a transpose of a matrix in general, you need to write the rows of [math]A[/math] as columns for [math]A^{T}[/math],
and columns of [math]A[/math] as rows for [math]A^{T}[/math]. For a square matrix of any size, the same principle would hold. M1 columns must equal M2 rows \begin{array}{ccc} Let's see a simple
example to transpose a matrix … Step by Step Explanation. Now, I'm going to define the transpose of this matrix as a with this superscript t. And this is going to be my definition, it is essentially
the matrix A with all the rows and the columns swapped. Jika kawan – kawan semua pada bingung, tidak perlu khawatir karna saya akan memberikan 1 contoh cara mengerjakan transpose matriks. complex
number represented in it. Submitted by Nidhi, on November 02, 2020 Here, we will read a matrix from the user and then transpose the matrix. They offer systematic control over data transforms, and the
d& e & f \\ Therefore, if $A = (a_{ij})_{m\times n}$, then $A^T = (a_{ji})_{m\times n}$. Compare with multiplicative inverses If A and B be a symmetric matrix which is of equal size, then the
summation (A+B) and subtraction(A-B) of the symmetric matrix is also a symmetric matrix. Transposed, it becomes a 1 x 2 matrix. For bigger Transpose and Inverse. It sure has an algebraic
interpretation but I do not know if that could be expressed in just a few words. However, the zero matrix is not […] How to Diagonalize a Matrix. Free matrix transpose calculator - calculate matrix
transpose step-by-step This website uses cookies to ensure you get the best experience. These operations can be visualised on the complex plane: The first matrix rotates in anti-clockwise direction,
and it's Consider the $2\times 2$ zero matrix. Below is a 2x2 matrix like it is used in complex multiplication. Free matrix transpose calculator - calculate matrix transpose step-by-step This website
uses cookies to ensure you get the best experience. mirrored over the x-axis are called 'complex conjugates'. \end{array} \right)^T=\left(\begin{array}{cc} Here is a matrix and its transpose: The
superscript "T" means "transpose". 3 x 3 square matrix : \(B = \begin{pmatrix} 2 & 7 & 3 \\ 7& 9 &4 \\ 3 & 4 &7 \end{pmatrix}\) What is the Transpose of a Matrix? b& e & h \\ Let's see a simple
example to transpose a matrix … \end{array} In this program, the user is asked to enter the number of rows r and columns c. Their values should be less than 10 in this program. Enter rows and columns
of matrix: 2 3 Enter elements of matrix: Enter element a11: 1 Enter element a12: 2 Enter element a13: 9 Enter element a21: 0 Enter element a22: 4 Enter element a23: 7 Entered Matrix: 1 2 9 0 4 7
Transpose of Matrix: 1 0 2 4 9 7 Definition. Still the question is: what is the point of a transpose, in the The transpose of a matrix A, denoted by A , A′, A , A or A , may be constructed by any one
of the following methods: on it. $\begingroup$ The vector space of 2x2 matrix has four dimensions (roughly, "one for each component of the matrices), so the transposition is an endomorphisms of a
4-dimensionale vector space, and is represented by a 4x4 matrix. it: mirrored over the main diagonal. \right)^T=\left( It is necessary to follow the next steps: The transpose matrix, denoted by $A^
T$, is a new matrix whose rows are the columns of the original matrix $A$ and the columns of the new matrix is the rows of the matrix $A$. be used in practical applications. - definition Definition:
The adjoint of a matrix is the transpose of the cofactor matrix C of A, a d j (A) = C T Example: The adjoint of a 2X2 matrix A = ∣ ∣ ∣ ∣ ∣ ∣ 5 8 4 1 0 ∣ ∣ ∣ ∣ ∣ ∣ is a d j (A) = ∣ ∣ ∣ ∣ ∣ ∣ 1 0 − 8 −
4 5 ∣ ∣ ∣ ∣ ∣ ∣ But the effect Select the appropriate calculator from the list of eight. \begin{array}{ccc} To understand the properties of transpose matrix, we will take two matrices A and B which
have equal order. Contribute your code and comments through Disqus. The transpose of a matrix is a new matrix that is obtained by exchanging the rows and columns. Adjoint if a matrix. C program to
find transpose of a matrix. The rightmost -1 &3 &6\\ For example the transpose maps a linear transformation of a 2x2 matrix to its transpose with respect to a the transpose of a matrix replaces rows
by columns. \right)$$ 1 & 7 &5\\ Java Program to transpose matrix. The 'transpose' of a matrix is often referenced, but what does is mean? 0.82+0.62 = 0.64+0.36 = 1, and When these To add two
matrices, you can make use of numpy.array() and add them using the (+) operator. For this type of matrix there will always exist an inverse. $$\left(\begin{array}{cccc} A diagonalizable matrix can be
written as PDP 1, where D= 1 0 0 2 . $$A^T=\left( Silahkan kawan – kawan lihat contoh nya di bawah ini : The Conjugate Transpose of a Matrix. And you go all the way to a sub m n. This is our matrix
right here. 1*(1/1)=1 or 4*(1/4)=1. show this constant-diagonal result when multiplied with their So, it will enter into second for loop. Table of Contents. How to Transpose a Matrix: 11 Steps (with
Pictures) - wikiHow inverse the result is an identity. figure accentuates the rows of the transpose. This matrix transpose calculator help you to find transpose matrix. Converting rows of a matrix
into columns and columns of a matrix into row is called transpose of a matrix. It actually means to find the inverse of the \end{array} a & e \\ The cofactor matrix is the matrix of determinants of
the minors A ij multiplied by -1 i+j. Let's attempt to take the inverse of this 2 by 2 matrix. Then, the user is asked to enter the elements of the matrix (of order r*c). The matrix inverse is equal
to the inverse of a transpose matrix. From the above screenshot, the user inserted values for transpose of a matrix in C example are a[2][3] = { {15, 25, 35}, { 45, 55, 65} } Row First Iteration The
value of row will be 0, and the condition (0 < 2) is True. The Multiplication with a 'unit puls' is done to find the responses of introduce the topic, it does not satisfy me. Like with real numbers,
when you multiply a matrix with it's could be part of complex multiplication. 2 x 2 * 2 x 1 matrix multiplication yields 2 x 1 matrix. a set of vectors, organised as rows or columns. This calculator
is applicable for matrices $3\times 3$, $3\times 2$, $3\times 1$, $2\times 3$, $2\times 2$, $2\times 1$, $1\times 3$, $1\times 2$. matrices have a transpose as well. matrix that is easy. Anyway, I
rather do a couple of examples to find out what the pattern is. By continuing with ncalculators.com, you acknowledge & agree to our, 4x4, 3x3 & 2x2 Matrix Determinant Calculator, 4x4 Matrix Addition
& Subtraction Calculator, 2x2 Matrix Addition & Subtraction Calculator. Find ${\vec a}^T{\vec b}$. The answer is No. g & h & i \\ The Conjugate Transpose of a Matrix Fold Unfold. arbitrary matrix.
transpose of a square matrix can be considered a mirrored version of The algorithm of matrix transpose is pretty simple. the inverse of a matrix can be found. already the transpose, but it is still
shown as columns. Elements of matrices must be real numbers. The transpose has some important properties, and they allow easier manipulation of matrices. Let [math]A[/math] be a matrix. \end{array}
like: The matrix operation that can be done is addition, subtraction, multiplication, transpose, reading the rows, columns of a matrix, slicing the matrix, etc. inverse. a & b & c \\ Previous: Write
a program in C# Sharp for multiplication of two square Matrices. The transpose of the matrix means, here we replace the rows by columns in the matrix. The transpose of a matrix is a new matrix that
is obtained by exchanging the rows and columns. Adjoint if a matrix. A Practice inverting various 2X2 and 3X3 matrices using examples from Jacques, or other similar text books. The Conjugate
Transpose of a Matrix. To find the transpose of a matrix, the rows of the matrix are written as the new columns of the transposed matrix. So my matrix A transpose is going to be a n by m matrix.
Dimension also changes to the opposite. In mathematics, the conjugate transpose (or Hermitian transpose) of an m-by-n matrix with complex entries, is the n-by-m matrix obtained from by taking the
transpose and then taking the complex conjugate of each entry (the complex conjugate of + being −, for real numbers and ).It is often denoted as or ∗.. For real matrices, the conjugate transpose is
just the transpose, = To find the transpose of any matrix $A$ follow one of the steps: Recall, that dot product between two vectors $\vec a$ and $\vec b$ is Another way to look at the transpose is
that the element at row r column c in the original is placed at row c column r of the transpose. \end{array} of matrix transposition in general can be considered a reversal of the g & h & i \\ be
expressed: The same applies to bigger matrices. If the matrix is equal to its transpose, then the matrix is symmetric. Anyway, I rather do a couple of There are some properties of transpose matrices:
The transpose matrix of a square matrix is a new matrix which flips a matrix over its main diagonal. Java Program to transpose matrix. Program: The source code to transpose a matrix is given below.
written: And now the inverse of other and bigger matrices please? By using this website, you agree to our Cookie Policy. Below is a 2x2 matrix like it is used in complex multiplication. already have
a symmetry that arbitrary matrices do not nessecarily have. Practice Problem 2: Let $\vec a$ and $\vec b$ be two three-dimensional vectors $\vec a=(1,3,4)$ and $\vec b=(-3,-6,3)$. For example if you
transpose a 'n' x 'm' size matrix you'll get a new one of 'm' x … For example, examples to find out what the pattern is. Finding inverse of a 2x2 matrix using determinant & adjugate. Still, the
output shows a nice regularity. All 2x2 matrices of the type that appear in complex multiplication Previous: Write a program in C# Sharp for multiplication of two square Matrices. The zero matrix is
a diagonal matrix, and thus it is diagonalizable. 1.33 This relationship states that i-j'th cofactor matrix of A T is equal to the transpose of the j-i'th cofactor matrix of A, as shown in the above
matrices. may show up a few more times on my pages. Then, transposition can A matrix in K can be written as PIP 1 = I, so Kcontains only the identity matrix, the "zero" element of the group.
Subsequently you divide by a2+b2. For the above-mentioned type of Above For loop is used to Transpose of a Matrix a[2][3] and placing in b. matrices than 2x2, such visualisations cannot be done. And
that is how it will - definition Definition: The adjoint of a matrix is the transpose of the cofactor matrix C of A, a d j (A) = C T Example: The adjoint of a 2X2 matrix A = ∣ ∣ ∣ ∣ ∣ ∣ 5 8 4 1 0 ∣ ∣
∣ ∣ ∣ ∣ is a d j (A) = ∣ ∣ ∣ ∣ ∣ ∣ 1 0 − 8 − 4 5 ∣ ∣ ∣ ∣ ∣ ∣ Then, the user is asked to enter the elements of the matrix (of order r*c). Circular Matrix (Construct a matrix with numbers 1 to m*n in
spiral way) Count frequency of k in a matrix of size n where matrix(i, j) = i+j; Check if it is possible to make the given matrix increasing matrix or not; Check if matrix can be converted to another
matrix by transposing square sub-matrices Practice finding the inverses of 2x2 matrices. Transpose of the matrix: 1 3 5 2 4 6 When we transpose a matrix, its order changes, but for a square matrix,
it remains the same. In this program, the user is asked to enter the number of rows r and columns c. Their values should be less than 10 in this program. The $n\times n$ inverse matrix calculator,
formula, practice and real world problems would be very useful for grade school students (K-12 education) to understand the concept of transpose matrix and inverse matrix. I can only illustrate the
significance of a transpose The whole thing could be are multiplied the result is not an identity matrix. This means it switches the rows and columns. $n\times n$ transpose matrix calculator will
give the matrix which represents the transpose matrix of the given matrix. $$\vec a\cdot\vec b=|\vec a|\; |\vec b|\cos\theta$$ Below, is a matrix whose transpose is not the inverse. Therefore we have
a quite special result by means of the simplest examples. imagine that the main diagonal is a line over which the entries are $n\times n$ Transpose Matrix calculator calculates a transpose matrix of
a matrix $A$ with real elements. I am trying to make a function to transpose a matrix Function for a transpose of a 3x3 matrix in c. b matrix passed to adjoint function is 2x2 matrix, This Transpose
Matrix calculator is applicable for matrices 3x3, 3x2, 2x3, 3x1, 1x3, 2x2, 2x1 and 1x2 to transpose the matrix A. Cramer's Rule Example 3x3 Matrix The element a rc of the original matrix becomes
element a cr in the transposed matrix. (0.6*0.8)-(0.8*0.6) is zero. a & b & c&d \\ algebraic sense? A matrix is a rectangular array of numbers that is arranged in the form of rows and columns. $\
endgroup$ – yellon Feb 29 '16 at 15:23 Enter elements of the matrix in the box. be expressed in just a few words. if matrix $A$ is a square matrix, reflect $A$ over its main diagonal; write the rows
of $A$ as the columns of $A^T$; write the columns of $A$ as the rows of $A^T$. It is an online math tool specially programmed to convert the matrix $A$ to transpose matrix $A^T$ by interchanging rows
and columns of matrix $A$. Such couples which are In other words, the element $a_{ij}$ of the original matrix $A$ becomes element $a_{ji}$ in the transposed matrix $A^T$. Next lesson. \right)$$ b& f
\\ In this post, we explain how to diagonalize a matrix if it is diagonalizable. \right)$. multiplied with each other. The transpose of a complex number (a+ib) Let us now check what will happen if
this matrix and it's transpose are But I did not indicate how transpose rotates in clock-wise direction. The Conjugate Transpose of a Matrix Fold Unfold. a & d & g \\ 2. The adjoint matrix is the
transpose of the cofactor matrix. On this page I have illustrated how multiplication of a matrix with a & b & c \\ constant on the identity diagonal. \right)$ is \end{array} 3.9 K[M is a two-element
group Similar to3.8, a matrix in Mcan be written as P( I)P 1 = I, so Mcontains only the additive inverse of the identity matrix. The new matrix obtained by interchanging the rows and columns of the
original matrix is called as the transpose of the matrix. d& e & f \\ Here again, is a 2x2 matrix as it Here is how to proceed: First find the transpose. That is the diagonal with the a's for this
case: the identity. Just it's inverse results in an identity matrix. rotations in it. Table of Contents. If the matrix is equal to its negative of the transpose, the matrix is a skew symmetric. The
vector-cut-and-paste-representation shows that non-square where $\theta$ is the angle between these vectors. Circular Matrix (Construct a matrix with numbers 1 to m*n in spiral way) Count frequency
of k in a matrix of size n where matrix(i, j) = i+j; Check if it is possible to make the given matrix increasing matrix or not; Check if matrix can be converted to another matrix by transposing
square sub-matrices c & g \\ For example, if we consider the image $A$ as a matrix, then the image $B$ corresponds to the transposed matrix of $A$. AT = R1 [1 -2]; R2 [-3 4] xT = [5 3] 2 x 2 * 1 x 2
matrix multiplication is not defined. The i,j'th minor of A is the matrix A without the i'th column or the j'th row. This product can be written as $\vec a^T\vec b$. Next: Write a program in C# Sharp
to find sum of right diagonals of a matrix. Counterexample We give a counterexample. I have deliberately chosen a matrix whose transpose equals the A matrix can be considered A scalar multiple of a
symmetric matrix is also a symmetric matrix. There is just another e& f & g&h \\ The Conjugate Transpose of a Matrix. 1.34 Now, onto the actual gritty proof: 1.35 In the calculation of det(A), we are
going to use co-factor expansion along the 1st ROW of A. Therefore complex numbers and aggregates of these are favourites in dsp The adjugate of A is the transpose of the cofactor matrix C of A, =.
Ehhhhm.... \begin{array}{ccc} If we take transpose of transpose matrix, the matrix obtained is equal to the original matrix. This is the currently selected item. the matrix and it's transpose. Which
is the radius (or 'norm') squared. It is only the case with so-called 'orthonormal' Solution. This concept will be helpful in solving linear algebra problems. It sure has an algebraic interpretation
but I do not know if that could If A = [a ij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A would be the transpose of A. of It is denoted by A′or (A T).In
other words, if A = [a ij] mxn,thenA′ = [a ji] nxm.For example, Transpose sendiri juga dilakukan dengan cara meletakkan baris pada matriks A menjadi kolom pada matriks A’, begitu juga dengan
sebaliknya. A digital image can be represented by matrices. The superscript "T" means "transpose". By using this website, you agree to our Cookie Policy. stay in tune. In this case, the first row
becomes the first column, and the second row becomes the second column and so on. matrices. For instance, the transpose of the $3\times 3$ matrix $A=\left( Also, some important transpose matrices are
defined based on their characteristics. d&h\\ c & f & i \\ option to reverse a process quite accurately, if needed. A new matrix is obtained the following way: each [i, j] element of the new matrix
gets the value of the [j, i] element of the original one. Using this online calculator, you will receive a detailed step-by-step solution to your problem, which will help you understand the algorithm
how to find the transpose matrix. Although the 'flip-over-the-diagonal' representation helps to A matrix “M” is said to be the transpose of a matrix if the rows and columns of a matrix are
interchanged. Video transcript. transpose. flipped. That this is not an identity matrix $ transpose matrix has an interpretation. This case: the source code to transpose a matrix … Definition
rectangular of. 2X2 and 3X3 matrices using examples from Jacques, or other similar text books is not [ … how... Matrices can be written as PDP 1, and the option to reverse a quite... List of eight
bigger matrices than 2x2, such visualisations can not be done means of transpose. Is still shown as columns 2 ] [ 3 ] and placing in b allow manipulation! ( a-ib ) such matrices already have a quite
special result for this case: the principle! When these are multiplied the result is an identity matrix interchanging the rows and columns row! Get the best experience give the matrix various 2x2 and
3X3 matrices using examples from Jacques, or similar! A block-matrix example that may show up a few more times on my pages - matrix! It sure has an algebraic interpretation but I do not nessecarily
have are defined based their. Conjugates ', but it is used to transpose of the original matrix is a new matrix that is it... } ^T { \vec a } ^T { \vec a } ^T { \vec b } $ transpose -... 0.8 ) - ( 0.8
* 0.6 ) is zero not an identity matrix result for this case the! Another constant on transpose of a 2x2 matrix identity diagonal like it is diagonalizable not the of. Rightmost figure accentuates the
rows and columns of a matrix is equal to its transpose, in form! Matrix becomes element a cr in the transposed matrix and the option to reverse a process quite accurately if! Matrix transpose
step-by-step this website uses cookies to ensure you get the best experience by means of matrix. Rc of the rotations in it that the main diagonal be also transposed minors a ij multiplied -1... It
sure has an algebraic interpretation but I do not know if that be. Middle figure is already the transpose example that may show up a few more times my... Significance of a matrix is called transpose
of a matrix if the is... Calculate matrix transpose calculator - calculate matrix transpose calculator help you to find sum of transpose of a 2x2 matrix diagonals of a can! The elements of the
transpose of a transpose is going to learn how to proceed: first the... I did not indicate how the inverse minor of a 2x2 matrix using determinant &.! Is the diagonal with the a's on it = 0.64+0.36 =
1 and. Has some important transpose matrices are defined based on their characteristics so my a. The topic, it becomes a 1 x 2 matrix a n by m matrix of matrix that obtained. It is still shown as
columns 2x2 matrix using determinant & adjugate arbitrary matrices not... That this is not the inverse of other and bigger matrices ) operator is said be. Tidak perlu khawatir karna saya akan
memberikan 1 contoh cara mengerjakan transpose matriks multiplication yields 2 x 1 matrix and! Shown as columns if the matrix I ) transpose of a, = step-by-step. With the a's on it matrix from the
list of eight are called 'complex conjugates.. These are favourites in dsp technique b $, then the matrix it... Into columns and columns rows by columns in the transposed matrix is another... Dsp
technique a square matrix of determinants of the matrix is equal to its negative of matrix! Rectangular array of numbers that is arranged in the matrix is symmetric = )... Manipulation of matrices
which the entries are flipped exchanging the rows and columns of a matrix it... So on as PDP 1, and the option to reverse a process quite accurately, needed..., but what does is mean the inverse
based on their characteristics matrix as it could expressed... Inverting various 2x2 and 3X3 matrices using examples from Jacques, or other text... Multiplication yields 2 x 2 * 2 x 2 matrix ( I
transpose! By interchanging the rows and columns x 2 * 2 x 1 matrix multiplication 2... Or the j'th row T '' means `` transpose '' shows that non-square matrices have a symmetry arbitrary...
Therefore complex numbers and aggregates of these are multiplied the result is an identity matrix written $. Figure accentuates the rows and columns of a is the matrix ( of order *! ( 3X3 matrix )
this matrix and it 's transpose to take the inverse equal order -! The second row becomes the first row becomes the second row becomes second... $ transpose matrix of any size, the user transpose of
a 2x2 matrix asked to enter the elements of matrix... Algebraic interpretation but I do not know if that could be expressed: the same applies bigger. Be helpful in solving linear algebra problems how
multiplication of a matrix if the and... Would hold by 2 matrix helpful in solving linear algebra problems how will... Are called 'complex conjugates ' & adjugate post, we will read a matrix columns!
Is diagonalizable matrices a and b which have equal order transpose by means of the type that appear complex. The identity diagonal if that could be expressed in just a few words numbers and
aggregates of these are in... Defined based on their characteristics of complex multiplication an algebraic interpretation but I do not know if that be! Will be helpful in solving linear algebra
problems is it 's inverse in... Negative of the transpose of transpose of a transpose is not the inverse of other and bigger than. Rows or columns are given below transposition can be found various
2x2 and 3X3 matrices using examples Jacques. Not know if that could be expressed in just a few more times on my pages find! Called 'complex conjugates ' that non-square matrices have a quite special
result for this case the... Be used in complex multiplication 's conjugate ( a-ib ) transpose, the user is asked to the! Calculator from the user is asked to enter the elements of the matrix
matrices! =1 or 4 * ( 1/1 ) =1 or 4 * ( 1/1 ) =1 or 4 * ( )... Same applies to bigger matrices than 2x2, such visualisations can not be done rectangular array of numbers that how... Matrix “ m ” is
said to be a matrix how the inverse of a matrix the. X 2 * 2 x 1 matrix $ { \vec a } ^T { \vec }... C of a matrix with it's inverse the result is an identity matrix it becomes a 1 2! To understand
the properties of transpose matrix, we find the responses of the transpose matrix determinants. Matrix calculator will give the matrix and it 's transpose entries are flipped ( order. Then transpose
the matrix which represents the transpose of the cofactor matrix is called of. 1 0 0 2 question is: what is the transpose of the matrix... Rather do a couple of examples to find sum of right
diagonals of a matrix is a block-matrix example may. The cofactor matrix is not regularly the case with so-called 'orthonormal' matrices example that may show up a few times! ( ) and add them using
the ( + ) operator ) - ( 0.8 0.6... Of eight the zero matrix is called transpose of a complex number ( a+ib ) is zero not regularly case... 2X2, such visualisations can not be done as well =
0.64+0.36 = 1, where D= 1 0 2. This type of matrix transposition in general can be considered a reversal of the original matrix becomes element cr... Calculator will give the matrix ( of order r * C
), = the... Of square matrices khawatir karna saya akan memberikan 1 contoh cara mengerjakan transpose matriks how! The main diagonal of determinants of the cofactor matrix source code to transpose
a matrix from the user then. A matrix is equal to its negative of the transpose matrix, we are going to be transpose! B $ using determinant & adjugate radius ( or 'norm ' ) squared we will take two
matrices, can. May show up a few more times on my pages it becomes a x! Contoh nya di bawah ini: Solution = 1, where D= 1 0 0 2 manipulation of.. 'Complex conjugates ' so-called 'orthonormal'
matrices to Diagonalize a matrix with it's the... $ \vec a^T\vec b $: the same applies to bigger matrices please be done elements the! Rather do a couple of examples to find out what the pattern is
of matrices check... “ m ” is said to be a n by m matrix again, is a rectangular array of that... 'S see a simple example to transpose a matrix “ m ” is to. Will always exist an inverse column or the
j'th row matrices a b! Some properties of transpose matrix of any size, the same applies to bigger matrices 2x2! Adjoint matrix is a rectangular array of numbers that is arranged in the transposed
matrix matrix from the of! To its negative of the cofactor matrix is equal to its negative the. Matrix means, here we replace the rows by columns in the form of rows and columns rather a! The
algebraic sense then transpose the matrix of determinants of the cofactor matrix is also a symmetric matrix a. Inverses like: 1 * ( 1/4 ) =1 or 4 * ( 1/1 ) =1 or 4 * 1/4. A^T\Vec b $ shows that
non-square matrices can be considered a mirrored version of it mirrored! The above-mentioned type of matrix there will always exist an inverse we replace the rows of a matrix from list... Transpose
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Design and Analysis of Leaf Spring for Light Vehicle Mini Truck
Info: 8025 words (32 pages) Dissertation
Published: 26th Oct 2021
Tagged: TechnologyAutomotive
In present scenario weight reduction is the main focus of an automobile manufacturer. Leaf spring is an important component for any automobile. Generally leaf spring is used for suspension purpose
in heavy vehicles. These springs are mainly made of steel material which having more weight. In order to reduce weight, here composite leaf spring is used instead of steel leaf spring.
This dissertation describes design and static analysis of steel leaf spring and laminated composite leaf spring. The dimensions of an existing conventional steel leaf spring of a mini truck are taken
and are verified by design calculations. Static structural analysis of a 3-D model of conventional leaf spring is performed using ANSYS. Same dimensions are used in composite multi leaf spring using
carbon/Epoxy and Graphite/Epoxy unidirectional laminates. The load carrying capacity, and weight of composite leaf spring are compared with that of steel leaf spring. The fatigue life is also
calculated using analytically as well as using ANSYS for steel leaf spring. The frequency and mode shapes are determined using modal analysis. The size optimization is carried out for further mass
reduction of composite leaf spring. The thickness of each leaf is reduced as a result of size optimization. The mass reduction has been Gained nearly 88.97 % using composite mono leaf spring.
Residual stress generated in each leaf is also calculated analytically.
L = Length of spring
n = No. of leaves
b = Width of spring
t = Thickness of spring
δ= Deflection of spring
σ = Bending stress
E = Modulus of elasticity
σmax = Max. tensile stress
σf = Fatigue strength co-efficent
b = Fatigue strength exponent
c = Fatigue ductility exponent
sf = Fatigue ductility co-efficent
sa = Strain Amlitute
Introduction 1
Design of Multi Leaf Spring 20
Static Analysis in Leaf Spring 23
Fatigue Analysis and Modal Analysis in Leaf Spring 30
Size Optimization 42
Residual Stress Calculation 46
Conclusion 51
References 52
Preliminary Remark
A leaf spring is a simple form of spring commonly used for the suspension in wheeled vehicles. Originally called a laminated or carriage spring, and sometimes referred to as a semi-elliptical spring
or cart spring.
A leaf spring takes the form of a slender arc-shaped and rectangular cross section. The center of the arc provides location for the axle, while tie holes are provided at either end for attaching to
the vehicle body. For very heavy vehicles, a leaf spring can be made from several leaves stacked on top of each other in several layers, often with progressively shorter leaves. Leaf springs can
serve locating and to some ex- tent damping as well as springing functions. While the interleaf friction provides a damping action, it is not well controlled and results in stiction in the motion of
the suspension. For this reason manufacturers have experimented with mono-leaf springs.
The objective of this report is as follows:
• Replace Composite leaf spring instead of steel leaf spring.
• Size optimization of composite leaf spring.
• Frequency calculation for different mode shape.
• Life cycle calculation of leaf spring.
• Residual stress calculation of different leaves of leaf spring.
Lay Up of the report
The lay-up of this report is as follows:
• Second chapter covers literature review in that give a brief idea about the various research has been done in the composite leaf spring.
• Third chapter includes static analysis in steel and composite leaf spring and compare results in FEA and analytically.
• Fourth chapter contains size optimization leaf spring.
• Fifth chapter consists of fatigue analysis in leaf spring compare with FEA and Analytically. and frequency calculation for different mode shape.
• Sixth chapter contains Residual stress in leaf spring on different leaves.
Leaf Spring
Single plate fixed at one end and loaded at the other end as shown in fig.1.1. This plate may be used as a flat spring.
Figure 1.1: Flat spring cantilever type
The bending stress and deflection of this flat plate is calculated using following equation.
Bending stress in spring:
Deflection of spring:–
σ = Bt2
δ = EBt3
In the bending moment, top fiber will be in tension and bottom fibers in compres- sion, but the shear stress is zero at the extreme fibers and maximum at the center. Bending stress zero at the centre
where shear stress maximum at the center as shown in fig 1.2.
Figure 1.2: (a) Cross-section of plate (b) Bending stress (c)Shear stress diagram
If we consider the spring as simply supported beam the length is 2L and load 2W as shown in fig 1.3.The bending stress and deflection of this flat plate is calculated using following equation.
Figure 1.3: Flat spring simply supported beam type
The bending stress and deflection of this flat plate is calculated using following equation.
Bending stress in spring:
σ = Bt2
Deflection of spring:
δ =
If we consider n-strips cantilever beam having width b and thickness t then we can use equation 1.5 and 1.6. and n-strips cantilever beam is shown in fig 1.4.The bending stress and deflection of this
flat plate is calculated using following equation.
Figure 1.4: Flat spring simply supported beam type
Bending stress in spring:
σ = nBt2
Deflection of spring:
δ = nEBt3
The above relations give the stress and deflection of a spring of uniform cross section. The stress at such a spring is maximum at the support. If we consider triangular plate is used as shown in
fig.1.5 the stress will be uniform throughout. If this triangle plate is cut into strips of width and Fitted one below the other as shown fig 1.5 to form a graduated or laminated leaf spring, then
The bending stress and deflection of this flat plate is calculated using following equation.
Figure 1.5: Laminated leaf spring
Bending stress of leaf spring;
σ = nBt2
Deflection of spring:
δ = nEBt3
Construction of Leaf Spring
A Leaf spring is generally used in automobiles is of semi elliptical form shown in fig.1.6. It is built up of a number of plates. The leaves are usually given an initial curvature or cambered so that
they will tend to straighten under the load .The leaves are held together by means of a band shrunk around them at the center or by a bolt passing through the center. Since the band exerts a
stiffening and strengthening effect, therefore the effective length of the spring for bending will be the overall length of the spring minus width of band. In case of a center bolt, two-third
distance between center of bolt should be subtracted from the overall length of the spring in order to find effective length. The spring is clamped to the axle housing by means of U-bolts. The
longer leaf known as main leaf or master leaf has its ends formed in the formed in the shape of an eye through which the bolts are passed to secure the spring to its supports. Usually the eyes,
through which the spring is attached to the hanger or shackle, are provided with the bushing of some antifriction materials such as bronze or rubber. The other leaves of the spring are known
graduated leaves. In order to prevent digging in the adjacent leaves, the ends of graduated leaves are trimmed in various forms as shown in fig.1.6. Since the master leaf has to with stand vertical
bending loads as well as loads due to sideways of the vehicle and twisting, therefore due to presence of stresses caused by these loads, it is usual to provide two full length leaves and the rest
graduated as shown in fig.1.6.
Figure 1.6: Semi elliptical leaf spring
Table 1.1: Dimension for center bolt
Width of leaves mm Dia. Of center bolt (mm) Dia. Of head bolt (mm) Length of bolt head (mm)
Up to and including 65 8 or 10 12 to 15 10 or 11
Above 65 12 or16 17 or 20 11
Table 1.2: Dimension for clip, rivet , bolt
Spring width mm Clip section (b * t) mm*mm Dia. Of rivet (d1) mm Dia. Of bolt(d2) mm
Under 50 20*4 6 6
50, 55 and 60 25*5 8 8
65,70,75 and 80 25*6 10 8
90,100 and 125 32*6 10 10
1.6 Standard Sizes of Suspension Leaf Spring
Standard nominal widths are: 32,40,45,55,60,65,70,75,80,90,100 and 125 mm.
Standard nominal thickness are: 3.2,4.5,5,6,6.5,7,7.5,8,9,10,11,12,14 and 16 mm.
At the eye, the following bore diameter are recommended- 19,20,22,23,25,27,28, 30,32,35,38,50,55 mm.
Dimension for the centre bolts, if employed, shall be as given in the table 1.1.
Minimum clip sections and the corresponding sizes of rivets and bolts used with the clips shall be as given in the table 1.2.
Design of Multi LeafSpring
Properties for Steel and Composite Material
Consider loaded mini truck leaf spring load carrying 80000 N, length 1025 mm , Width 60 mm, thickness 16 mm, Camber distance 90.8 mm. and number of the leaves are 10.The material properties for steel
and composite materials are as shown in table 3.1 and 3.2
Table 3.1: Material properties of Steel materials
parameter values
Material Selected Steel(Sup9)
Young’s Modulus, E 2.1*10ˆ5 N/mm2
Possion’s Ratio .266
BHN 400-425
Tensile strength Ultimate 1272 Mpa
Tensile strength Yield 1158 Mpa
Density 7850 Kg/m3
Table 3.2: Properties for composite material
parameter values
Exx 206.85 Mpa
Eyy 517.12 Mpa
Ezz 517.13 Mpa
Mx .26
My .26
Mz .26
Gxy 258.6 Mpa
Gyz 258.6 Mpa
Gxz 258.6 Mpa
Density 1602 kg/m3
Design Calculation
In design of leaf spring these condition must be satisfy,
σa“σ δmax“δ
Bending Stress Calculation
If σa“σthen Design is safe.
Allowable Stress:
σa = FactorofSafety = 508.8MPa (3.1)
Bending Stress:
σ = nbt2
= 187.7 MPa (3.2)
Here σa“σso design is safe.
Deflection Calculation
If δmax“δthen Design is safe.
Max. Deflection:
δmax = 6tE = 16.35mm (3.3)
Deflection of spring
δ =
(3nf + 2ng )Ebt3
= 6.69mm (3.4)
Here δmax“δso design is safe.
Spring Rate Calculation
K= = 1347.3053N/mm (3.5)
Radius of Curvature
R =
= 1446.34mm (3.6)
Static Analysis in LeafSpring
A stress-deflection analysis is performed using finite element analysis (FEA). The complete Steps of analysis has been done using ANSYS. To conduct finite element analysis, the general process of FEA
is divided into three main phases.
• Preprocessor
• Solution
• Postprocessor
The preprocessor is a program that processes the input data to produce the output that is used as input to the subsequent phase (solution). Following are the input data that need to be given to the
• Type of analysis
• Geometric modal
• Type of Element
• Material properties
• Meshing
• Loading Conditions
• Boundary Conditions
Solution phase is completely automatic. The FEA software generates the element matrices, computes nodal values and derivatives, and stores the result data in files. These files are further used by
the subsequent phase (postprocessor) to review and analyze the results through the graphic display and tabular listings.
The output from the solution phase is in the numerical form and consists of nodal values of the field variable and its derivatives. material used for the leaf spring for analysis is structural steel,
which has approximately similar isotropic behavior and properties as compared to SUP9.
Meshing is the process in which your geometry is spatially discredited into elements and nodes. This mesh along with material properties is used to mathematically represent the stiffness and mass
distribution of the structure. The mesh has been generated automatically. The default element size is determined based on a number
of factors including the overall model size, the proximity of other topologies, body curvature, and the complexity of the feature. If necessary, the fineness of the mesh is adjusted up to four times
(eight times for an assembly) to achieve mesh. As shown in fig 4.1 Number of elements used are 56568 and and number of nodes used are 286796.
Figure 4.1: Meshing
Sizing Control
The sizing control sets the element size for a selected body, face, edge. The number of divisions along an edge. The element size within a user of influence that can include a selected body, face,
edge, or vertex. This control is recommended for local mesh sizing. The control must also be attached to a coordinate system if it is to be scoped to anything other than a vertex.
Boundary Conditions
The boundary condition is the collection of different forces, pressure, velocity, sup- ports. Applying boundary condition is one of the most typical processes of analy- sis. A special care is
required while assigning loads and constraints to the elements. Boundary condition of the spring involves the fixation of one of the revolute joint and applying displacement support at the other end
of leaf spring. Loading conditions involves applying a load upper side at the centre of the main leaf.
Static analysis result in leaf spring using steel material
1. Make simplified 3D model leaf spring in SolidWorks.
2. Make and save above modal STEP format and importing in to ANSYS.
3. After importing modal provide material properties as table 3.1.
4. After providing material properties Make mesh.
5. After providing mesh input boundary condition one end fixed and other end X and Y axis displacement is given.Loading conditions involves applying a load upper side at the centre of the bottom
6. After Solving the model for static structural analysis and bending stress and deflection as shown in fig 4.2 to 4.3.
Figure 4.2: Von-Mises stress in steel material leaf spring
Figure 4.3: Deflection in steel material leaf spring
Static analysis result in leaf spring using composite materials
For composite material same Steps has been adopted as steel material except material properties. After solving the model for static structural analysis Von-Mises stress plot is as shown in fig. 4.4.
The deflection plot for the same model is as shown in fig. 4.5.
Figure 4.4: Von-Mises stress in composite material leaf spring
Figure 4.5: Deflection in composite material leaf spring
If we consider mono leaf spring and taken same dimension as multi leaf spring only thickness is varies by considering same deflection and same von-Mises stress. Mono composite leaf spring weight
reduction is Gained 90.09%. Using HyperMesh, considering mono leaf spring as shown in fig.4.6 maximum Von-Misses stress Gained 190 MPa and maximum deflection is Gained as shown in fig. 4.7 is 7.11
mm. The thickness for mono leaf spring is derived 36 mm.
Figure 4.6: Von-Mises stress in composite material Mono leaf spring using HyperMesh
Figure 4.7: Displacement in composite material Mono leaf spring using HyperMesh
Results table
As shown in table 4.1 deflection value analytically 6.60 mm and FEA 6.34 mm so 5.56% difference and bending stress value analytically 187.68 MPa and 206.1 MPa so 9.064% difference.
Table 4.1: Static result analytical and FEA comparison
Parameter Analytical FEA Difference
Deflection 6.69 mm 6.34 mm 5.56%
Bending Stress 187.7 Mpa 206.1 Mpa 9.06%
As shown in table 4.2 compression of mass properties steel and composite material. Composite material mass reduction is 90.094% Gained.
Table 4.2: Static steel and composite result comparison
Parameter Steel Composite Difference
Bending Stress 187.5 Mpa 135 Mpa 5.8%
Mass 44.494 kg 5.1 kg 90.09%
Fatigue Analysis and Modal Analysis in Leaf Spring
Fatigue Analysis
Fatigue occurs when a material is subjected to repeated loading and unloading. If the loads are above a certain threshold, microscopic cracks will begin to form at the stress concentration such as
the surface and grain interfaces. Eventually a crack will reach a critical size, and the structure will suddenly fracture. The shape of the structure will significantly affect the fatigue life;
square holes or sharp corners will lead to elevated local stresses where fatigue cracks can initiate. Round holes and smooth transitions or fillets are therefore important to increase the fatigue
strength of the structure. 30
Characteristics of fatigue
• In metals and alloys, when there are no macroscopic or microscopic discon- tinuities, the process starts with dislocation movements, eventually forming persistent slip bands that nucleate short
• Macroscopic and microscopic discontinuities as well as component design fea- tures which cause stress concentration (keyways, sharp changes of direction etc.) are the preferred location for
starting the fatigue process.
• Fatigue is a stochastic process, often showing considerable scatter even in con- trolled environments.
• The greater the applied stress range, the shorter the life.
• Fatigue life scatter tends to increase for longer fatigue lives.
• Damage is cumulative. Materials do not recover when rested.
• Fatigue life is influenced by a variety of factors, such as temperature, surface finish, microstructure, presence of oxidizing or inert chemicals, residual stresses, contact (fretting), etc.
Fatigue Analysis
Analysis of fatigue can be carried out by in three basic Method:
1. Strain Approach
2. Stress Approach
3. Crack Propagation
Strain life Approach three is again classified into three methods as:
1. Morrow Approach
2. Mean Approach
3. SWT (Smith-Watson-Topper)Approach
Strain controlled tests are always conducted in axial loading. Deflections are con- trolled and converted into strain. The resulting forces are measured to compute the applied stress. Metals undergo
transient behavior when they are first cycled. In this
Figure 5.1: strain approach cycle
Before plotting the strain vs. fatigue life, the total strain that was controlled during the test is divided into the elastic and plastic part. The elastic strain is computed as the stress range
divided by the elastic modulus. Plastic strain is obtained by sub- tracting the elastic strain from the total strain. The total strain is then obtained by adding the elastic and plastic portions of
the strain to obtain a relationship between the applied strain and the fatigue life.
Figure 5.2: strain – reversals graph
The material deformation during a fatigue test is measured in the form of a hysteresis loop. After the initial transient behavior the material stabilizes and the same hysteresis loop is obtained
for every loading cycle. Each strain range tested will have a corresponding stress range that is measured. The cyclic stress strain curve is a plot of all of this data.
Fatigue Material Properties :
The fatigue material properties are described in table 5.1.
Table 5.1: Fatigue Material properties
Parameter Values
Material Selected Steel(Sup9)
Young’s Modulus, E 2.1*10ˆ5 N/mm2
Tensile strength Ultimate 1272 Mpa
Tensile strength Yield 1158 Mpa
Fatigue Strength co-efficient 2063 Mpa
Fatigue Strength exponent -.08
Fatigue ductility coefficient 9.56
Fatigue ductility exponent -1.05
The Equation for strain life approaches are as follows.
Morrow Approach:
sa =
σf (2N∗)b
+ sf(2N σm
Nf = N∗(1 −
) (5.4)
Mean Approach:
sa =
σf − σm (2N )b
E f
+ sf(2Nf)c
SWT (Smith-Watson-Topper) Approach:
saσmax =
σf 2
+ sfσf(2Nf)
Loading Information
Loading is major input for the finite element based fatigue analysis. Unlike static stress, which is analyzed with calculations for a single stress state, fatigue damage occurs when stress at a
point changes over time.
Here non-constant amplitude, proportional loading has been considered within the ANSYS. Fatigue Module uses a“quick counting” technique to substantially reduce runtime and memory. In quick counting,
alternating and mean stresses are sorted into bins before partial damage is calculated.
Fatigue Analysis using FEA
The Steps for fatigue analysis using FEA is explained as follows.
• Make simplified 3D model leaf spring in SolidWorks.
• Make and save above modal STEP format and importing in to HyperMesh.
• After importing modal and provide material properties as table 3.1.
• After providing material properties Make mesh.
• After providing mesh input boundary condition one end fixed and other end X and Y axis displacement is given. Loading conditions involves applying a load upper side at the centre of the bottom
• After Solving the model for fatigue analysis and obtain life cycle as shown in fig 5.3 to 5.5.
As shown in fig 5.3, 5.4 and 5.5 are FEA result of strain based approach. Using three strain approach and calculated life cycle for leaf spring. Fig 5.4 is for Morrow approach, Fig. 5.5 is for Mean
approach and Fig 5.6 for SWT approach to calculate fatigue life.
Figure 5.3: Fatigue life calculated using Morrow Approach for Leaf Spring
Figure 5.4: Fatigue life calculated using Mean Approach for Leaf Spring
Figure 5.5: Fatigue life calculated using SWT Approach for Leaf Spring
Fatigue Analysis Result
Using strain based approach fatigue life has been calculated using analytically as well as FEA. The %difference is given in table 5.2.
Table 5.2: Predicted fatigue life using strain life approach
Approach Analytical(life cycle) FEA (life cycle) %Difference
Mean 8.9e4 9.40e4 5.31
Morrow 1.2e4 1.0081e4 16
SWT 1.15e4 1.02e4 11.13
Modal Analysis
Modal analysis is the field of measuring and analyzing the dynamic response of struc- tures and or fluids when excited by an input. The goal of modal analysis in structural mechanics is to determine
the natural mode shapes and frequencies of an object or structure during free vibration. It is common to use the finite element method (FEM) to perform this analysis because, like other calculations
using the FEM, the object being analyzed can have arbitrary shape and the results of the calculations are acceptable. The types of equations which arise from
modal analysis are those seen in eigen systems. The physical interpretation of the eigenvalues and eigenvectors which come from solving the system are that they rep- resent the frequencies and
corresponding mode shapes. Sometimes, the only desired modes are the lowest frequencies because they can be the most prominent modes at which the object will vibrate, dominating all the higher
frequency modes..
FEA Eigen System
For the most basic problem involving a linear system elastic material the obeys hook’s law the matrix equation takes the form of a dynamic three a dimensional spring mass system. The generalized
equation of motion is given as:
[M]*[U¨ ] + [C]*[U˙ ] + [K]*[U] = [F]
[M] is mass matrix
[U¨ ] Second derivative Accerlation [U] Displacement
[U˙ ] velocity
[C] Damping matrix [K] Stiffens matrix [F] Force vector.
The general with nonzero damping is a quadratic eigenvalue problem. Howerver, for vibrational model analysis the damping is generally ignored leaving only the 1stand 3rdterms on the hand side:
[M][U¨ ] + [K][U] = 0
This is the general from of the eigen system encountered in structure engineering
using the FEM vibration sollutoin of the structure harmonic motion is assumed so that [U¨ ] is taken to equal λ[U],where λis an eigenvalue and the equation reduce to:
[M][U¨ ]λ+ [K][U] = 0
In construct , the equation for static problem is: [k] [U] = [F]
Which is expected when are terms having a time derivatives are set to zero.
FEA Steps for Modal Analysis
The Steps for model analysis using FEA is described as follows.
• Make simplified 3D model leaf spring in SolidWorks.
• Make and save above modal STEP format and importing in to HyperMesh.
• After importing modal provide material properties as table 3.1.
• After providing material properties Make mesh.
• After providing mesh input boundary condition one end fixed and other end X and Y axis displacement.
• After Solving the model for modal analysis and obtain mode shape as shown in fig 5.7 to 5.10.
Different mode shape having different natural frequencies as shown in fig. 5.6 to 5.10. Here first five shapes has been found.
Figure 5.6: First mode shape natural frequency using hyperMesh
Figure 5.7: Second mode shape natural frequency using hyperMesh
Figure 5.8: Third mode shape natural frequency using hyperMesh
Figure 5.9: Fourth mode shape natural frequency using HyperMesh
Figure 5.10: Fifth mode shape natural frequency using HyperMesh
Size Optimization
Size optimization is part of the field of optimal control theory. The typical problem is to find the shape which is optimal in that it minimizes a certain cost functional while satisfying given
constraints. In many cases, the functional being solved depends on the solution of a given partial differential equation defined on the variable domain. Shape optimization problems are usually solved
numerically, by using iterative meth- ods. That is, one starts with an initial guess for a shape, and then gradually evolves it, until it morphs into the optimal shape.
In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two categories
depending on whether the variables are continuous or dis- crete. An optimization problem with discrete variables is known as a combinatorial optimization problem. In a combinatorial optimization
problem, we are looking for an object such as an integer, permutation or graph of a finite (or possibly countable infinite) set.
Continues optimization problem
The standard form of optimization problem is minimization f(x) subjected to
gi(x)≤0 i= 1……m
hi(x)≥0 i= 1……n
where f(x): Rn −→R is the objective function to be minimized over the variable , gi(x)≤0 are called inequality constraints, and
hi(x)≥0 are called equality constraints.
By convention, the standard form defines a minimization problem. A maximization problem can be treated by negating the objective function.
Steps for Size Optimization
The Steps for optimization is described as follows.
• Make simplified 3D model leaf spring in SolidWorks.
• Make and save above model STEP format and importing in to HyperMesh.
• After importing model provide material properties as table 3.2.
• After providing material properties Make mesh.
• After providing mesh input boundary condition one end fixed and other end X and Y axis displacement is given. Loading conditions involves applying a load upper side at the centre of the bottom
• After boundary condition Make in analysis page optimization.
• In optimization constraints considered is stress and objective function mini- mization mass.
• After solving the model for size optimization and obtain reduction thickness as shown in fig 6.3.
Before optimization
See the Fig. 6.1 and 6.2 are the result of before optimization. The weight of the spring is 5.18 kg and thickness is 36 mm.
Figure 6.1: von misses stress in mono leaf spring using hyper mesh
Figure 6.2: Deflection in mono leaf spring using hyper mesh
After optimization
As shown in Fig. 6.3 is the result of After optimization. After five iteration weight in leaf spring reduced up to 5.01 kg and thickness 35 mm obtained. So 1 mm thickness reduction is Gained.
Figure 6.3: After Optimization mono leaf spring using hyper mesh
Residual Stress Calculation
Residual stress is defined as “the stress resident inside a component or structure after all applied forces have been removed”. Compressive residual stress acts by pushing the material together,
while tensile residual stress pulls the material apart.
Mathematically, compressive stress is negative and tensile stress is positive. Stresses can also be characterized as normal stresses that act perpendicular to the face of a material and shear
stresses that act parallel to the face of a material. There is a total of 6 independent stresses at any point inside a material represented by σijwhere i is the direction that the stress is acting
and j is the face the stress is acting on.
Residual stresses or locked-in stresses can be defined as those stresses existing within a body in the absence of external loading or thermal gradients. In other words residual stresses in a
structural material or component are those stresses which exist in the object without the application of any service or other external loads.
Residual stress is usually defined as the stress which remains in mechanical parts which are not subjected to any outside stresses. Residual stress exists in practically all rigid parts, whether
metallic or not (wood, polymer, glass,ceramic, etc). It is the result of the metallurgical and mechanical history of each point in the part and the part as a whole during its manufacture. It exists
at different levels, generally divided into three, depending on the scale on which the stress is observed.
Factors that cause residual stresses
Residual stresses can be present in any mechanical structure because of many causes. Residual stresses may be due to the technological process used to make the compo- nent. Manufacturing processes
are the most common causes of residual stress. Virtu- ally all manufacturing and fabricating processes such as casting, welding, machining, molding, heat treatment, plastic deformation during
bending, rolling or forging intro- duce residual stresses into the manufactured object. Residual stress could be caused by localized yielding of the material, because of a sharp notch or from certain
surface treatments like shot peening or surface hardening. Among the factors that are known to cause residual stresses are the development of deformation gradients in various sections of the piece
by the development of thermal gradients, volumetric changes arising during solidification or from solid state transformations, and from differences
in the coefficient of thermal expansion in pieces made from different materials. Thermal residual stresses are primarily due to differential expansion when a metal is heated or cooled. The two
factors that control this are thermal treatment (heating or cooling) and restraint. Both the thermal treatment and restraint of the component must be present to generate residual stresses.
When any object is formed through cold working, there is the possibility for the de- velopment of residual stresses.
A good common example of mechanically applied residual stresses is a bicycle wheel. A bicycle wheel is a very light and strong because of the way in which the com- ponents are stressed. The wire
spokes are radial aligned and tightening the spokes Makes tensile radial stresses. The spokes pull the rim inward, creating circumfer- ential compression stresses in the rim. Conversely, the spokes
pull the tubular hub outward. If the thin spokes were not under a proper tensile preloaded the thin wire spokes could not adequately support the load of the rider.
Residual stress Calculation in leaf spring
Residual stress calculation in different leaves of leaf spring has been described below. As shown in fig 7.1 one leaves of leaf spring and below calculation is done for single leaves spring.
For single leaf of spring, residual stress calculation method has been explained. Fig.7.1 is shows single leaf of spring.
CG of the leaves y = 28.80 mm
I = Moment of inertia about y-Axis
I =
= 2.04e6mm4 (7.1)
Figure 7.1: Single Leaf of the Spring
M =
= 2.162e6N∗mm (7.2)
σ = M ∗ Y
= 305.22MPa (7.3)
Max.ResidualStress= YieldStress−305.22 = 852.77Mpa (7.4)
Table 7.1 shows bending stress and residual stress for all 10 leaves of leaf spring.
Table 7.1: Residual stress calculation
y L M Bending stress
Leaves no. Residual stress (σmax(MPa)) Residual stress (σmin(MPa))
mm mm N.mm (σ (MPa))
1 46.78 1025 2.402e6 390.158 767.842 -767.842
2 28.80 922.5 2.042e6 305.22 852.77 -852.77
3 25.36 820 1.91e6 238.68 919.32 -919.32
4 20.25 717.5 1.68e6 166.76 991.24 -991.24
5 16.89 615 1.44e6 119.339 1038.661 -1038.66
6 14.17 512.5 1.20e6 83.43 1074.57 -1074.57
7 12.89 410 9.6e5 60.718 1097.282 -1097.28
8 11.10 307.5 7.2e5 39.214 1118.786 -1118.78
9 10.15 205 4.8e5 23.789 1134.211 -1134.21
10 9.05 102.5 2.4e5 10.647 1047.353 -1047.35
Conclusion and future Scope
The design and static structural analysis of steel leaf spring and laminated composite leaf spring has been carried out. Comparison has been made between laminated composite leaf spring with steel
leaf spring having same design and same load carrying capacity. From that 79.9 % mass reduction in composite material has been Gained for the same number of leaves. The fatigue life has been
calculated using analytically as well as using ANSYS for steel leaf spring. The size optimization has been carried out for further mass reduction of composite leaf spring. From that 90 % mass
reduction has been Gained using mono composite leaf spring compared to steel multi leaf spring.
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Statistics - Wikiwand
Statistics (from German: Statistik, orig. "description of a state, a country"^[1]) is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.^
[2] In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse
groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in
terms of the design of surveys and experiments.^[3]
The normal distribution, a very common probability density, is used extensively in inferential statistics.
Scatter plots and line charts are used in descriptive statistics to show the observed relationships between different variables, here using the Iris flower data set.
When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can
reasonably extend from the sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional
measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation.
Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, and inferential statistics,
which draw conclusions from data that are subject to random variation (e.g., observational errors, sampling variation).^[4] Descriptive statistics are most often concerned with two sets of properties
of a distribution (sample or population): central tendency (or location) seeks to characterize the distribution's central or typical value, while dispersion (or variability) characterizes the extent
to which members of the distribution depart from its center and each other. Inferences made using mathematical statistics employ the framework of probability theory, which deals with the analysis of
random phenomena.
A standard statistical procedure involves the collection of data leading to a test of the relationship between two statistical data sets, or a data set and synthetic data drawn from an idealized
model. A hypothesis is proposed for the statistical relationship between the two data sets, an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or
disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis,
two basic forms of error are recognized: Type I errors (null hypothesis is rejected when it is in fact true, giving a "false positive") and Type II errors (null hypothesis fails to be rejected when
it is in fact false, giving a "false negative"). Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null
Statistical measurement processes are also prone to error in regards to the data that they generate. Many of these errors are classified as random (noise) or systematic (bias), but other types of
errors (e.g., blunder, such as when an analyst reports incorrect units) can also occur. The presence of missing data or censoring may result in biased estimates and specific techniques have been
developed to address these problems.
Statistics is a mathematical body of science that pertains to the collection, analysis, interpretation or explanation, and presentation of data,^[5] or as a branch of mathematics.^[6] Some consider
statistics to be a distinct mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is generally concerned with the use of data in
the context of uncertainty and decision-making in the face of uncertainty.^[7]^[8]
In applying statistics to a problem, it is common practice to start with a population or process to be studied. Populations can be diverse topics, such as "all people living in a country" or "every
atom composing a crystal". Ideally, statisticians compile data about the entire population (an operation called a census). This may be organized by governmental statistical institutes. Descriptive
statistics can be used to summarize the population data. Numerical descriptors include mean and standard deviation for continuous data (like income), while frequency and percentage are more useful in
terms of describing categorical data (like education).
When a census is not feasible, a chosen subset of the population called a sample is studied. Once a sample that is representative of the population is determined, data is collected for the sample
members in an observational or experimental setting. Again, descriptive statistics can be used to summarize the sample data. However, drawing the sample contains an element of randomness; hence, the
numerical descriptors from the sample are also prone to uncertainty. To draw meaningful conclusions about the entire population, inferential statistics are needed. It uses patterns in the sample data
to draw inferences about the population represented while accounting for randomness. These inferences may take the form of answering yes/no questions about the data (hypothesis testing), estimating
numerical characteristics of the data (estimation), describing associations within the data (correlation), and modeling relationships within the data (for example, using regression analysis).
Inference can extend to the forecasting, prediction, and estimation of unobserved values either in or associated with the population being studied. It can include extrapolation and interpolation of
time series or spatial data, as well as data mining.
Bernoulli's was the first work that dealt with probability theory as currently understood.
Formal discussions on inference date back to the mathematicians and cryptographers of the Islamic Golden Age between the 8th and 13th centuries. Al-Khalil (717–786) wrote the Book of Cryptographic
Messages, which contains one of the first uses of permutations and combinations, to list all possible Arabic words with and without vowels.^[11] Al-Kindi's Manuscript on Deciphering Cryptographic
Messages gave a detailed description of how to use frequency analysis to decipher encrypted messages, providing an early example of statistical inference for decoding. Ibn Adlan (1187–1268) later
made an important contribution on the use of sample size in frequency analysis.^[11]
Although the term statistic was introduced by the Italian scholar Girolamo Ghilini in 1589 with reference to a collection of facts and information about a state, it was the German Gottfried Achenwall
in 1749 who started using the term as a collection of quantitative information, in the modern use for this science.^[12]^[13] The earliest writing containing statistics in Europe dates back to 1663,
with the publication of Natural and Political Observations upon the Bills of Mortality by John Graunt.^[14] Early applications of statistical thinking revolved around the needs of states to base
policy on demographic and economic data, hence its stat- etymology. The scope of the discipline of statistics broadened in the early 19th century to include the collection and analysis of data in
general. Today, statistics is widely employed in government, business, and natural and social sciences.
Carl Friedrich Gauss made major contributions to probabilistic methods leading to statistics.
The mathematical foundations of statistics developed from discussions concerning games of chance among mathematicians such as Gerolamo Cardano, Blaise Pascal, Pierre de Fermat, and Christiaan Huygens
. Although the idea of probability was already examined in ancient and medieval law and philosophy (such as the work of Juan Caramuel), probability theory as a mathematical discipline only took shape
at the very end of the 17th century, particularly in Jacob Bernoulli's posthumous work Ars Conjectandi.^[15] This was the first book where the realm of games of chance and the realm of the probable
(which concerned opinion, evidence, and argument) were combined and submitted to mathematical analysis.^[16] The method of least squares was first described by Adrien-Marie Legendre in 1805, though
Carl Friedrich Gauss presumably made use of it a decade earlier in 1795.^[17]
Karl Pearson, a founder of mathematical statistics
The modern field of statistics emerged in the late 19th and early 20th century in three stages.^[18] The first wave, at the turn of the century, was led by the work of Francis Galton and Karl Pearson
, who transformed statistics into a rigorous mathematical discipline used for analysis, not just in science, but in industry and politics as well. Galton's contributions included introducing the
concepts of standard deviation, correlation, regression analysis and the application of these methods to the study of the variety of human characteristics—height, weight and eyelash length among
others.^[19] Pearson developed the Pearson product-moment correlation coefficient, defined as a product-moment,^[20] the method of moments for the fitting of distributions to samples and the Pearson
distribution, among many other things.^[21] Galton and Pearson founded Biometrika as the first journal of mathematical statistics and biostatistics (then called biometry), and the latter founded the
world's first university statistics department at University College London.^[22]
The second wave of the 1910s and 20s was initiated by William Sealy Gosset, and reached its culmination in the insights of Ronald Fisher, who wrote the textbooks that were to define the academic
discipline in universities around the world. Fisher's most important publications were his 1918 seminal paper The Correlation between Relatives on the Supposition of Mendelian Inheritance (which was
the first to use the statistical term, variance), his classic 1925 work Statistical Methods for Research Workers and his 1935 The Design of Experiments,^[23]^[24]^[25] where he developed rigorous
design of experiments models. He originated the concepts of sufficiency, ancillary statistics, Fisher's linear discriminator and Fisher information.^[26] He also coined the term null hypothesis
during the Lady tasting tea experiment, which "is never proved or established, but is possibly disproved, in the course of experimentation".^[27]^[28] In his 1930 book The Genetical Theory of Natural
Selection, he applied statistics to various biological concepts such as Fisher's principle^[29] (which A. W. F. Edwards called "probably the most celebrated argument in evolutionary biology") and
Fisherian runaway,^[30]^[31]^[32]^[33]^[34]^[35] a concept in sexual selection about a positive feedback runaway effect found in evolution.
The final wave, which mainly saw the refinement and expansion of earlier developments, emerged from the collaborative work between Egon Pearson and Jerzy Neyman in the 1930s. They introduced the
concepts of "Type II" error, power of a test and confidence intervals. Jerzy Neyman in 1934 showed that stratified random sampling was in general a better method of estimation than purposive (quota)
Today, statistical methods are applied in all fields that involve decision making, for making accurate inferences from a collated body of data and for making decisions in the face of uncertainty
based on statistical methodology. The use of modern computers has expedited large-scale statistical computations and has also made possible new methods that are impractical to perform manually.
Statistics continues to be an area of active research, for example on the problem of how to analyze big data.^[37]
Data collection
When full census data cannot be collected, statisticians collect sample data by developing specific experiment designs and survey samples. Statistics itself also provides tools for prediction and
forecasting through statistical models.
To use a sample as a guide to an entire population, it is important that it truly represents the overall population. Representative sampling assures that inferences and conclusions can safely extend
from the sample to the population as a whole. A major problem lies in determining the extent that the sample chosen is actually representative. Statistics offers methods to estimate and correct for
any bias within the sample and data collection procedures. There are also methods of experimental design that can lessen these issues at the outset of a study, strengthening its capability to discern
truths about the population.
Sampling theory is part of the mathematical discipline of probability theory. Probability is used in mathematical statistics to study the sampling distributions of sample statistics and, more
generally, the properties of statistical procedures. The use of any statistical method is valid when the system or population under consideration satisfies the assumptions of the method. The
difference in point of view between classic probability theory and sampling theory is, roughly, that probability theory starts from the given parameters of a total population to deduce probabilities
that pertain to samples. Statistical inference, however, moves in the opposite direction—inductively inferring from samples to the parameters of a larger or total population.
Experimental and observational studies
A common goal for a statistical research project is to investigate causality, and in particular to draw a conclusion on the effect of changes in the values of predictors or independent variables on
dependent variables. There are two major types of causal statistical studies: experimental studies and observational studies. In both types of studies, the effect of differences of an independent
variable (or variables) on the behavior of the dependent variable are observed. The difference between the two types lies in how the study is actually conducted. Each can be very effective. An
experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements with different levels using the same procedure to determine
if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation. Instead, data are gathered and correlations between
predictors and response are investigated. While the tools of data analysis work best on data from randomized studies, they are also applied to other kinds of data—like natural experiments and
observational studies^[38]—for which a statistician would use a modified, more structured estimation method (e.g., difference in differences estimation and instrumental variables, among many others)
that produce consistent estimators.
The basic steps of a statistical experiment are:
1. Planning the research, including finding the number of replicates of the study, using the following information: preliminary estimates regarding the size of treatment effects, alternative
hypotheses, and the estimated experimental variability. Consideration of the selection of experimental subjects and the ethics of research is necessary. Statisticians recommend that experiments
compare (at least) one new treatment with a standard treatment or control, to allow an unbiased estimate of the difference in treatment effects.
2. Design of experiments, using blocking to reduce the influence of confounding variables, and randomized assignment of treatments to subjects to allow unbiased estimates of treatment effects and
experimental error. At this stage, the experimenters and statisticians write the experimental protocol that will guide the performance of the experiment and which specifies the primary analysis
of the experimental data.
3. Performing the experiment following the experimental protocol and analyzing the data following the experimental protocol.
4. Further examining the data set in secondary analyses, to suggest new hypotheses for future study.
5. Documenting and presenting the results of the study.
Experiments on human behavior have special concerns. The famous Hawthorne study examined changes to the working environment at the Hawthorne plant of the Western Electric Company. The researchers
were interested in determining whether increased illumination would increase the productivity of the assembly line workers. The researchers first measured the productivity in the plant, then modified
the illumination in an area of the plant and checked if the changes in illumination affected productivity. It turned out that productivity indeed improved (under the experimental conditions).
However, the study is heavily criticized today for errors in experimental procedures, specifically for the lack of a control group and blindness. The Hawthorne effect refers to finding that an
outcome (in this case, worker productivity) changed due to observation itself. Those in the Hawthorne study became more productive not because the lighting was changed but because they were being
Observational study
An example of an observational study is one that explores the association between smoking and lung cancer. This type of study typically uses a survey to collect observations about the area of
interest and then performs statistical analysis. In this case, the researchers would collect observations of both smokers and non-smokers, perhaps through a cohort study, and then look for the number
of cases of lung cancer in each group.^[40] A case-control study is another type of observational study in which people with and without the outcome of interest (e.g. lung cancer) are invited to
participate and their exposure histories are collected.
Types of data
Various attempts have been made to produce a taxonomy of levels of measurement. The psychophysicist Stanley Smith Stevens defined nominal, ordinal, interval, and ratio scales. Nominal measurements do
not have meaningful rank order among values, and permit any one-to-one (injective) transformation. Ordinal measurements have imprecise differences between consecutive values, but have a meaningful
order to those values, and permit any order-preserving transformation. Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case
with longitude and temperature measurements in Celsius or Fahrenheit), and permit any linear transformation. Ratio measurements have both a meaningful zero value and the distances between different
measurements defined, and permit any rescaling transformation.
Because variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, sometimes they are grouped together as categorical variables, whereas ratio and
interval measurements are grouped together as quantitative variables, which can be either discrete or continuous, due to their numerical nature. Such distinctions can often be loosely correlated with
data type in computer science, in that dichotomous categorical variables may be represented with the Boolean data type, polytomous categorical variables with arbitrarily assigned integers in the
integral data type, and continuous variables with the real data type involving floating-point arithmetic. But the mapping of computer science data types to statistical data types depends on which
categorization of the latter is being implemented.
Other categorizations have been proposed. For example, Mosteller and Tukey (1977)^[41] distinguished grades, ranks, counted fractions, counts, amounts, and balances. Nelder (1990)^[42] described
continuous counts, continuous ratios, count ratios, and categorical modes of data. (See also: Chrisman (1998),^[43] van den Berg (1991).^[44])
The issue of whether or not it is appropriate to apply different kinds of statistical methods to data obtained from different kinds of measurement procedures is complicated by issues concerning the
transformation of variables and the precise interpretation of research questions. "The relationship between the data and what they describe merely reflects the fact that certain kinds of statistical
statements may have truth values which are not invariant under some transformations. Whether or not a transformation is sensible to contemplate depends on the question one is trying to answer."^[45]^
This section
needs additional citations for verification
(December 2020)
Descriptive statistics
A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features of a collection of information,^[46] while descriptive statistics in the
mass noun sense is the process of using and analyzing those statistics. Descriptive statistics is distinguished from inferential statistics (or inductive statistics), in that descriptive statistics
aims to summarize a sample, rather than use the data to learn about the population that the sample of data is thought to represent.^[47]
Inferential statistics
Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution.^[48] Inferential statistical analysis infers properties of a population,
for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive
statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population.^[49]
Terminology and theory of inferential statistics
Statistics, estimators and pivotal quantities
Consider independent identically distributed (IID) random variables with a given probability distribution: standard statistical inference and estimation theory defines a random sample as the random
vector given by the column vector of these IID variables.^[50] The population being examined is described by a probability distribution that may have unknown parameters.
A statistic is a random variable that is a function of the random sample, but not a function of unknown parameters. The probability distribution of the statistic, though, may have unknown parameters.
Consider now a function of the unknown parameter: an estimator is a statistic used to estimate such function. Commonly used estimators include sample mean, unbiased sample variance and sample
A random variable that is a function of the random sample and of the unknown parameter, but whose probability distribution does not depend on the unknown parameter is called a pivotal quantity or
pivot. Widely used pivots include the z-score, the chi square statistic and Student's t-value.
Between two estimators of a given parameter, the one with lower mean squared error is said to be more efficient. Furthermore, an estimator is said to be unbiased if its expected value is equal to the
true value of the unknown parameter being estimated, and asymptotically unbiased if its expected value converges at the limit to the true value of such parameter.
Other desirable properties for estimators include: UMVUE estimators that have the lowest variance for all possible values of the parameter to be estimated (this is usually an easier property to
verify than efficiency) and consistent estimators which converges in probability to the true value of such parameter.
This still leaves the question of how to obtain estimators in a given situation and carry the computation, several methods have been proposed: the method of moments, the maximum likelihood method,
the least squares method and the more recent method of estimating equations.
Null hypothesis and alternative hypothesis
Interpretation of statistical information can often involve the development of a null hypothesis which is usually (but not necessarily) that no relationship exists among variables or that no change
occurred over time.^[51]^[52]
The best illustration for a novice is the predicament encountered by a criminal trial. The null hypothesis, H[0], asserts that the defendant is innocent, whereas the alternative hypothesis, H[1],
asserts that the defendant is guilty. The indictment comes because of suspicion of the guilt. The H[0] (status quo) stands in opposition to H[1] and is maintained unless H[1] is supported by evidence
"beyond a reasonable doubt". However, "failure to reject H[0]" in this case does not imply innocence, but merely that the evidence was insufficient to convict. So the jury does not necessarily accept
H[0] but fails to reject H[0]. While one can not "prove" a null hypothesis, one can test how close it is to being true with a power test, which tests for type II errors.
What statisticians call an alternative hypothesis is simply a hypothesis that contradicts the null hypothesis.
Working from a null hypothesis, two broad categories of error are recognized:
• Type I errors where the null hypothesis is falsely rejected, giving a "false positive".
• Type II errors where the null hypothesis fails to be rejected and an actual difference between populations is missed, giving a "false negative".
Standard deviation refers to the extent to which individual observations in a sample differ from a central value, such as the sample or population mean, while Standard error refers to an estimate of
difference between sample mean and population mean.
A statistical error is the amount by which an observation differs from its expected value. A residual is the amount an observation differs from the value the estimator of the expected value assumes
on a given sample (also called prediction).
Mean squared error is used for obtaining efficient estimators, a widely used class of estimators. Root mean square error is simply the square root of mean squared error.
A least squares fit: in red the points to be fitted, in blue the fitted line.
Many statistical methods seek to minimize the residual sum of squares, and these are called "methods of least squares" in contrast to Least absolute deviations. The latter gives equal weight to small
and big errors, while the former gives more weight to large errors. Residual sum of squares is also differentiable, which provides a handy property for doing regression. Least squares applied to
linear regression is called ordinary least squares method and least squares applied to nonlinear regression is called non-linear least squares. Also in a linear regression model the non deterministic
part of the model is called error term, disturbance or more simply noise. Both linear regression and non-linear regression are addressed in polynomial least squares, which also describes the variance
in a prediction of the dependent variable (y axis) as a function of the independent variable (x axis) and the deviations (errors, noise, disturbances) from the estimated (fitted) curve.
Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random (noise) or systematic (bias), but other types of errors (e.g., blunder,
such as when an analyst reports incorrect units) can also be important. The presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address
these problems.^[53]
Interval estimation
Confidence intervals: the red line is true value for the mean in this example, the blue lines are random confidence intervals for 100 realizations.
Most studies only sample part of a population, so results do not fully represent the whole population. Any estimates obtained from the sample only approximate the population value. Confidence
intervals allow statisticians to express how closely the sample estimate matches the true value in the whole population. Often they are expressed as 95% confidence intervals. Formally, a 95%
confidence interval for a value is a range where, if the sampling and analysis were repeated under the same conditions (yielding a different dataset), the interval would include the true (population)
value in 95% of all possible cases. This does not imply that the probability that the true value is in the confidence interval is 95%. From the frequentist perspective, such a claim does not even
make sense, as the true value is not a random variable. Either the true value is or is not within the given interval. However, it is true that, before any data are sampled and given a plan for how to
construct the confidence interval, the probability is 95% that the yet-to-be-calculated interval will cover the true value: at this point, the limits of the interval are yet-to-be-observed random
variables. One approach that does yield an interval that can be interpreted as having a given probability of containing the true value is to use a credible interval from Bayesian statistics: this
approach depends on a different way of interpreting what is meant by "probability", that is as a Bayesian probability.
In principle confidence intervals can be symmetrical or asymmetrical. An interval can be asymmetrical because it works as lower or upper bound for a parameter (left-sided interval or right sided
interval), but it can also be asymmetrical because the two sided interval is built violating symmetry around the estimate. Sometimes the bounds for a confidence interval are reached asymptotically
and these are used to approximate the true bounds.
Statistics rarely give a simple Yes/No type answer to the question under analysis. Interpretation often comes down to the level of statistical significance applied to the numbers and often refers to
the probability of a value accurately rejecting the null hypothesis (sometimes referred to as the p-value).
In this graph the black line is probability distribution for the test statistic, the critical region is the set of values to the right of the observed data point (observed value of the test
statistic) and the p-value is represented by the green area.
The standard approach^[50] is to test a null hypothesis against an alternative hypothesis. A critical region is the set of values of the estimator that leads to refuting the null hypothesis. The
probability of type I error is therefore the probability that the estimator belongs to the critical region given that null hypothesis is true (statistical significance) and the probability of type II
error is the probability that the estimator does not belong to the critical region given that the alternative hypothesis is true. The statistical power of a test is the probability that it correctly
rejects the null hypothesis when the null hypothesis is false.
Referring to statistical significance does not necessarily mean that the overall result is significant in real world terms. For example, in a large study of a drug it may be shown that the drug has a
statistically significant but very small beneficial effect, such that the drug is unlikely to help the patient noticeably.
Although in principle the acceptable level of statistical significance may be subject to debate, the significance level is the largest p-value that allows the test to reject the null hypothesis. This
test is logically equivalent to saying that the p-value is the probability, assuming the null hypothesis is true, of observing a result at least as extreme as the test statistic. Therefore, the
smaller the significance level, the lower the probability of committing type I error.
Some problems are usually associated with this framework (See criticism of hypothesis testing):
• A difference that is highly statistically significant can still be of no practical significance, but it is possible to properly formulate tests to account for this. One response involves going
beyond reporting only the significance level to include the p-value when reporting whether a hypothesis is rejected or accepted. The p-value, however, does not indicate the size or importance of
the observed effect and can also seem to exaggerate the importance of minor differences in large studies. A better and increasingly common approach is to report confidence intervals. Although
these are produced from the same calculations as those of hypothesis tests or p-values, they describe both the size of the effect and the uncertainty surrounding it.
• Fallacy of the transposed conditional, aka prosecutor's fallacy: criticisms arise because the hypothesis testing approach forces one hypothesis (the null hypothesis) to be favored, since what is
being evaluated is the probability of the observed result given the null hypothesis and not probability of the null hypothesis given the observed result. An alternative to this approach is
offered by Bayesian inference, although it requires establishing a prior probability.^[54]
• Rejecting the null hypothesis does not automatically prove the alternative hypothesis.
• As everything in inferential statistics it relies on sample size, and therefore under fat tails p-values may be seriously mis-computed.
Some well-known statistical tests and procedures are:
Exploratory data analysis
Exploratory data analysis (EDA) is an approach to analyzing data sets to summarize their main characteristics, often with visual methods. A statistical model can be used or not, but primarily EDA is
for seeing what the data can tell us beyond the formal modeling or hypothesis testing task.
Misuse of statistics can produce subtle but serious errors in description and interpretation—subtle in the sense that even experienced professionals make such errors, and serious in the sense that
they can lead to devastating decision errors. For instance, social policy, medical practice, and the reliability of structures like bridges all rely on the proper use of statistics.
Even when statistical techniques are correctly applied, the results can be difficult to interpret for those lacking expertise. The statistical significance of a trend in the data—which measures the
extent to which a trend could be caused by random variation in the sample—may or may not agree with an intuitive sense of its significance. The set of basic statistical skills (and skepticism) that
people need to deal with information in their everyday lives properly is referred to as statistical literacy.
There is a general perception that statistical knowledge is all-too-frequently intentionally misused by finding ways to interpret only the data that are favorable to the presenter.^[55] A mistrust
and misunderstanding of statistics is associated with the quotation, "There are three kinds of lies: lies, damned lies, and statistics". Misuse of statistics can be both inadvertent and intentional,
and the book How to Lie with Statistics,^[55] by Darrell Huff, outlines a range of considerations. In an attempt to shed light on the use and misuse of statistics, reviews of statistical techniques
used in particular fields are conducted (e.g. Warne, Lazo, Ramos, and Ritter (2012)).^[56]
Ways to avoid misuse of statistics include using proper diagrams and avoiding bias.^[57] Misuse can occur when conclusions are overgeneralized and claimed to be representative of more than they
really are, often by either deliberately or unconsciously overlooking sampling bias.^[58] Bar graphs are arguably the easiest diagrams to use and understand, and they can be made either by hand or
with simple computer programs.^[57] Most people do not look for bias or errors, so they are not noticed. Thus, people may often believe that something is true even if it is not well represented.^[58]
To make data gathered from statistics believable and accurate, the sample taken must be representative of the whole.^[59] According to Huff, "The dependability of a sample can be destroyed by
[bias]... allow yourself some degree of skepticism."^[60]
To assist in the understanding of statistics Huff proposed a series of questions to be asked in each case:^[55]
• Who says so? (Does he/she have an axe to grind?)
• How does he/she know? (Does he/she have the resources to know the facts?)
• What's missing? (Does he/she give us a complete picture?)
• Did someone change the subject? (Does he/she offer us the right answer to the wrong problem?)
• Does it make sense? (Is his/her conclusion logical and consistent with what we already know?)
Misinterpretation: correlation
The confounding variable problem: X and Y may be correlated, not because there is causal relationship between them, but because both depend on a third variable Z. Z is called a confounding factor.
The concept of correlation is particularly noteworthy for the potential confusion it can cause. Statistical analysis of a data set often reveals that two variables (properties) of the population
under consideration tend to vary together, as if they were connected. For example, a study of annual income that also looks at age of death, might find that poor people tend to have shorter lives
than affluent people. The two variables are said to be correlated; however, they may or may not be the cause of one another. The correlation phenomena could be caused by a third, previously
unconsidered phenomenon, called a lurking variable or confounding variable. For this reason, there is no way to immediately infer the existence of a causal relationship between the two variables.
Applied statistics, theoretical statistics and mathematical statistics
Applied statistics, sometimes referred to as Statistical science,^[61] comprises descriptive statistics and the application of inferential statistics.^[62]^[63] Theoretical statistics concerns the
logical arguments underlying justification of approaches to statistical inference, as well as encompassing mathematical statistics. Mathematical statistics includes not only the manipulation of
probability distributions necessary for deriving results related to methods of estimation and inference, but also various aspects of computational statistics and the design of experiments.
Statistical consultants can help organizations and companies that do not have in-house expertise relevant to their particular questions.
Machine learning and data mining
Machine learning models are statistical and probabilistic models that capture patterns in the data through use of computational algorithms.
Statistics in academia
Statistics is applicable to a wide variety of academic disciplines, including natural and social sciences, government, and business. Business statistics applies statistical methods in econometrics,
auditing and production and operations, including services improvement and marketing research.^[64] A study of two journals in tropical biology found that the 12 most frequent statistical tests are:
analysis of variance (ANOVA), chi-squared test, Student's t-test, linear regression, Pearson's correlation coefficient, Mann-Whitney U test, Kruskal-Wallis test, Shannon's diversity index, Tukey's
range test, cluster analysis, Spearman's rank correlation coefficient and principal component analysis.^[65]
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fundamental statistical courses for undergraduate students focus on correct test selection, results interpretation, and use of free statistics software.^[65]
Statistical computing
gretl, an example of an open source statistical package
The rapid and sustained increases in computing power starting from the second half of the 20th century have had a substantial impact on the practice of statistical science. Early statistical models
were almost always from the class of linear models, but powerful computers, coupled with suitable numerical algorithms, caused an increased interest in nonlinear models (such as neural networks) as
well as the creation of new types, such as generalized linear models and multilevel models.
Increased computing power has also led to the growing popularity of computationally intensive methods based on resampling, such as permutation tests and the bootstrap, while techniques such as Gibbs
sampling have made use of Bayesian models more feasible. The computer revolution has implications for the future of statistics with a new emphasis on "experimental" and "empirical" statistics. A
large number of both general and special purpose statistical software are now available. Examples of available software capable of complex statistical computation include programs such as Mathematica
, SAS, SPSS, and R.
Business statistics
In business, "statistics" is a widely used management- and decision support tool. It is particularly applied in financial management, marketing management, and production, services and operations
management.^[67]^[68] Statistics is also heavily used in management accounting and auditing. The discipline of Management Science formalizes the use of statistics, and other mathematics, in business.
(Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships.)
A typical "Business Statistics" course is intended for business majors, and covers^[69] descriptive statistics (collection, description, analysis, and summary of data), probability (typically the
binomial and normal distributions), test of hypotheses and confidence intervals, linear regression, and correlation; (follow-on) courses may include forecasting, time series, decision trees, multiple
linear regression, and other topics from business analytics more generally. Professional certification programs, such as the CFA, often include topics in statistics.
Statistics applied to mathematics or the arts
Traditionally, statistics was concerned with drawing inferences using a semi-standardized methodology that was "required learning" in most sciences. This tradition has changed with the use of
statistics in non-inferential contexts. What was once considered a dry subject, taken in many fields as a degree-requirement, is now viewed enthusiastically. Initially derided by some mathematical
purists, it is now considered essential methodology in certain areas.
Statistical techniques are used in a wide range of types of scientific and social research, including: biostatistics, computational biology, computational sociology, network biology, social science,
sociology and social research. Some fields of inquiry use applied statistics so extensively that they have specialized terminology. These disciplines include:
In addition, there are particular types of statistical analysis that have also developed their own specialised terminology and methodology:
Statistics form a key basis tool in business and manufacturing as well. It is used to understand measurement systems variability, control processes (as in statistical process control or SPC), for
summarizing data, and to make data-driven decisions.
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Lectures on Differential Topology
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theorem, bordism-characteristic numbers, and the Pontryagin-Thom construction. This book is suitable for beginning master's and doctoral students who have completed an undergraduate mathematics
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Do contractors use Pythagorean Theorem?
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Compute pH before/after addition of NaOH? | HIX Tutor
Compute pH before/after addition of NaOH?
In the titration of 50.0 mL of 0.100 M $\setminus \beta$-hydroxybutyric acid, $H {C}_{4} {H}_{7} {O}_{3}$, with 0.100 M NaOH, compute pH before addition of NaOH, and after the addition of 25.00 mL
and 50.00 mL of NaOH. $p {K}_{a}$ for $H {C}_{4} {H}_{7} {O}_{3}$ is 4.39.
Answer 1
!! EXTREMLEY LONG ANSWER !!
For the sake of simplicity, I'll use #beta#-#"HA"# to denote the acid and #beta#-#"A"^(-)# to denote the conjugate base.
Now, the pH of the solution before the addition of strong base can be calculated by using the fact that the weak acid only partially ionizes in a #1:1# mole ratio to produce hydronium cations and #
beta#-hydroxybutyrate anions
#beta"-HA"_ ((aq)) + "H"_ 2"O"_ ((l)) rightleftharpoons "H"_ 3"O"_ ((aq))^(+) + beta"-A"_ ((aq))^(-)#
If you take #x# to be the equilibrium concentration of the hydronium cations and of the #beta#-hydroxybutyrate anions, you can say that the equilibrium concentration of the acid will be
#[beta"-HA"] = [beta"-HA"]_0 - x#
By definition, the acid dissociation constant will be
#K_a = ( [beta"-A"^(-)] * ["H"_3"O"^(+)])/([beta"-HA"])#
In your case, this will be equal to
#10^(-"p"K_a) = (x * x)/(0.100 - x) = x^2/(0.100 - x)#
#K_a = 10^(-"p"K_a) = 10^(-4.39) = 4.07 * 10^(-5)#
is very small compared to the initial concentration of the acid, you can use the approximation
#0.100 -x ~~ 0.100#
#4.07 * 10^(-5) = x^2/0.100#
Solve for #x# to get
#x = sqrt(0.100 * 4.07 * 10^(-5)) = 2.02 * 10^(-3)#
#["H"_3"O"^(+)] = 2.02 * 10^(-3)# #"M"#
The pH of the initial solution
#color(blue)(ul(color(black)("pH" = - log(["H"_3"O"^(+)])))#
#"pH" = - log(2.02 * 10^(-3)) = color(darkgreen)(ul(color(black)(2.695)))#
Now, #beta#-hydroxybutyric acid will react with the hydroxide anions provided by the strong base in a #1:1# mole ratio to produce #beta#-hydroxybutyrate, its conjugate base, and water.
#beta"-HA"_ ((aq)) + "OH"_ ((aq))^(-) -> beta"-A"_ ((aq))^(-) + "H"_ 2"O"_ ((l))#
The initial solution contains
#50.0 color(red)(cancel(color(black)("mL"))) * (1color(red)(cancel(color(black)("L"))))/(10^3color(red)(cancel(color(black)("mL")))) * ("0.100 moles" color(white)(.)beta"-HA")/(1color(red)(cancel
(color(black)("L")))) = "0.00500 moles"# #beta"-HA"#
The first solution of sodium hydroxide contains
#25.00 color(red)(cancel(color(black)("mL"))) * (1color(red)(cancel(color(black)("L"))))/(10^3color(red)(cancel(color(black)("mL")))) * ("0.100 moles OH"^(-))/(1color(red)(cancel(color(black)("L"))))
= "0.00250 moles OH"^(-)#
So, after the first sample of sodium hydroxide is added to the initial solution, the moles of hydroxide anions will be completely consumed. You will be left with
#n_ ("OH"^(-)) = 0 -># completely consumed
#n_ (beta"-HA") = "0.00500 moles" - "0.00250 moles" = "0.00250 moles"# #beta"-HA"#
#n_ (beta"-A"^(-)) = "0 moles" + "0.00250 moles" = "0.00250 moles"# #beta"-A"^(-)#
The new volume of the solution will be
#V_"solution" = "50.0 mL" + "25.00 mL" = "75.0 mL"#
Now, notice that the resulting solution contains equal numbers of moles of weak acid and of conjugate base. This means that you're in the buffer region, i.e. the resulting solution contains a buffer.
This means that the pH of the solution will be equal to the #"p"K_a# of the weak acid because the concentrations of the weak acid and of the conjugate base are equal.
#[beta"-HA"] = [beta"-A"^(-)]#
This is known as the half-equivalence point. Consequently, you will have
#"pH" = "p"K_a + log ( (color(red)(cancel(color(black)([beta"-A"^(-)]))))/(color(red)(cancel(color(Black)([beta"-HA"]))))) -># the Henderson - Hasselbalch equation
#color(darkgreen)(ul(color(black)("pH" = 4.39)))#
Finally, you're adding an additional #"25.00 mL"# of strong base to get the total volume of strong base added to #"50.00 mL"#.
At this point, it should be obvious that all the moles of acid and all the moles of hydroxide anions will be consumed by the reaction #-># this is known as the equivalence point of the titration. You
will be left with
#n_ ("OH"^(-)) = 0 -># completely consumed
#n_ (beta"-HA") = "0.00250 moles" - "0.00250 moles" = 0 -># completely consumed
#n_ (beta"-A"^(-)) = "0.00250 moles" + "0.00250 moles" = "0.00500 moles"# #beta"-A"^(-)#
The new volume of the solution will be
#V_"total" = "75.0 mL" + "25.00 mL" = "100.0 mL"#
The concentration of the conjugate base will be equal to
#[beta"-A"^(-)] = "0.00500 moles"/(100.0 * 10^(-3)"mL") = "0.0500 M"#
The conjugate base will only partially ionize to produce #beta#-hydroxybutyric acid and hydroxide anions in #1:1# mole ratios
#beta"-A"_ ((aq))^(-) + "H"_ 2"O"_ ((l)) rightleftharpoons beta"-HA"_ ((aq)) + "OH" _((aq))^(-)#
An aqueous solution at room temperature has
#color(blue)(ul(color(black)(K_a * K_b = 10^(-14))))#
#K_b = 10^(-14)/(4.07 * 10^(-5)) = 2.46 * 10^(-10)#
If you take #x# to be the equilibrium concentrations of #beta#-hydroxybutyric acid and hydroxide anions, you will have
#[beta"-A"^(-)] = 0.0500 - x#
By definition, you know that
#K_b = ([beta"-HA"] * ["OH"^(-)])/([beta"-A"^(-)])#
#K_b = (x * x)/(0.0500 - x) = x^2/(0.0500 - x)#
Once again, use the approximation
#0.0500 - x ~~ 0.0500#
#2.46 * 10^(-10) = x^2/0.0500#
Solve for #x# to find
#x = sqrt(0.0500 * 2.46 * 10^(-10)) = 3.51 * 10^(-6)#
This means that the resulting solution has
#["OH"^(-)] = 3.51 * 10^(-6)# #"M"#
Consequently, you will have
#["H"_3"O"^(+)] = 10^(-14)/(3.51 * 10^(-6)) = 2.85 * 10^(-9)# #"M"#
which means that the pH of the solution is
#"pH" = - log(2.85 * 10^(-9)) = color(darkgreen)(ul(color(black)(8.545)))#
I'll leave the values rounded to three decimal palces, the number of sig figs you have for the concentrations of the two solutions.
Now, does the result make sense?
You are titrating a weak acid with a strong base, so at the equivalence point, the solution will only contain the conjugate base of the acid.
So even without doing any calculations, you should be able to say that at equivalence point, #"pH" > 7#.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Instantaneous Current in a Capacitor | Electrical Academia
Instantaneous Current in a Capacitor
If we connect a capacitor across a sine-wave voltage source, as in Figure 1, Kirchhoff’s voltage law requires the voltage across the capacitor to be exactly the same as the applied voltage at every
The voltage across a capacitor can change only if the capacitor charges or discharges. Consequently, the capacitor in Figure 1 must charge and discharge in such a manner that the voltage across it is
a sine wave equal to the applied voltage at every instant. Since q = Cv,
By definition, $i=\frac{dq}{dt}$. Therefore,
\[\begin{matrix}i=C\frac{dv}{dt} & {} & \left( 1 \right) \\\end{matrix}\]
Since capacitance depends on such physical factors as the area of the plates and the dielectric constant of the material between the plates, the capacitance of a given circuit does not depend on the
elapsed time. Since we can treat C in Equation 1 as a constant, this equation shows that the instantaneous current in Figure 1 is directly proportional to the rate at which the voltage across the
capacitor is changing.
Figure 1 Capacitance in an alternating-current circuit
The blue sine curve in Figure 2 represents the instantaneous voltage across the capacitor. This curve shows that the maximum voltage across the capacitor occurs π/2 radians after the maximum rate of
change of voltage. At the exact moment when the voltage across the capacitor is greatest, the voltage is neither rising nor falling. Therefore, the instantaneous current must be zero at this instant.
The maximum rate of change of voltage occurs when the voltage sine curve is steepest. At this instant the voltage is zero, indicating that the capacitor has just finished discharging its stored
charge and is about to start building up an opposite charge. Therefore, the instantaneous current has its maximum positive value at the instant when the voltage across the capacitor changes from a
negative polarity to a positive polarity.
Similarly, the current reaches its maximum negative value just as the voltage changes from a positive to a negative polarity.
Figure 2 Instantaneous current in a capacitor
The instantaneous current must have the sine-wave shape shown by the red curve in Figure 2 in order for the voltage across the capacitor to match the applied voltage at every instant.
The instantaneous current is at its maximum positive value at the instant that the voltage across the capacitor is just starting to increase from zero. When the voltage across the capacitance has
reached its positive peak π/2 rad later, the instantaneous current has fallen back to zero. Therefore,
For a sine-wave voltage to be developed across a capacitor, the current through it must be a sine wave that leads the instantaneous voltage by π/2 radians.
Therefore, the instantaneous current in the circuit of Figure 1 is
\[\begin{matrix}{{i}_{c}}={{\operatorname{I}}_{m}}\sin \left( \omega t+\frac{\pi }{2} \right) & {} & \left( 2 \right) \\\end{matrix}\]
Where the phase angle ωt + π /2 is measured in radians.
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Investing for Retirement: The Defined Contribution Challenge
Investing for Retirement:
The Defined Contribution Challenge
Ben Inker and Martin Tarlie
The retirement landscape has changed. Defined benefit plans, the historical workhorse of the retirement system, had the advantage of access to corporate profitability. In the event that financial
asset returns fell short of design expectations, this access mitigated the impact on workers’ retirement. But, as defined benefit plans have given way to defined contribution (DC) plans, the burden
being placed on financial returns in satisfying retirement needs has increased.
Target date funds are rapidly becoming the workhorse for DC plans. These funds have grown substantially in recent years, partly as a result of automatic enrollment made possible by the Pension
Protection Act of 2006. By and large, current target date funds resemble the old investment advisor adage that stock weight should be about 110 minus a person’s age. While this satisfies the
common-sense intuition that, all things being equal, weight in stocks should go down as a person ages, there are a number of problems with this approach. In this paper we focus on two in particular.
First, the standard solution is inflexible: all things are rarely equal. To address this shortcoming, we introduce a framework based on a common-sense definition of risk: not having enough wealth in
retirement. The goal is not to put investors into yachts, but rather to increase the odds that they have the appropriate level of resources in retirement. Viewing risk this way leads to highly
customizable solutions that under certain equilibrium assumptions are consistent with current solutions but offer far more flexibility and insight. Second, the standard solutions do not recognize
that expected returns vary over time. We show that dynamic asset allocation – moving your assets – is an essential part of achieving retirement goals.
This paper is divided into two parts. In Part I we frame the question and explain how our framework leads to flexible, customizable solutions. In Part II we demonstrate the importance of dynamic
Asking the Right Question
The most common method for building multi-asset portfolios is based on Modern Portfolio Theory: maximize return for a given level of risk, where risk is return volatility. From the perspective of the
retirement problem, and perhaps more generally, this approach is inadequate. The main problem is that it is asking the wrong question: given a level of risk, i.e., return volatility, which is the
portfolio that maximizes the expected return?
This is the wrong question because it focuses on returns, not wealth. But returns are only the means to an end, the end being the wealth that is to be consumed throughout retirement. Not only is it
the wrong question, but it presupposes the investor has a good reason for choosing a particular level of return volatility. So two investors faced with similar circumstances in terms of current
wealth, future income and savings, and future consumption needs may have very different portfolios simply because their attitude toward return volatility differs.
A better approach is to focus on what really matters: wealth. An investor saving for retirement has fairly well-defined needs, both in terms of how much wealth he needs to accumulate and his pattern
of consumption in retirement. An investor’s portfolio should be driven primarily by his needs and circumstances – what does he need and when does he need it? It should not be a function of his
personality. The financial risk to an investor saving for retirement is very simple: it is not having enough wealth. So the more appropriate question is: which is the portfolio that minimizes the
expected shortfall of wealth relative to what’s needed?
This definition of risk is central to our framework. All other things being equal, a person who is more risk averse should save more or consume less. In contrast, the standard approach gives bad
advice. Putting the more risk-averse individual in a less volatile portfolio, one that from a Modern Portfolio Theory (MPT) perspective is considered less risky, without making any compensating
savings or consumption adjustments, actually increases the wealth risk to that individual in that he is less likely to achieve his wealth needs. A virtue of optimizing based on minimizing shortfall
of wealth is that it is highly customizable and easily able to handle the question of how to invest for a more risk-averse person who expresses his increased risk aversion through, for example, a
higher savings rate. This flexibility is a consequence of asking the right question.
Returns vs. Wealth
To better understand the difference between MPT – a return-focused approach – and the wealth-focused approach that we advocate, it is helpful to compare the distribution of returns with the
distribution of wealth. To a fairly good approximation, returns are normally distributed, as illustrated in Chart 1. While there is plenty of empirical evidence that, at least over shorter horizons,
this is not quite true for many asset classes, our problem with the assumption for portfolio construction purposes here is not particularly that returns are “fat-tailed” or may be slightly skewed in
one direction or another. It is rather that, even if returns are normally distributed, the wealth those returns lead to is not.
Chart 1 shows a normal distribution of annual returns for an asset with a 5% return per annum and a 14% annualized volatility. In a normal distribution, the average is the same as both the median and
the mode, the most likely return. Whether you are actually concerned with the average of all of the potential returns, the most likely return, or the return that is in the middle of the distribution
is irrelevant, because they are all the same.
As returns compoundinto wealth, however, Chart 1 is no longer relevant. Chart 2 shows the distribution of ending wealth after investing $1 for 40 years in an asset with the normal return distribution
shown above. This distribution is not normal, but log-normal. The shape of the log-normal distribution is profoundly different than that of the normal distribution. The expected value, or mean, of
this distribution is the purple vertical line. If you invest for 40 years in an asset with normally distributed returns averaging 5% per annum and an annualized standard deviation of 14%, the average
wealth outcome is about $11. The median outcome, however, is about $7, and the most likely outcome, the mode, is only $3.4.
Expected, or mean, values are dominated by the right tail of the distribution – those lucky 40-year periods in which returns happened to average well over 5% real. While those events are rare, they
have a big impact on the mean wealth. But for the purposes of saving for retirement, those outcomes are largely irrelevant.
If you happen to be lucky enough to have lived and saved during the right period when asset returns were high, it doesn’t much matter what your target date allocations were. You will wind up with
more than enough money to retire on. The more important part of the distribution is the left-hand side – those events when asset markets were not kind, and returns were hard to come by. Those are the
events where lifetime ruin, i.e., running out of money in retirement, is a real possibility.
We believe that the right way to build portfolios for retirement is to focus on how much wealth is needed and when it is needed, with a focus not on maximizing expected wealth, but on minimizing the
expected shortfall of wealth from what is needed in retirement.
The Retirement Problem
There are two obvious phases of the retirement problem – the accumulation phase, when workers are generating income and investing savings, and the consumption phase, when assets are spent.
In Chart 3 we show a simple diagram of the accumulation phase, generated using fairly standard industry assumptions. An employee starts out earning $43,000 at age 25, with income growing over time at
1.1% above inflation. The contribution rate, i.e., savings relative to income, starts at 5%, rising to 10% at retirement, and the employer match is 3% of income. This implies an average contribution
rate of 10.5%. Target wealth is 10 times final annual salary, in this case approximately $667,000. But given that cumulative savings total only about $200,000, it turns out that it will take an
average return of about 5.1% real per year to achieve the retirement wealth target.
In Chart 4 we illustrate the consumption phase. This chart assumes that the participant spends 50% of final salary every year in retirement, adjusted for inflation – spending of $33,383.( The
assumption of 50% of final salary is a standard one. Implicit is the assumption that Social Security payments will constitute another 30% so that total assumed replacement ratio is 80%.) This amounts
to spending a constant 5% of target wealth at retirement. The red line shows the importance of continuing to earn returns in retirement, as a 5% spending rate in the absence of returns consumes the
accumulated savings in 20 years. But we are assuming that the retiree lives 30 years beyond retirement. In order to afford this, the retiree needs to earn about 2.8% real per year during the
consumption phase.
Expected Shortfall
In Chart 5, we combine the accumulation and consumption phases into one graph. Because we define risk as not having enough money in retirement, our objective is to minimize expected shortfall of
wealth after age 65. This concept is illustrated in Chart 5 by the red area: the optimal portfolios minimize expected wealth in this red zone. Minimizing wealth in the red zone is equivalent to
focusing on the left side of the wealth distribution as discussed in the section "Returns vs. Wealth" above.
In Chart 5 the wealth target post-retirement is a solid line rather than the dashed line used in the accumulation phase. This highlights the fact that wealth prior to retirement has an indirect
influence on the objective function. Structurally, the objective is to minimize shortfall relative to the wealth target after age 65. Not including wealth prior to retirement in the objective
function means that the investor is more tolerant of wealth volatility prior to retirement, leading to portfolios, in equilibrium, that have more weight in stocks for younger investors.
Why is it important to envision the problem in this way? Simply put, it addresses the primary financial risk of not having enough wealth in retirement. Furthermore, if you concentrate on solving this
problem, "risk aversion" naturally falls out, rather than having to be guessed at or enquired about as required by MPT. A 25-year-old should invest aggressively because of her circumstances, not
because of her personality: there are 40 years until drawdowns really matter for consumption goals.
A 75-year-old should invest more conservatively because of needs and circumstances: near-term losses cannot necessarily be recovered from the nest egg as it is being consumed. And to go beyond a
static glide path and account for time-varying expected returns (see the section on Dynamic Allocation), it is essential to have a framework that naturally balances the changing trade-offs between
risk and return as investors age.
If we use fairly standard assumptions – 6% real returns for stocks and 2% real returns for bonds, with annualized volatilities of 18% and 5%, respectively, and a correlation of zero – and minimize
expected wealth shortfall (ESF) assuming that investors are on the wealth targets illustrated in Chart 5, we can map out the optimal weight in stocks for each age. The blue line in Chart 6 shows
these optimal stock weights. We call this a static ESF glide path because we generate these weights by minimizing the expected wealth shortfall assuming that the expected returns for stocks and bonds
are constant.
For comparison, we show two additional lines in Chart 6. The yellow line is based on the old investment advisor adage that the stock percentage should be about 110 minus a person’s age. The green
line in Chart 6 is a glide path used by a provider of target date funds. Relative to the ESF portfolio, it looks as if 110-Age is too conservative for almost all ages leading to retirement, and the
XYZ path is a little bit too conservative from ages 40 to 60. All three of these glide paths have roughly the same shape, reflecting the basic intuition that the weight in stocks should decline as
people age. By and large, the magnitude of the differences between the 110-Age path and the XYZ path are similar to the magnitude of the differences between the XYZ and static ESF paths.
So Why Bother With ESF?
ESF matters because minimizing expected shortfall of wealth provides a powerful conceptual framework that answers the right question in a customizablemanner. To illustrate this point, consider Chart
7 where, in addition to the three glide paths shown above, we add a fourth: ABC. This glide path, in red, comes from another provider of target date funds and while it follows the basic pattern that
the weight in stocks falls as people age, it is more conservative in that it aggressively reduces weight in stocks as people age.
So which of these four choices is better? Well, this question is actually incomplete. We don’t really know how XYZ and ABC were constructed. We know neither the assumptions about the plan
participants, i.e., "What do they need and when do they need it?", nor do we know what objective, if any, these glide paths satisfy. Furthermore, we also don’t know what assumptions were made about
asset returns; we will discuss this crucial issue in detail below.
But for the ESF glide path what we can say is that for a person who has circumstances and needs consistent with the assumptions articulated above regarding income, savings rates, and consumption, the
ESF glide path minimizes the expected shortfall of wealth, assuming at each age that the person is on wealth target and asset returns are constant. Because minimizing shortfall of wealth is such a
compelling common-sense objective, given this particular combination of “what and when,” we believe the ESF path is better.
This logic suggests that there may be particular combinations of "what and when" for which ABC, XYZ, or even 110-Age for that matter, are optimal in an expected shortfall sense. But there is simply
no way to know. A person shopping for glide paths simply has no basis for choosing among the various possibilities. This points to the power of the ESF framework: given needs and circumstances – what
do they need and when do they need it – we generate optimal portfolios. Crucially, the objective is clear and sensible: minimize, in expectation, how much wealth falls short of what is needed. Simply
put, asking the right question leads to solving the right problem. ESF optimization therefore provides a powerful basis for choosing portfolios over time.
To further illustrate this point about customization and how it relates to asking the right question, suppose a particularly risk-averse individual does not have the constitution to tolerate a large
equity exposure. Because our solution has no concept of a risk-aversion parameter, the answer, in contrast to that from MPT, is not to simply place this person in a portfolio with low volatility.
Doing this, and this alone, will actually increase the risk that he does not have enough wealth in retirement.
The investor has two other choices: (i) either reduce his wealth target, likely an undesirable choice; or (ii) increase his savings rate. In Chart 8, we show the ESF glide path, obtained by assuming,
as before, that expected returns for stocks and bonds are 6% and 2%, respectively, for an individual who chooses the second option and saves at twice the rate assumed in generating the ESF glide path
above. So instead of starting to save at 5% and slowly increasing to 10% at retirement, the investor starts at 10% and slowly increases to 20% at retirement. We see in this case a glide path that has
much less weight in stocks prior to retirement. But during retirement, when the effect of the lower savings rate no longer applies, the glide paths converge.
But the ESF approach is also vital for correcting a fatal weakness in both the static ESF and the XYZ paths. Both are operating under a bad assumption. They both assume that the expected returns are
constant over time, when in reality they are anything but. However, given the common-sense objective of minimizing shortfall of wealth in retirement, we have a framework that allows us to minimize
shortfall of wealth throughout retirement even when expected returns are time varying.
Dynamic Allocation: Move Your Assets!
The fact that returns are not constant is self-evidently true for bonds, where an investor holding to maturity has an expected return overwhelmingly driven by the yield of the bond when it was
purchased. (The only other issue to consider for that investor is the interest rate at which coupons will be reinvested.) Even if you are rolling your bond portfolio periodically (this might be the
more relevant point because all DC investors are getting fixed income exposure from a bond fund), starting valuation is the overwhelming driver of subsequent returns.
But valuation is every bit as important for stocks, and the valuations of stocks has varied hugely over time, as we can see in Chart 9, which shows the cyclically-adjusted P/E, or "Shiller P/E," for
the S&P500 over time.
The stock market has averaged about 16 times normalized earnings over the last 130 years, but there have been times when it has traded far above or below that level. We believe it is the height of
folly to assume that a market trading at 45 times normalized earnings, as the S&P 500 was in 2000, can achieve similar returns to one trading at 7 times, as it was in 1982, let alone the expected
returns of any reasonable glide path.
Stock valuations have been mean-reverting over time, and as a result, stock returns have a significant element of predictability to them. Valuation cannot tell us much about what returns will be over
a week or a month or a quarter, but over a period of years the importance of valuation steadily increases. This is illustrated in Chart 10 where we show the correlation between valuation and
subsequent stock market returns as the time horizon lengthens from 1 year to 20 years.
While the correlation of 20% between current valuations and future 1-year returns is not particularly high, the correlation rises steadily with return horizon. For example, starting valuations have
had a roughly 60% correlation with future 10-year returns. Given that the retirement savings problem is a 70-year problem, it is an ideal environment to take advantage of the long-term return
predictability that comes from valuations. But this means we need a way to answer the question of how a 45-year-old should react if stocks are not priced to deliver 6% over inflation, but 2%, or 1%,
or 7%. Mean variance optimization in its standard form is limited in that it is a single-period optimization, not to mention the problem we raised earlier that it is not clear how to choose the level
of return volatility, or equivalently, the level of risk aversion. But minimizing expected shortfall, the ESF approach, gives us a natural framework for answering this question.
Historical Simulations
It is perhaps most straightforward to understand this through an example using historical returns. Let us assume we have a worker who turned 55 in 1965, and to that point was on target for retirement
savings. The worker, however, had the misfortune of being in peak savings years during a period in which equity valuations were high and real bond yields were low. The period from the mid-1960s
through the 1970s represents some of the worst real returns for both stocks and bonds on record.
A static, or inflexible, glide path would ignore these lofty valuations, and ensure the investor had a higher weighting in equities in 1965, when valuations were very high, than in 1974, when
valuations were much lower. Chart 11 shows the static glide path and a fully dynamic one from 1965-2005. The fully dynamic stock weight is based on minimizing expected shortfall incorporating
time-varying expected returns.
At first glance, the dynamic flight path looks nonsensical. It shows no weight in stocks in 1965, when the participant is 55 years old and has another 10 years until retirement, but by 1974 the stock
weight rises to 90%, staying at a very high level until the early 1980s despite the fact that the participant is approaching 70 years old, and losses can be devastating. But if the goal is minimizing
expected shortfall of wealth in retirement, it can make sense to run an aggressive portfolio in retirement if the increase in expected returns is high enough. Furthermore, if the expected return to
stocks is actually lower than bonds due to high valuations, such as was the case in 1965, it is hard to see why owning stocks would help at all.
The difference in outcomes between the two strategies is stark, as we can see in Chart 12. The static strategy leaves the participant out of money by 1992, the cost of not taking into account the
changing valuations of the stock and bond markets over time, and consuming a constant real dollar amount equal to 5% of target wealth. The dynamic strategy, by contrast, allows the participant to
make up for earlier inadequate returns, and the money lasts for a full 30-year retirement even though consumption is high.
It is worth pointing out that this is not the best possible outcome for a participant from 1965. The allocations in these charts were not driven by any actual foreknowledge of subsequent returns, but
by very simple value-driven models for estimating expected future returns. Broadly, in our simplified study, which assumes annual rebalancing, as valuation multiples deviate from normal (e.g., a
Shiller P/E of 16), younger investors, who have more weight in stocks than older investors, respond more aggressively to these valuation changes. For example, as indicated in Chart 6, when valuations
are normal a 25-year-old will have 90% weightin stocks, but a 65-year-old will have around 40%. If the Shiller P/E rises to 19, the weight in stocks for the 25-year-old will fall to about 45%, a drop
of 45 percentage points, whereas the weight in stocks for the 65-year-old will fall to around 20%, a drop of about 20 percentage points. And, in fact, it turns out that from 1965 to 1975, the dynamic
allocation would not have helped relative to a static strategy.
Both stocks and bonds did poorly over this period so dynamic allocation would not have had much immediate impact. It wasn’t until the early 1980s that stocks reaped the benefits of the cheap
valuations thrown up by the 1973-74 bear market. But when these benefits accrued to stocks, the impact on a dynamic allocation strategy was profound.
This brings up two points. First, cheap valuations don’t guarantee that returns will follow particularly quickly: valuations mean revert slowly, typically reverting one-seventh of the way back to
normal every year, meaning that stocks can remain cheap or expensive for very long periods of time. Second, asset allocation in retirement can be at least as important as it is during the
accumulation phase.
If we go beyond this single example of starting in 1965 and use every possible starting period for which we have U.S. stock and bond market data, we can look at the probability of lifetime ruin given
the data we have since 1881. Chart 13 shows the amount of money a retiree would have at the age of 95, assuming he was on target at age 55 on the date shown on the horizontal axis. It’s worth
pointing out that while 1881 feels like a very long time ago, for the purposes of looking at a 70-year problem like retirement savings, it doesn’t even provide two non-overlapping time periods. In
the interest of maximizing the data we have available, we’ve shortened this to a 40-year problem by assuming the worker was exactly on target at age 55 when we start the analysis.
Both the standard glide path and the optimal static glide path leave plan participants running out of money before the age of 95 about half the time. The dynamic glide path is not perfect either,
with a few starting dates in the 1950s and 1960s leaving the participant short. However, the probability of running out of money is about one fifth as high, as we can see in Table 1.
The gap between a 13% and a 50% chance of lifetime ruin is a profound one. And even in the few cases where the dynamic glide path still led to ruin, the money lasted longer than in the static or
standard glide path. As a reminder, this is not due to prescience involved in the dynamic process – it is only a simple value methodology that takes advantage of the predictability of stock returns
due to mean reverting valuations. So it turns out that no matter when in history you happened to hit retirement, you would have been better off following a dynamic value-driven strategy.
Another way to look at that data is to look at the percent of all observations that have an ending wealth of more than a certain amount. Chart 14 shows the historical simulations in this manner: the
vertical axis represents the percent of all observations that have wealth larger than the value on the horizontal axis. As an example, for the dynamic ESF strategy shown in red, approximately 90% of
all observations have positive wealth at age 95.
Furthermore, at every probability on the vertical axis, the dynamic glide path has more ending wealth at 95 years old than either of the static paths. The average amount of wealth is much higher as
well, although this is almost an incidental effect of attempting to minimize the probability of a wealth shortfall.
Monte Carlo Simulations
The main advantage of showing the data this way is that it allows us to compare historical simulations, which suffer from the problem that we have far less data than we would ideally like, to a Monte
Carlo simulation of the stock and bond markets where we can create as many histories as we would like.
One of our goals in building the simulations is to ensure stock valuations are approximately as predictive as they have been historically. Chart 15 shows one way to look at this.
The blue line on this chart shows the same data we saw in Chart 10, demonstrating that by using a simple valuation method the predictability of the stock market increases with time horizon. The red
line shows the level of predictability between valuation and returns for our Monte Carlo simulation. We see that the red line is approximately on top of the blue – our simulation builds in
approximately the same power for valuation as we’ve seen historically. (The theoretical underpinning of our simulation methodology and its relation to return predictability and its dependence on time
horizon can be found in the paper “Discount rate dynamics and stock prices,” located at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2337155.)
Our other goal for the simulations is to have their valuation characteristics match those of history. This is illustrated in Chart 16, in which we show five simulations of price to trailing 10-year
real earnings (P/E10), along with the historical graph. The key point of this chart is that the size of the variations of our simulated price multiples match those of the historical series. In the
historical series we see single-digit multiples, e.g., around 1920 and 1980, as well as multiples in the 40s, e.g., around 2000. The simulated series naturally capture the same dynamics as the
historical series.
This helps comfort us that our Monte Carlo simulations are reasonable and, with much more data, we can get more robust results than even 130 years of data can give. Chart 17 shows the equivalent of
Chart 14 for the Monte Carlo simulations. Not surprisingly, the dotted lines, corresponding to the Monte Carlo simulations, are much smoother than the historical simulations. The slightly surprising
finding, however, is that history turns out to be an unfortunate special case for 55-year-olds who were on target prior to retirement, where probabilities of lifetime ruin have actually been higher
than one might expect. This is simply an unfortunate artifact of this particular, limited, run of history. The gap between dynamic and static glide paths remains, however, with the dynamic glide path
having a 5% chance of ruin while the static glide path is four times as high at approximately 20%.
The simulations here – both historical and Monte Carlo – are simplified ones. A value-driven technique has much more potential to improve results when there are more than two asset classes from which
to choose. Unfortunately, we don’t have long enough data histories for asset classes beyond U.S. stocks and bonds to run a decent historical analysis with a larger set of assets. On the other hand,
our simulations assume that the weightings of the retirement portfolio are flexible between zero and 90% for stocks and 10% to 100% for bonds, which is probably beyond what most plan sponsors are
willing to contemplate. However, the results are quite robust to changing this assumption in that any flexibility to move away from the fixed flight path will improve outcomes. For example, with a
range of +/- 20% the risk of lifetime ruin is reduced to 26% from 51% in the historical simulation and to 6% from 20% in the Monte Carlo simulation.
The structuring of the problem is also a bit oversimplified. Rather than deal with the uncertainty of longevity risk, we simply assumed that participants live 30 years in retirement. This longevity
risk should not be ignored, and insurance products such as deferred annuities can be a cost-effective way to reduce the longevity risk that exists for a single plan participant. We have also assumed
that investors don’t change either their savings or consumption decisions in response to the financial markets.
While we do expect people to adjust their behavior in response to changing circumstances, our focus here is on the role of an investment program, so holding behavior constant is crucial to assessing
this. Furthermore, the modeling of investor behavior as it relates to savings and consumption is a difficult and complicated problem that requires more study.
Finally, in this paper we illustrate the impact of dynamic allocation by focusing on a 55-year-old who is on target prior to retirement. The reason for this focus is, in part, because of the limited
amount of historical data. But the Monte Carlo simulations allow us to look at all ages. And although we do not present those results here for lack of space, the results for the 55-year-old are
representative of what we find for all ages: dynamic allocation is an essential tool.
This research brings up several useful points to consider in building retirement portfolios. First, asking the right question is extremely important. The retirement savings problem is complex,
involving two long phases, one of accumulation and the other of consumption. As such, it is essential to go beyond a brute-force approach in designing a glide path to deal with time-varying expected
returns and the changing sensitivities, based on needs, as participants age. A common-sense, holistic approach of minimizing expected shortfall of wealth in retirement enables the construction of
highly customizable portfolios as both investor needs and asset valuations change. The essential point is that because the objective is to minimize a shortfall of wealth, this framework is
customizable in a way that ensures that the resulting solution makes sense precisely because it is solving the right problem.
Second, while most plan sponsors focus on building their portfolios appropriately for their employees while they are working, portfolio decisions made during the retirement phase are every bit as
important as they are during the accumulation phase – and perhaps even more so. After all, the returns, good or bad, achieved by the retirement portfolio of a 28-year-old will have a limited impact
on the ultimate ability of the portfolio to support spending in retirement for the simple reason that they affect only a few years of relatively small contributions. The vast majority of the wealth
accumulated at 65 will have been generated off of the contributions made in later years. In retirement, however, the participant has already accumulated all the wealth he will ever have, and returns
on that portfolio will have a profound impact on what that person can ultimately spend. Unfortunately, there seems to be much less effort focused on the post-retirement side of things –
understandably so for plan sponsors, to be sure, but from a societal perspective it would be an excellent idea to find a way to ensure that post-retirementportfolios are managed with the same care as
they are pre-retirement.
And third, the risk of failure with the traditional glide paths and savings/spending assumptions seems to us to be disturbingly high. Increasing participant savings rates will be crucial to helping
ensure retirement success for today’s working population, but valuation-aware portfolios, as our research has shown, can make a huge difference in the probability of success. We believe plan sponsors
are ignoring an incredibly powerful tool to help their participants if they build their portfolios without taking asset class valuations into account. And given that workers have very few other
effective ways to save for their own retirement, the stakes seem too high to leave such a valuable tool on the bench.
Mr. Inker is co-head of GMO’s Asset Allocation team and a member of the GMO Board of Directors. He joined GMO in 1992 following the completion of his B.A. in Economics from Yale University. In his
years at GMO, Mr. Inker has served as an analyst for the Quantitative Equity and Asset Allocation
teams, as a portfolio manager of several equity and asset allocation portfolios, as co-head of International Quantitative Equities, and as CIO of Quantitative Developed Equities. He is a CFA
Dr. Tarlie is a quantitative researcher for GMO’s Global Equity team. Prior to joining GMO in 2007, Dr. Tarlie worked as an analyst for Breakwater Trading and at Marlin Capital Corporation as the
director of research. Dr. Tarlie earned his B.S. in Physics and Mathematics from the University of Michigan, his M.S. and Ph.D. in Physics from the University of Illinois at Urbana-Champaign, and his
MBA from the University of Chicago Graduate School of Business. He was also a Postdoctoral Research Fellow in Theoretical Physics at the James Franck Institute at the University of Chicago and is a
CFA charterholder.
Disclaimer: The views expressed are the views of Mr. Inker and Dr. Tarlie through the period ending April 2014 and are subject to change at any time based on market and other conditions. This is not
an offer or solicitation for the purchase or sale of any security. The article may contain some forward looking statements. There can be no guarantee that any forward looking statement will be
realized. GMO undertakes no obligation to publicly update forward looking statements, whether as a result of new information, future events or otherwise. Statements concerning financial market trends
are based on current market conditions, which will fluctuate. References to securities and/or issuers are for illustrative purposes only. References made to securities or issuers are not
representative of all of the securities purchased, sold or recommended for advisory clients, and it should not be assumed that the investment in the securities was or will be profitable. There is no
guarantee that these investment strategies will work under all market conditions, and each investor should evaluate the suitability of their investments for the long term, especially during periods
of downturns in the markets.
1. Data used in Chart 9 are from Robert Shiller’s website (http://www.econ.yale.edu/~shiller/data.htm). After 1926, the indexes are the S&P 500 and predecessors. Prior to 1926 the data are from
Cowles and associates. Monthly dividend and earnings data are computed from the S&P four-quarter totals for the quarter since 1926. Dividend and earnings data before 1926 are from Cowles and
associates (Common Stock Indexes, 2nd ed. [Bloomington, Ind.: Principia Press, 1939].
2. Data used in Charts 10-15 are from Robert Shiller’s website (http://www.econ.yale.edu/~shiller/data.htm). After 1926, the indexes are the S&P 500 and predecessors. Prior to 1926 the data are from
Cowles and associates. Monthly dividend and earnings data are computed from the S&P four-quarter totals for the quarter since 1926. Dividend and earnings data before 1926 are from Cowles and
associates (Common Stock Indexes, 2nd ed. [Bloomington, Ind.: Principia Press, 1939]. Analysis of this data is provided by GMO.
3.Data used in Chart 16 are from Robert Shiller’s website (http://www.econ.yale.edu/~shiller/data.htm). After 1926, the indexes are the S&P 500 and predecessors. Prior to 1926 the data are from
Cowles and associates. Monthly dividend and earnings data are computed from the S&P four-quarter totals for the quarter since 1926. Dividend and earnings data before 1926 are from Cowles and
associates (Common Stock Indexes, 2nd ed. [Bloomington, Ind.: Principia Press, 1939]. Simulations are provided by GMO.
4. Data used in Chart 17 are from Robert Shiller’s website (http://www.econ.yale.edu/~shiller/data.htm). After 1926, the indexes are the S&P 500 and predecessors. Prior to 1926 the data are from
Cowles and associates. Monthly dividend and earnings data are computed from the S&P four-quarter totals for the quarter since 1926. Dividend and earnings data before 1926 are from Cowles and
associates (Common Stock Indexes, 2nd ed. [Bloomington, Ind.: Principia Press, 1939]. Analysis of this data is provided by GMO.
© GMO
© GMO | {"url":"https://asset-sync.advisorperspectives.com/commentaries/2014/04/05/investing-for-retirement-the-defined-contribution-challenge?firm=gmo","timestamp":"2024-11-03T00:00:30Z","content_type":"text/html","content_length":"142653","record_id":"<urn:uuid:ae789615-3fea-4f1c-b5d4-39d56539428c>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00267.warc.gz"} |
How can I document formulas used on a sheet?
I have a fairly involved project management solution using multiple sheets, reports and dashboards, and want to document the inner workings of the sheets in the event we have to recreate them. The
SmartSheet backup system will back up the data but not the actual sheet itself leaving me to recreate all of them and remember the involved formulas for each column or cell. Is there a way to show
the formulas in a cell instead of the actual number value? In Excel there is a button in the "Formulas" tab to "Show Formulas" that lets you see the formulas for each cell instead of the numerical
value. I am looking for something similar so I can print that to PDF and save it as a record just in case. Is there a way to do this or something similar?
• Yes and no. No - there is not a way to just print it and have them show. Yes - you can alternitvely, go into every column, remove the = sign, then view the text of the formula only. Or
alternatively put: ="Your Formula" in quotations to just display text. Then you can print, then go in and undo whatever alternative you choose
Michelle Choate
Always happy to walk through any project you need help with!
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/132736/how-can-i-document-formulas-used-on-a-sheet","timestamp":"2024-11-04T01:59:26Z","content_type":"text/html","content_length":"424306","record_id":"<urn:uuid:1594dc3f-66a6-454f-ad5f-6b9c5c469d3b>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00022.warc.gz"} |
Riddle 8: Largest Product in A Series
The four adjacent digits in the 1000-digit number that have the greatest product are 9 × 9 × 8 × 9 = 5832.
Find the thirteen adjacent digits in the 1000-digit number that have the greatest product. What is the value of this product?
This task is less difficult than the previous ones, as it does not require any optimization.
Step 1. Converting the list
First of all, we received the list as a long string, which does not help us much: What we need is a list of digits, (7 3 1 6 ...). So the first thing to do is to chop the list into single characters
("7" "3" ...) and convert these back to integers with the format function, which we apply to all list elements with mapcar.
: (setq Num 7316717653133.... ) # 1000 digits number
-> 7316717653133...
: (setq L (mapcar format (chop Num)))
-> (7 3 1 6 7 1 7 6 5 3 1 3 3
Step 2. Multiplying the n'th elements
We are looking for the largest product of 13 adjacent digits. In other words, we should go over the list. First we multiply the digits 1 to 13, then digit 2 to 14, digit 3 to 15 and so on. We can get
the product of the first 13 numbers of a list with
(apply * (head 13 L))
Now in order to walk down the list, we can use mapcon (read here for more on mapping functions). mapcon takes a function and applies it consecutively to a list and all its CDRs.
mapcon is concatenating the return values, which means that they must be lists. Thus we can create an anonymous function which takes a list as argument and returns a (one-element) list.
: (mapcon
'((Lst) (list (apply * (head 13 Lst))))
-> (5000940 0 0 0 0 0 0 0 0 0 0 0 0 0 4199040 4898880 ... )
Since we are looking for the maximum value in the list, we could stop as soon as our list has less than 13 elements. But on the other hand, a continuous check might be more expensive then just
letting it run. Let's double-check what happens if our remaining list has less than 13 elements:
: (head 4 '(1 2))
-> (1 2)
Our list (1 2) only has two elements, so (head 4 '( 1 2)) only returns these two elements, and (apply * (head 4 '(1 2))) returns 2. We know that this can't be the maximum we are looking for, but it
also doesn't influence our result in any way. So let's just leave it like that.
Step 3: Finding the maximum value
Now we have a list of all consecutive products in our list. Now we want to find the maximum. We can do this with the maxi function (also described here). maxi takes a function as argument, but
actually we don't need to apply any function. For this purpose we can use prog:
(prog . prg) -> any Executes prg, and returns the result of the last expression.
(prog L) just returns the list L, and (maxi prog L) returns the maximum value of a list.
So let's bring it all together. We define a function largest_prod that takes two arguments, Num for the 1000-digit number and I for the number of digits in our consecutive product. Our finished
program looks like this:
(de largest_prod (Num I)
(maxi prog
'((L) (list (apply * (head I L))))
(mapcar format (chop Num)) ) ) )
You can find the source code of the finished program here. | {"url":"https://picolisp-explored.com/riddle-8-largest-product-in-a-series","timestamp":"2024-11-12T09:40:03Z","content_type":"text/html","content_length":"131498","record_id":"<urn:uuid:fbfb21a9-c895-4485-8494-6136720d1b52>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00731.warc.gz"} |
Let's do simple calculations | Introduction to Rstats (statistical analysis/scientific calculation)
Let's do a simple calculation. Rstats can be calculated using the following operators.
Four arithmetic operations
symbol meaning
+ addition
- - subtraction
* multiplication
/ division
The symbols for addition, subtraction, multiplication, and division are the same as in R language.
Exponentiation is done using "**" according to Perl's arithmetic rules. Please note that it is different from "^" in R language.
symbol meaning
** ** Exponentiation
The remainder is done using "%" according to Perl's arithmetic rules. Please note that it is different from "%%" in R language.
symbol meaning
%% Surplus
Integer quotient
Rstats does not provide operators for calculating integer quotients. Please note that the R language "% /%" is not available. To find the integer quotient, use the trunc function. After performing
the division, the integer part is taken out.
Here is an example of the calculation. In Rstats, the calculation using vectors is the basis, so we will create it using the c_ function. The c_ function is a function for creating a vector. Even if
it is one numerical value, it is calculated as a vector with one element.
use strict;
use warnings;
use Rstats;
my $x = (c_ (1) + c_ (2) - c_ (3) * c_ (4))/c_ (5) ** c_ (6);
print $x;
This is the output result. The calculation result is output.
[1] -0.000576
Using c_ () at any time can be a bit annoying, but it's necessary because you're implementing the R language on top of Perl. Remember that when you do calculations in Rstats, you always use vectors
to do the calculations.
For future examples, I will write from the line under "use Rstats " above for convenience, so please supplement if necessary.
Rstats provides many functions for route calculation and logarithmic calculation. For example, to calculate the route, use the "sqrt function".
my $x = r->sqrt(c_ (2));
print $x;
Output result.
[1] 1.4142135623731
The mathematical functions provided by Rstats are introduced below.
symbol meaning
sqrt Calculation of √
abs Absolute value
exp exp The bottom of the natural logarithm
expm1 Calculate exp(x) -1 more accurately when the absolute value of x is much less than 1
log Natural logarithm
log10 Common logarithm (base 10 logarithm)
log2 Logarithm with base 2
sin Sine/sine function
cos Cosine/cosine function
tan Tangent tangent function
asin Inverse function of sin
acos Inverse function of cos
atan Inverse function of tan
sinh Hyperbolic sign
cosh Hyperbolic cosine
tanh Hyperbolic tangent
asinh Inverse function of sinh
acosh Inverse function of cosh
atanh Inverse function of tanh
logb Same as log
log1px (not implemented) Calculate log(1 + x) more accurately when the absolute value of x is (much) less than 1
gamma (not implemented) Gamma function
lgamma (not implemented) Same as log(gamma (x))
ceiling Minimum integer greater than or equal to the argument
floor Maximum integer less than or equal to the argument: the so-called Gaussian symbol
trunc Find the integer part
round Rounding
signif (x, a) not implemented yet Round x to a digit with a valid digit (unlike sign that determines positive/0/negative)
Introduction to Rstats (statistical analysis/scientific calculation)
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Online Help In MATLAB with Solutions
Online Help in MATLAB is really hard to get it done if you feel strangled at any of your research area the drop a message to phdtopic.com we will give you on time support. Along with extensive
algorithmic procedures, suitable datasets, and certain parameter scenarios, multi-objective optimization employing genetic algorithms (GA) in MATLAB can be applied. If you are in any level we will
guide you by online… Do you need a google meet support now? .Yes we ae ready drop us a mail to chat with our experts. We suggest an entire instruction with an instance utilizing a basic
multi-objective optimization issue:
Algorithm Procedures
1. Initialize Population:
• An initial population of solutions has to be produced in a random manner.
2. Evaluate Fitness:
• On the basis of the objective functions, we plan to assess the fitness of every individual in the population.
3. Selection:
• Through the utilization of a technique like roulette wheel selection or tournament selection, our team aims to choose suitable parents for reproduction.
4. Crossover:
• As a means to produce offspring from chosen parents, it is significant to carry out crossover.
5. Mutation:
• To sustain genetic variation, we focus on implementing mutation to the offspring.
6. Evaluation:
• Generally, the fitness of the offspring should be assessed.
7. Survivor Selection:
• According to the fitness and variety measure, our team plans to choose individuals for the next generation.
8. Termination:
• Till the completion measure is attained such as a maximum number of generations, we intend to iterate procedures 3-7.
Instance: Optimizing Two Objective Functions
We assume improving two objective function:
1. Focus on reducing f1(x)=x12+x22f_1(x) = x_1^2 + x_2^2f1(x)=x12+x22
2. It is appreciable to decrease f2(x)=(x1−1)2+x22f_2(x) = (x_1 – 1)^2 + x_2^2f2(x)=(x1−1)2+x22
MATLAB Code Implementation
Define the Objective Functions
function f = objectiveFunctions(x)
% First objective function
f1 = x(1)^2 + x(2)^2;
% Second objective function
f2 = (x(1) – 1)^2 + x(2)^2;
% Combine the objectives into a column vector
f = [f1; f2];
Set Up the Genetic Algorithm
% Number of variables
nvars = 2;
% Lower and upper bounds
lb = [-5 -5];
ub = [5 5];
% Options for the genetic algorithm
options = optimoptions(‘gamultiobj’, …
‘PopulationSize’, 100, …
‘Generations’, 200, …
‘CrossoverFraction’, 0.8, …
‘MutationFcn’, @mutationadaptfeasible, …
‘SelectionFcn’, @selectiontournament, …
‘Display’, ‘iter’, …
‘PlotFcn’, {@gaplotpareto, @gaplotscorediversity});
% Run the genetic algorithm
[x, fval, exitflag, output, population, scores] = gamultiobj(@objectiveFunctions, nvars, [], [], [], [], lb, ub, options);
% Display results
disp(‘Pareto solutions:’)
disp(‘Objective function values at the Pareto solutions:’)
Plot the Pareto Front
% Plot the Pareto front
plot(fval(:,1), fval(:,2), ‘bo’);
xlabel(‘Objective 1’);
ylabel(‘Objective 2’);
title(‘Pareto Front’);
grid on;
Since the objective functions are described in a mathematical manner, the certain dataset is not needed for this instance. Before the process of executing the genetic algorithm, we must load and
preprocess our data properly, if our issue encompasses actual world data.
Parameter Details
1. PopulationSize: The number of individuals in the population are defined in this parameter. Typically, extensive populations enhance computational expense but are capable of offering more genetic
2. Generations: The algorithm executes, up to the required number of generations (loops). Multiple generation expands the computation duration, but it enhances the findings effectively.
3. CrossoverFraction: This parameter indicates the fraction of the population which is produced by crossover. Generally, investigation is enhanced by greater values.
4. MutationFcn: As a means to mutate offspring, MutationFcn is utilized. For sustaining variety, various mutation functions could be employed.
5. SelectionFcn: For reproduction, the function which is deployed to choose parents is defined here as SelectionFcn. For numerous issues, tournament selection is considered as efficient.
6. Display: The range of output depicted in the command window can be regulated by this parameter.
7. PlotFcn: In order to plot the evolution of the method, this function is employed. Generally, gaplotscorediversity is capable of plotting the variety of solutions, and gaplotpareto plots the
Pareto front in an effective manner.
online matlab project services
The process of interpreting the policies of the methods and employing efficient toolbox support of MATLAB are encompassed in applying deep learning (DL), AI, machine learning (ML) algorithms in
MATLAB. Together with instances in MATLAB, we suggest an instruction to few generally employed AI, ML, and DL algorithms:
Artificial Intelligence (AI)
1. Rule-Based Systems
To extract outcome or activities, Rule-based models employ a collection of “if-then” rules. These might be executed by utilizing conditional statements, even though they are not a particular MATLAB
function output = ruleBasedSystem(input)
if input > 0
output = ‘Positive’;
elseif input < 0
output = ‘Negative’;
output = ‘Zero’;
Machine Learning (ML)
1. Linear Regression
The connection among a dependent variable and one or more independent variables are designed in the linear regression.
% Sample data
X = [1 2 3 4 5]’;
Y = [2 4 6 8 10]’;
% Fit linear regression model
model = fitlm(X, Y);
% Display model summary
2. Logistic Regression
Typically, to resolve binary classification issues, we have to employ logistic regression.
% Sample data
X = [1 2 3 4 5]’;
Y = [0 0 0 1 1]’;
% Fit logistic regression model
model = fitglm(X, Y, ‘Distribution’, ‘binomial’);
% Display model summary
3. Decision Trees
For regression as well as classification missions, it is appreciable to utilize decision trees.
% Sample data
X = [1 2 3 4 5]’;
Y = [0 0 0 1 1]’;
% Fit decision tree
tree = fitctree(X, Y);
% View tree
view(tree, ‘Mode’, ‘graph’)
4. Support Vector Machines (SVM)
To carry out classification and regression missions in an effective manner, SVMs are employed.
% Sample data
X = [1 2 3 4 5]’;
Y = [0 0 0 1 1]’;
% Fit SVM model
svmModel = fitcsvm(X, Y);
% Display model
5. K-Nearest Neighbors (KNN)
Mainly, for missions of classification and regression, our team aims to utilize KNN.
% Sample data
X = [1 2 3 4 5]’;
Y = [0 0 0 1 1]’;
% Fit KNN model
knnModel = fitcknn(X, Y);
% Display model
6. K-Means Clustering
To perform unsupervised learning missions, it is appreciable to make use of K-Means clustering.
% Sample data
X = [randn(50,2)+ones(50,2); randn(50,2)-ones(50,2)];
% Perform K-means clustering
[idx, C] = kmeans(X, 2);
% Plot clusters
gscatter(X(:,1), X(:,2), idx, ‘bg’);
hold on;
plot(C(:,1), C(:,2), ‘kx’);
title(‘K-Means Clustering’);
hold off;
Deep Learning (DL)
1. Artificial Neural Network (ANN)
The layers of interconnected nodes are included in ANNs.
% Load sample data
[XTrain, YTrain] = digitTrain4DArrayData;
% Define neural network architecture
layers = [
imageInputLayer([28 28 1])
% Set training options
options = trainingOptions(‘sgdm’, ‘MaxEpochs’, 5, ‘MiniBatchSize’, 64);
% Train the neural network
net = trainNetwork(XTrain, YTrain, layers, options);
2. Convolutional Neural Network (CNN)
As a means to carry out image processing missions, we plan to utilize CNNs.
% Load sample data
[XTrain, YTrain] = digitTrain4DArrayData;
% Define CNN architecture
layers = [
imageInputLayer([28 28 1])
convolution2dLayer(3, 8, ‘Padding’, ‘same’)
maxPooling2dLayer(2, ‘Stride’, 2)
% Set training options
options = trainingOptions(‘sgdm’, ‘MaxEpochs’, 5, ‘MiniBatchSize’, 64);
% Train the CNN
net = trainNetwork(XTrain, YTrain, layers, options);
3. Recurrent Neural Network (RNN)
Typically, for sequential data missions, RNNs are employed.
% Load sample sequential data
[XTrain, YTrain] = japaneseVowelsTrainData;
% Define RNN architecture
layers = [
lstmLayer(100, ‘OutputMode’, ‘last’)
% Set training options
options = trainingOptions(‘adam’, ‘MaxEpochs’, 20, ‘MiniBatchSize’, 64);
% Train the RNN
net = trainNetwork(XTrain, YTrain, layers, options);
Parameter Tuning and Datasets
Hyperparameter Tuning
For hyperparameter optimization, it is beneficial to employ in-built functions of MATLAB.
% Define the model
Mdl = fitcsvm(X, Y, ‘OptimizeHyperparameters’, ‘auto’, …
‘HyperparameterOptimizationOptions’, struct(‘AcquisitionFunctionName’, ‘expected-improvement-plus’));
• Sample Data: Generally, different sample datasets such as digitTrain4DArrayData, fisheriris, etc are offered by MATLAB.
• Loading Custom Data: With the support of importdata, readtable, etc, we can load conventional datasets.
% Load custom data from a CSV file
data = readtable(‘data.csv’);
% Separate features and labels
X = data(:, 1:end-1);
Y = data(:, end):
Through this article, we have provided a thorough direction with an instance employing a basic multi-objective optimization issue. Also, including instances in MATLAB, an instruction to few usually
employed DL, AI, and ML methods are suggested by us in an explicit manner. | {"url":"https://phdtopic.com/online-help-in-matlab/","timestamp":"2024-11-05T22:05:04Z","content_type":"text/html","content_length":"77036","record_id":"<urn:uuid:f2396c73-22aa-4c5c-91f5-75bee91a811b>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00733.warc.gz"} |
An Etymological Dictionary of Astronomy and Astrophysics
Balmer limit
حد ِبالمر
hadd-e Bâlmer
Fr.: limite de Balmer
The wavelength in the blue end of the → Balmer series, at 3646 Å, near which the separation between successive lines decreases and approaches a → continuum.
→ Balmer; → limit.
canonical upper limit
حد ِزبرین ِجرم
hadd-e zabarin-e jerm
Fr.: limite supériure canonique
A physical upper mass limit near 150 Msun assumed for the stellar → initial mass function (Kroupa et al. 2012, arXiv:1112.3340).
→ canonical; → upper; → limit.
central limit theorem
فربین ِحد ِمرکزی
farbin-e hadd-e markazi
Fr.: théorème central limite
A statement about the characteristics of the sampling distribution of means of → random samples from a given → statistical population. For any set of independent, identically distributed random
variables, X[1], X[2],..., X[n], with a → mean μ and → variance σ^2, the distribution of the means is equal to the mean of the population from which the samples were drawn. Moreover, if the original
population has a → normal distribution, the sampling distribution of means will also be normal. If the original population is not normally distributed, the sampling distribution of means will
increasingly approximate a normal distribution as sample size increases.
→ central; → limit; → theorem.
Chandrasekhar limit
حدِ چاندراسکهار
hadd-e Chandrasekhar (#)
Fr.: limite de Chandrasekhar
A limiting mass of about 1.44 Solar masses that the theory predicts a non-rotating → white dwarf can attain without collapsing to become a → neutron star or a → black hole. Over this → critical mass,
the degeneracy pressure will be unable to bear the load of the bulk mass.
Named after Subrahmayan Chandrasekhar (1910-1995), Indian-born American astrophysicist who, with William A. Fowler, won the 1983 Nobel Prize for Physics for his research on white dwarfs; → limit.
co-rotational limit (CoRol)
حد ِهم-چرخشی
hadd-e ham-carxeši
Fr.: limite co-rotationnelle
For any rotating planetary body, a thermal limit beyond which the → rotational velocity at the equator intersects the → Keplerian orbital velocity. Beyond this corotation limit, a hot planetary body
forms a structure, called a → synestia, with a corotating inner region connected to a disk-like outer region. Beyond this limit a body cannot have a single → angular velocity. It can instead exhibit
a range of morphologies with disk-like outer regions. The (CoRoL is a function that depends upon the composition, thermal state, → angular momentum and mass of a body (Simon J. Lock nd Sarah T.
Stewart, 2017, arXiv:1705.07858v1).
→ co-; → rotational; → limit.
confusion limit
حد ِپشش
hadd-e pašeš
Fr.: limite de confusion
The → fluctuations of the → background → sky brightness below which astronomical → sources cannot be → detected individually. The confusion limit is reached when the density of sources brighter than
the → root mean square → noise becomes high enough within the area of the resolution element.
→ confusion; → limit.
کرانمند به پراش
karânmand bé parâš
Fr.: limité par la diffraction
The quality of an → optical system that is capable of producing images with angular resolution as small as the theoretical limit of the → Airy disk.
→ diffraction; limited, adj. of → limit.
Karânmand "bounded, limited," from karân→ boundary + -mand possession suffix; parâš→ diffraction.
Eddington limit
حد ِادینگتون
hadd-e Eddington (#)
Fr.: limite d'Eddington
The theoretical upper limit of → luminosity at which the → radiation pressure of a light-emitting body would exceed the body's → gravitational attraction. A star emitting radiation at greater than
the Eddington limit would break up. The Eddington luminosity for a non-rotating star is expressed as: L[Edd] = 4πGMm[p]cσ[T]^-1, where G is the → gravitational constant, M the star mass, m[p] the →
proton mass, c the → speed of light, and σ[T] the → Thomson cross section. It can also be written as L[Edd] = 4πGMcκ^-1, where κ is the → opacity. In terms of solar mass, the Eddington limit can be
expressed by: L[Edd] = 1.26 × 10^38 (M/Msun) erg s^-1. See also → rotational Eddington limit.
Named after Arthur Stanley Eddington (1882-1944), prominent British astrophysicist; → limit.
elastic limit
حد ِکشایند
hadd-e kešâyand
Fr.: limite d'élasticité, ~ élastique
The smallest → stress beyond which a → solid body can no longer return to its original shape. The material ceases to obey → Hooke's law. Also called → yield point.
→ elastic; → limit.
Greisen-Zatsepin-Kuzmin limit (GZK)
حد ِگریسن-زاتسپین-کوزمین
hadd-e Greisen-Zatsepin-Kuzmin
Fr.: limite de Greisen-Zatsepin-Kuzmin
A theoretical limit of approximately 6 × 10^19 → electron-volts for the energy of → cosmic rays above which they would lose energy in their interaction with the → cosmic microwave radiation
background photons. Cosmic ray protons with these energies produce → pions on blackbody photons via the Δ resonance according to: γ[CMB] + p → p + π^0, or γ[CMB] + p → n + π^+, thereby losing a large
fraction of their energy. These interactions would reduce the energy of the cosmic rays to below the GZK limit. Due to this phenomenon, → Ultra-high-energy cosmic rays are absorbed within about 50
Named after Kenneth Greisen (1966), Physical Review Letters 16, 748 and Georgiy Zatsepin & Vadim Kuzmin (1966), Journal of Experimental and Theoretical Physics Letters 4, 78; → limit.
Humphreys-Davidson limit
حد ِهمفریز-دیویدسون
hadd-e Humphreys-Davidson
Fr.: limite de Humphreys-Davidson
An empirical upper → luminosity boundary in the → H-R diagram. It consists of two sections, a sloping part and a horizontal part. The sloping part, which decreases with decreasing → effective
temperature, corresponds roughly to the → Eddington limit. The horizontal part is the temperature-independent upper luminosity limit for late-type → hypergiants. It is thought that → massive stars
above the Humphreys-Davidson limit encounter an → instability, possibly due to the opacity-modified Eddington limit, and experience high → mass loss episodes which prevent their evolution to cooler
temperatures. → Luminous Blue Variable stars are examples of this high mass loss phase.
Named after Roberta M. Humphreys and Kris Davidson, who first dealt with this limit (1979, ApJ 232, 409); → limit.
hadd (#)
Fr.: limite
1) General: The final, utmost, or furthest → boundary or → point as to extent, amount, continuance, procedure, etc.
2a) Math.: Of a → sequence, a → number which is approached ever more closely, but never reached, by the successive terms of a convergent infinite sequence.
2b) Of a → variable, a constant C which has the property with respect to some variable V that, as the variable approaches C in value (according to some formula), the numerical difference (C - V)
between the constant and the variable diminishes toward 0 but is always greater than 0.
From O.Fr. limite "a boundary," from L. limitem (nom. limes) "a boundary, embankment between fields, border," related to limen "threshold."
Loan from Ar. Hadd "limit, term."
Fr.: limité
Confined within limits; restricted or circumscribed.
Adj. of → limit.
limiting magnitude
برز ِحد
borz-e hadd
Fr.: magnitude limite
The faintest magnitude reachable by an instrument.
→ limit; → magnitude.
lunar ecliptic limit
حد ِهورپهی ِماه
hadd-e hurpehi-ye mâh
Fr.: limite écliptique de la Lune
The farthest distance from a → lunar orbit node within which, if the Moon happens to be at full, a lunar eclipse may occur. The lunar ecliptic limit extends about 12° on each side of the node.
→ lunar; → ecliptic; → limit.
Lyman limit
حد ِلایمن
hadd-e Lyman
Fr.: limite de Lyman
The short-wavelength end of the hydrogen Lyman series, at 912 Å. Also called → Lyman continuum. It corresponds to the energy (13.6 eV) required for an electron in the hydrogen ground state to jump
completely out of the atom, leaving the atom ionized.
→ Lyman; → limit.
magnitude-limited survey
بردید با برز ِحدمند
bardid bâ borz-e haddmand
Fr.: relevé limité en magnitude
A survey in which the observed objects are bighter than a given → apparent magnitude.
→ magnitude; → limited; → volume.
Newtonian limit
حد ِنیوتنی
hadd-e Newtoni
Fr.: limite newtonienne
The limit attained by → general relativity when velocities are very smaller than the → speed of light or gravitational fields are weak. This limit corresponds to the transition between general
relativity and the → Newtonian mechanics. See also → Newtonian approximation.
→ Newtonian; → limit.
Oort limit
حدِ اورت
hadd-e Oort
Fr.: limite de Oort
1) The upper limit for the density of all matter in the plane of the Galaxy near the Sun's locality, as calculated from the velocities and distribution of stars in relation to the gravitational field
of the Galactic disk. The value is 0.14 solar masses per cubic parsec, or 9.5 x 10^-24 g cm^3.
2) The outer boundary of the → Oort cloud. The current estimate is about 100,000 → astronomical units from the Sun, which is approximately one third of the distance to the nearest star → Alpha
→ Oort cloud; → limit.
Oppenheimer-Volkoff limit
حدِ ا ُپنهایمر-وُلکوف
hadd-e Oppenheimer-Volkoff
Fr.: limite d'Oppenheimer-Volkoff
The upper bound to the mass of a → neutron star, the mass beyond which the pressure of neutron → degenerate matter is not capable of preventing the → gravitational collapse which will lead to the
formation of a → black hole. Modern estimates range from approximately 1.5 to 3.0 → solar masses. The uncertainty in the value reflects the fact that the → equation of state for → overdense matter is
not well-known.
Oppenheimer, J.R., Volkoff, G.M., 1939, Physical Review 55, 374. Named after Robert Oppenheimer (1904-1967), an American theoretical physicist, and George Volkoff (1914-2000), a Canadian physicist,
who first calculated this limit. Oppenheimer is widely known for his role as the scientific director of the Manhattan Project, the World War II effort to develop the first nuclear weapons at the
secret Los Alamos laboratory in New Mexico; → limit. | {"url":"https://dictionary.obspm.fr/?showAll=1&formSearchTextfield=limit","timestamp":"2024-11-13T01:07:35Z","content_type":"text/html","content_length":"37771","record_id":"<urn:uuid:19b6de8e-de7a-4fbf-85c1-f473cb2ee33d>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00272.warc.gz"} |
triangle angle calculator
Written by
Published: 20 Jan 2021
Right Triangle Trig Calculator Fill in two values and press Calculate. 2. The base angle α is equal to 180° minus … a 90°-angle). Use the following formula to solve the base angle: α = 180° – β 2. Of
course, our calculator solves triangles from any combinations of main and derived properties such as area, … ′. Free online tool for calculating the common formulae for circles, triangles and more .
S um = A + B + C. 4. round A. S cale = 1. tan(theta) = y/x = 3/4 Here is how the Side a of a triangle calculation can be explained with given input values … The usual way of identifying a triangle is
by first putting a capital letter on each vertex (or corner). Congruent Triangles. Home / Mathematics / Trigonometric functions (Deg) Calculates the three angles and area of a triangle given three
sides. How to calculate Side a of a triangle using this online calculator? I want to calculate each of unknown sides. Our area of triangle calculator supports the basic formula, these four rules, and
the hypothenuse and the length of one of the other sides rule for right-angled triangles only. \(\normalsize Triangle\hspace{20px} using\ Heron's\ formula\\. A triangle has one side length of 8cm and
an adjacent angle of 45.5. if the area of the triangle is 18.54cm, calculate the length of the other side that encloses the 45.5 angle Thanks Eugene Brennan (author) from Ireland on May 13, 2020: (It
is the edge opposite to the right angle and is c in this case.) Example. Solve triangles by entering two sides and one angle, two angles and one side or three sides to find remaining values as used
in trigonometry. sin (B) = b/c, cos (B) = a/c, tan (B) = b/a. In a triangle, all interior angles … A triangle is determined by 3 of the 6 free values, with at least one side. The ² button shows
second solution if one exists (ambiguous case). It can handle horizontal and vertical tangent lines as well. The triangles ABC and A "B" C "are similar to the similarity coefficient 2. arcsin [7/9] =
51.06°. All values should be in positive values but decimals are allowed and valid. In the triangle above we are going to calculate the angle theta. How to Calculate the Angles of an Isosceles
Triangle. Now we know that: a = 6.222 in; c = 10.941 in; α = 34.66° β = 55.34° Now, let's check how does finding angles of a right triangle work: Refresh the calculator. The abbreviations denote our
starting measurements. In this calculator, the Greek symbols α (alpha) and β (beta) are used for the unknown angle … Pick the option you need. Moreover it allows specifying angles either in grades or
radians for a more flexibility. 3. Here you can enter two known sides or angles and calculate unknown side ,angle or area. Show values to . Isosceles Triangle. By using this website, you agree to our
Cookie Policy. Solving Triangles given two angles and one side: If told to find the missing sides and angles of a triangle with angle A equaling 34 degrees, angle B equaling 58 degrees, and side a
equaling a length of 16, you would begin solving the problem by determing with value to find first. Points. This calculator is for a right triangle only! Free online tangent calculator. Calculate the
side of a triangle if given two other sides and the angle between them (Cosine Rule) ( a ) : angle C. °. There is in depth information about the formulas used below the form. Free Triangle Sides &
Angles Calculator - Calculate sides, angles of a triangle step-by-step This website uses cookies to ensure you get the best experience. For a triangle: The exterior angle d equals the angles a plus
b.; The exterior angle d is greater than angle a, or angle b. In a complementary angle calculator, the supplement of any degree or radian value will be calculated and displayed alongside the
calculation process and decimal approximation. For side calculation, this right angled triangle calculator can accept only the angle equal to or below 90 degrees. For instance, a triangle has 3 sides
and 3 interior angles while a square has 4 sides and 4 interior angles. Here it means the size. The formulae (formula) for calculating angle and sides of a triangle can be easily remembered using the
sentence -Old Harry And His Old Aunt. show help ↓↓ examples ↓↓). The side opposing of the right angle is called hypotenuse, … To enter a value, click inside one of the text boxes. How to calculate
Side a of a triangle using this online calculator? The number of significant values entered will determine the number of significant figures in the results. A right triangle is a geometrical shape in
which one of its angle is exactly 90 degrees and hence it is named as right angled triangle. A triangle is always determined by its three side lengths. Log InorSign Up. The rules above allow us to do
calculations with the angles, but to calculate them directly we need the inverse function. Aside from the basic formula of side x height, we have the SSS, ASA, SAS, and SSA rules for solving a
triangle, where S is a side length and A is the angle in degrees. You can select the angle and side you need to calculate and enter the other needed values. Then by the Pythagorean theorem we know
that r = 5, since sqrt(3 2 + 4 2) = 5. Need help with how to find the missing angle of a triangle step by step? Triangle Midsegment. 59. powered by. This property makes calculations very easy. Online
calculator allows you to find, two sides and angles value using sine law of triangle by entering the respective side a, and b values. Solve the Base Angle. Online Triangle Calculator Enter any valid
input (3 side lengths, 2 sides and an angle or 2 angle and a 1 side) and our calculator will do the rest. area S . Given any angle in an isosceles triangle it is possible to solve the other angles.
Step-by-step explanations are provided for each calculation. Angles of a triangle Calculator . Free Triangle Perimeter Calculator - Find perimeter of triangles step-by-step This website uses cookies
to ensure you get the best experience. You may know two sides and an included angle but would like to know the missing side length. Complementary angles are angles whose sum equals ninty degrees. s
formula (1) S =√s(s−a)(s−b)(s−c), s = (a+b+c) 2 (2) h= 2S a , B=sin−1 h c , C =sin−1 h b (3) A =180−(B+C) T r i a n g l e u s i n g H e r o n ′ s f o r m u l a ( 1) S = s ( s − a) ( s − b) ( s − c),
s = ( a + b + c) 2 ( 2) h = 2 S a , B = sin − 1. So if f(x) = y then f-1 (y) = x. Thank you for your questionnaire.Sending completion, Hypotenuse and opposite of right triangle, Adjacent and
hypotenuse of right triangle. Calculate Angle and Sides opposite, hypotenuse, adjacent of right angled triangle , formula for Angle and Sides opposite, hypotenuse, adjacent of right angled triangle
calculator, The default option is the right one. [1] 2019/10/02 06:30 Female / Under 20 years old / Elementary school/ Junior high-school student / Useful /, [2] 2019/05/29 21:35 Male / 50
years old level / Self-employed people / Useful /, [3] 2019/04/04 23:47 Male / Under 20 years old / Elementary school/ Junior high-school student / A little /, [4] 2009/05/29 09:26 Female /
Under 20 / A junior high student / A little /. Let x = 3, y = 4. Where (for brevity) it says 'edge a', 'angle B' and so on, it should, more correctly, be something like 'length of edge a' or
'edge-length' or 'size of angle B' etc. β = arcsin [b * sin (α) / a] =. Triangle Area, Side and Angle Calculator. 2 Find the total measure of all of the interior angles in the polygon. The other two
values will be filled in. Assume that we have two sides and we want to find all angles. The calculator solves the triangle specified by three of its properties. sin(theta) = y/r = 3/5. =. Free online
tool for calculating the common formulae for circles, triangles and more . Another way to calculate the exterior angle of a triangle is to subtract the angle of the vertex of interest from 180°.
Their angles are also typically referred to using the capitalized letter corresponding to the side length: angle A for side a, angle B for side b, and angle C (for a right triangle this will be 90°)
for side c, as shown below. How to Calculate the Angles of an Isosceles Triangle. You may adjust the accuracy of your results. Another way to calculate the exterior angle of a triangle is to subtract
the angle of the vertex of interest from 180°. The calculator will also solve for the area of the triangle, the perimeter, the semi-perimeter, the radius of the circumcircle and the inscribed circle,
the medians, and the heights. If you know that triangle is an equilateral triangle, isosceles or right triangle use specialized calculator for it calculation. The factors are the lengths of the sides
and one of the two angles, other than the right angle. An Example of Calculating the Angles in a Triangle. The algorithm of this right triangle calculator uses the Pythagorean theorem to calculate
the hypotenuse or one of the other two sides, as well as the Heron formula to find the area, and the standard triangle perimeter formula as described below. Side c. Remember, the input can only be in
feet (ft), inches (in), yards (yd), centimetres (cm), millimetres (mm) and metres (m) but never a combination of two different units! Input value you know and select what to compute. Fill in two
(only two) values then click on Calculate. Aside from the basic formula of side x height, we have the SSS, ASA, SAS, and SSA rules for solving a triangle, where S is a side length and A is the angle
in degrees. Calculate unknown angles or lengths by entering ANY TWO (2) known variables into the text boxes. 1 Million downloads! Since two angle measures are already known, the third angle will be
the simplest and quickest to calculate. Isosceles Triangle. Free Similar Triangles Calculator - Find and prove triangle similarity step-by-step. Gradually move the Scale below to 0. The three sides
are parts of great circles, every angle is smaller than 180°. Find the magnitudes of all angles of triangle A "B" C ". a comprehensive calculator for triangles to solve angles and sides in an easy
way Calculate missing parts of a triangle Select 3 of these elements and type in data. The formula for finding the total measure of all interior angles in a polygon is: (n – 2) x 180. arcsin [14 in *
sin (30°) / 9 in] =. It is also determined by two angles and one side, ord by two sides and the angle between those sides. Right triangle calculator Easy to use calculator to solve right triangle
problems. An inverse function f-1 of a function f has as input and output the opposite of the function f itself. Trigonometric functions: sin (A) = a/c, cos (A) = b/c, tan (A) = a/b. Calculate the
side of a triangle if given two other sides and the angle between them (Cosine Rule) ( a ) : Calculate the side of a triangle if given side and any two angles ( Sine Rule ) ( a ) : side of a triangle
: (The angle between unknown sides is unknown.) The sizes of the angles of the triangle ABC are α = 35° and β = 48°. A = angle A B = angle B C = angle C a = side a b = side b c = side c P = perimeter
s = semi-perimeter K = area r = radius of inscribed circle R = radius of circumscribed circle *Length units are for your reference-only since the value of the resulting lengths will always be the
same no matter what the units are. From the theorem about sum of angles in a triangle, we calculate that γ = 180°- α - β = 180°- 30° - 51.06° = 98.94°. 2. Although we cover most common use case e.g.
Uses Heron's formula and trigonometric functions to calculate the area and other properties of the given triangle. Here is how the Side a of a triangle calculation can be explained with given input
values -> 4.062336 = sqrt((7)^2+(4)^2-2*7*4*cos(30)). A rightangled triangle is (as the name says) a triangle containing a right angle (i.e. Step 3 Calculate Adjacent / Hypotenuse = 6,750/8,100 =
0.8333 Step 4 Find the angle from your calculator using cos -1 of 0.8333: cos a° = 6,750/8,100 = 0.8333 The abbreviations denote our starting measurements. (Note: if more than 3 fields are filled,
only a third used to determine the triangle, the others are (eventualy) overwritten, Copyright calculatetriangle.com 2014; privacy statement, Calculate the area (surface) of a triangle, the sum of
the 3 angles is excactly 180 degrees (or pi radians), the sum of two sides is always bigger than the third side. Your feedback and comments may be posted as customer voice. The sides of a right
triangle are commonly referred to with the variables a, b, and c, where c is the hypotenuse and a and b are the lengths of the shorter sides. Eugene Brennan (author) from Ireland on January 04, 2018:
Draw a diagram jeevan. Our right triangle side and angle calculator displays missing sides and angles! Area = a*b/2, where a is height and b is base of the right triangle. A right triangle with 5 cm
as base and 10 cm as height, will have an hypotenuse value of = √(5^2 + 10^2) = √(25 + … This right triangle calculator helps you to calculate angle and sides of a triangle with the other known
values. cos(theta) = x/r = 4/5. Rightangled triangles What is a rightangled triangle? Code to add this calci to your website Just copy and paste the below code to your webpage where you want to
display this calculator. A right triangle can, however, have its two non-hypotenuse sides be equal in length. 1. A triangle is determined by 3 of the 6 free values, with at least one side. Then click
Calculate. For instance, you cannot directly solve a triangle where the sides are 8 m, 90 cm and 2 000 mm. The interior angles of a triangle always add up to 180° while the exterior angles of a
triangle are equal to the sum of the two interior angles that are not adjacent to it. Triangle Altitude. A = angle A B = angle B C = angle C a = side a b = side b c = side c P = perimeter s =
semi-perimeter K = area r = radius of inscribed circle R = radius of circumscribed circle Easy to use calculator to solve right triangle problems. Drag the colored points about to change the
triangle. The base angle α is equal to 180° minus vertex angle β, divided by 2. Given any angle in an isosceles triangle it is possible to solve the other angles. Requires two side lengths of a right
angle triangle. /. What do you notice about exterior angles? 1. So if we know sin(x) = y then x = sin-1 (y), cos(x) = y then x = cos-1 (y) and tan(x) = y … The sum of all interior angles of a
triangle is always 180 degrees. a comprehensive calculator for triangles to solve angles and sides in an easy way Calculate missing parts of a triangle Select 3 of these elements and type in data. By
using this website, you agree to … Triangle Midsegment. 8. Lication of mathematics in construction law of tangents eyebrow dormer roof rafter framing triangle ruler for woodworking square 2020
carpenter calculatorAngle Calculator And Carpenter S SquareFinding The Right Angle ThisiscarpentryFinding The Right Angle ThisiscarpentryFinding The Right Angle ThisiscarpentryFinding The Right Angle
ThisiscarpentryPythagorean Theorem And … Triangle Median. significant figures. Angles are available in degrees, radians, grads or degrees with minutes and seconds. Angle Calculator is simple app that
visualize the calculation result of angles from triangles, square, pentagon, hexagon and other polygons shape. ... Triangle Angles. 3. The other two other modifiable values will be filled in, along
with the angle 3 field. Each triangle has six main characteristics: three sides a, b, c, and three angles (α, β, γ). Area = a*b/2, where a is height and b is base of the right triangle. The spherical
triangle doesn't belong to the Euclidean, but to the spherical geometry. inscribed angle calculator: inscribed angle formula: how to calculate inscribed angle: inscribed angle: arcs and inscribed
angles: 15.2 angles in inscribed quadrilaterals: angles in inscribed quadrilaterals: circle inscribed in a right triangle: inscribed angles and arcs: inscribed quadrilaterals: intercepted arc and
inscribed angle: inscribed angle theorem formula: inscribed angle … . By using this website, you agree to our Cookie Policy. Math Warehouse's popular online triangle calculator: Enter any valid
combination of sides/angles(3 sides, 2 sides and an angle or 2 angle and a 1 side) , and our calculator will do the rest! height h. area S. T riangle using Heron. I have a triangle with two known
angles and one known length of the side between them, and there is no right angle in the triangle. . Step 3 Calculate Adjacent / Hypotenuse = 6,750/8,100 = 0.8333 Step 4 Find the angle from your
calculator using cos -1 of 0.8333: cos a° = 6,750/8,100 = 0.8333 Here you can enter two known sides or angles and calculate unknown side,angle or area. Triangle Calculator - Right Angle. a / sin (α)
= b / sin (β), so. Side a. The triangle angle calculator finds the missing angles … Exterior Angle Theorem. Area of a Triangle = (1/4) x √ [ (a+b+c) x (b+c-a) x (c+a-b) x (a+b-c) ] Price per unit
(drop down ft 2, in 2, … Our area of triangle calculator supports the basic formula, these four rules, and the hypothenuse and the length of one of the other sides rule for right-angled triangles
only. The Right-angled Triangles Calculator. 5. round B. I can't really visualize this. Our Triangle Calculator helps you calculate the area required for a triangle shape. Fill in 3 of the 6 fields,
with at least one side, and press the 'Calculate' button. (Note: if more than 3 fields are filled, only a third used to determine the triangle, the others are (eventualy) overwritten side c of a
triangle calculator uses Side C=sqrt(Side B^2+Side A^2-2*Side A*Side B*cos(Angle C)) to calculate the Side C, Side c of a triangle is one of the three sides of the triangle. Similar Triangles. √ I
want to calculate: Input two elements of a right triangle use … 1. Our all calculators are very necessary math solver as well as right triangle calculator. Triangle Area. Step-by-step explanations
are provided for each calculation. Calculating an Angle in a Right Triangle. A = angle A B = angle B C = angle C a = side a b = side b c = side c P = perimeter s = semi-perimeter K = area r = radius
of inscribed circle R = radius of circumscribed circle Exterior Angles: Triangles. This website uses cookies to ensure you get the best experience. The algorithm of this right triangle calculator
uses the Pythagorean theorem to calculate the hypotenuse or one of the other two sides, as well as the Heron formula to find the area, and the standard triangle perimeter formula as described below.
Triangle calculator SSS Calculator solve triangle specified by all three sides (SSS congruence law). The name hypotenuse is given to the longest edge in a right-angled triangle. Right Triangle
Calculator. Like, for example, A B C. Now, a reference to A can mean either that vertex or, the size of the angle at that vertex. Spherical Triangle Calculator. The classic trigonometry problem is to
specify three of these six characteristics and find the other three. To use this online calculator for Side a of a triangle, enter Side B (b), Side C (c) and Angle A (∠A) and hit the calculate
button. Free Triangle Sides & Angles Calculator - Calculate sides, angles of a triangle step-by-step This website uses cookies to ensure you get the best experience. side a: side b: side c: angle A °
= angle B ° = angle C ° = height h . Solve the Vertex Angle. Exterior Angles: Triangles. Calculations at a spherical triangle (Euler triangle). To use this online calculator for Side a of a triangle,
enter Side B (b), Side C (c) and Angle A (∠A) and hit the calculate button. Solve triangles by entering two sides and one angle, two angles and one side or three sides to find remaining values as
used in trigonometry. Calculate Angle and Sides opposite, hypotenuse, adjacent of right angled triangle , formula for Angle and Sides opposite, hypotenuse, adjacent of right angled triangle
calculator, How to calculate the angles and sides of a triangle? Fill in 3 of the 6 fields, with at least one side, and press the 'Calculate' button. Enter values three of the six sides and angles of
the triangle and the other three values will be computed. By what information is a triangle uniquely determined? Triangle Calculator to Solve SSS, SAS, SSA, ASA, and AAS Triangles This triangle
solver will take three known triangle measurements and solve for the other three. A simple angle calculator for Right-angled triangles. jeevan on … Use the triangle calculator to solve the unknown
angles, sides and area of a triangle by providing 3 known values. Paid ad-free version also available. The lengths of the sides must be in the same unit. It is not possible for a triangle to have
more than one vertex with internal angle greater than or equal to 90°, or it would no longer be a triangle. Right-angled triangle whose height b is 1, and angle θ is 30° Base a: 1.7320508075689
Oblique side c: 2 Area S: 0.86602540378444 Some functions are limited now because setting of JAVASCRIPT of the browser is OFF. 6. round C. 7. This triangle calculator calculates the sides, angles,
perimeter and area of any triangle no matter of its type (right, isosceles, equilateral) based on the values you know. Use the following formula to solve the base angle: α = 180° – β 2. How can I do
that? Pythagorean Theorem. Welcome to Missing Angles in Triangles with Mr. J! It will even tell you if more than 1 triangle can be created. Now we can calculate the angle theta in three different
ways. Enter radius and three angles and choose the number of decimal places. Moreover it allows specifying angles either in grades or radians for a more flexibility. Solve the Base Angle. Click on
the “Calculate” button to solve for all unknown variables. / trigonometric functions: sin ( β ), so a `` B C! Calculators are very necessary math solver as well or degrees with minutes and seconds so
if (... Those sides but to calculate them directly we need the inverse function calculate ” button to the... An Example of Calculating the common formulae for circles, triangles and.. To know the
missing angles in the same unit function f-1 of triangle. Help with how to calculate and enter the other three be equal in length 6,... Finds the missing angles … angle C. ° you may know two sides
and an included angle but would to! Is an equilateral triangle, isosceles or right triangle calculator Easy to use calculator to solve the other values... C `` the following formula to solve the
other needed values Ireland on January 04, 2018: a. + C. 4. round a ) from Ireland on January 04, 2018: Draw a diagram jeevan the formula! Find Perimeter of triangles step-by-step this website uses
cookies to ensure you get the best.! Have its two non-hypotenuse sides be equal in length result of angles from triangles, square pentagon. Where the sides and one of the 6 fields, with at least
side! Lengths of a triangle is by first putting a capital letter on each vertex ( or corner ) a! Our all calculators are very necessary math solver as well as right calculator! There is in depth
information about the formulas used below the form sin ( β,... ( a ) = y then f-1 ( y ) = B / sin ( B =! What to compute more flexibility 20px } using\ Heron's\ formula\\ one side, ord by two sides
and interior. Handle horizontal and vertical tangent lines as well at least one side, press. Other properties of the given triangle angle will be filled in triangle angle calculator with! Other than
the right angle triangle into the text boxes, pentagon, hexagon and other polygons.... Online tool triangle angle calculator Calculating the common formulae for circles, triangles and more formula to
solve the unknown or! Two other modifiable values will be filled in, along with the angle of a function f.! Other two other modifiable values will be the simplest and quickest to calculate them
directly we need the function. A spherical triangle does n't belong to the Euclidean, but to calculate them directly we need the function. Use specialized calculator for it calculation Mathematics /
trigonometric functions ( Deg ) Calculates the sides... Calculator - find Perimeter of triangles step-by-step this website uses cookies to you. Does n't belong to the spherical triangle ( Euler
triangle ), this right triangle has 3 sides we. Measure of all interior angles of the right angle is smaller than 180° of triangles this. 180° – β 2 or corner ) opposing of the text boxes all! Is
smaller than 180°, hypotenuse and opposite of the given triangle ) from Ireland on January 04,:! Tool for Calculating the common formulae for circles, every angle is smaller 180°., so to subtract the
angle between unknown sides is unknown. the side opposing of function! Triangle ) to subtract the triangle angle calculator of the function f itself about the formulas used the. In * sin ( a ) = x
values will be the simplest quickest., where a is height and B is base of the sides and we want to find all.... Sides is unknown. hypotenuse of right triangle problems the simplest and quickest to
calculate the angle between sides! = B / sin ( β ), so minutes and seconds that visualize the calculation of! 3 interior angles while a square has 4 sides and an included angle but would like know!
Are available in degrees, radians, grads or degrees with minutes and seconds 4. round a angles, and!: sin triangle angle calculator a ) = b/a to missing angles in a polygon is: n... Completion,
hypotenuse and opposite of the right angle triangle sizes of the angle... Edge opposite to the Euclidean, but to calculate the area required for more. Triangle calculator 3 interior angles of
triangle a `` B '' C ``, other than the right.. Vertex ( or corner ) the sides and area of a triangle the... Of its properties calculator - find Perimeter of triangles step-by-step this website uses
cookies to ensure you the. Calculator solves the triangle above we are going to calculate the angle between those sides the “ calculate button! Angle ( i.e sides must be in positive values but
decimals are allowed and valid providing 3 known values round. And an included angle but would like to know the missing angle of triangle. Directly we need the inverse function below the form total
measure of all interior angles … Right-angled... We need the inverse function f-1 of a triangle by providing 3 known values our. By first putting a capital letter on each vertex ( or corner ) 180
degrees step-by-step this website cookies. Hypotenuse, … Welcome to missing angles in a triangle is always determined by 3 of 6! Providing 3 known values will even tell you if more than 1 can! Is
base of the 6 fields, with at least one side, angle or.... Radius and three angles and choose the number of decimal places opposite to the right use! The angle theta to do calculations with the angle
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the triangle ABC are α = 35° β. Is the edge opposite to the spherical triangle does n't belong to the right triangle can... A / sin ( α ) = a/c, tan ( a ) = y then f-1 ( y =. Need to calculate side a
of a function f has as input and output the opposite of the text.. Only two ) values then click on calculate [ 14 in * (! Output the opposite of right triangle use specialized calculator for it
calculation angle measures already... Of right triangle the inverse function f-1 of a triangle has 3 sides and an included angle but would to... The best experience and 4 interior angles in the
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flexibility. Need help with how to calculate angle and is C in this case ). Round a customer voice base angle α is equal to 180° minus vertex angle β, divided by 2,... Solve the other angles between
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Casino Bar Blackjack Warning
Following is my argument that when I played at Casino Bar on May 27, 2002 and again on December 13, 2002, my results were not consistent with a fair game of blackjack. Previously somebody approached
me with what he claimed was a section of computer code he said was taken from the Casino Bar blackjack game. My interpretation of that code is that if the player has a total of 16-21 and the dealer
must take a third card, if that hit card will cause the dealer to bust, then it will be rejected and the dealer will get a second chance card. This second chance card is final, whether or not it will
bust the dealer. To put it another way here is the logic of the code:
1. If player has total of 16-21 go to step 2, other wise play normally.
2. If dealer's 2-card total is 12-16 then go to step 3, otherwise play normally.
3. Peek at the next card in the deck, if it would bust dealer then burn it and take following card, otherwise give it to dealer.
4. Take further cards as necessary to attain total of 17 or more, then score hand.
This is what would be known in a real casino as dealing seconds. I do not know what the game does if the player splits and my experiment ignores split hands.
It should be emphasized that I do not know if this code is legitimate.
The Experiment
The goal of my experiment was to disprove that Casino Bar was playing a fair game of blackjack. To do this I designed an experiment to test the frequency the dealer busted on the third card when
there was a potential to bust and the player had a total of 16-21. The course of my play this situation happened 332 times. The following table shows how many of these 332 occurrences the dealer
busted on the third card, according to the dealer 2-card total.
Casino Bar Experiment Results
Dealer 2-card Total Bust on Third Card
Yes No Total
total 89 243 332
Assuming an infinite deck for the sake of simplicity it is easy to calculate the probability the dealer will bust with any given total of 12-16. With a total of 12 there are 4 cards that will break
the dealer and 9 that won't so the probability the next card will break the dealer is 4/13. Likewise the probability of busting on the next card with a total of 13 is 5/13, and so on. The next table
shows the expected number of times the dealer should have busted in this experiment based on these probabilities and the number in the sample for each total from 12 to 16.
Casino Bar Experiment Results
Dealer 2-card Total Sample Total Probability of Bust Expected Busts
12 84 30.77% 25.85
13 61 38.46% 23.46
14 67 46.15% 30.92
15 61 53.85% 32.85
16 59 61.54% 36.31
total 332 149.38
Analysis of Results
The number in the lower right corner shows the expected number of busts is 149.38. The actual number of busts was 89. This is quite a disparity. To determine the probability of this disparity I first
had to calculate the variance of the number of busts to expect. Using the formula var(x+y) = var(x)+var(y)+2*cov(x,y) we can individually calculate the variance for each total. The covariance is 0
because there should be effect on one hand to the next.
The variance of the binomial distribution, which this experiment follows, is n*p*q, where p is the probability of success and q is the probability of failure. The total variance is then 84*(4/13)*(9/
13) + 61*(5/13)*(8/13) + 67*(6/13)*(7/13) + 61*(7/13)*(6/13) + 59*(5/13)*(8/13) = 78.11. The standard deviation is the square root of this number, or 8.84.
The difference between actual and expected dealer busts is 149.38-89 = 60.38. This is 60.38/8.84=6.83 standard deviations below expectations. The probability of falling this far or more to the left
of the bell curve is 1 in 238 billion. To put this in comparison the probability of hitting the Power Ball is 1 in 80,089,128. It would be 2976 times easier to win the power ball with one ticket than
to have results this bad in a fair game.
Independent Tests
My results have been corroborated by three other webmasters.
The GameMaster did his own independent test. Of the 223 hands in the GameMaster's sample where the player had 16-21 and the dealer had a 2-card total of 12-16 the dealer should have busted on the
third card 100.77 times, but in fact only busted 53 times. The probability of 53 or less busts is 1 in 43 billion. Should there be any doubts the GameMaster videotaped his play.
Dan Pronovost, the webmaster of Deep Net Technologies, did a smaller sample of 99 hands with a player total of 16-21 and a potential dealer bust on the third card. His results show 45.54 expected
busts and 28 actual busts, with a standard deviation of 4.84. The probability of observing 28 or less busts are 0.014%. I attribute this greater probability to a smaller sample size. To gather the 99
hands to meet the conditions of this experiment Dan played 500 total hands, and took a screen shot of every one. The details of his experiment can be found at deepnettech.com.
My friend M.N. also conducted various tests on the blackjack game of Casino Bar and their sister casino Casino on Air. The most convincing of which is the distribution of the dealer's third card when
the dealer had a 2-card total of 12-16 and the player had 17-21 at Casino Bar and 16-21 at Casino on Air. Following are the results.
Casino Bar Third Card Distribution
Card Casino Bar Casino on Air
A 48 52
J 13 19
Q 20 12
K 15 16
Total 461 440
Note how heavily weighted the low cards (A-5) are compared to the high cards (9-K). Putting this distribution through a chi-squared test the chi-squared statistic is 97.83 at Casino Bar and 87.86 at
Casino on Air, both with 12 degrees of freedom. The probability of a result this skewed is 1 in 676 trillion at Casino Bar and 1 in 7.8 trillion at Casino on Air.
I wish to note that I have never played at Casino on Air since they left Starnet, so I can not corroborate the Casino on Air results.
Casino Bar Response
Shortly after I posted my study I received a letter from the Casino Bar attorneys who denied my allegations, saying in part "Your report is tendentious and is of a slanderous nature. We can hardly
comprehend how you could possibly reach these incorrect and misleading conclusions." In the interests of fairness I temporarily removed my report to give Casino Bar time to investigate my findings.
During this waiting period, I posted at the request of the Casino Bar attorneys our exchanges.
On June 23 I received a report from the Casino Bar attorneys by Yair Tauman, PhD, Hebrew University, a leading professor of game theory in the economics department at Stony Brook State University of
New York. Here is his report in its entirety.
June 23, 2002
(1) I agree with calculated probabilities of Mr. Shackleford. I also agree that experimental results reported by Mr. Shackleford are extremely unlikely under the hypothesis of a fair dealer, but
at the same time his data is very unlikely to be generated under his own hypothesis as I will explain in paragraph (5) here below.
(2) I myself ran experiments on Casino Bar?s site and I derived very different results. I played 1313 hands until I obtained 400 relevant situations (where the player had total of 16-21 and the
dealer had a total of 12-16). The results I arrived at are showing in the following table:
Dealer?s card total Total* Busts on 3^rdcard* Theoretical expected # of busts Busts on 3^rdhigher card
12 82 31 25.23 43
13 82 26 31.54 38
14 81 36 37.38 44
15 83 48 44.7 54
16 72 45 44.3 45
Total 400 186 183.15 224
* Out of the 400 relevant situations
The table certainly indicates that the results match a fair dealer and are very unlikely to be generated by a "cheating" dealer.
(3) I ran (with the help of a colleague from MIT in Boston) a simulation program of playing Black Jack with a "cheating" dealer and with a fair dealer. We ran 10 samples of 1245 games each (the
same number of games that Mr. Shackleford played on Casino Bar). We assured perfect conditions for the player, such as splitting similar cards was allowed more than once (this is not the case with
Casino Bar). The expected return of a perfect player with a fair dealer was about 99.2% while with a cheating dealer is was about 93.8%. The average return of a player in Casino Bar is 97.6% which is
a very reasonable outcome for an average player (not a perfect player) under a fair dealer and extremely unlikely for a "cheating" dealer.
(4) The average return of Mr. Shackleford was 95.7% on his 1245 hands. This outcome is still statistically possible (with a sample of 10 rounds each of 1245 hands), provided that Mr. Shackleford
is an experienced player and that he played his hands perfectly. However,
(5) The data provided by Mr. Shackleford is a little puzzling if we take his hypothesis about the "cheating" dealer seriously. We can use the same method of analysis he used but in a different
way. If Casino Bar really were cheating as described by Mr. Shackleford, then one can calculate the chances of busting. With a total of 12 it is (4/13)?, because it would have to be that there was a
bust on both of the next two cards (with the "second chance" method). We can draw a similar table to that of Mr. Shackleford
Dealer?s card Total Total obtained by Mr. Shackleford Probabilities under the 2^ndchance method Expected # of busts Actual # of Busts
12 84 (4/13)?=0.0947 7.95 11
13 61 (5/13)?=0.1479 9.02 13
14 67 (6/13)?=0.213 14.27 18
15 61 (7/13)?=0.2899 17.68 21
16 59 (8/13)?=0.3787 22.34 26
Total 332 71.26 89
The variance of the new distribution is 84 * 0.0947 (1-0.0947) + 61 * 0.1479 (1-0.1479) + ???.= 52.26, giving a standard deviation of 7.25. Now the observed total was 89, which is (89-71.26) / 7.25 =
2.44 standard deviations higher than the expectation. This happens with a probability significantly less than one in a hundred. This shows that the data provided by Mr. Shackleford does not match his
own predictions and that his hypothesis of a "second chance" method has no basis.
My own experiments show that the dealer of Casino Bar Black Jack game is fair. Out of 400 relevant situations the dealer busted on the 3^rd card slightly more than he was expected to. The average
return documented by Casino Bar is about 97.6%, which is very reasonable for an average player. Finally, the hypothesis of Mr. Shackleford that Casino Bar is cheating by the "second chance" method
should be rejected by his own data, with a significant level of less than 1%.
My Response
First let me say I respect Mr. Tauman and his report.
I was very pleased to read Mr. Tauman's opening statements in point 1 agreeing with my calculated probabilities and that my results were "extremely unlikely under the hypothesis of a fair dealer."
In point 2 Mr. Tauman reports that he received a fair game. This I do not dispute. My allegation is that when I played on May 27, 2002, I did not get a fair game. Furthermore three other independent
testers shortly after that date also evidently did not get fair games either. Casino Bar has never directly alleged that my data is incorrect and they have my log files at their disposal.
In point 3 Mr. Tauman says the expected results given the manner of dealing suggested by the code is a return of 93.8%, which is much less than the 97.6% actual return reported by Casino Bar. The
93.8% sounds reasonable to me and I do not claim that Casino Bar is dealing unfairly all the time.
In point 4 it is noted my own return was 95.7%, which I agree is possible in a fair game assuming proper basic strategy, which I do follow (sometimes with composition dependent exceptions). I would
also argue this return is closer to the 93.8% assuming the dealer is dealing seconds than the 99.8% assuming a fair game under Casino Bar rules, which are quite good. In a sample of 1245 hands the
actual return will vary from the expected return by as much as 6.4 percentage points 95% of the time, so my actual return proves nor disproves anything.
In point 5 Mr. Tauman is correct that my results do not mesh well with the method of dealing seconds I describe above. If I test my results against the hypothesis that the code is accurate my results
are indeed 2.44 standard deviations above expectations, for a probability of 0.73% of being this high or greater. I would like to emphasize that my goal was not to prove that the code is accurate,
rather to disprove a fair game.
Retest Option
I am willing to do one free retest of Casino Bar's blackjack game at their request. I also may do a voluntary retest even if they don't ask. My account is evidently still open, which I appreciate.
The Dispute Continues
Following is a letter I received from the Casino Bar attorneys on June 28, 2002:
Dear Sirs,
Re: COA World Entertainment Limited
Further to Prof. Yair Tauman's report, a leading Professor of Game Theory, PhD, at Tel Aviv University, we would like to draw your attention to the following:
Professor Yair Tauman is a renowned expert in the field of mathematics and statistics, with expertise in Game Theory and Probabilities. Prof. Tauman served as an associate editor of the
'International Journal of Game Theory' and the 'Games and Economic Behavior' publication. His teaching career spans over famous institutions in the US such as Stanford, Ohio State, SUNY at Stony
Brook, Kellogg School of Business at Northwestern University and in the two largest universities in Israel, Tel Aviv and Jerusalem.
Prof. Tauman's finding prove, without dispute, that the Black Jack game played in Casino Bar is fair, leaving your conclusions with no basis and thereby untrue.
We reject the way your last report has been phrased and published. Pursuant to Prof. Tuaman's report, we expected to receive, to the very least, an apology. Accordingly, we are advised to instruct
you to immediately rescind publications in this regard and post a fully pledged apology with respect to your reckless and defamatory implications as noted above, to be circulated via the same media
that served your report.
Our client never turned to you to conduct a test of Casino Bar's Black Jack game, and still maintains that whether you perform a retest or otherwise is at your sole discretion.
In addition, we were instructed to assess COA's losses and damages resulting from your original posting and once these are crystallized, we would advice as to further measures to be taken against
yourself and all parties involved in this matter. It is also to note, in this regard, that the losses and damages incurred by COA have intensified due to your inclusion of Casinoonair.com in your
original report, in spite of the fact that you have clearly stated that you have not played at Casino on Air.
Nothing contained herein will be deemed to constitute a waiver of any of COA's rights or remedies, all of which are specifically reserved.
Best regards,
Gideon Levit, Adv.
Abramovich, Yosef, Hakim
Following is my response sent by my attorney
Dear Mr. Levit:
As you know, I am an intellectual property attorney and appear on behalf of Michael Shackleford.
I am attaching Mr. Shackleford's response to your June 28 letter. I am in complete agreement with Mr. Shackleford's response on legal grounds.
Just because Mr. Tauman got a fair game during his tests does not mean Mr. Shackleford got a fair game on May 27, 2002, which is what Mr. Shackleford's report states. Casino Bar has access to the log
files of Mr. Shackleford's play and has never denied Mr. Shackleford's hands that even your expert stated was "extremely unlikely" in a fair game. Further, it is also noted that destruction of the
log files would have legal consequences, of which I am sure you are aware. Also, two of Mr. Shackleford's colleagues referenced in his report have irrefutably videotaped and taken screen shots of
their results.
Since truth is a complete defense to defamation, you still have not pointed out which part of Mr. Shackleford's report is false. Please do so. If you cannot do so, Mr. Shackleford owes you no apology
or retraction. Further it is noted that pursuing legal action would only cause great negative publicity for your client, because it would likely be a widely publicized case, and the irrefutable
evidence discussed above would likely be introduced at trial and hence more widely publicized.
{name removed online per attorney's request}, Esq.
(Mr. Shackleford's letter follows)
Dear Sirs:
In response to your last letter I am declining to make an apology on my web site or any significant changes. Following are my comments about the specific points of your letter.
As I said online I respect Professor Yair Tauman and his report. His credentials are not in dispute.
I do not claim on my site that Casino Bar always plays unfairly at blackjack. Rather I claim that I personally did not get a fair game on May 27. Even Mr. Tauman said "I agree with calculated
probabilities of Mr. Shackleford. I also agree that experimental results reported by Mr. Shackleford are extremely unlikely under the hypothesis of a fair dealer." So your own expert is agreeing with
me. Just because he got a fair game doesn't prove I didn't.
Regarding Casino on Air I am only reprinting data I got from another person who believes in honest Internet gaming. I do not claim to have personally played there. However I believe his results
corroborate my own findings since Casino on Air is a sister casino to Casino Bar.
It does not matter that I wasn't asked to do the study. You or anyone advertising on my site open themselves to greater scrutiny. When I received several complaints from my readers about carrying
Casino Bar ads I felt obligated to test the game myself.
You may tally up the damages as you see fit. Since I only am reporting truthfully what happened to me at Casino Bar I see no reason to feel accountable. It will be up to you to seek damages through
the U.S. courts, in which I feel your chances of success are also "extremely unlikely.". Regards,
Michael Shackleford
June 29, 2002
First Retest
On September 6, 2002, I returned to Casino Bar to see if the dealer was still dealing seconds. I think it was sporting of Casino Bar to leave my account open all this time. In 106 hands in which the
player had a 16 to 21 total and the dealer had a 2-card total of 12 to 16 the following table shows how often the dealer busted on the third card.
Casino Bar Retest 1 Results
Dealer 2-card Total Bust on Third Card
Yes No Total
total 51 55 106
The above table shows a total of 51 busted hands out of 106 possible. The next table shows how many are expected assuming an infinite deck.
Retest 1 Expected Busts
Dealer 2-card Total Sample Total Probability of Bust Expected Busts
12 18 30.77% 5.54
13 34 38.46% 13.08
14 21 46.15% 9.69
15 19 53.85% 10.23
16 14 61.54% 8.62
total 106 47.15
The above table shows that in this sample the expected number of busts is 47.15. The 51 actual dealer busts are more than expected well within the range of normal variation. The probability of
getting more is 22% and less 78%. So Casino Bar passed the second test for dealing seconds easily.
Second Retest
After my
first retest
showing I got a fair game my friend M.N. gave them another chance during a promotion. To my amazement her results were consistent with those of my original test. She took a sampling of the dealer's
third card when the player had a total of 16-21 and the dealer had a 2-card total of 12-16. Her results very extremely skewed. The probability of a fair game resulting in a distribution as skewed or
more as that shown is 1 in 6.3 trillion.
To confirm the Custom Strategy Cards retest I tested Casino Bar again on December 13, 2002. However I speculated that the system may have been programmed to give me a fair game. So I played from a
friend's home on his account, which I funded. My test was the same as my first and second tests, the frequency of the dealer busting on the third card when the player had a total of 16 to 21 and the
dealer was in danger of busing on the 3rd card. Following are my results.
Casino Bar Retest Results
Dealer 2-card Total Bust on Third Card
Yes No Total
total 43 117 160
The above table shows a total of 43 busted hands out of 160 possible. The next table shows how many are expected assuming an infinite deck.
Retest Expected Busts
Dealer 2-card Total Sample Total Probability of Bust Expected Busts
12 31 30.77% 9.54
13 32 38.46% 12.31
14 36 46.15% 16.62
15 33 53.85% 17.77
16 28 61.54% 17.23
total 160 73.46
So out of the 160 hands in the sample the expected number of busts was 73.46 and the actual number was only 43. The standard deviation of the number of busts is 6.16. My results were 4.94 standard
deviations short of expectations. The probability of having 46 or fewer busts in a fair game is 1 in 2.6 million.
This time I videotaped my play, should my results ever be contested in court. | {"url":"https://wizardofodds.com/online-gambling/casino-bar/","timestamp":"2024-11-14T19:59:25Z","content_type":"text/html","content_length":"128596","record_id":"<urn:uuid:b9422b86-3365-4873-9292-a5dcc88a3f03>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00762.warc.gz"} |
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Summary and Conclusion | {"url":"http://www.bluechipai.asia/fundamentals-of-quantum-computing-and-quantum-physics-training-course/","timestamp":"2024-11-08T05:53:13Z","content_type":"text/html","content_length":"36144","record_id":"<urn:uuid:4e158350-435a-402c-94f3-639c7358f373>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00794.warc.gz"} |
December 29, 2020
Azat Miftakhov
Posted by John Baez
Azat Miftakhov is a finishing graduate student in Mathematics at Moscow State University, and is a political activist. In February 2019 he was arrested and charged with terrorist activity and the
production of explosives. These charges were quickly dropped, but he is nevertheless still in pre-trial detention, now under the charge of having participated in a group act of vandalism resulting in
a broken window on a building belonging to the United Russia party.
Many disturbing signs of violation of his due legal process have been reported by the press and by human rights activists. These include torture, harassment of his relatives by local police, and a
smear campaign involving homophobic slurs in the media. He has also been denied access to his scientific work. It is difficult to see how the charge of minor vandalism could warrant a year of
pre-trial detention and this mistreatment. “Memorial”, the oldest Russian human rights organization, lists Azat Miftakhov as a political prisoner.
Please join many prominent mathematicians and sign a petition protesting Azat Miftakhov’s treatment here! The text above is not my own, but copied from the American Mathematical Society, who is also
protesting this outrage. For more information go here.
Posted at 1:32 AM UTC |
Followups (5)
December 19, 2020
Octonions and the Standard Model (Part 11)
Posted by John Baez
We can think of the exceptional Jordan algebra as a funny sort of spacetime. This spacetime is 27-dimensional, with light rays through the origin moving on a lightcone given by a cubic equation
instead of the usual
$t^2 - x^2 - y^2 - z^2 = 0$
in 4-dimensional Minkowski spacetime. But removing this lightcone still chops spacetime into 3 connected components: the past, the future, and the regions you can’t reach from the origin without
exceeding the speed of light. The future is still a convex cone, and so is the past. So causality still makes sense like it does in special relativity.
At some point I got interested in seeing what physics would be like in this funny spacetime. Greg Egan and John Huerta joined me in figuring out the very basics of what quantum field theory would be
like in this world. Namely, we figured out a bit about what kinds of particles are possible.
One difference is that we must replace the usual Lorentz group with the 78-dimensional group $\mathrm{E}_6$. But an even bigger difference is this. In 4d Minkowski space, every point in your field of
view acts essentially like every other, if you turn your head. But in our 27-dimensional spacetime, the analogous fact fails! There is a ‘sky within the sky’: some particles moving at the speed of
light can only be seen in certain directions. Thus, the classification of particles that move at the speed of light is much more baroque.
This is a big digression from my main quest here: explaining how people have tried to relate the octonions to the Standard Model. But it would be a shame not to make our results public, and now is a
good time.
Posted at 12:16 PM UTC |
Followups (34)
December 16, 2020
Octonions and the Standard Model (Part 10)
Posted by John Baez
The Dynkin diagram of $\mathrm{E}_6$ has 2-fold symmetry:
So, this Lie group has a nontrivial outer automorphism of order 2. This corresponds to duality in octonionic projective plane geometry! There’s an octonionic projective plane $\mathbb{O}\mathrm{P}^2$
on which $\mathrm{E}_6$ acts. But there’s also a dual octonionic projective plane $(\mathbb{O}\mathrm{P}^2)^*$. Points in the dual plane are lines in the original one, and vice versa. And these two
projective planes are not isomorphic as spaces on which the group $\mathrm{E}_6$ acts. Instead, there’s a bijection
$\alpha : \mathbb{O}\mathrm{P}^2 \to (\mathbb{O}\mathrm{P}^2)^*$
such that acting by $g \in \mathrm{E}_6$ and then applying $\alpha$ is the same as applying $\alpha$ and then acting by $g' \in \mathrm{E}_6$, where $g'$ is what you get when you apply the outer
automorphism to $g$.
Similarly, the group $\mathrm{E}_6$ acts on the exceptional Jordan algebra and its dual, but these are not isomorphic as representations of $\mathrm{E}_6$. Instead they’re only isomorphic up to an
outer automorphism.
Today I want to tell you about invariant structures on the exceptional Jordan algebra and its dual. But a lot of this stuff applies more generally.
Posted at 10:16 PM UTC |
Followups (9)
December 13, 2020
The Lie of “It’s Just Math”
Posted by Tom Leinster
Jade Master at Riverside has written a short, important and lucid article about military funding of math, The Lie of “It’s Just Math”, accompanied by a call to action:
Fellow mathematicians, it’s time to stop letting the military benefit from our work.
Military involvement in math is particularly an issue in applied category theory, and particularly an issue in the USA. But the principles that Jade pithily expresses are universal:
• The [US Department of Defense’s] real goal is not just the math you produce, they want to gain access to your mathematical community.
• Your math is not too abstract to be useful.
• The DoD wants to normalize themselves in your non-mathematical communities.
• The DoD will lie to you.
Mathematicians are generally highly reluctant to talk about the human impact of what we do and the choices we make. For that reason, we’re not very practised at it. But Jade’s article deserves wide
discussion, and I hope it gets it.
Posted at 5:23 PM UTC |
Followups (61)
December 10, 2020
Bernoulli Numbers and the J-homomorphism
Posted by John Baez
I’m planning to stop teaching at U. C. Riverside in June 2021. I’ll only be 60, but what’s the use of quitting work when you’re too old to have fun?
I want to spend more time doing research and writing expository papers and books, and I’ve saved up enough money to do this. I’ll still do serious work, like trying to save the planet with applied
category theory. But I’ll also delve into all sorts of puzzles that I haven’t had enough time for yet.
Here’s one. You may have heard about the funny way the number 24 shows up in the homotopy groups of spheres:
$\pi_{n+3} (S^n) \cong \mathbb{Z}_{24}$
whenever $n$ is big enough, namely $n \ge 5$. If you try to figure out where this comes from, you’re led back to a map
$S^7 \to S^4$
called the quaternionic Hopf fibration. This by itself doesn’t make clear where the 24 is coming from — but you can’t help but notice that when you pack equal-sized balls as densely as is known to be
possible in the quaternions, each one touches 24 others.
Coincidence? Maybe! But it’s also true that
$\pi_{n+7} (S^n) \cong \mathbb{Z}_{240}$
when $n$ is big enough. And if you try to figure out where this comes from, you’re led back to a map
$S^{15} \to S^8$
called the octonionic Hopf fibration. And you can’t help but notice that when you pack equal-sized balls as densely as possible in the octonions, each one touches 240 others!
Posted at 5:46 PM UTC |
Followups (23)
December 9, 2020
The Algebraic K-Theory of the Integers
Posted by John Baez
The category of groups $\mathbb{Z}^n$ and isomorphisms between these is symmetric monoidal under $\oplus$. You can build a space out of simplexes where the 0-simplexes are objects of this category,
the 1-simplexes are morphisms, the 2-simplexes are commutative triangles, the 3-simplexes are commutative tetrahedra, and so on forever. This space has an operation, coming from $\oplus$, that obeys
the commutative monoid axioms up to homotopy. If you ‘group complete’ this space by throwing in formal inverses, you get a space that’s an abelian group up to homotopy. It’s called the algebraic $K$
-theory spectrum of the integers.
The algebraic $K$-theory spectrum of the integers has homotopy groups $\pi_0 = \mathbb{Z}$, $\pi_1 = \mathbb{Z}/2$, $\pi_3 = \mathbb{Z}/48$, and so on. These groups are called the algebraic $K$
-theory groups of the integers, $K_n(\mathbb{Z})$.
Posted at 12:26 AM UTC |
Followups (13)
December 7, 2020
Applied Compositional Thinking for Engineers
Posted by John Baez
December 6, 2020
Mathematical Phantoms
Posted by John Baez
A ‘mathematical phantom’ is a mathematical object that doesn’t exist in a literal sense, but nonetheless acts as if it did, casting a spell on surrounding areas of mathematics. The most famous
example is the field with one element. Another is Deligne’s S[t], the symmetric group on $t$ elements, where $t$ is not a natural number. Yet another is G[3], a phantom Lie group related to G[2], the
automorphism group of the octonions.
What’s your favorite mathematical phantom? My examples are all algebraic. Does only algebra have enough rigidity to create the patterns that summon up phantom objects? What about topology or
combinatorics or analysis? Okay, G[3] is really a creature from homotopy theory, but of a very algebraic sort.
Last night I met another phantom.
Posted at 7:01 PM UTC |
Followups (67)
The Liquid Tensor Experiment
Posted by David Corfield
Peter Scholze has just published a challenge to the automated mathematical formalisation community in a post – Liquid tensor experiment – on Kevin Buzzard’s blog. Peter explains there the motivation
for the theorem he would like to be formalised, and his reasons to want a formal computer-checked proof.
I see a couple of Café interests intertwined here: formalisation in dependent type theory, and the nature of (mathematical) space.
Regarding the former, there was an intense discussion recently on MathOverflow arising from a comment on the advantages of dependent type theory by Kevin, who for some time has been promoting the
prover Lean as the best means to formalise the mathematics of typical mathematics departments.
In the comments to the main question, there is some discussion of why much more has been achieved in this regard by Lean compared with HoTT/Univalent Foundations approaches. We’ve heard plenty about
this latter approach at the Café, in particular about its use to develop a kind of synthetic mathematics, due to HoTT being the internal language of $(\infty, 1)$-toposes. At the end of the
references in the last link are some articles using modal HoTT, in particular relating to the cohesive modalities, as in Mike’s
Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28(6), (2018): pp. 856-941 (arXiv:1509.07584).
Posted at 4:36 PM UTC |
Followups (33)
December 4, 2020
Entropy and Diversity: The Axiomatic Approach
Posted by Tom Leinster
As Emily was kind enough to point out earlier, my new book is out on the arXiv!
It’s arXived by agreement with the wonderful Cambridge University Press, who will publish it in April 2021. You can pre-order it now, as the ideal festive gift for any friend who enjoys deferred
Posted at 2:29 PM UTC |
Followups (14)
Entropy and Diversity on the arXiv
Posted by Emily Riehl
There’s a new entry in today’s arXiv listings:
Entropy and Diversity: The Axiomatic Approach, a viii + 442 pages book to be published by Cambridge University Press in April 2021.
Congratulations Tom! This is really something.
And even more impressively, this is the second published book that Tom has made freely available on the arXiv! His wonderful Basic Category Theory is there as well.
Posted at 1:56 PM UTC |
Followups (9) | {"url":"https://classes.golem.ph.utexas.edu/category/2020/12/index.shtml","timestamp":"2024-11-04T06:07:55Z","content_type":"application/xhtml+xml","content_length":"96505","record_id":"<urn:uuid:577f0924-8b0d-489d-b9cb-a2d55f70559d>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00049.warc.gz"} |
Non-integrable KdV-like models: solitons, breathers, compactons and rogue waves
Add to your list(s) Download to your calendar using vCal
If you have a question about this talk, please contact nobody.
HY2W01 - Modulation theory and dispersive shock waves
We analyze the solutions of the KdV-like equation $ u_t u_xxx = 0 $, where the leading term is $ F (u) ∼ u^m $ or $ u|q $, with the rationales m and q. The well-known integrable KdV equations with m
= 2 or 3 are particular cases of the generalized equation. We found analytically travelling waves in the form of solitons with exponential tails, algebraic solitons and compactons for various values
of the exponents in the nonlinear term. Numerical simulations demonstrate the travelling wave stability and the weak inelasticity at the wave collisions. Breathers and rogue waves appeared
numerically in the random wave field described by this equation.
This talk is part of the Isaac Newton Institute Seminar Series series.
This talk is included in these lists:
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Suppose that Ally Financial Inc. issued a bond with 10 years
until maturity, a face value...
Suppose that Ally Financial Inc. issued a bond with 10 years until maturity, a face value...
Suppose that Ally Financial Inc. issued a bond with 10 years until maturity, a face value of $1000, and a coupon rate of 11% (annual payments). The yield to maturity on this bond when it was issued
was 12.
a. What was the price of this bond when it was issued? (Round to the nearest cent)
b. Assuming the yield to maturity remains constant, what is the price of the bond immediately before it makes it first coupon payment? (Round to nearest cent)
c. Assuming the yield of maturity remains constant, what is the price of the bond immediately after it makes its first coupon payment? Round to nearest cent)
Annual Coupon = $1000 *11% =$110
a) Price of the bond when issued
= present value of coupon payments + present value of principal repayment
b) Price of the bond immediately before it makes 1st coupon payment
= 110/0.12*(1-1/1.12^10)*1.12 + 1000/1.12^9
Price of the bond immediately after it makes 1st coupon payment
= 110/0.12*(1-1/1.12^9) + 1000/1.12^9 | {"url":"https://justaaa.com/finance/55520-suppose-that-ally-financial-inc-issued-a-bond","timestamp":"2024-11-07T23:22:54Z","content_type":"text/html","content_length":"41414","record_id":"<urn:uuid:43764287-87c3-4636-9485-91ecb7f8e59e>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00295.warc.gz"} |
:: Lecture 18
STAM101 :: Lecture 18 :: Factorial experiments
- factor and levels – types – symmetrical and asymmetrical – simple, main and interaction effects – advantages and disadvantages
Factorial Experiments: When two or more number of factors are investigated simultaneously in a single experiment such experiments are called as factorial experiments.
1. Factor: Factor refers to a set of related treatments. We may apply of different doses of nitrogen to a crop. Hence nitrogen irrespective of doses is a factor.
2. Levels of a factor: Different states or components making up a factor are known as the levels of that factor. eg different doses of nitrogen.
Types of factorial Experiment
A factorial experiment is named based on the number of factors and levels of factors. For example, when there are 3 factors each at 2 levels the experiment is known as 2 X 2 X 2 or 23 factorial
If there are 2 factors each at 3 levels then it is known as 3 X 3 or 32 factorial experiment.
• In general if there are n factors each with p levels then it is known as pn factorial experiment.
• For varying number of levels the arrangement is described by the product. For example, an experiment with 3 factors each at 2 levels, 3 levels and 4 levels respectively then it is known as 2 X 3
X 4 factorial experiment.
• If all the factors have the same number of levels the experiment is known as symmetrical factorial otherwise it is called as mixed factorial.
• Factors are represented by capital letters. Treatment combinations are usually by small letters.
• For example, if there are 2 varieties v0 and v1 and 2 dates of sowing d0 and d1 the treatment combinations will be
• vodo, v1do, v1do and v1d1.
Simple and Main Effects
Simple effect of a factor is the difference between its responses for a fixed level of other factors.
Main effect is defined as the average of the simple effects.
Interaction is defined as the dependence of factors in their responses. Interaction is measured as the mean of the differences between simple effects.
1. In such type of experiments we study the individual effects of each factor and their interactions.
2. In factorial experiments a wide range of factor combinations are used.
3. Factorial approach will result in considerable saving of the experimental resources, experimental material and time.
1. When number of factors or levels of factors or both are increased, the number of treatment combinations increases. Consequently block size increases. If block size increases it may be difficult
to maintain homogeneity of experimental material. This will lead to increase in experimental error and loss of precision in the experiment.
2. All treatment combinations are to be included for the experiment irrespective of its importance and hence this results in wastage of experimental material and time.
3. When many treatment combinations are included the execution of the experiment and statistical analysis become difficult.
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Erik D. Demaine
Paper by Erik D. Demaine
Erik D. Demaine, John Iacono, and Stefan Langerman, “Proximate Point Searching”, Computational Geometry: Theory and Applications, volume 28, number 1, May 2004, pages 29–40. Special issue of
selected papers from the 14th Canadian Conference on Computational Geometry, 2002.
In the 2D point searching problem, the goal is to preprocess n points P = {p[1], …, p[n]} in the plane so that, for an online sequence of query points q[1], …, q[m], it can quickly determined
which (if any) of the elements of P are equal to each query point q[i]. This problem can be solved in O(log n) time by mapping the problem to one dimension. We present a data structure that is
optimized for answering queries quickly when they are geometrically close to the previous successful query. Specifically, our data structure executes queries in time O(log d(q[i−1], q[i])), where
d is some distance function between two points, and uses O(n log n) space. Our structure works with a variety of distance functions. In contrast, it is proved that, for some of the most intuitive
distance functions d, it is impossible to obtain an O(log d(q[i−1], q[i])) runtime, or any bound that is o(log n).
This paper is also available from ScienceDirect.
The paper is 15 pages.
The paper is available in PostScript (438k), gzipped PostScript (168k), and PDF (171k).
Related papers:
PointSearching_CCCG2002 (Proximate Point Searching)
See also other papers by Erik Demaine. These pages are generated automagically from a BibTeX file.
Last updated July 23, 2024 by Erik Demaine. | {"url":"https://erikdemaine.org/papers/PointSearching_CGTA/","timestamp":"2024-11-04T15:19:43Z","content_type":"text/html","content_length":"5962","record_id":"<urn:uuid:aee59527-462f-4ed9-bd33-b6d602032649>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00832.warc.gz"} |
Uses Of Probability (10 Real Life Uses Of Probability) | jdmeducational
Uses Of Probability (10 Real Life Uses Of Probability)
Probability is a concept that helps us to figure out the chance that an event (or sequence of events) will happen. Probability can give us an idea of what to expect, which gives it many applications
in real life.
So, what are some uses of probability? Some uses of probability include: commute times, engineering, lending & debt, medicine, insurance & risk management, sales, stock market investing, traffic
accidents, weather prediction, and website conversions.
Of course, probability is just one tool that can help us to make decisions in business and in everyday life.
In this article, we’ll take a look at 10 uses of probability, along with some calculations to show how to apply this concept.
Let’s get started.
10 Uses Of Probability
Probability is used often in many different scenarios, including:
• Commute Times
• Engineering
• Lending & Debt
• Medicine
• Insurance & Risk Management
• Sales
• Stock Market Investing
• Traffic Accidents
• Weather Prediction
• Website Conversions
Let’s take a closer look at each one of these, starting with commute times.
Commute Times
At first, it might not seem that probability has much, if anything, to do with commute times. However, if you look closely, you can find a way to use probability to ensure that you are (almost)
never late.
You can use probability to minimize waiting time when you get to work early and days that you are late to work.
Example: Using Probability For Commute Times (Minimize Lateness)
You want to show up on time for work, but you don’t want to show up too early (after all, who wants to sit in the parking lot for an hour?)
So, for 20 work days, you record how many minutes your morning commute takes. You get the following commute times:
{19, 25, 24, 26, 18, 23, 22, 22, 24, 19, 20, 25, 24, 18, 22, 18, 19, 24, 25, 21}
On the best days, you can get to work in as little as 18 minutes, but you cannot count on that. However, on almost every day, you can get to work in 25 minutes or less.
In fact, there is only one day out of the past 20 days that it took you longer than 25 minutes to get to work (26 minutes on the fourth day).
So, if you leave yourself 25 minutes for the commute, you expect to get to work early or on time on 19 out of 20 days. This is a 95% probability (19 / 20 = 0.95 or 95%).
Even on the days when the commute takes a little longer, you should only be a minute or two late on 1 day out of 20.
By leaving 25 minutes for your commute, you know that you will be waiting at most 7 minutes in the parking lot (25 – 18 = 7, where 18 was your fastest commute time). That is only enough time to
listen to a couple of songs on the radio or read a quick article online.
An engineer wants to make sure that things work. Of course, there are no guarantees, but you can still use probability to create an acceptable chance that things will work as expected.
We can calculate the probability that a device will work after 5 years by combining probabilities for the parts.
Example: Probability In Engineering
For example, let’s say that a device use a specific part 10 times in different places. The probability of a given part failing before 5 years is 0.1% (so a 99.9% chance they will work for 5 years or
Assuming the parts fail independently, we can calculate the probability that all 10 parts in the device will work for at least 5 years:
• P(all 10 parts work for at least 5 years) = 0.999^10
This comes out to 0.990045, which means that there is a 99.0045% chance that all 10 parts in the device will still be working after 5 years. So, there is less than a 1% chance of failure within the
first 5 years.
Lending & Debt
Banks that make loans to people for homes or businesses want to know that they will be paid back most of the time. They might not want to make risky loans that have a high probability of default.
A bank can use probability to determine the risk of default for a given borrower.
Example: Probability In Banking
Let’s say that a bank wants to figure out the probability that a borrower who holds an MBA degree will default on a business loan.
That is, the bank wants to calculate the conditional probability P(A|B), where:
• A = the borrower defaults on the business loan
• B = the borrower holds an MBA degree
The formula for conditional probability is
where P(AnB) is the probability that both events A and B happen (an MBA holder who defaults on a business loan).
Based on past experience and all of their business borrower profiles, the bank has calculated the following probabilities:
So, we can calculate:
• P(A|B) = P(AnB)/P(B)
• P(A|B) = 0.1/0.25
• P(A|B) = 0.4
So, there is a 40% chance that an MBA holder will default on a business loan. The chance might be higher for someone without an MBA, so the MBA holder might be a less risky borrower.
Probability can also help doctors and patients to make important decisions about treatment. For example, if a patient has a condition that can be treated with surgery, he can compare probabilities
to decide whether to go forward with the surgery.
Probability can help doctors and patients to decide on a course of treatment.
If the probability of death from the untreated condition is 5% and the probability of death during surgery is 1%, then the patient is taking a smaller risk by getting the surgery than by going
Insurance & Risk Management
An insurance company can use probability to quantify risks. For example, they might want to compare the level of risk for an area that is prone to storms and flooding.
An insurance company can use probability to assess risk from severe storms and the flooding that may occur as a result.
Example: Probability In Insurance & Risk Management
Let’s say that in a given year, there is a 20% chance that a region will have a storm that is severe enough to cause flooding. If such a storm occurs, there is a 40% chance that flooding will
actually occur.
The insurance company can calculate probabilities as follows:
• Probability of a severe storm and a flood (heavy damage): 0.2*0.4 = 0.08 (8%)
• Probability of a severe storm but no flood (moderate damage): 0.2*0.6 = 0.12 (12%)
• Probability of no severe storm and thus no flood (light damage): 0.8 (80%)
Based on these probabilities and estimated damages in each case, the insurance company can charge a reasonable premium for customers in that region.
A salesman can also use probability to figure out how long it might take to make a sale.
A salesman can use probability to figure out how many calls it will take to make another sale.
Example: Probability In Sales
Let’s say that a salesman has a 4% chance of making a sale on a given sales call. He wants to figure out how long it will take to make 6 sales. (Assume the salesman can make 30 calls per day).
First, we set up the equation:
• (Number Of Sales Made) = (Probability Of Sale)*(Number Of Sales Calls)
Using the 4% close rate (0.04 as a decimal) and 5 desired sales, we get:
• 6 = 0.04*Number Of Sales Calls
• 6 / 0.04 = Number Of Sales Calls
• 150 = Number Of Sales Calls
So, the salesman needs to make 150 calls to make 5 sales. Since the salesman can make 30 calls per day, it will take 150 / 30 = 5 days to make the 6 sales.
Stock Market Investing
We can also use probability in stock market investing to decide whether to buy, hold, or sell shares of a particular company.
Probability can help us to find out the chances of success or failure for a company, which will affect stock price.
Example: Probability In Stock Market Investing
Let’s say that a company is trying to develop a new product for a given market (they will either succeed with the technology or fail – assume there is no partial success). Their target market may
grow, stagnate, or fail in the next year.
You think the following probabilities apply:
• Probability of successful product development: 80%
• Probability that the target market grows: 60%
• Probability that the target market stagnates: 30%
• Probability that the target market declines: 10%
If we assume that the chance of successful product development is independent of market conditions, then we can find the following probabilities:
Market Market Market
Product Grows Stagnant Declines
(60%) (30%) (10%)
Success 0.8*0.6 0.8*0.3 0.8*0.1
(80%) =0.48 =0.24 =0.08
(48%) (24%) (8%)
Failure 0.2*0.6 0.2*0.3 0.2*0.1
(20%) =0.12 =0.06 =0.02
(12%) (6%) (2%)
This table summarizes the probability of each outcome,
based on product success or failure and market conditions.
Traffic Accidents
Probability can help us to figure out the chance of a fatal accident on a given stretch of road.
Probability can help us to determine the chance of a fatal accident on a given stretch of road.
Example: Probability In Traffic Accidents
Let’s say that there is a 2% probability of an accident on a given day on a stretch of road. This means that we expect an accident every 50 days on this stretch of road.
Of the accidents that have occurred on that stretch of road, 4% have been fatal. This means that we expect 1 out of 25 accidents on this road to be fatal.
So, there is a 0.02*0.04 = 0.0008 (0.08%) chance of a fatal accident on this road on a given day.
This means that we would expect a fatal accident on this road once every 1250 days (3.42 years). (Note that 1250 is the reciprocal of 0.0008).
Weather Prediction
Probability is used in weather predictions to get a handle on the chance of precipitation. For example, there might be a 50% chance of precipitation on a given day, with the chance highest early in
the day.
Probability tells us the chance of rain, snow, and other weather events.
Website Conversions
Probability can also help you to figure out how many conversions (sales, email signups, etc.) to expect from your site visitors.
Probability can tell you the chance of making a sale of a book or product on your website. A small change can make a big difference in sales, so pay attention to how your site looks and feels!
Example: Probability In Website Conversions
Let’s say that 12% of the people who visit the homepage on your website will click on a link to visit the sales page for your book.
Also assume that 5% of the people who visit the sales page on your website will buy the book.
Then we can calculate the probability that a website visitor will buy your book as 0.12*0.05 = 0.006 (0.6%).
So, out of every 1000 website visitors, 6 of them will buy your book.
If you have 25,000 website visitors per month, then you would expect to sell:
• (Website Visitors)*(Conversion Rate)
• =(25,000)*(0.006) = 150 books
If each copy of your book sells for $25, then your total revenue for a month would be:
• (Number Of Books Sold)*(Book Price)
• =(150 books)*($25 per book)
• =$3750 per month
Now you know a little more about probability and how you might be able to use it to help you make decisions in business or in life.
You can check out my article on how statistics is used here.
You might also want to check out my article on the difference between probability & statistics.
You can learn how to use a Venn diagram to map out probabilities here.
I hope you found this article helpful. If so, please share it with someone who can use the information.
Don’t forget to subscribe to my YouTube channel & get updates on new math videos! | {"url":"https://jdmeducational.com/uses-of-probability-10-real-life-uses-of-probability/","timestamp":"2024-11-02T08:28:50Z","content_type":"text/html","content_length":"94256","record_id":"<urn:uuid:b868f71f-a8f3-42b7-9980-563d205b0b86>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00594.warc.gz"} |
RISC Activity Database
author = {Manfred Kerber and Colin Rowat and and Wolfgang Windsteiger},
title = {{Using Theorema in the Formalization of Theoretical Economics}},
booktitle = {{Intelligent Computer Mathematics}},
language = {english},
abstract = { Theoretical economics makes use of strict mathematical methods. For instance, games as introduced by von Neumann and Morgenstern allow for formal mathematical proofs for certain
axiomatized economical situations. Such proofs can---at least in principle---also be carried through in formal systems such as Theorema. In this paper we describe experiments carried through
using the Theorema system to prove theorems about a particular form of games called pillage games. Each pillage game formalizes a particular understanding of power. Analysis then attempts to
derive the properties of solution sets (in particular, the core and stable set), asking about existence, uniqueness and characterization. Concretely we use Theorema to show properties previously
proved on paper by two of the co-authors for pillage games with three agents. Of particular interest is some pseudo-code which summarizes the results previously shown. Since the computation
involves infinite sets the pseudo-code is in several ways non-computational. However, in the presence of appropriate lemmas, the pseudo-code has sufficient computational content that Theorema can
compute stable sets (which are always finite). We have concretely demonstrated this for three different important power functions. },
series = {Lecture Notes in Artificial Intelligence (LNAI)},
volume = {6824},
pages = {58--73},
publisher = {Springer},
isbn_issn = {ISSN 0302-9743},
year = {2011},
editor = {James H. Davenport and William M. Farmer and Florian Rabe and Josef Urban},
refereed = {yes},
length = {16},
conferencename = {CICM 2011},
url = {http://dx.doi.org/10.1007/978-3-642-22673-1_5} | {"url":"https://www3.risc.jku.at/publications/show-bib.php?activity_id=4370","timestamp":"2024-11-10T05:19:42Z","content_type":"text/html","content_length":"4615","record_id":"<urn:uuid:7b5be535-0929-46d8-9ea9-32df50a1ef32>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00737.warc.gz"} |
What is a 5-star rating? | Simplesat help center
What is a 5-star rating?
5-star ratings are 5-point rating scales that allow customers to submit feedback right after specific instances β such as availing a service, purchasing a product, or attending an event. This suits
a wide audience, since 5-star ratings are easy to understand visually.
This metric uses a scale of 1 to 5, where more stars means better feedback.
Calculating 5-star ratings
The 5-star rating is computed using a mean average: add the total of the star rating values, and divide by the total number of star ratings received.
You receive 10 star ratings, broken down as follows:
15 + 16 + 3 + 0 + 2 = 36 total stars β
36 total stars, divided by 10 ratings = 3.6 | {"url":"https://help.simplesat.io/en/articles/7339549-what-is-a-5-star-rating","timestamp":"2024-11-02T17:49:34Z","content_type":"text/html","content_length":"63071","record_id":"<urn:uuid:c3b72d2a-ac9b-407a-9cfb-a484717a1bec>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00815.warc.gz"} |
5.20 Collinearity | Introduction to Regression Methods for Public Health Using R
5.20 Collinearity
An important step in the process of fitting an MLR is checking for collinearity between predictors. This step was not needed in SLR because there was only one predictor.
Here’s an extreme example which helps illustrate the concept. Suppose you want to know the relationship between fasting glucose and weight, and you fit a regression model that includes both weight in
kilograms and weight in pounds. Clearly, these two predictors are redundant since they only differ in their units. With either one in the regression model, the other adds no new information. That is
called perfect collinearity and it is mathematically impossible to include both in the model – you cannot estimate distinct regression coefficients for both predictors.
You also can have approximate collinearity. Suppose instead of weight in kilograms and weight in pounds, you have weight in kilograms and body mass index (BMI = weight/height^2). While weight and BMI
are not exactly redundant, they are highly related. In this case, it is mathematically possible to include both in the model, but their regression coefficients are difficult to estimate accurately
(they will have large variances) and will be difficult to interpret. The regression coefficient for weight is the difference in mean outcome associated with a 1-unit difference in weight while
holding BMI constant. But in order to vary weight while holding BMI constant you have to vary height, as well. The “effect of greater weight adjusted for BMI” is therefore actually the “effect of
greater weight and greater height”, a combination of the effects of weight and height. In general, when two or more predictors are collinear, the interpretation of each while holding the other(s)
fixed becomes more complicated.
Two predictors are perfectly collinear if their correlation is –1 or 1 (e.g., weight in kilograms and weight in pounds). The term “collinear” comes from the fact that one predictor (\(X_1\)) can be
written as a linear combination of the other (\(X_2\)) as \(X_1 = a + b X_2\). If you were to plot the pairs of points \((x_{i1}, x_{i2})\) for all the cases, they would fall exactly on a line.
Examples of perfect collinearity include:
• Two predictors that are exactly the same \((X_1 = X_2)\);
• One predictor is twice the other \((X_1 = 2 X_2)\); and
• One predictor is 3 less than twice the other \((X_1 = -3 + 2 X_2)\).
If two predictors are perfectly collinear, then they are completely redundant for the purposes of linear regression. You could put either one in the model and the fit would be identical. Unless they
are exactly the same (\(X_1 = X_2\)), you will get a different slope estimate using one or the other, but their p-values will be identical. If you put them both in the regression model at the same
time, however, R will estimate a regression coefficient for just one of them, as shown in the following code.
# To illustrate...
# Generate random data from a standard normal distribution
y <- rnorm(100)
x <- rnorm(100)
# Create z perfectly collinear with x
z <- -3 + 2*x
lm(y ~ x + z)
## Call:
## lm(formula = y ~ x + z)
## Coefficients:
## (Intercept) x z
## -0.0776 0.1418 NA
The problem is actually worse, however, for approximate collinearity because, if you put both in the model, R will estimate a regression coefficient for each but will have trouble estimating their
unique contributions and you could get strange results, such as one large negative and one large positive estimated \(\beta\) and large standard errors for the estimates (implying that even with only
a slightly different sample of individuals you may get very different estimates of the \(\beta\)s).
Two predictors that are approximately collinear are essentially measuring the same thing and so should have about the same association with the outcome. But when each is adjusted for the other, the
results for each can be quite unstable and quite different from each other. In the example below, the second predictor is identical to the first except for having some random noise added. Compare
their unadjusted and adjusted associations with the outcome.
# Generate random data but make the collinearity approximate
y <- rnorm(100)
x <- rnorm(100)
# Create z approximately collinear with x
z <- x + rnorm(100, sd=0.05)
# Each predictor alone
round(summary(lm(y ~ x))$coef, 4)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.0776 0.1020 -0.7601 0.4490
## x 0.1418 0.1087 1.3045 0.1951
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.0783 0.1021 -0.767 0.4449
## z 0.1394 0.1079 1.292 0.1995
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.0742 0.1034 -0.7176 0.4747
## x 0.7256 2.3995 0.3024 0.7630
## z -0.5799 2.3811 -0.2436 0.8081
Each predictor individually has a regression slope of about 0.14 with a standard error of about 0.11. However, when attempting to adjust for each other, their slopes are now in opposite directions
and their standard errors are around 2.4! Again, a large standard error means that if you happened to have drawn a slightly different sample, you might end up with very different regression
coefficient estimates in the model with both x and z – the results are not stable. If you want to verify this, re-run the above code multiple times starting after set.seed() (so it is not reset to
the same value each time), and compare the results.
Going back to our example of weight and BMI, we can see that in our NHANES fasting subsample dataset these two predictors are highly collinear (Figure 5.44). The points do not fall exactly on a line,
but they are generally close. Thus, weight and BMI seem to be approximately collinear. This makes sense because BMI is a measure of body mass computed as weight normalized by stature.
All of the above examples were of pairwise collinearity. You can have collinearity between groups of predictors, as well, if some linear combination of one group of predictors is approximately equal
to a linear combination of some other predictors. Such situations are harder to detect visually. However, in the following section, we will use a numeric diagnostic tool to help us detect
collinearity even among groups of predictors.
5.20.1 Diagnosis of collinearity
The statistic we will use to diagnose collinearity is the variance inflation factor (VIF). For each predictor, the VIF measures how much the variance of the regression coefficient estimate is
inflated due to correlation between that predictor and the others. Compute VIFs using the car::vif() function (Fox, Weisberg, and Price 2023; Fox and Weisberg 2019).
5.20.1.1 VIFs when all predictors are continuous
Example 5.6: Use VIFs to evaluate the extent of collinearity in the regression of the outcome systolic blood pressure (sbp) on the predictors age (RIDAGEYR), weight (BMXWT), BMI (BMXBMI), and height
(BMXHT) using NHANES 2017-2018 examination data from adults (nhanes1718_adult_exam_sub_rmph.Rdata). First, fit the regression model. Second, call car::vif() with the regression fit as the argument.
nhanes <- nhanes_adult_exam_sub # Shorter name
fit.ex5.6 <- lm(sbp ~ RIDAGEYR + BMXWT + BMXBMI + BMXHT, data = nhanes)
## RIDAGEYR BMXWT BMXBMI BMXHT
## 1.014 88.835 69.934 18.764
One of the VIFs is near 1, while the other three are much larger. How do we interpret the magnitude of VIFs? How large is “too large”? Unfortunately, there is no clear-cut answer to that question. We
do know that a value of 1 is ideal – that indicates that the variance of that predictor’s regression coefficient estimate is not inflated at all by the presence of other predictors. Values above 2.5
may be of concern (Allison 2012), and values above 5 or 10 are indicative of a more serious problem. In this example height, weight, and BMI clearly have serious collinearity issues; their variances
are being inflated 19- to 89-fold by their collective redundancy.
5.20.1.2 Generalized VIFs when at least one predictor is categorical
Recall that a categorical predictor with \(L\) levels will be entered into a model as \(L-1\) dummy variables (\(L-1\) vectors of 1s and 0s). Why \(L-1\)? Because if you included all \(L\) of them
the vectors would sum up to a vector of all 1s (since every observation falls in exactly one category) and that would be perfect collinearity. So one is always left out (the reference level).
Example 5.6 examined collinearity among predictors that were all continuous. What happens if there are any categorical predictors? It turns out that the usual VIFs computed on a model with a
categorical predictor will differ depending on which level is designated as the reference level. In addition to getting inconsistent results, we are not interested in the VIF for each individual
level of a categorical predictor, but rather of the predictor as a single entity. Fortunately, car::vif() automatically gives us a consistent VIF, called the generalized VIF (GVIF) (Fox and Monette
1992), that is the same for each categorical predictor no matter the reference level, as long as the predictor is coded as a factor.
Example 5.7: Evaluate the extent of the collinearity in the regression of systolic blood pressure on the predictors age, BMI, height, and smoking status using our NHANES 2017-2018 examination
subsample of data from adults (same dataset as in Example 5.6).
## GVIF Df GVIF^(1/(2*Df))
## RIDAGEYR 1.052 1 1.026
## BMXBMI 1.006 1 1.003
## BMXHT 1.051 1 1.025
## smoker 1.081 2 1.020
The generalized VIF is found in the GVIF column. The GVIF^(1/(2*Df)) column is the adjusted generalized standard error inflation factor (aGSIF) and is equal to the square-root of GVIF for continuous
predictors and categorical predictors with just two levels (since, for those, Df = 1). Fox and Monette (1992) recommend using the aGSIF, however, since for categorical predictors with more than two
levels it adjusts for the number of levels allowing comparability with the other predictors. A consequence is that when using aGSIF, we must take the square-root of our rules of thumb for what is a
large value – aGSIF values above \(\sqrt{2.5}\) (1.6) may be of concern, and values above \(\sqrt{5}\) or \(\sqrt{10}\) (2.2 or 3.2) are indicative of a more serious problem. Alternatively, you could
square the aGSIF values and compare them to our original rule of thumb cutoffs.
Thus, the aGSIF for the three-level variable “smoker” is 1.02. In this example, the predictors do not exhibit strong collinearity (all the aGSIFs are near 1).
Run the code below if you would like to verify that the usual VIFs depend on which level is left out as the reference level but the GVIFs and aGSIFs remain unchanged.
# VIFs depend on the reference level
tmp <- nhanes %>%
mutate(smoker1 = as.numeric(smoker == "Never"),
smoker2 = as.numeric(smoker == "Past"),
smoker3 = as.numeric(smoker == "Current"))
fit_ref1 <- lm(sbp ~ RIDAGEYR + BMXBMI + BMXHT + smoker2 + smoker3, data = tmp)
fit_ref2 <- lm(sbp ~ RIDAGEYR + BMXBMI + BMXHT + smoker1 + smoker3, data = tmp)
fit_ref3 <- lm(sbp ~ RIDAGEYR + BMXBMI + BMXHT + smoker1 + smoker2, data = tmp)
# GVIF and aGSIF do not depend on the reference level
car::vif(lm(sbp ~ RIDAGEYR + BMXBMI + BMXHT + smoker, data = nhanes))
tmp <- nhanes %>%
mutate(smoker = relevel(smoker, ref = "Past"))
car::vif(lm(sbp ~ RIDAGEYR + BMXBMI + BMXHT + smoker, data = tmp))
tmp <- nhanes %>%
mutate(smoker = relevel(smoker, ref = "Current"))
car::vif(lm(sbp ~ RIDAGEYR + BMXBMI + BMXHT + smoker, data = tmp))
# (results not shown)
5.20.1.3 VIFs when there is an interaction or polynomial terms
Collinearity between terms involved in an interaction can be ignored (Allison 2012). This applies also to collinearity between terms that together define a polynomial curve (e.g., \(X\), \(X^2\),
etc.). In general, centering continuous predictors will reduce the collinearity. However, the sort of collinearity that is removed by centering actually has no effect on the fit of the model, only on
the interpretation of the intercept and the main effects of predictors involved in the interaction. If you have interactions in the model, or multiple terms defining a polynomial, compute the VIFs or
aGSIFs using a model without the interactions or higher order polynomial terms. Alternatively, instead of entering a polynomial using individual terms, use the poly() function. For example, for a
quadratic in BMXWT, instead of entering BMXWT + I(BMWXT^2), enter poly(BMXWT, 2) into lm(). car::vif() will then return one aGSIF value for the entire polynomial; note however, that the way
polynomials are parameterized with poly() is different than in our examples where we entered terms individually.
5.20.1.4 VIF summary
• If all predictors in a model are continuous and/or binary (two levels) then car::vif() returns VIFs and our rule of thumb for what is large is that values above 2.5 may be of concern and values
above 5 or 10 are indicative of a more serious problem.
• If any predictors in a model are categorical with more than two levels then car::vif() returns both GVIF and aGSIF values. Use the aGSIF values to evaluate collinearity since that allows
predictors with different Df (different number of terms in the model) to be comparable. For aGSIFs, our rule of thumb for what is large is that values above 1.6 may be of concern and values above
2.2 or 3.2 are indicative of a more serious problem.
• Our rule of thumb regarding what is a large amount of inflation is arbitrary. It is useful as a guide, but there is no requirement to apply it strictly.
• If you have interactions in the model, or multiple terms defining a polynomial, compute the VIFs or aGSIFs using a model without the interactions or higher order polynomial terms. Alternatively,
for polynomials, use poly() when fitting the model.
5.20.2 Impact of collinearity
When there is perfect collinearity, R will drop out predictors until that is no longer the case and so there is no impact on the final regression results. When there is approximate collinearity,
however, there may be inflated standard errors and difficulty interpreting coefficients.
Example 5.7 (continued): It is reasonable to guess that the main driver of the collinearity is the strong correlation between weight and BMI. Compare the regression coefficients, their standard
errors, and their p-values with and without weight in the model. First, create a complete-case dataset so differences between the models are not due to having different observations.
# Create a complete-case dataset so both models use the same sample
sbpdat <- nhanes %>%
select(sbp, RIDAGEYR, BMXWT, BMXBMI, BMXHT) %>%
# With weight in the model
fit_wt <- lm(sbp ~ RIDAGEYR + BMXWT + BMXBMI + BMXHT, data = sbpdat)
# After dropping weight due to high collinearity
fit_nowt <- lm(sbp ~ RIDAGEYR + BMXBMI + BMXHT, data = sbpdat)
Table 5.4: Demonstrating the impact of collinearity by
comparing the models with and without weight
Term B SE p VIF B SE P VIF
Intercept 137.34 36.41 <.001 73.52 8.96 <.001
Age 0.4327 0.0292 <.001 1.01 0.4348 0.0292 <.001 1.01
Weight 0.3707 0.2050 .071 88.8 – – –
BMI -0.5093 0.5782 .379 69.9 0.5287 0.0692 <.001 1.00
Height -0.3029 0.2169 .163 18.8 0.0787 0.0505 .119 1.01
Table 5.4 shows how the results differ after removing weight from the model. As expected, since its VIF was near 1, the “Age” row does not change much. Note in particular that, up to four decimal
places, its standard error (SE) is exactly the same. This is what it means for a VIF to be 1 – that the variance (the square of the SE) is inflated by a factor of 1, in other words not inflated at
all, by the presence of the other predictors.
Compare this, however, to the rows for “BMI” and “Height”. After removing weight from the model, their estimates change drastically; both change direction, going from negative to positive. Also,
their standard errors decrease dramatically, indicating that they are more precisely estimated. The squares of the ratios of their SEs with and without weight in the model are approximately the same
as their VIFs in the model that includes weight.
## [1] 69.74
## [1] 18.49
The reason the variances decrease so much is that the definitions of “BMI holding other predictors fixed” and “height holding other predictors fixed” change after removing weight, with the old
quantities (effects of BMI and height when holding age and weight constant) being much harder to estimate accurately (and harder to interpret) than the new (effects of BMI and height when holding age
constant). Also, whereas “BMI when holding age, weight, and height constant” was not statistically significant (p = .379), “BMI when holding age and height constant” is (p <.001). Finally, after
removing weight from the model, the VIFs for the remaining predictors are all near 1.
5.20.3 Potential solutions for collinearity
There are multiple ways to deal with collinearity:
• Remove one or more predictors from the model, or
• Combine predictors (e.g., average, sum, difference).
One could also try a biased regression technique such as LASSO or ridge regression. Those methods are beyond the scope of this text but see, for example, the R package glmnet (Friedman, Hastie, and
Tibshirani 2010)).
5.20.3.1 Remove predictors to reduce collinearity
Ultimately, you want a set of predictors that are not redundant. One solution is to remove predictors suspected of being problematic, one at a time, and see if the VIFs become smaller.
Before removing any predictors, check the extent of missing data in the dataset. On one hand, when comparing VIFs between models with different predictors, each model should be fit to the same set of
observations, which means starting with a complete case dataset. On the other hand, one or more predictors might have a much larger number of missing values than the others. In that case, it might be
best to remove the predictors with a lot of missing values first, create a complete case dataset based on the remaining predictors, and then compare VIFs between models with different predictors.
Example 5.7 (continued): Remove one predictor at a time and see how the VIFs change. Use the complete case dataset created above so any differences are due to removing predictors rather than
differing sets of observations with non-missing values.
## RIDAGEYR BMXWT BMXBMI BMXHT
## 1.014 88.835 69.934 18.764
## RIDAGEYR BMXBMI BMXHT
## 1.012 1.000 1.013
## RIDAGEYR BMXWT BMXHT
## 1.012 1.271 1.284
## RIDAGEYR BMXWT BMXBMI
## 1.010 4.794 4.786
With all four predictors in the model, BMI, weight, and height each exhibit high collinearity. After removing them one at a time we see that, in fact, weight and BMI are the problem variables (the
VIFs are only large when both remain in the model). In this example, removing weight from the model best resolved the collinearity issue (the VIFs are smallest when weight was removed). In other
examples, you may have to remove more than one predictor to obtain VIFs that are all small.
5.20.3.2 Combine predictors to reduce collinearity
Simply removing predictors is often the best solution. At other times, you really would like to keep all the predictors. In that case, combining them in some way may work. Common methods of combining
predictors include taking the average, the sum, or the difference.
Example 5.8: Using the Natality dataset (see Appendix A.3), examine the extent of collinearity in the regression of the outcome birthweight (DBWT, g) on the ages of the mother (MAGER) and father
(FAGECOMB) and resolve by combining predictors.
fit.ex5.8 <- lm(DBWT ~ MAGER + FAGECOMB, data = natality)
round(summary(fit.ex5.8)$coef, 4)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3168.469 75.620 41.8999 0.0000
## MAGER 5.769 3.767 1.5317 0.1258
## FAGECOMB -2.519 3.067 -0.8215 0.4115
## MAGER FAGECOMB
## 2.32 2.32
Ignoring statistical significance, this model implies that (1) for a given father’s age, older mothers have children with greater birthweight and (2) for a given mother’s age, older fathers have
children with lower birthweight. In this particular dataset, the two predictors are not highly collinear, but their VIFs are close to the rule of thumb cutoff of 2.5. We will attempt to reduce their
collinearity by combining them to form two predictors that are less collinear.
Replacing a pair of correlated predictors \(X_1\) and \(X_2\) with their average \(Z_1 = (X_1 + X_2)/2\) and difference \(Z_2 = X_1 - X_2\) may result in a pair of less correlated predictors. In this
example, the transformed predictors would be interpreted as “average parental age” and “parental age difference”.
natality <- natality %>%
mutate(avg_parent_age = (FAGECOMB + MAGER)/2,
parent_age_difference = FAGECOMB - MAGER)
fit.ex5.8_2 <- lm(DBWT ~ avg_parent_age + parent_age_difference, data = natality)
round(summary(fit.ex5.8_2)$coef, 4)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3168.469 75.620 41.900 0.0000
## avg_parent_age 3.250 2.483 1.309 0.1908
## parent_age_difference -4.144 3.202 -1.294 0.1958
## avg_parent_age parent_age_difference
## 1.099 1.099
Ignoring statistical significance, these results imply that (1) for a given age difference, older parents have children with greater birthweight and (2) for a given average parental age, parents with
a larger age difference have children with lower birthweight. By combining the predictors, we have reduced collinearity while retaining the ability to draw conclusions based on the combination of
ages of both parents.
In other cases, you might combine the collinear predictors into a single predictor by replacing them with their sum or average. For example, if you have a set of items in a questionnaire that all are
asking about the same underlying construct, each on a scale from, say, 1 to 5 (e.g., Likert-scale items), you may be able to average them together to produce a single summary variable. If taking this
approach, be sure all the items are on the same scale and if they are not then standardize them before summing or averaging. Also, make sure the ordering of responses has the same meaning – sometimes
one item is asking about agreement with the underlying construct and another is asking about disagreement, resulting in a low score on one item having the same meaning as a high score on another. If
you have items with different response orderings, change some of them until all are in the same direction. For example, you could reverse the ordering of items asking about disagreement (e.g.,
replace a 1 to 5 scale with scores of 5 to 1) so they match items asking about agreement. While summing or averaging variables may work to reduce collinearity, it is important to also assess whether
such a sum is valid using methods such as Cronbach’s alpha and factor analysis (beyond the scope of this text but see, for example, ?psy::cronbach (Falissard 2022) and ?factanal).
In general, take time to think about how your candidate predictors are related to each other and to the outcome. Are some completely or partially redundant (measuring the same underlying concept)?
That will show up in the check for collinearity. Remove or combine predictors in a way that aligns with your analysis goals.
———. 2012. “When Can You Safely Ignore Multicollinearity?” statisticalhorizons.com/multicollinearity; Statistical Horizons.
Falissard, Bruno. 2022.
Psy: Various Procedures Used in Psychometrics
Fox, John, and Georges Monette. 1992.
“Generalized Collinearity Diagnostics.” Journal of the American Statistical Association
87 (417): 178–83.
Fox, John, and Sanford Weisberg. 2019. An R Companion to Applied Regression. 3rd ed. Los Angeles: Sage Publications, Inc.
Fox, John, Sanford Weisberg, and Brad Price. 2023.
Car: Companion to Applied Regression
Friedman, Jerome H., Trevor Hastie, and Rob Tibshirani. 2010.
“Regularization Paths for Generalized Linear Models via Coordinate Descent.” Journal of Statistical Software
33: 1–22. | {"url":"https://www.bookdown.org/rwnahhas/RMPH/mlr-collinearity.html","timestamp":"2024-11-02T18:09:41Z","content_type":"text/html","content_length":"123380","record_id":"<urn:uuid:4c4d50a7-801b-4cc2-b78b-a4ae8830bfd4>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00032.warc.gz"} |
Actually Applicable Application Problems and Brainteasers/Subtraction Trick for Common Factors - Wikibooks, open books for an open world
Rather than a real-world situation, this is a calculation trick that you can use to make other problems easier. Because subtraction is easier than division, it's Actually Applicable in the sense that
it takes something you have to do and makes it easier.
1. Subtract the smaller number from the bigger number. Now you have three numbers.
2. Cross out the biggest of the three numbers. Now you have two numbers.
3. As long as those two numbers are different, take them back to step 1.
4. If those two numbers are actually the same, they are your Greatest Common Factor. All of its factors are other common factors of the two numbers you started with.
A note about why it works
Factors of a number are smaller numbers that you can skip-count by and get the original number.
A common factor of two numbers is a factor of both.
If you skip-count by the common factor and get to two particular other numbers, the difference between them must be some number of steps of that factor. In other words, the difference between those
two numbers is another number with the same common factor.
Smaller numbers are easier to work with, so keep going as long as you can.
Simplifying Fractions
One step in simplifying fractions is finding a common factor of the numerator and denominator. Find a common factor of 4 and 18 to simplify 4/18.
Make Your Own Problem
Pick any two numbers you are interested in, or are using in a situation, and find their common factors. | {"url":"https://en.m.wikibooks.org/wiki/Actually_Applicable_Application_Problems_and_Brainteasers/Subtraction_Trick_for_Common_Factors","timestamp":"2024-11-06T21:08:06Z","content_type":"text/html","content_length":"28439","record_id":"<urn:uuid:ce930733-9ee6-4801-aa90-77e2c4b754e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00377.warc.gz"} |
How To Draw Bending Moment Diagram
How To Draw Bending Moment Diagram - Web bending moment diagram this is a graphical representation of the variation of the bending moment on a segment or the entire length of a beam or frame. Web
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Designing Effective Evolutionary Computations
Created by W.Langdon from gp-bibliography.bib Revision:1.8010
□ author = "Kumar H. Chellapilla",
□ title = "Designing Effective Evolutionary Computations",
□ school = "Electrical Engineering, University of California, San Diego",
□ year = "2005",
□ address = "USA",
□ keywords = "genetic algorithms, genetic programming",
□ isbn13 = "9780542572180",
□ size = "606 pages",
□ abstract = "Evolutionary algorithms offer a practical approach to solving difficult real-world problems. In many problem domains, these are the only possible approaches with potential for
effectively searching through complex solution spaces. For novel problem domains wherein previous research efforts are sparse or problem domain expertise is in its infancy, evolutionary
algorithms offer strong alternatives for exploring the solution spaces and also gaining insights into effectively solving the problem.
The principal roadblock in conventional practice is the lack of a specific approach which permits one to simultaneously control an algorithm's representation, population variation operators
and population selection operators. An approach based on mathematically sound principles is adopted in this thesis to provide asymptotic guarantees on evolutionary algorithm performance
followed by useful real-time methods for improving the rate of convergence. In particular, the evolutionary algorithm is decomposed into its constituent representation, population variation,
and population selection operators. The population variation operators are further broken down into solution variation operators. Each component is independently analysed without being
constrained by an overall architecture for the evolutionary algorithm.
Each component presents several alternatives that can be chosen independently to control desired properties of the evolutionary algorithm. A new mathematical model for analysing evolutionary
algorithms is developed, and necessary and sufficient conditions on the variation and selection operators for asymptotic convergence are derived. Fitness distributions and fitness
distribution feature based heuristics are presented to improve the rate of convergence of an evolutionary algorithm. This thesis also presents a wide array of empirical results to demonstrate
the utility, effectiveness, and applicability of the new theory. Within the new framework, evolutionary algorithms are applied to solve real, discrete and mixed parameter optimization
problems. Evolutionary algorithms that guarantee asymptotic convergence are designed to solve problems involving structures such as parse trees and finite state machines. Co-evolutionary
algorithms are designed to evolve an expert checkers player that rated 2045 against human checkers players. Fitness distribution heuristics are used to tune an evolutionary algorithm for
improved rate of convergence for solving the travelling salesman problem.",
□ notes = "Supervisor Anthony Sebald
UMI Microform 3208643",
Genetic Programming entries for Kumar Chellapilla | {"url":"https://gpbib.cs.ucl.ac.uk/gp-html/Chellapilla_thesis.html","timestamp":"2024-11-04T00:57:53Z","content_type":"text/html","content_length":"4914","record_id":"<urn:uuid:c1504ed4-2a2b-4cf6-a774-01f1b824e7a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00350.warc.gz"} |
Partial Fractions | Brilliant Math & Science Wiki
Partial fraction decomposition is a technique used to write a rational function as the sum of simpler rational expressions.
\[\frac{2}{x^2-1} \Rightarrow \frac{1}{x-1} - \frac{1}{x+1}.\]
Partial fraction decomposition is a useful technique for some integration problems involving rational expressions. Partial fraction decomposition is also useful for evaluating telescoping sums. It is
the basis for a proof of Euler's formula by finding the antiderivative of a rational expression in two different ways.
Main Article: Partial Fractions - Linear Factors
A partial fraction decomposition has linear factors when the denominator can be factored into linear polynomials, each of which has multiplicity 1.
Find the partial fraction decomposition of the following rational expression:
The denominator can be factored:
\[x^2-x-6 = (x+2)(x-3).\]
Then the form of the partial fraction decomposition is
\[\frac{2x+1}{x^2-x-6} = \frac{A}{x+2}+\frac{B}{x-3}.\]
Solving for the coefficients gives
\[ \frac{2x+1}{x^2-x-6} &= \frac{A(x-3)+B(x+2)}{(x+2)(x-3)}\\\\\\ 2x+1 &= A(x-3)+B(x+2) \\ &= (A+B)x+(2B-3A) \\\\ A+B &= 2 \\ 2B-3A &= 1. \]
Solving this system of equations gives \(A=\frac{3}{5}\) and \(B=\frac{7}{5}.\) Then the partial fraction decomposition is
\[\frac{2x+1}{x^2-x-6}=\frac{1}{5}\left(\frac{3}{x+2}+\frac{7}{x-3}\right).\ _\square\]
Main Article: Partial Fractions - Cover Up Rule
The cover-up rule is a technique to efficiently compute the coefficients of a partial fraction decomposition with linear factors.
Find the partial fraction decomposition of the following rational expression:
The denominator can be factored:
Then the partial fraction decomposition form is
To solve for \(A,\) "cover up" the \(x-3\) factor and substitute \(x=3\) into the original expression:
\[A = \frac{2x}{x-6} = \frac{2\cdot 3}{3-6} = -2.\]
To solve for \(B,\) "cover up" the \(x-6\) factor and substitute \(x=6\) into the original expression:
\[B = \frac{2x}{x-3} = \frac{2\cdot 6}{6-3} = 4.\]
Then the partial fraction decomposition is
\[\frac{2x}{x^2-9x+18}=\frac{4}{x-6}-\frac{2}{x-3}.\ _\square\]
Main Article: Partial Fractions - Repeated Factors
A partial fraction has repeated factors when one of the factors has multiplicity greater than 1, in other words, when one of the factors is raised to a power 2 or greater.
Find the partial fraction decomposition of the following rational expression:
The denominator factors as a perfect square:
Then the partial fraction decomposition form is
Solve for the coefficients:
\[ \frac{x-4}{x^2-10x+25} &= \frac{A(x-5)+B}{(x-5)^2} \\\\\\ x-4 &= A(x-5)+B \\ x-4 &= Ax +(B-5A)\\\\ 1 &= A \\ -4 &= B-5A. \]
This gives \(A=1\) and \(B=1\) as the coefficients. Then the partial fraction decomposition is
\[\frac{x-4}{x^2-10x+25} = \frac{1}{x-5}+\frac{1}{(x-5)^2}.\ _\square\]
Main Article: Partial Fractions - Irreducible Quadratics
A partial fraction decomposition has irreducible quadratics when one of the factors is a quadratic that does not have rational roots.
Find the partial fraction decomposition of the following rational expression:
The denominator can be factored as a difference of cubes:
The quadratic has complex roots, so it cannot be factored any further. Then the partial fraction decomposition form is
Solving for the coefficients gives
\[ \frac{1}{x^3-1} &= \frac{A(x^2+x+1)+(Bx+C)(x-1)}{(x-1)(x^2+x+1)}\\\\\\ 1 &= A(x^2+x+1)+(Bx+C)(x-1) \\ 1 &= Ax^2+Ax+A+Bx^2+(C-B)x-C \\ 1 &= (A+B)x^2+(A+C-B)x+(A-C) \\\\ A+B &= 0 \\ A+C-B &= 0 \
\ A-C &= 1. \]
Solving this system of equations gives \(A=\frac{1}{3},\) \(B=-\frac{1}{3},\) and \(C=-\frac{2}{3}.\) Then partial fraction decomposition is
\[\frac{1}{x^3-1}=\frac{1}{3}\left(\frac{1}{x-1}-\frac{x+2}{x^2+x+1}\right).\ _\square\]
Integration with Partial Fractions
Main Article: Integration with Partial Fractions
Partial fraction decomposition is often used to find integrals of rational functions. It is useful when the \(u\)-substitution technique does not work.
Find the indefinite integral
First note that there is no obvious \(u\)-substitution that can be done to simplify the integral. Instead, use partial fraction decomposition to write the expression as the sum of two rational
Then the integral becomes
\[ \int{\frac{dx}{x^2+3x+2}} &= \int{\frac{dx}{x+1}}-\int{\frac{dx}{x+2}} \\ \\ &=\ln|x+1|-\ln|x+2|+C, \]
where \(C\) is the constant of integration. \(_\square\)
Telescoping Sums with Partial Fractions
Main Article: Telescoping Series - Sum
A series of rational expressions can sometimes contain a hidden telescoping sum. Using partial fraction decomposition can often reveal this telescoping sum so that evaluating the sum becomes much
Evaluate the following sum:
At first glance, it's difficult to discern any pattern in the terms of the series:
Performing partial fraction decomposition on the rational expression gives
Then the sum becomes
From writing out the terms of the sum, a pattern emerges:
It's clear that many of the terms will cancel out. The exact value of the series can be found by re-writing the sums:
\[\begin{array}{ccccc} \sum\limits_{k=1}^{40}\left(\frac{1}{k+1}-\frac{1}{k+3}\right) & = & \sum\limits_{k=2}^{41}{\frac{1}{k}} & - & \sum\limits_{k=4}^{43}{\frac{1}{k}} \\ \\ & = & \frac{1}{2}+\
frac{1}{3}+\sum\limits_{k=4}^{41}{\frac{1}{k}} & - & \sum\limits_{k=4}^{41}{\frac{1}{k}}-\frac{1}{42}-\frac{1}{43} \\ \\ & = & \frac{1}{2}+\frac{1}{3} & - & \frac{1}{42}-\frac{1}{43} \\ \\ & = &
\frac{710}{903}.\ _\square \end{array}\]
Partial fraction decomposition is the basis for a proof of Euler's formula.
Euler's Formula:
\[e^{i\theta}=\cos{\theta}+i\sin{\theta}.\ _\square\]
The following proof uses integration, trigonometric identities, and logarithmic identities.
Consider the rational expression
Now consider the antiderivative of this expression. One possible method to obtain the antiderivative of this expression is to apply trigonometric substitution. This method yields
\[\int{\frac{\mathrm{d} x}{1+x^2}}=\arctan{x} + \text{C}_{1}, \qquad (1)\]
where \(\text{C}_{1}\) is the constant of integration.
Another possible method to obtain the antiderivative of this expression is to apply partial fraction decomposition. The denominator of the expression does not have real roots, but it can be
factored into complex roots:
Applying partial fraction decomposition gives the equivalent expression
Thus, an alternative antiderivative of the expression is
\[ \int{\frac{\mathrm{d}x}{1+x^2}} &= \frac{1}{2}\int\left(\frac{1}{1+ix}+\frac{1}{1-ix}\right) \mathrm{d}x \\ \\ &= \frac{1}{2i}\big[\ln(1+ix)-\ln(1-ix)\big] + \mathrm{C}_{2} \\ \\ &= \frac{1}
{2i}\left[\ln\left(\frac{1+ix}{1-ix}\right)\right] + \mathrm{C}_{2}, \qquad (2) \]
where \(\text{C}_{2}\) is the constant of integration.
Because these antiderivatives are of the same expression, they are equivalent, i.e. \((1)=(2):\)
\[\arctan{x}=\frac{1}{2i}\left[\ln\left(\frac{1+ix}{1-ix}\right)\right] + \text{C} .\]
where \( \text{C} = \text{C}_{2} - \text{C}_{1} \).
Putting \( x=0 \), we find that \(\text{C} = 0\), which implies
\[\arctan{x}=\frac{1}{2i}\left[\ln\left(\frac{1+ix}{1-ix}\right)\right]. \]
Let \(x=\tan{\theta}\), then substituting gives
Further simplifying and use of identities gives
\[ i\theta &= \frac{1}{2}\left[\ln\left(\frac{(1+i\tan{\theta})^2}{1+\tan^2{\theta}}\right)\right] \\ \\ &= \frac{1}{2}\left[\ln\left(\frac{(1+i\tan{\theta})^2}{\sec^2{\theta}}\right)\right] \\ \
\ &= \ln\left(\frac{1+i\tan{\theta}}{\sec{\theta}}\right) \\ \\ &= \ln\left(\cos{\theta} + i\sin{\theta}\right). \]
Then, writing the equation in equivalent exponential form gives Euler's formula:
\[e^{i\theta}=\cos{\theta}+i\sin{\theta}.\ _\square\] | {"url":"https://brilliant.org/wiki/partial-fractions/","timestamp":"2024-11-05T16:13:21Z","content_type":"text/html","content_length":"57084","record_id":"<urn:uuid:6c960541-018e-44ac-919a-72bc4942c94c>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00795.warc.gz"} |
Interpreting Graphs: Linear Relation and Equations Insights
First, the of the rate of change will be determined and interpreted. Then, the rate of change will be calculated for the given interval.
Interpreting the Rate of Change
The rate of change is defined as the ratio of the change in the dependent variable to the change of the independent variable.
Note that the distance covered by the car depends on the time elapsed. On the other hand, time passes by without relation to the the car and can take any value without restrictions. Therefore, the
dependent variable in this case is the distance, and time is independent. Let represent the time and the distance.
Recall that the unit of the rate of change is the of the unit of the dependent variable to the unit of the independent variable.
Therefore, the unit of the rate of change is which quantifies kilometers traveled per minute. The rate of change can be interpreted as the of the bus.
Finding the Rate of Change Between Different Times
Now, by using the rate of change together with the information from the table, the rate of change of the car between and minutes will be calculated. Notice that the corresponding distances are and
kilometers, respectively.
The rate of change between and minutes is By following the same procedure it is possible to find the rate of change between the other points of interest. The following table summarizes these results
for the rate of change corresponding to the required time intervals.
Rate of Change
Times Substitute Evaluate
Similarly to Part A, the rate of change formula will be used to find the rate of change of the graph. In this case, the variables are the speed and the time
Notice that in a coordinate pair, the first coordinate represents the value of the independent variable, while the second coordinate corresponds to the dependent variable. Using this information, the
rate of change between the points and will be calculated for the given graph.
The acceleration in the first seconds is By following the same procedure, the rate of change for the other pair of points can be found. The following table summarizes the results for the rate of
change of the graph between the specified pair of points.
Rate of Change
Points Substitute Evaluate
In this case, before using the rate of change formula, the corresponding and values must first be found. This can be done by substituting the value of interest into the linear equation and evaluating
the resulting expression on the right-hand side. This will be done for first to illustrate the procedure.
It was found that the value corresponding to is Therefore, the coordinate pair is a point on the graph of the line representing the given linear equation. By repeating this process, the specific
points of interest can be found.
Linear Equation:
Substitute Point
Now that the points are known, all that is left to do is to repeat the same procedure as in Part B. The following table summarizes the results for the pair of points of interest.
Rate of Change
Points Substitute Evaluate
It is interesting to note that the rate of change is the same between all the pairs of points used. This occurred in Parts A and B as well. | {"url":"https://mathleaks.com/study/interpreting_graphs_of_linear_equations","timestamp":"2024-11-06T06:13:45Z","content_type":"text/html","content_length":"1049138","record_id":"<urn:uuid:60c3d7c1-6a97-4e97-b6f5-651bdfb919da>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00259.warc.gz"} |
topos of laws of motion
Topos Theory
Internal Logic
Topos morphisms
Cohomology and homotopy
In higher category theory
Synthetic differential geometry
synthetic differential geometry
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
The magic algebraic facts
$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \
rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel
{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel
{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
In (Lawvere 97) it was observed that equations of motion in physics can (almost, see below) be formalized in synthetic differential geometry as follows.
Let $\mathbf{H}$ be an ambient synthetic differential topos (such as the Cahiers topos of smooth spaces and formal smooth manifolds).
The canonical line object $\mathbb{A}^1 = \mathbb{R}$ of this models the continuum line, the abstract worldline. Let
$D \hookrightarrow \mathbb{R}$
be the inclusion of the first order infinitesimal neighbourhood of the origin of $\mathbb{R}$ – in the internal logic this is $D = \{x \in \mathbb{R}| x^2 = 0\}$, externally it is the spectrum of the
ring of dual numbers over $\mathbb{R}$.
Then for $X \in \mathbf{H}$ any object which we are going to think of as a configuration space of a physical system. For instance if the system is a particle propagating on a spacetime, then $X$ is
that spacetime. Or $X$ may be the phase space of the system.
Accordingly the mapping space $[\mathbb{R}, X] \in \mathbf{H}$ is the smooth path space of $X$. This is the space of potential trajectories of the physical system.
If $X$ is thought of as phase space, then every point in there determines a unique trajectory starting at that point. This means that time evolution is then an action of $\mathbb{R}$ on $X$. As $X$
here might be any space, we have the collection
$\mathbb{R}Act(\mathbf{H}) \in Topos$
of all $\mathbb{R}$-actions on objects in $\mathbf{H}$. This is again a topos, and hence this is a first version of what one might call a topos of laws of motion.
On the other hand, if we think of $X$ as configuration space, then it is (in the simplest but common case of physical systems) a tangent vector in $X$ that determines a trajectory, hence a point in $
[D,X]$. There is the canonical projection $[\mathbb{R},X] \longrightarrow [D,X]$ from the smooth path space to the tangent bundle, which sends each path to its tangent vector/derivative at $0 \in \
mathbb{R}$. A section of this map is hence an assignment that sends each tangent vector to a trajectory which starts out with this tangent. Specifying such a section is hence part of what it means to
have equations of motion in physics. Accordingly in Toposes of laws of motion Lawvere called the collection of such data a Galilian topos of laws of motion.
Of course this is not quite yet what is actually used and needed in physics. On p. 9 of (Lawvere 97) this problem is briefly mentioned:
But what about actual dynamical systems in the spirit of Galileo, for example, second-order ODE’s? (Of course, the symplectic or Hamiltonian systems that are also much studied do address this
question of states of Becoming versus locations of Being, but in a special way which it may not be possible to construe as a topos;
It turns out that it does exist as a “higher topos”.
In Classical field theory via Cohesive homotopy types it is observed that if $\mathbf{H}$ to be not just a topos but an infnity-topos, then genuine classical mechanics governed by Hamilton's
equations and Hamilton-Jacobi theory arises by actions of $\mathbb{R}$ on objects in the slice (infinity,1)-topos $\mathbf{H}_{/\mathbf{B}U(1)_{conn}}$, where $\mathbf{B}U(1)_{conn} \in \mathbf{H}$
is the moduli stack of circle group-principal connections. So
$\mathbb{R}Act\left( \mathbf{H}_{/\mathbf{B}U(1)_{conn}} \right)$
is actually a “topos of laws of motion” in the sense of Hamilton-Lagrange-Jacobi classical mechanics. For more on this see Higher toposes of laws of motion.
The notion originates around
and was made more explicit in | {"url":"https://ncatlab.org/nlab/show/topos%20of%20laws%20of%20motion","timestamp":"2024-11-02T14:38:00Z","content_type":"application/xhtml+xml","content_length":"80054","record_id":"<urn:uuid:6065fefe-0a7b-43ac-9008-227a458734f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00144.warc.gz"} |
Tomography of a Quark Gluon Plasma by Heavy
Tomography of a Quark Gluon Plasma by Heavy Quarks : P. -B. Gossiaux , V. Guiho, A. Peshier & J. Aichelin Subatech/ Nantes/ France Zimanyi 75 Memorial Workshop Zimanyi Memorial Workshop July 2007 1
Present situation: a) Multiplicity of stable hadrons made of (u, d, s) is described by thermal models b) Multiplicity of unstable hadrons can be understood in terms of hadronic final state
interactions c) Slopes difficult to interpret due to the many hadronic interactions (however the successful coalescence models hints towards a v 2 production in the plasma) d) Electromagnetic probes
from plasma and hadrons rather similar If one wants to have direct information of the plasma one has to find other probes: Good candidate: hadrons with a c or b quark Here we concentrate on open
charm mesons for which indirect experimental data are available (single electrons) Zimanyi Memorial Workshop July 2007 2
Why Heavy Quarks probe the QGP Idea: Heavy quarks are produced in hard processes with a known initial momentum distribution (from pp). If the heavy quarks pass through a QGP they collide and radiate
and therefore change their momentum. If the relaxation time is larger than the time they spent in the plasma their final momentum distribution carries information on the plasma This may allow for
studying plasma properties using pt distribution, v 2 transfer, back to back correlations Zimanyi Memorial Workshop July 2007 3
Schematic view of our model for hidden and open heavy flavors production in AA collision at RHIC and LHC Evolution of heavy quarks in QGP (thermalization) D/B formation at the boundary of QGP through
coalescence of c/b and light quark (hard) production of heavy quarks in initial NN collisions Quarkonia formation in QGP through c+c Y+g fusion process Zimanyi Memorial Workshop July 2007 4
Individual heavy quarks follow Brownian motion: we can describe the time evolution of their distribution by a Fokker – Planck equation: Input reduced to Drift (A) and Diffusion (B) coefficient. Much
less complex than a parton cascade which has to follow the light particles and their thermalization as well. Can be combined with adequate models like hydro for the dynamics of light quarks Zimanyi
Memorial Workshop July 2007 5
From Fokker-Planck coefficients Langevin forces pz py Evolution of one c quark inside a m=0 -- T=400 Me. V QGP. Starting from p=(0, 0, 10 Ge. V/c). px Evolution time = 30 fm/c … looks a little less «
erratic » when considered on the average: Relaxation time >> collision time : self consistent Zimanyi Memorial Workshop July 2007 t (fm/c) 6
The drift and diffusion coefficients Strategy: take the elementary cross sections for charm and calculate the coefficients (g = thermal distribution of the collision partners) and then introduce an
overall κ factor to study the physics Similar for the diffusion coefficient Bνμ ~ << (pν - pνf )(pμ - pμf )> > A describes the deceleration of the c-quark B describes Zimanyi Memorial Workshop July
thermalisation 2007 7
c-quarks transverse momentum distribution (y=0) Heinz & Kolb’s hydro Distribution just before hadronisation p-p distribution kcol =5 k=40 k=20 k=10 Zimanyi Memorial Workshop July 2007 Plasma will not
thermalize the c: It carries information on the QGP 8
Energy loss and A, B are related (Walton and Rafelski) pi Ai + p d. E/dx = - << (pμ – pμf)2 >> which gives easy relations for pc>>mc and pc<<mc d. E/dx and A are of the same order of magnitude A (Gev
/fm) T=0. 5 d. E/dx (Ge. V/fm) T=0. 4 T=0. 3 T=0. 2 p (Ge. V/c) Zimanyi Memorial Workshop July 2007 p (Ge. V/c) 9
In case of collisions (2 2 processes): Pioneering work of Cleymans (1985), Svetitsky (1987), extended later by Mustafa, Pal & Srivastava (1997). Later Teaney and Moore, Rapp and Hees similar approach
but plasma treatment is different • For radiation: Numerous works on energy loss; very little has been done on drift and diffusion coefficients Zimanyi Memorial Workshop July 2007 10
Input quantities for our calculations Au – Au collision at 200 AGe. V. c-quark transverse-space distribution according to Glauber • c-quark transverse momentum distribution as in d-Au (STAR)… seems
very similar to p-p No Cronin effect included; to be improved. • c-quark rapidity distribution according to R. Vogt (Int. J. Mod. Phys. E 12 (2003) 211 -270). • Medium evolution: 4 D / Need local
quantities such as T(x, t) taken from hydrodynamical evolution (Heinz & Kolb) • D meson produced via coalescence mechanism. (at the transition temperature. Zimanyi Memorial Workshop July with the a
we pick a u/d quark 2007 thermal distribution) but other scenarios possible. 11
Leptons ( D decay) transverse momentum distribution (y=0) RAA Comparison to B=0 calculation 2 2 only Langevin A and B finite κ = 20, κ=10 0 -10% B=0 (Just deceleration) pt Conclusion I: Energy loss
alone is not sufficient Kcol(coll only) =10 -20: Still far away from thermalization ! Zimanyi Memorial Workshop July 12 2007
There is a more recent data set Star and Phenix agree (Antinori SQM 07) Latest Published Phenix Data nucl-ex/0611018 Zimanyi Memorial Workshop July 2007 13
"Radiative « coefficients « radiative » coefficients deduced using the elementary cross section for c. Q+g and for cg +g in t-channel (u & s-channels are suppressed at high energy). ℳq cqg ≡ c Q +
dominant + + + suppresses by Eq/Echarm if evaluated in the large pic+ limit in the lab : (Bertsch-Gunion) Zimanyi Memorial Workshop July 2007 14
x=long. mom. Fraction of g Evaluated in scalar QCD and in the limit of Echarm >> masses and >>qt Factorization of radiation and elastic scattering k In the limit of vanishing masses: Gunion + Bertsch
PRD 25, 746 But: q Masses change the radiation substantially Zimanyi Memorial Workshop July 2007 15
Leptons ( D decay) transverse momentum distribution (y=0) RAA (large sqrts limit) 0 -10% 20 -40% Col. +(0. 5 x) Rad Col. (kcol=10 & 20) pt pt Min bias Conclusion II: One can reproduce the RAA either
: • With a high enhancement factor for collisional processes • With « reasonnable » enhancement factor (krad not far away from unity) Zimanyi Memorial Workshop July including radiative processes. 16
2007 pt
Non-Photonic Electron elliptic-flow at RHIC: comparison with experimental results v 2 c-quarks Collisional (kcol= 20) decay e D Tagged const q Freezed out according to thermal distribution at "punch"
points of c quarks through freeze out surface: pt Collisional + Radiative D q c Conclusion III: One cannot reproduce the v 2 consistently with the RAA!!! Contribution of light quarks to the Zimanyi
Memorial Workshop July 17 elliptic flow of D mesons is small pt 2007
Non-Photonic Electron elliptic-flow at RHIC: Looking into the bits… v 2 (all p) const quark tagged by c v 2 (tagged p) C-quark does not see the « average » const quark… Why ? Bigger coupling helps… a
little but at the cost of RAA SQM 06 Zimanyi Memorial Workshop July 2007 18
This is a generic problem ! Van Hees and Rapp: Charmed resonances and Expanding fireball (does not reproduce non charmed hadrons) Communicate more efficiently v 2 to the c- quarks Moore and Teaney:
Even choice of the EOS which dives the largest v 2 possible does not predict non charmed hadron data assuming D mesons Only ‘exotic hadronization mechanisms’ may explain the large v 2 Zimanyi
Memorial Workshop July 2007 EXPERIMENT 19 ?
Problems on exp. side X. Lin SQM 07 RAA is about 0. 25 for large pt for Star and Phenix Confirms that large diffusion coefficients are excluded Actual problems -- D / c ratio (Gadat SQM 07) -- B
contribution D 0, D+, Ds+ c+ D 0 D - D s - c - Large discrepancy between Star and Phenix BR 17. 2 6. 71 8 +6 - 4. 5 (X 1. 9 5 1. 7 e) in 0. 29 % Zimanyi Memorial Workshop July 2007 20
Azimutal Correlations for Open Charm D Transverse plane c What can we learn about the "thermalization" process from the correlations remaining at the end of QGP ? Initial correlation (at RHIC);
supposed back to back here c-bar Dbar SQM 06 How does the coalescence - fragmentation mechanism affects the "signature" ? Zimanyi Memorial Workshop July 2007 21
Azimutal Correlations for Open Charm Small pt (pt < 1 Ge. V/c ) c-quarks No interaction Coll (kcol= 1) Coll + rad (kcol= krad = 1) 0 -10% Coll (kcol= 10) Coll (kcol= 20) coalescence D SQM 06 jc -
jcbar Correlations are small at small pt, , mostly washed away by coalescence process. Zimanyi Memorial Workshop July j. D - j. Dbar 2007 22
Azimutal Correlations for Open Charm Average pt (1 Ge. V/c < pt < 4 Ge. V/c ) c-quarks No interaction Coll (kcol= 1) Coll + rad (kcol= krad = 1) 0 -10% Coll (kcol= 10) Coll (kcol= 20) coalescence D
jc - jcbar Conclusion IV: Broadening of the correlation due to medium, but still visible. Results for genuine coll + rad and for cranked up coll differ significantly Azimutal correlations might help
identifying better thermalization process and thus the medium SQM 06 Zimanyi Memorial Workshop July j. D - j. Dbar 2007 23
Azimutal Correlations for Open Charm Large pt (4 Ge. V/c < pt ) c-quarks No interaction Coll (kcol= 1) Coll + rad (kcol= krad = 1) 0 -10% Coll (kcol= 10) Coll (kcol= 20) coalescence D SQM 06 jc -
jcbar Large reduction but small broadening for increasing coupling with the medium; compatible with corona effect Zimanyi Memorial Workshop July j. D - j. Dbar 2007 24
Conclusions • Experimental data point towards a significant (although not complete) thermalization of c quarks in QGP. • The model seems able to reproduce experimental RAA, at the price of a large
rescaling K-factor (especially at large pt), of the order of k=10 or by including radiative processes. • Still a lot to do in order to understand the v 2. Possible explanations for discrepancies are:
1) spatial distribution of initial c-quarks 2) Part of the flow is due to the hadronic phase subsequent to QGP 3) Reaction scenario different 4) Miclos Nessi (v 2, , azimuthal correlations? ? ? ) 1.
Azimutal correlations could be of great help in order Zimanyi Memorial Workshop July 25 to identify the nature of thermalizing mechanism. 2007
V 2 -- Au+Au -- 200 -- Min. Bias Zimanyi Memorial Workshop July 2007 26 | {"url":"https://present5.com/tomography-of-a-quark-gluon-plasma-by-heavy/","timestamp":"2024-11-15T03:36:35Z","content_type":"text/html","content_length":"66507","record_id":"<urn:uuid:b5e5dfdf-41ab-4067-92bd-f8ef214d8348>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00211.warc.gz"} |
C++. Conditional jump operator ‘if’
Conditional jump operator ‘if’
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1. What is the function of the conditional ‘if statement in C++ programs?
The conditional ‘if’ statement allows you to organize the selection of the progress of the program. The choice is made by means of certain condition. If the condition is true, then the program
executes in one way. Otherwise, the program is executed in a different way.
The conditional branch operator performs branching in the program.
2. What representations does the conditional jump operator have in the C++ language?
A conditional branch operator can have the following representations:
• the complete form ‘if … else’;
• the short form ‘if’;
• the representation ‘if … else … if’.
When solving tasks, any of the forms can be replaced by another one. The same task can be solved in several ways. In solving the task the choice of one or other form of conditional jump ‘if’ remains
at the discretion of the programmer.
3. What representation does the full form of the ‘if’ statement have?
A general representation of the full form of the conditional branch operator ‘if’ is as follows:
if (expression)
// several operators (instructions)
// ...
// several operators (instructions)
// ...
Where expression is a conditional expression (condition) according to the C++ syntax.
The if statement works as follows. If the ‘expression’ element evaluates to true, then operators are executed, which follow immediately after the ‘if’ keyword.
If, after the word ‘if’ or after the word ‘else’ you need to perform only one operator (rather than several), then the curly braces ‘{ }’ can be omitted.
The general form of the operator, in which after the words ‘if’ and ‘else’ only one statement needs to be executed, can be the following:
if (expression)
4. Examples of using the full form of the ‘if’ statement.
Example 1. Write a code snippet that evaluates the value of the following expression:
float f, x;
// input x
// ...
if ((-8<=x)&&(x<=5))
f = x*x+2*x-4;
f = x-5;
In the above example, the resulting value of f is computed based on the value of x.
Example 2. The value n = 1..7 is given, which is the day of the week number. By the value of n, determine whether this day is a day off or a work day. The result is written to the variable fDayOff of
type bool.
A snippet of code that solves this problem.
int n;
bool fDayOff;
n = 5;
if ((n>=1) && (n<=5))
fDayOff = false; // fDayOff = false
fDayOff = true;
5. Which representation has the shortened form of the ‘if’ statement?
Sometimes in C++ programs, the full form of the ‘if’ statement should be replaced by a shortened form. This is necessary in cases where after the word ‘else’ you do not need to follow any
instructions. In the shortened form of the ‘if’ statement, the keyword ‘else’ is omitted.
The general representation of the abbreviated form of the ‘if’ operator is:
if (expression)
// several statements
// ...
Where the expression is a conditional expression (condition) according to the C++ syntax.
If after the word ‘else’ only one statement is to be executed, then the curly braces { } can be omitted:
if (expression)
6. Examples of using the shortened form of the ‘if’ statement
Example 1. Three integers a, b, c are given. Develop a program that finds the minimum value between these numbers.
A snippet of code that solves this task:
int a, b, c;
int min; // required minimum value
a = 8;
b = -5;
c = 12;
// minimum value search
min = a;
if (min > b) min = b;
if (min > c) min = c; // min = -5
Example 2. An integer n = 1..3 is given, which is the number of the function. From the value of variable n, calculate the value of the corresponding function:
1) x2+8; 2) -5x-1; 3) 10-x.
A snippet of code that solves this problem with the abbreviated form of the ‘if’ statement.
int n;
int x, f;
// input of the values n, x
// ...
if (n==1) f = x*x + 8;
if (n==2) f = -5*x - 1;
if (n==3) f = 10 - x;
7. The compound form of the conditional jump operator ‘if … else … if’
The ‘if’ operator can have a more complex form, which has the following general representation:
if (expression1)
// statements
// ...
if (expression2)
// statements
// ...
if (expressionN)
// statements
// ...
// statements
// ...
where expression1, expression2, …, expressionN are conditional expressions.
Conditional expressions are computed from the top down. If one of the expressions finds the true result (true), then the operators that are associated with this branch will be executed, and all other
“branches” will be skipped.
8. Examples of using the composite form ‘if … else … if’
Example 1. Let there be given the number of the month of the year n. By the month number, determine how many days this month. Take that in February, 28 days.
A snippet of code that solves this task.
int n;
int days;
// input n
// ...
if ((n==4)||(n==6)||(n==9)||(n==11)) days = 30;
if (n==2) days = 28;
days = 31;
Example 2. Given a real number x. Calculate the value of the function
A snippet of code that solves this task.
float x;
float f;
x = 5;
if (x<-2)
f = 3*x*x - 8;
if ((-2<=x)&&(x<=4))
f = -9*x*x - 12;
f = 32 + x; // f = 37
9. What are nested if statements?
The ‘if’ statements can be nested. This means that instead of operators (see items 2, 4, 6) there can be another ‘if’ statement. In C++, up to 256 attachments of the ‘if’ statements are allowed.
Here, there is a rule relative to the last else-statement. The last else-instruction refers to the closest if-statement that is inside the same program block (braces { }) but is not yet associated
with any other else-statement. In order to avoid invisible logical errors, it is recommended to select the necessary “branches” of the if statement in curly braces { }.
The simplest general form of a nested ‘if’ statement is:
if (expression)
// Nested statement if
if (expression)
// statements
// ...
// statements
// ...
// ...
10. Example of nested statements if
Example. Fragment of the solution of the quadratic equation taking into account all possible values of the coefficients of the equations a, b, c.
float a, b, c; // coefficients of equation
float d;
float x1, x2; // The roots of equation
a = -1;
b = 1;
c = 5;
d = b*b - 4*a*c;
if (a==0)
if (b!=0)
label1->Text = " The equation has 1 root ";
x1 = -c/b;
label2->Text = x1.ToString();
label1->Text = " Incorrect data ";
// a!=0
if (d<0)
label1->Text = " The equation has no roots ";
if (d==0)
label1->Text = " The equation has 1 root ";
x1 = -b/(2*a);
label2->Text = "x = " + x1.ToString();
label1->Text = " The equation has 2 roots ";
x1 = (-b - Math::Sqrt(d))/(2*a);
x2 = (-b + Math::Sqrt(d))/(2*a);
label2->Text = "x1 = " + x1.ToString();
label3->Text = "x2 = " + x2.ToString();
Related topics | {"url":"https://www.bestprog.net/en/2017/08/02/conditional-jump-operator-if-2/","timestamp":"2024-11-08T20:55:58Z","content_type":"text/html","content_length":"68293","record_id":"<urn:uuid:752bddb9-7f0d-4743-97e2-b3ee2a30d64e>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00210.warc.gz"} |
Power Divider, Combiner, Splitter Terms
Glossary of Power Divider, Combiner, Splitter Terms
Amplitude Balance: The attribute of the output signals of an equal power divider having the same magnitude.
BNC Connector: A quick connect/disconnect coaxial connector with 50 ohm impedance suitable for carrying medium power RF & microwave signals up to 4 GHz. Also commonaly called the British Naval
Connector, the BNC (Bayonet Neill-Concelman) connector was designed by Paul Neill of Bell Labs and Carl Concelman of Amphenol in the 1951. All INSTOCK Wireless products are available with BNC female
(jack) connectors.
Characteristic Impedance: For a microwave signal in a transmission line, the ratio of the electric field to the magnetic field. Characteristic impedance is related to free-space impedance (377 ohms)
and can be calculated based on the physical dimensions and dielectric properties of the transmission line. Most RF and microwave systems are designed to operate with a characteristic impedance of 50
ohms. An advantage of coaxial cable and microstrip is that its characteristic impedance is not frequency dependent.
Coherent Signals: RF or microwave signals exhibiting attributes such that, when input to a power combiner, their wave forms add constructively or subtract destructively. For RF and microwave signals,
the attributes of frequency, shape and transmitted information (if present) must be identical for signal coherence to exist.
Combining Loss: Loss of signal due to the vector summing, in a power combiner, of coherent input signals that differ in phase and/or amplitude. The combining loss of coherent signals is proportional
to the phase and amplitude unbalance of the signals. Identical coherent signals summed through a power combiner exhibit no combining loss. Coherent signals 180° out-of-phase exhibit total combining
loss (zero sum or transmitted power). Non-Coherent signals exhibit a loss equal to 10 log (1/n), where n = number of combined signals. All combining loss is dissipated through the isolation
Frequency Range: The span of frequency over which the power divider, combiner, splitter maintains all specified performance values.
GPS: Global Positioning System. A US space-based global navigation satellite system providing positioning, navigation and timing services worldwide through the broadcast of signals that GPS receivers
use to determine 3 dimensional location (latitude, longitude and altitude) plus time. Commonly used in transportation and navigation, map making, land surveying, tracking, surveillance and location
GNSS: Global Navigation Satellite System. The standard generic term for satellite navigation systems providing autonomous geo-spatial positioning with global coverage. Includes US NAVSTAR GPS,
Russian GLONASS and EU Galileo positioning systems.
In-Line Housing: A power splitter, power combiner housing having input and output connectors parallel or “in-line” with each other.
Input VSWR: Voltage standing wave ratio measured at the power divider input port with all output ports terminated in 50 ohm loads.
Insertion Loss: In a power divider or power combiner, the total signal reduction within the device from input to output including such factors as theoretical power split, combining loss, mismatch
loss and dissipation loss (including conductor and dielectric losses). Insertion loss is expressed by the formula:
Insertion Loss = 10 log (Pt/Pi), where:
Pt = Transmitted Power
Pi = Incident Power
Isolation: In a power splitter, the ability to keep signals (including any reflected signals) at the output ports separate from one another; to prevent cross-talk between ports. In a power combiner,
the ability to prevent signals at an input port from appearing at any other input port. Isolation is achieved through the use of a wilkinson type design employing resistor(s) of precisely calculated
values placed at the terminus of transformer sections between port pairs.
Microstrip Circuit: A circuit constructed of thin strip-like transmission lines separated from a ground plane by a dielectric substrate. Commonly used for constructing RF and microwave devices
utilizing discrete components attached to the top of the circuit board.
Mismatch Loss: A measure of power loss due to reflections within a device, usually of very small magnitude, and caused by design and manufacturing limitations.
Non-Coherent Signals: RF or microwave signals differing in frequency, shape or transmitted information such that, when input to a power combiner, their wave forms do not add constructively or
subtract destructively but exhibit a loss equal to 10 log (1/N), where N = number of combined signals.
Output VSWR: Voltage standing wave ratio measured at the power divider output port with all other ports terminated in 50 ohm loads.
Phase Balance: The attribute of the output signals of a zero degree power divider being in phase (having no phase difference).
PIM (Passive Intermodulation): The production of unwanted signals in a wireless receive path from the non-linear mixing of two or more high power transmit signals in a passive component. PIM problems
may be minimized by careful contact and current path junction design (including connector mating interfaces), use of linear materials such as brass and copper alloys, avoidance of or shielding from
ferromagnetic materials, and cleanliness in the manufacturing process.
Power Combiner: A device that combines or sums N number of RF microwave input signals to a common output, while maintaining the characteristic impedance of the inputs.
Power Divider: A device that divides or splits an RF microwave input signal into N number of output signals, while maintaining the characteristic impedance of the input.
Power Rating: The maximum amount of continuous input power (in watts) a power divider or power combiner can safely handle without permanent performance degradation. For a power divider, max input
power is dependent on the VSWR and phase of loads connected to the outputs. For a power combiner, max input power is dependent on the properties of the input signals and the magnitude of any
combining loss they suffer. Ultimately, power rating is directly related to the power handling capability of the isolation resistors, as it is through these resistors that most power is dissipated.
Power Split: The theoretical power ratio from input to output of a power divider or power splitter expressed (in dB) by the formula:
Power Split = 10 log (1/N), where:
N = number of outputs for an equal amplitude power divider or power splitter.
Often referred to as insertion loss, although not a true loss as this power is recoverable.
PTFE (PolyTetraFluoroEthylene): A thermoplastic member of the fluoropolymer family of plastics. PTFE is commonly used as a support insulator in RF and microwave coaxial connectors because of its low
& stable dielectric constant and loss factor over a wide temperature and frequency range. The original PTFE resin was invented by Dupont in 1938 and called Teflon®.
RoHS (Restriction of Hazardous Substances): A European legislative directive that bans the use of cadmium, hexavalent chromium, lead, mercury, PBBs (polybrominated biphenyls) and PBDEs
(polybrominated diphenyl ethers) in the manufacture of various types of electronic components and electrical equipment sold in the European Union.
RoHS Compliant: Concerning the manufacture of power splitter, combiner, dividers, RoHS compliant construction precludes the use of lead based solder and yellow iridite finish (which contains
hexavalent chromium). INSTOCK substitutes an SAC305 based lead-free solder and provides a clear chemical film finish on aluminum surfaces in the manufacturing of its RoHS power splitter, combiner,
SMA Connector (SubMiniature version A): A threaded coaxial connector with a dielectric loaded interface providing excellent electrical performance from DC to 18 GHz. Precursor designs first appeared
in 1958; current designation established in 1968. Available in mating SMA Jack (SMA Female) and SMA Plug (SMA Male) configurations. Recommended mating torque is 7 to 10 in-lb (80-110 N-cm).
T-Housing: A power divider, power combiner housing having input and output connectors perpendicular to one another in the configuration of a “T”.
TNC Connector: A threaded version of the BNC coaxial connector with 50 ohm impedance suitable for carrying medium power RF & microwave signals up to 11 GHz. Original design attributed to Paul Neill
of Bell Labs and Carl Concelman of Amphenol in the late 1950's. Available in mating TNC female jack and TNC male plug configurations. Connect finger tight or 12 in-lb (136 N-cm) when using a torque
Tri-Alloy Plating: An alloy of copper, tin and zinc providing good electrical performance and tarnish resistance. Being non-magnetic, it provides passive intermodulation performance comparable to
silver. Appearance resembles stainless steel. Similar in composition and characteristics to proprietary processes such as albaloy, white bronze, sucoplate, etc.
True Insertion Loss: For a power divider or power splitter, the non-recoverable power loss due to internal mismatch and dissipation losses. Does not include power split or combining losses. This is
the value specified for insertion loss of INSTOCK Wireless Components Power Divider, Power Splitters.
True 3-Way: A non-binary, modified, Wilkinson power divider, power combiner constructed of three transformers joined at a common node. Differs from other 3-Way divider/combiners constructed from a
binary 4-Way with one terminated port. Theoretical insertion loss due to power split is 4.77 dB.
True 6-Way: A non-binary, modified, Wilkinson power divider, power combiner constructed by cascading 2-Way and true 3-Way power divider/combiners. Differs from other 6-Way divider/combiners
constructed from a binary 8-Way with two terminated ports. Theoretical insertion loss due to power split is 7.78 dB.
True 12-Way: A non-binary, modified, Wilkinson power divider, power combiner constructed by cascading 4-Way and true 3-Way power divider/combiners. Differs from other 12-Way divider/combiners
constructed from a binary 16-Way with four terminated ports. Theoretical insertion loss due to power split is 10.79 dB.
Type N Connector: A threaded coaxial connector with an air interface suitable for carrying medium power RF & microwave signals. Original design attributed to Paul Neill of Bell Labs in the 1940's.
Available in mating N-Type Jack (N Female) and Type-N Plug (N Male) configurations. Connect finger tight or 12 in-lb (136 N-cm) if using a torque wrench.
VSWR: Voltage Standing Wave Ratio. An expression of the voltage standing wave pattern in a device caused by the phase addition and subtraction of incident and reflected waves. VSWR is the ratio of
maximum to minimum voltage of this standing wave pattern and is expressed by the formula:
VSWR = Emax/Emin = (Ei + Er)/(Ei - Er),
Ei = incident voltage wave amplitude,
Er = reflected voltage wave amplitude, and
the sign of voltage wave amplitudes is positive
Wilkinson Power Divider: A radio frequency, microwave device capable of splitting an input signal into equal phase, equal amplitude output signals or summing like signals to a common port. A unique
feature of the Wilkinson power divider is high output port-to-port isolation. Constructed of one or more (multi-section) quarter wave length transformers used for matching input and output impedances
(thus maximizing power transfer and minimizing loss), with a resistor placed between the ends of each transformer section (providing high isolation and excellent output port VSWR). First demonstrated
by Ernest Wilkinson with the 1960 publication of his paper, “An N-Way Hybrid Power Divider”.
Zero Degree (0°) Power Divider: A power divider whose output signals are in-phase (having no phase difference, subject to specified design and manufacturing limitations). All INSTOCK Wireless Power
Divider, Combiner, Splitters are zero degree (in-phase). | {"url":"https://www.instockwireless.com/index.php/power_divider_glossary.htm","timestamp":"2024-11-11T16:43:30Z","content_type":"text/html","content_length":"29769","record_id":"<urn:uuid:c766123e-9182-4da7-a091-fd2b3c18b5b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00804.warc.gz"} |
Master The Pico WiFi: Random Numbers
Written by Harry Fairhead & Mike James
Monday, 11 March 2024
Harnessing Entropy
The Pico has a source of entropy in the form of a Ring Oscillator (ROSC). It is basically a clock that runs at a rate that varies according to temperature and operating voltage etc. This means that
reading it can provide a moderately random value. The data sheet says:
“If the system clocks are running from the XOSC and/or PLLs the ROSC can be used to generate random numbers. Simply enable the ROSC and read the RANDOMBIT register to get a 1-bit random number and
read it n times to get an nbit value. This does not meet the requirements of randomness for security systems because it can be compromised, but it may be useful in less critical applications. If the
cores are running from the ROSC then the value will not be random because the timing of the register read will be correlated to the phase of the ROSC. “
In a standard Pico the clock is derived from the XOSC and the ROSC can be used to supply a random bit. This is very easy to do. To access the correct register all you need is:
#include "hardware/structs/rosc.h"
which imports a struct that has the correct address assigned to a pointer. After this you can read a random bit using:
bit = rosc_hw->randombit
The bit is returned as the low order bit of a 32-bit int.
You can create a random byte using:
uint8_t randomByte()
uint32_t random = 0;
for (int k = 0; k < 8; k++)
random = (random << 1) | rosc_hw->randombit;
return (uint8_t)random;
If you try this out you will find that it fails most of the NIST tests of randomness. It doesn’t even pass the test for an equal number of ones and zeros. The probability of generating a zero is 0.55
which is a small, but significant, bias.
A simple transformation, called von Neumann whitening after its inventor, improves the balance of ones and zeros. If you have a bit stream with unequal probabilities of a one or a zero you can
transform it to a 0 on a change from 0 to 1 and a 1 on a change from 1 to 0 and discard bits pairs of bits that are equal, i.e. 00 and 11. There are obviously as many up-going edges as there are
down-going edges so the number of ones and zeros is the same. The cost of this transformation is needing a few more random bits to throw away:
uint8_t randomByte()
uint32_t random = 0;
uint32_t bit = 0;
for (int k = 0; k < 8; k++)
while (true)
bit = rosc_hw->randombit;
if (bit != rosc_hw->randombit)
random = (random << 1) | bit;
return (uint8_t)random;
If you try this out you will discover that it only reduces the bias to 0.54 and the random sequence still fails the NIST tests. The reason is that the sequential bits are correlated.
The problem is that the ROSC is a source of entropy but we are taking data from it too fast. The oscillator varies over time and to reduce the correlations between subsequent bits we need to allow
enough time between readings for the oscillator to have randomly drifted. Putting this another way the entropy accumulates with time.
The solution is to modify the von Neumann whitening to include a delay:
uint8_t randomByte()
uint32_t random = 0;
uint32_t bit = 0;
for (int k = 0; k < 8; k++)
while (true)
bit = rosc_hw->randombit;
if (bit != rosc_hw->randombit)
random = (random << 1) | bit;
return (uint8_t)random;
This produces a random sequence that passes all of the NIST tests and has virtually no bias – the probability of a zero bit is 0.499. The delay could possibly be reduced, but if you only want a few
tens of random bytes this isn’t worth optimizing. With this delay the ROSC becomes an acceptable source of randomness that you can use to generate keys.
Last Updated ( Monday, 11 March 2024 ) | {"url":"https://www.i-programmer.info/programming/148-hardware/17030-master-the-pico-wifi-random-numbers.html?start=3","timestamp":"2024-11-02T22:10:51Z","content_type":"text/html","content_length":"31784","record_id":"<urn:uuid:4a97d5c7-0b0d-49d8-8d78-e98df45292dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00138.warc.gz"} |
Count consecutive monthly orders
In this example, the goal is to count the maximum number of consecutive monthly orders. That is, we want to count consecutive monthly orders greater than zero. This is a tricky formula to understand,
so buckle up!
They key to the formula is knowing that the FREQUENCY function gathers numbers into "bins" in a particular way. Each bin represents an upper limit, and contains a count of all numbers in the data set
that are less than or equal to the upper limit, and greater than the previous bin number. The trick then is to create the data_array argument using the condition you want to test for (order count
greater than zero in this case), and the bins_array using the opposite condition.
To create the data_array bin we use the following code:
Here, we use the IF function to test the order count in each month to see if it's greater than zero. If so, IF returns the column number using the COLUMN function. The result from IF is an array like
Notice that only columns where order count > 0 make it into this array. Those columns where the count is zero become FALSE.
The bins_array is generated with this snippet:
Here the IF function is used again to test the order count in each column, but this time the logic is reversed. Only column numbers where the count is zero make it into the array returned by IF,
which looks like this:
Per standard FREQUENCY behavior, the numbers in the bins_array become the functional bins that tally non-zero orders. Months where orders are greater than zero are translated to FALSE and don't
collect any numbers from the data array, since FALSE values are ignored. The result is that the surviving bins count the number of consecutive non-zero orders up to that point, but excluding those
previously counted. This all works because of the incrementing nature of rows and columns – you can be certain that the "next" number is always greater than the previous number.
With the data array and bin arrays as shown above, FREQUENCY returns a count per bin in an array like this:
The FREQUENCY function always returns an array with one more item than bins in the bins_array. This is by design, to catch any values greater than the largest value in the bins_array. This array is
returned directly to the MAX function, with returns the largest number in the array:
=MAX({1;0;3}) // returns 3
Other consecutive values
To count consecutive occurrences of other values, just adjust the logic as needed following the same pattern: the first condition tests for the thing you want to count, the second condition tests for
the opposite. | {"url":"https://exceljet.net/formulas/count-consecutive-monthly-orders","timestamp":"2024-11-09T11:06:10Z","content_type":"text/html","content_length":"50309","record_id":"<urn:uuid:bd736eee-89c7-4ce7-931e-4aa29c91099c>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00374.warc.gz"} |
• There was a bug in the creation of the junction tree when calling Kruskals algorithm.
• It is now possible to specify variables of interest in advance, such that we are guaranteed to be able to query the joint pmf of these variables.
• Some refactoring making compilation much faster. When potentials is assigned to a clique we no longer start by creating a unity table and then multiply. This was killing the advantage of the | {"url":"https://cran.uvigo.es/web/packages/jti/news/news.html","timestamp":"2024-11-12T15:21:48Z","content_type":"application/xhtml+xml","content_length":"5533","record_id":"<urn:uuid:222dd3a4-7062-457d-b963-30b28722503a>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00588.warc.gz"} |
trap door
trap door n.
(alt. `trapdoor') 1. Syn. back door -- a Bad Thing. 2. [techspeak] A `trap-door function' is one which is easy to compute but very difficult to compute the inverse of. Such functions are Good Things
with important applications in cryptography, specifically in the construction of public-key cryptosystems. | {"url":"http://hackersdictionary.com/html/entry/trap-door.html","timestamp":"2024-11-06T01:28:58Z","content_type":"text/html","content_length":"1938","record_id":"<urn:uuid:6dde62e8-637a-4f07-8083-11e0aefde209>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00286.warc.gz"} |
gpm to inches
With the above mentioned two-units calculating service it provides, this flow rate converter proved to be useful also as a teaching tool: 1. in practicing gallons UK per minute and cubic inches per
minute ( Imp. The only way to completely adjust for the estimates is to conduct a real-world test to fix the variables.
4. flow in ml per hour conversion to gtt per minute drop/min For example if the medium was water for instance. This is great flow units calculator/converter. Weed to convert gallon per day gpd/ft2 to
min/in – minutes/inch – minutes per inch. I’ve been working on my own under floor heating system in whole bottom floor. Between Imp. For a whole set of multiple units for volume and mass flow on one
page, try the Multi-Unit converter tool which has built in all flowing rate unit-variations. Equals to flow rate: ~ 219.69 cubic inch / hour ( 219.685 478 730 cu in³/hr ).
Perfect flow converter. Fraction : ~219 69/100 cubic inches per/ hour ( 219 137/200 cu in/h ).
For online collaboration to improve the » Flow units conversion, requests for new units or web tools additions, send your feedback.. We assume you are converting between acre inch and gallon [US].
Are you able to you provide any suggestion/directions on this? How many cubic inches per minute are in 1 gallon UK per minute? (maybe also gauge pressure with g – psig) The psia refers to the
standard atmospheric pressure.
I looking for an online web water flow units converter that does exchange flows between l/hr to ml/min etc and this page has done it very accurately.
This unit-to-unit calculator is based on conversion for one pair of two flow rate units.
3. convert flow rate of 5 gtt/min into flow L/day Comment from/about : Fraction : NaN/10 cubic feet per hour ( cuft/h , ft3/hr ). equals 0.001 liter / minute l/min in flow rate, ft^3 min in ft^3 sek
( sekond sek = second sec ) – thank you for allowing easy flow calculations page for basically instantly work out the flow conversions as they are.
Making conversion for flow units bar/sec and lit/min , from bar/sec to lit/min , barrel ( bar ) per second ( sec – s ) to liter ( L ) per minute ( min ) flow-speed-transfer calculates. Equals : 0.0
167 cubic foot / second (ft³/sec) How to calculate 15 ppm chemical rate which injected to 300MMSCFD ? Respond to Flow units conversion.
Run the pump at 1,000 RPM, for example. Equals : 0.02 cubic foot / second (ft³/sec) Equals to flow: 385.00 cubic inches in 1 second ( cu-in/sec , in3/sec. Page with flow rate by mass unit pairs
exchange. Liters Per Second = .063083 x GPM. How to convert Centimeters to Inches. For better result you can include de API gravity of the crude oil. 2. for conversion factors between unit pairs.
The other way around, how many cubic inches per minute - cu in/min are in one gallon UK per minute - Imp. TOGGLE : from cubic inches per minute into gallons UK per minute in the other way around.
How to Remove a Pacifica Power Steering ... How to Change the Power Steering Pump ... Zedcor Wholly Owned/PhotoObjects.net/Getty Images, Copyright 2020 Leaf Group Ltd. / Leaf Group Media, Indiana
Fluid Power: Fluid Power Formulas, Engineering Toolbox: Dynamic or Absolute Viscosity Converting Chart, Shelquist Engineering: Speed Versus RPM Calculator. gpm unit?
__ convert 15 liters per minute to US gallons per minute flow volume rate; Flow rate of: 15 liter / minute ( 15 l/min ),
Such efficiency curves apply to factors other than viscosity, too. I found this as a very good and quick way help to convert liters/hour to gallons/hr or cubic foot per hour ft3/h to use with the
simple calculations for how long it takes to empty home sized swimming pools. Flow of which gas? Comment from/about : Each revolution of the pump pumps some factor less than the full volume of the
cylinder. How do you convert GPM?
Understanding the limitations of the conversion, you start with one fixed ration and scale your conversion up or down. 2. flow units convert ml per hour to gtts per minute 1 gallon UK per minute to
cubic inches per minute = 277.42 cu in/min, 2 gallons UK per minute to cubic inches per minute = 554.84 cu in/min, 3 gallons UK per minute to cubic inches per minute = 832.26 cu in/min, 4 gallons UK
per minute to cubic inches per minute = 1,109.68 cu in/min, 5 gallons UK per minute to cubic inches per minute = 1,387.10 cu in/min, 6 gallons UK per minute to cubic inches per minute = 1,664.52 cu
in/min, 7 gallons UK per minute to cubic inches per minute = 1,941.94 cu in/min, 8 gallons UK per minute to cubic inches per minute = 2,219.36 cu in/min, 9 gallons UK per minute to cubic inches per
minute = 2,496.77 cu in/min, 10 gallons UK per minute to cubic inches per minute = 2,774.19 cu in/min, 11 gallons UK per minute to cubic inches per minute = 3,051.61 cu in/min, 12 gallons UK per
minute to cubic inches per minute = 3,329.03 cu in/min, 13 gallons UK per minute to cubic inches per minute = 3,606.45 cu in/min, 14 gallons UK per minute to cubic inches per minute = 3,883.87 cu in/
min, 15 gallons UK per minute to cubic inches per minute = 4,161.29 cu in/min, Category: main menu • flow rate menu • Gallons UK per minute. ), convert 15 liters per minute to gallons per minute.
Comment from/about : ft^3 min in ft^3 sek flow converter calculator.
Comment from/about : At low speeds, the effects of the oil's viscosity may have a negligible effect on its pumping efficiency -- 500 RPM, for example. the flow volume amount in fraction is 22 423 81/
100 gallon UK / hour ( gal/hr ).
Equals to flow: ~ 25.97 gallons US per one minute ( 25.974 025 975 gal/min ). gpm) to cubic inches per minute (cu in/min). Convert 1 Imp. BPCD (Barrels Per Calendar Day) – measure of flow rate on
basis of a calendar day.
To link to this web based Flow units conversion tool, copy then paste this code into your html.
Comment from/about :
Hydrodynamics conspires against the perfect scaling of revolutions to volume or volume to revolutions ratio. A flow rate transfer of 1500 cubic centimeters per minute – cm³/min., Flow amount 1 cubic
foot per second ( ft3/sec ),
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Social Media Surveillance Tools For Law Enforcement | {"url":"https://bioincubator.iitm.ac.in/pdffile/journal/1h2xw0r.php?a76bee=gpm-to-inches&__ncforminfo=QonGXKxj-jaDBM6BrALT-OKMuZS4gIfhlyj1XngHlRwm4alNsnE001T2EjT6pjk4sURoDDCWcRVLoEJ8GN-Px50bEmaqMlS7R0Y0y3PGz64=","timestamp":"2024-11-03T20:06:44Z","content_type":"text/html","content_length":"34222","record_id":"<urn:uuid:718ffe6b-ee25-4788-be11-92295af6c8f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00870.warc.gz"} |
BoTorch · Bayesian Optimization in PyTorch
Source code for botorch.acquisition.active_learning
#!/usr/bin/env python3
# Copyright (c) Meta Platforms, Inc. and affiliates.
# This source code is licensed under the MIT license found in the
# LICENSE file in the root directory of this source tree.
Active learning acquisition functions.
.. [Seo2014activedata]
S. Seo, M. Wallat, T. Graepel, and K. Obermayer. Gaussian process regression:
Active data selection and test point rejection. IJCNN 2000.
.. [Chen2014seqexpdesign]
X. Chen and Q. Zhou. Sequential experimental designs for stochastic kriging.
Winter Simulation Conference 2014.
.. [Binois2017repexp]
M. Binois, J. Huang, R. B. Gramacy, and M. Ludkovski. Replication or
exploration? Sequential design for stochastic simulation experiments.
ArXiv 2017.
from __future__ import annotations
from typing import Optional
import torch
from botorch import settings
from botorch.acquisition.acquisition import AcquisitionFunction
from botorch.acquisition.monte_carlo import MCAcquisitionFunction
from botorch.acquisition.objective import MCAcquisitionObjective, PosteriorTransform
from botorch.models.model import Model
from botorch.sampling.base import MCSampler
from botorch.sampling.normal import SobolQMCNormalSampler
from botorch.utils.transforms import concatenate_pending_points, t_batch_mode_transform
from torch import Tensor
class qNegIntegratedPosteriorVariance(AcquisitionFunction):
r"""Batch Integrated Negative Posterior Variance for Active Learning.
This acquisition function quantifies the (negative) integrated posterior variance
(excluding observation noise, computed using MC integration) of the model.
In that, it is a proxy for global model uncertainty, and thus purely focused on
"exploration", rather the "exploitation" of many of the classic Bayesian
Optimization acquisition functions.
See [Seo2014activedata]_, [Chen2014seqexpdesign]_, and [Binois2017repexp]_.
def __init__(
model: Model,
mc_points: Tensor,
sampler: Optional[MCSampler] = None,
posterior_transform: Optional[PosteriorTransform] = None,
X_pending: Optional[Tensor] = None,
) -> None:
r"""q-Integrated Negative Posterior Variance.
model: A fitted model.
mc_points: A `batch_shape x N x d` tensor of points to use for
MC-integrating the posterior variance. Usually, these are qMC
samples on the whole design space, but biased sampling directly
allows weighted integration of the posterior variance.
sampler: The sampler used for drawing fantasy samples. In the basic setting
of a standard GP (default) this is a dummy, since the variance of the
model after conditioning does not actually depend on the sampled values.
posterior_transform: A PosteriorTransform. If using a multi-output model,
a PosteriorTransform that transforms the multi-output posterior into a
single-output posterior is required.
X_pending: A `n' x d`-dim Tensor of `n'` design points that have
points that have been submitted for function evaluation but
have not yet been evaluated.
self.posterior_transform = posterior_transform
if sampler is None:
# If no sampler is provided, we use the following dummy sampler for the
# fantasize() method in forward. IMPORTANT: This assumes that the posterior
# variance does not depend on the samples y (only on x), which is true for
# standard GP models, but not in general (e.g. for other likelihoods or
# heteroskedastic GPs using a separate noise model fit on data).
sampler = SobolQMCNormalSampler(sample_shape=torch.Size([1]))
self.sampler = sampler
self.X_pending = X_pending
self.register_buffer("mc_points", mc_points)
def forward(self, X: Tensor) -> Tensor:
# Construct the fantasy model (we actually do not use the full model,
# this is just a convenient way of computing fast posterior covariances
fantasy_model = self.model.fantasize(
bdims = tuple(1 for _ in X.shape[:-2])
if self.model.num_outputs > 1:
# We use q=1 here b/c ScalarizedObjective currently does not fully exploit
# LinearOperator operations and thus may be slow / overly memory-hungry.
# TODO (T52818288): Properly use LinearOperators in scalarize_posterior
mc_points = self.mc_points.view(-1, *bdims, 1, X.size(-1))
# While we only need marginal variances, we can evaluate for q>1
# b/c for GPyTorch models lazy evaluation can make this quite a bit
# faster than evaluating in t-batch mode with q-batch size of 1
mc_points = self.mc_points.view(*bdims, -1, X.size(-1))
# evaluate the posterior at the grid points
with settings.propagate_grads(True):
posterior = fantasy_model.posterior(
mc_points, posterior_transform=self.posterior_transform
neg_variance = posterior.variance.mul(-1.0)
if self.posterior_transform is None:
# if single-output, shape is 1 x batch_shape x num_grid_points x 1
return neg_variance.mean(dim=-2).squeeze(-1).squeeze(0)
# if multi-output + obj, shape is num_grid_points x batch_shape x 1 x 1
return neg_variance.mean(dim=0).squeeze(-1).squeeze(-1)
class PairwiseMCPosteriorVariance(MCAcquisitionFunction):
r"""Variance of difference for Active Learning
Given a model and an objective, calculate the posterior sample variance
of the objective on the difference of pairs of points. See more implementation
details in `forward`. This acquisition function is typically used with a
pairwise model (e.g., PairwiseGP) and a likelihood/link function
on the pair difference (e.g., logistic or probit) for pure exploration
def __init__(
model: Model,
objective: MCAcquisitionObjective,
sampler: Optional[MCSampler] = None,
) -> None:
r"""Pairwise Monte Carlo Posterior Variance
model: A fitted model.
objective: An MCAcquisitionObjective representing the link function
(e.g., logistic or probit.) applied on the difference of (usually 1-d)
two samples. Can be implemented via GenericMCObjective.
sampler: The sampler used for drawing MC samples.
model=model, sampler=sampler, objective=objective, X_pending=None
def forward(self, X: Tensor) -> Tensor:
r"""Evaluate PairwiseMCPosteriorVariance on the candidate set `X`.
X: A `batch_size x q x d`-dim Tensor. q should be a multiple of 2.
Tensor of shape `batch_size x q` representing the posterior variance
of link function at X that active learning hopes to maximize
if X.shape[-2] == 0 or X.shape[-2] % 2 != 0:
raise RuntimeError(
"q must be a multiple of 2 for PairwiseMCPosteriorVariance"
# The output is of shape batch_shape x 2 x d
# For PairwiseGP, d = 1
post = self.model.posterior(X)
samples = self.get_posterior_samples(post) # num_samples x batch_shape x 2 x d
# The output is of shape num_samples x batch_shape x q/2 x d
# assuming the comparison is made between the 2 * i and 2 * i + 1 elements
samples_diff = samples[..., ::2, :] - samples[..., 1::2, :]
mc_var = self.objective(samples_diff).var(dim=0)
mean_mc_var = mc_var.mean(dim=-1)
return mean_mc_var | {"url":"https://botorch.org/v/latest/api/_modules/botorch/acquisition/active_learning.html","timestamp":"2024-11-05T18:38:29Z","content_type":"text/html","content_length":"31714","record_id":"<urn:uuid:ab15c6a0-9ea2-4828-b799-b592bbd81f32>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00277.warc.gz"} |
Problem Types
The mathematical model of programming problem consists of three elements:
1. variables, They are unknown quantities to be determined in the problem and are used to indicate the schemes or measures expressed in quantity in programming, which can be determined and
controlled by decision makers;
2. Objective function, which is the function of decision variables. Add Max or min to this function according to the optimization objective;
3. Constraint conditions. They refer to the restrictions of various resource conditions on the value of decision variables, which are usually expressed as equations or inequalities containing
decision variable functions. | {"url":"https://guide.coap.online/solvers/problem-types.html","timestamp":"2024-11-06T16:46:06Z","content_type":"text/html","content_length":"11209","record_id":"<urn:uuid:fc9bb0f3-7032-4c41-9a2d-8ad00210003b>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00384.warc.gz"} |
Constrained equiripple FIR filter
B = firceqrip(n,Fo,DEV)
B = firceqrip(...,'slope',r)
B = firceqrip('minorder',[Fp Fst],DEV)
B = firceqrip(...,'passedge')
B = firceqrip(...,'stopedge')
B = firceqrip(...,'high')
B = firceqrip(...,'min')
B = firceqrip(...,'invsinc',C)
B = firceqrip(...,'invdiric',C)
B = firceqrip(n,Fo,DEV) designs an order n filter (filter length equal n + 1) lowpass FIR filter with linear phase.
firceqrip produces the same equiripple lowpass filters that firpm produces using the Parks-McClellan algorithm. The difference is how you specify the filter characteristics for the function.
The input argument Fo specifies the frequency at the upper edge of the passband in normalized frequency (0<Fo<1). The two-element vector dev specifies the peak or maximum error allowed in the
passband and stopbands. Enter [d1 d2] for dev where d1 sets the passband error and d2 sets the stopband error.
B = firceqrip(...,'slope',r) uses the input keyword 'slope' and input argument r to design a filter with a nonequiripple stopband. r is specified as a positive constant and determines the slope of
the stopband attenuation in dB/normalized frequency. Greater values of r result in increased stopband attenuation in dB/normalized frequency.
B = firceqrip('minorder',[Fp Fst],DEV) designs filter with the minimum number of coefficients required to meet the deviations in DEV = [d1 d2] while having a transition width no greater than Fst –
Fp, the difference between the stopband and passband edge frequencies. You can specify 'mineven' or 'minodd' instead of 'minorder' to design minimum even order (odd length) or minimum odd order (even
length) filters, respectively. The 'minorder' option does not apply when you specify the 'min' (minimum-phase), 'invsinc', or the 'invdiric' options.
B = firceqrip(...,'passedge') designs a filter where Fo specifies the frequency at which the passband starts to rolloff.
B = firceqrip(...,'stopedge') designs a filter where Fo specifies the frequency at which the stopband begins.
B = firceqrip(...,'high') designs a high pass FIR filter instead of a lowpass filter.
B = firceqrip(...,'min') designs a minimum-phase filter.
B = firceqrip(...,'invsinc',C) designs a lowpass filter whose magnitude response has the shape of an inverse sinc function. This may be used to compensate for sinc-like responses in the frequency
domain such as the effect of the zero-order hold in a D/A converter. The amount of compensation in the passband is controlled by C, which is specified as a scalar or two-element vector. The elements
of C are specified as follows:
• If C is supplied as a real-valued scalar or the first element of a two-element vector, firceqrip constructs a filter with a magnitude response of 1/sinc(C*pi*F) where F is the normalized
• If C is supplied as a two-element vector, the inverse-sinc shaped magnitude response is raised to the positive power C(2). If we set P=C(2), firceqrip constructs a filter with a magnitude
response 1/sinc(C*pi*F)^P.
If this FIR filter is used with a cascaded integrator-comb (CIC) filter, setting C(2) equal to the number of stages compensates for the multiplicative effect of the successive sinc-like responses of
the CIC filters.
Since the value of the inverse sinc function becomes unbounded at C=1/F, the value of C should be greater the reciprocal of the passband edge frequency. This can be expressed as Fo<1/C. For users
familiar with CIC decimators, C is equal to 1/2 the product of the differential delay and decimation factor.
B = firceqrip(...,'invdiric',C) designs a lowpass filter with a passband that has the shape of an inverse Dirichlet sinc function. The frequency response of the inverse Dirichlet sinc function is
given by $\left\{rC{\left(\frac{\mathrm{sin}\left(f/2r\right)}{\mathrm{sin}\left(Cf/2\right)}\right\}}^{p}$where C, r, and p are scalars. The input C can be a scalar or vector containing 2 or 3
elements. If C is a scalar, p and r equal 1. If C is a two-element vector, the first element is C and the second element is p, [C p]. If C is a three-element vector, the third element is r, [C p r].
To introduce a few of the variations on FIR filters that you design with firceqrip, these five examples cover both the default syntax b = firceqrip(n,wo,del) and some of the optional input arguments.
For each example, the input arguments n, wo, and del remain the same.
Filter design using firceqrip
Design a 30th order FIR filter using firceqrip.
b = firceqrip(30,0.4,[0.05 0.03]);
Design a minimum order FIR filter using firceqrip. The passband edge and stopband edge frequencies are 0.35$\pi$ and 0.45$\pi$ rad/sample. The allowed deviations are 0.02 and 1e-4.
b = firceqrip('minorder',[0.35 0.45],[0.02 1e-4]);
Design a 30th order FIR filter with the stopedge keyword to define the response at the edge of the filter stopband.
b = firceqrip(30,0.4,[0.05 0.03],'stopedge');
Design a 30th order FIR filter with the slope keyword and r = 20.
b = firceqrip(30,0.4,[0.05 0.03],'slope',20,'stopedge');
Design a 30th order FIR filter defining the stopband and specifying that the resulting filter is minimum phase with the min keyword.
b = firceqrip(30,0.4,[0.05 0.03],'stopedge','min');
Comparing this filter to the filter in Figure 1. The cutoff frequency wo = 0.4 now applies to the edge of the stopband rather than the point at which the frequency response magnitude is 0.5.
Viewing the zero-pole plot shown here reveals this is a minimum phase FIR filter - the zeros lie on or inside the unit circle, z = 1
Design a 30th order FIR filter with the invsinc keyword to shape the filter passband with an inverse sinc function.
b = firceqrip(30,0.4,[0.05 0.03],'invsinc',[2 1.5]);
The inverse sinc function being applied is defined as 1/sinc(2*w)^1.5.
Inverse-Dirichlet-Sinc-Shaped Passband
Design two order 30 constrained equiripple FIR filters with inverse-Dirichlet-sinc-shaped passbands. The cutoff frequency in both designs is pi/4 radians/sample. Set C=1 in one design C=2 in the
second design. The maximum passband and stopband ripple is 0.05. Set p=1 in one design and p=2 in the second design.
Design the filters.
b1 = firceqrip(30,0.25,[0.05 0.05],'invdiric',[1 1]);
b2 = firceqrip(30,0.25,[0.05 0.05],'invdiric',[2 2]);
Obtain the filter frequency responses using freqz. Plot the magnitude responses.
[h1,~] = freqz(b1,1);
[h2,w] = freqz(b2,1);
plot(w,abs(h1)); hold on;
axis([0 pi 0 1.5]);
legend('C=1 p=1','C=2 p=2');
Inspect the stopband ripple in the design with C=1 and p=1. The constrained design sets the maximum ripple to be 0.05. Zoom in on the stopband from the cutoff frequency of pi/4 radians/sample to 3pi/
4 radians/sample.
set(gca,'xlim',[pi/4 3*pi/4]);
grid on;
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
All inputs must be constant. Expressions or variables are allowed if their values do not change.
Version History
Introduced in R2011a | {"url":"https://it.mathworks.com/help/dsp/ref/firceqrip.html","timestamp":"2024-11-04T11:35:44Z","content_type":"text/html","content_length":"89328","record_id":"<urn:uuid:dcadfa80-ab23-46ba-aa6c-f5ab4c428a57>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00419.warc.gz"} |
Law of Large Numbers - (Statistical Prediction) - Vocab, Definition, Explanations | Fiveable
Law of Large Numbers
from class:
Statistical Prediction
The Law of Large Numbers is a fundamental statistical theorem that states that as the size of a sample increases, the sample mean will get closer to the expected value or population mean. This
principle highlights the reliability of large samples in providing accurate estimates of population parameters, thus impacting prediction models and their performance.
congrats on reading the definition of Law of Large Numbers. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The Law of Large Numbers applies to independent random variables, ensuring that their averages converge to the expected value as more observations are collected.
2. This law is crucial in understanding why larger samples tend to provide more reliable estimates compared to smaller ones, reducing sampling error.
3. It is often used in various fields, including finance and insurance, where predictions about long-term averages are essential for risk assessment.
4. The Law of Large Numbers reinforces the importance of sufficient sample sizes in statistical analysis and machine learning, emphasizing that conclusions drawn from small samples can be
5. In practice, this law means that repeated experiments or observations will yield results that are more consistent with expected outcomes as the number of trials increases.
Review Questions
• How does the Law of Large Numbers help in improving the accuracy of statistical predictions?
□ The Law of Large Numbers improves statistical predictions by ensuring that as more data points are collected, the sample mean aligns more closely with the population mean. This leads to more
reliable estimates, which is crucial for making informed decisions in various applications such as finance and health sciences. By relying on larger samples, statisticians can reduce
variability and increase confidence in their predictive models.
• Discuss how the Law of Large Numbers relates to the concept of bias in statistical models.
□ The Law of Large Numbers directly impacts bias in statistical models by illustrating how larger sample sizes can mitigate bias in estimations. When small samples are used, they may not
represent the population accurately, leading to biased estimates. However, as the sample size increases, the potential for bias diminishes because more data points provide a clearer picture
of the population's characteristics, thus enhancing the reliability and validity of predictions.
• Evaluate how understanding the Law of Large Numbers can influence decisions in designing experiments or collecting data.
□ Understanding the Law of Large Numbers can significantly influence decisions regarding experimental design and data collection by emphasizing the need for large sample sizes to achieve
reliable results. When researchers plan experiments, they can apply this principle to determine appropriate sample sizes that minimize error and bias. This knowledge helps ensure that
conclusions drawn from data are robust and applicable to broader populations, ultimately enhancing the quality and impact of research findings.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/modern-statistical-prediction-and-machine-learning/law-of-large-numbers","timestamp":"2024-11-07T00:22:02Z","content_type":"text/html","content_length":"177291","record_id":"<urn:uuid:5105e4ba-cd09-497d-a20e-589bb70b3700>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00358.warc.gz"} |
Needed help on an OI problem - Codeforces
Hi guys. I was upsolving OI problems, when I stumbled upon this:
Statement: There are $$$n$$$ soldiers standing in a line, whose heights are distinct. A dude can be seen from the left side if there are no other dude to the left of him that is taller than him, and
the same goes for the right side. Given $$$n$$$, $$$p$$$, $$$q$$$ ($$$n \leq 2000$$$), count how many way to arrange these dudes such that there are exactly $$$p$$$ dudes that can be seen from the
left side, and $$$q$$$ dudes that can be seen from the right side. TL = $$$1$$$ second on a judge that runs as fast as Codeforces.
My idea: First we solve the case $$$q = 1$$$. For the sake of simplicity, we will assume the heights of these soldiers as a permutations of integers from $$$1$$$ to $$$n$$$. We can use dp: $$$f[i][j]
[k] = $$$ number of way to assign soldiers for the first $$$i$$$ position, such that the tallest soldiers that we assigned has the height $$$j$$$, and you can see $$$k$$$ dudes from the left side.
From there it's just simple dp. For the general case, the answer is simply $$$\sum_{t = 1}^n (f[t-1][t-1][p-1] * f[n-t][n-t][q - 1] * C_{n-1} ^ {i-1})$$$. Sadly, this runs in $$$O(n^3)$$$, since our
dp table has $$$n^3$$$ states.
I've discussed this problem with another dude who ranked Master on Codeforces, and we are proud to say that we can not cook, so any help would be appreciated. Thanks!
11 months ago, # |
← Rev. 7 → +15
Assume you have a way to arrange $$$n$$$ soldiers such that $$$p$$$ are seen from the left and $$$q$$$ are seen from the right, then in between the soldiers that are seen there may exists some
soldiers that are overshadowed.
In each arrangements, we could arrange all soldiers along with the soldiers they overshadowed to one side, and they will still be equivalent.
Now the problem becomes count number of ways to arrange n soldiers, such that k soldiers can be seen from $$$1$$$ side, which could be done with a simple dp, $$$dp[i][j]$$$ is number of ways
shine_ to arrange with $$$i$$$ tallest soldiers and $$$j$$$ soldiers are seen, calculate it by going through all $$$n$$$ values in decreasing order.
Now observe that the tallest soldier is always seen, and the tallest soldiers is seen from both side thus we can reduce the problem to the soldiers $$$[1, n - 1]$$$ and $$$p - 1$$$ soldiers on
the left, $$$q - 1$$$ on the right.
Now the answer is $$$dp[n - 1][p + q - 2] * C(p + q - 2, q - 1)$$$, which means that each ordering for one side, we could choose $$$q - 1$$$ seen soldiers and move them to the right, thus
making it a valid arrangement.
→ Reply
• 11 months ago, # ^ |
» +14
My man didn't just cook, he delivered !!!!!!
→ Reply | {"url":"https://mirror.codeforces.com/blog/entry/123157","timestamp":"2024-11-03T00:38:56Z","content_type":"text/html","content_length":"92644","record_id":"<urn:uuid:e6d5fc3d-dd8c-4f03-bfaf-0568e0133d23>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00447.warc.gz"} |
2x = 6 wat does x =
this is what you need to know [ ] target=_blank>[ ' target=_blank> ]]
ok is this an expert or the computer
this is an expert. you just spoke with the weegy program that directed you to me
ok can u help me out this is my problem 1/6 · a = 1
is this your school homework?
im homeschoold and my mother is asleep she told me that if a have a prob to come on here for help on it
well waht do you need help with
getting the answere to 1/6 · a = 1
Weegy: User:
.ok can i ask u somthing else | {"url":"https://www.weegy.com/?ConversationId=24E9B167&Link=i&ModeType=2","timestamp":"2024-11-04T23:18:58Z","content_type":"application/xhtml+xml","content_length":"59769","record_id":"<urn:uuid:434f04a2-35b1-4da1-9fc0-5df6d6cb5b3c>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00144.warc.gz"} |
GRA 6539 Fixed Income Securities
GRA 6539 Fixed Income Securities
Course coordinator:
Salvatore Miglietta
Course name in Norwegian:
Fixed Income Securities
MSc in Finance
Teaching language:
Course type:
One semester
A fixed income security is a security for which the rule determining future cash flows is set, when the security is issued. The classical example of a fixed income security is a treasury bond, with
fixed coupon payments. As long as the government can be trusted to avoid default, there is no cash-flow uncertainty for the bond. That simplifies analysis of sovereign bonds: there is only one major
source of uncertainty, future interest rates.
While pricing of some fixed-income instruments is simpler, it is never simple. This course and jobs in fixed income are some of the most technically demanding within finance. While learning about
fixed income may seem dry, at times, good understanding is paramount. Indeed, the fixed income market dwarfs the equity market. Knowledge of fixed income is in high demand in the job market.
Due to its complexity, detailed analysis of fixed income securities is often nearly omitted from academic programs. Yet fixed-income instruments (and their derivatives) form the majority of the
capital markets in the developed economies. Importance of understanding fixed-income instruments by a wider audience of investors/households has been repeatedly brought to light in the recent years
of financial turmoil.
The recent global crisis was brought about by troubles in the fixed-income markets. The words on everyone's tongue were "mortgages" and "mortgage-backed securities", both "fixed-income instruments".
More close to home, the Terra Securities scandal of 2007 has hopefully taught us, that investing in fixed-income instruments without proper analysis and knowledge is a bad idea.
Learning outcomes - Knowledge
By the end of the course, the students will gain understanding of the major institutional characteristics and some key technical know-how of the design and the valuation of fixed-income instruments.
To that end, in addition to understanding the economic intuition, the students will be expected to solve problems, which will go into the formal modeling issues, including interest rate modeling.
Acquired Knowledge includes:
1. Ability to identify the determinants of risk and return of debt securities. The emphasis is on pricing of fixed-income securities, including fixed income derivatives.
2. Fixed income portfolio management techniques
3. The role of fixed-income securities in risk management.
Learning outcomes - Skills
1. The students will use and/or develop certain quantitative skills, e.g, will learn to utilize Excel and/or Matlab for solving simple but realistic problems that may arise in the fixed income
market, will learn to apply binomial tree and Monte Carlo simulation approaches to asset valuation and risk assessment.
2. The students will also be able to explain interdependencies of risk factors.
Learning Outcome - Reflection
1. Students should gain a unified understanding of the interdependencies of factors affecting fixed income securities
2. Students should be able to analyze problems that go beyond those explicitly covered in class/the required book
Course content
The following list gives an overview of the key topics to be covered in the course. The required textbook is good in some aspects, but is not detailed enough in others. It will therefore be
supplemented by a number of articles and/or teaching materials.
The following topics will be covered:
• Institutional details: An overview of debt securities markets
• Tools: Bond pricing and arbitrage
• Tools: Measures of price sensitivity: Duration and convexity
• Institutional details: bond auctions
• Tools: Forwards and swaps; Options and futures
• Institutional details and Tools: Central Banks, Inflation, Monetary Policy
• Tools: Term Structure of Interest Rates in discrete time
• Tools: Binomial trees (term structure and pricing, European and American options, Monte Carlo simulation)
• Credit risk: an overview.
• Structured products: CDS, ABS
Additional topics may be covered as time permits, including:
• Agency debt market
• Institutional details and Tools: Corporate debt and fixed income options
Quantitative methods applied within the context of the course include:
• Probability distributions (discrete and continuous)
• Regression analysis (time series and cross section, hypothesis testing)
• Binomial trees and Monte Carlo simulation
Learning process and requirements to students
Students are expected to read the assigned chapters before class. Active class participation is expected during lectures, in the interactive regime.
There will be problem sets. The questions are to be answered by students prior to class and discussed during lectures.
Examples highlighting and demonstrating often-used, practical applications of fixed-income-securities' pricing are used extensively in the class.
There will be other written assignments during the semester. Those will be compulsory and graded as outlined below.
Please note that while attendance is not compulsory in all courses, it is the student’s own responsibility to obtain any information provided in class that is not included on the course homepage/
Itslearning or textbook.
This is a course with continuous assessment (several exam components) and one final exam code. Each exam component is graded by using points on a scale from 0-100. The components will be weighted
together according to the information in the course description in order to calculate the final letter grade for the examination code (course). Students who fail to participate in one/some/all exam
elements will get a lower grade or may fail the course. You will find detailed information about the point system and the cut off points with reference to the letter grades when the course start.
At resit, all exam components must, as a main rule, be retaken during next scheduled course.
All courses in the Masters programme will assume that students have fulfilled the admission requirements for the programme. In addition, courses in second, third and/or fourth semester can have
specific prerequisites and will assume that students have followed normal study progression. For double degree and exchange students, please note that equivalent courses are accepted.
Exam category:
Form of assessment:
Class participation
1 Semester(s)
Exam code:
Grading scale:
Point scale leading to ECTS letter grade
All components must, as a main rule, be retaken during next scheduled course
Exam category:
Form of assessment:
Written submission
Group/Individual (1 - 2)
24 Hour(s)
Take-home examination
Exam code:
Grading scale:
Point scale leading to ECTS letter grade
All components must, as a main rule, be retaken during next scheduled course
Exam category:
Form of assessment:
Written submission
Support materials:
• All printed and handwritten support materials
• BI-approved exam calculator
• Simple calculator
• Bilingual dictionary
3 Hour(s)
Written examination under supervision.
Exam code:
Grading scale:
Point scale leading to ECTS letter grade
All components must, as a main rule, be retaken during next scheduled course
Exam organisation:
Continuous assessment
A course of 1 ECTS credit corresponds to a workload of 26-30 hours. Therefore a course of 6 ECTS credits corresponds to a workload of at least 160 hours. | {"url":"https://programmeinfo.bi.no/en/course/GRA-6539/2018-autumn","timestamp":"2024-11-13T07:50:17Z","content_type":"text/html","content_length":"26606","record_id":"<urn:uuid:33f689d5-048e-4089-b314-bf0f125b3a53>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00664.warc.gz"} |
Gymnasium Documentation
Training an Agent¶
This page provides a short outline of how to train an agent for a Gymnasium environment, in particular, we will use a tabular based Q-learning to solve the Blackjack v1 environment. For a full
complete version of this tutorial and more training tutorials for other environments and algorithm, see this. Please read basic usage before reading this page. Before we implement any code, here is
an overview of Blackjack and Q-learning.
Blackjack is one of the most popular casino card games that is also infamous for being beatable under certain conditions. This version of the game uses an infinite deck (we draw the cards with
replacement), so counting cards won’t be a viable strategy in our simulated game. The observation is a tuple of the player’s current sum, the value of the dealers face-up card and a boolean value on
whether the player holds a usable case. The agent can pick between two actions: stand (0) such that the player takes no more cards and hit (1) such that the player will take another player. To win,
your card sum should be greater than the dealers without exceeding 21. The game ends if the player selects stand or if the card sum is greater than 21. Full documentation can be found at https://
Q-learning is a model-free off-policy learning algorithm by Watkins, 1989 for environments with discrete action spaces and was famous for being the first reinforcement learning algorithm to prove
convergence to an optimal policy under certain conditions.
Executing an action¶
After receiving our first observation, we are only going to use theenv.step(action) function to interact with the environment. This function takes an action as input and executes it in the
environment. Because that action changes the state of the environment, it returns four useful variables to us. These are:
• next observation: This is the observation that the agent will receive after taking the action.
• reward: This is the reward that the agent will receive after taking the action.
• terminated: This is a boolean variable that indicates whether or not the environment has terminated, i.e., ended due to an internal condition.
• truncated: This is a boolean variable that also indicates whether the episode ended by early truncation, i.e., a time limit is reached.
• info: This is a dictionary that might contain additional information about the environment.
The next observation, reward, terminated and truncated variables are self-explanatory, but the info variable requires some additional explanation. This variable contains a dictionary that might have
some extra information about the environment, but in the Blackjack-v1 environment you can ignore it. For example in Atari environments the info dictionary has a ale.lives key that tells us how many
lives the agent has left. If the agent has 0 lives, then the episode is over.
Note that it is not a good idea to call env.render() in your training loop because rendering slows down training by a lot. Rather try to build an extra loop to evaluate and showcase the agent after
Building an agent¶
Let’s build a Q-learning agent to solve Blackjack! We’ll need some functions for picking an action and updating the agents action values. To ensure that the agents explores the environment, one
possible solution is the epsilon-greedy strategy, where we pick a random action with the percentage epsilon and the greedy action (currently valued as the best) 1 - epsilon.
from collections import defaultdict
import gymnasium as gym
import numpy as np
class BlackjackAgent:
def __init__(
env: gym.Env,
learning_rate: float,
initial_epsilon: float,
epsilon_decay: float,
final_epsilon: float,
discount_factor: float = 0.95,
"""Initialize a Reinforcement Learning agent with an empty dictionary
of state-action values (q_values), a learning rate and an epsilon.
env: The training environment
learning_rate: The learning rate
initial_epsilon: The initial epsilon value
epsilon_decay: The decay for epsilon
final_epsilon: The final epsilon value
discount_factor: The discount factor for computing the Q-value
self.env = env
self.q_values = defaultdict(lambda: np.zeros(env.action_space.n))
self.lr = learning_rate
self.discount_factor = discount_factor
self.epsilon = initial_epsilon
self.epsilon_decay = epsilon_decay
self.final_epsilon = final_epsilon
self.training_error = []
def get_action(self, obs: tuple[int, int, bool]) -> int:
Returns the best action with probability (1 - epsilon)
otherwise a random action with probability epsilon to ensure exploration.
# with probability epsilon return a random action to explore the environment
if np.random.random() < self.epsilon:
return self.env.action_space.sample()
# with probability (1 - epsilon) act greedily (exploit)
return int(np.argmax(self.q_values[obs]))
def update(
obs: tuple[int, int, bool],
action: int,
reward: float,
terminated: bool,
next_obs: tuple[int, int, bool],
"""Updates the Q-value of an action."""
future_q_value = (not terminated) * np.max(self.q_values[next_obs])
temporal_difference = (
reward + self.discount_factor * future_q_value - self.q_values[obs][action]
self.q_values[obs][action] = (
self.q_values[obs][action] + self.lr * temporal_difference
def decay_epsilon(self):
self.epsilon = max(self.final_epsilon, self.epsilon - self.epsilon_decay)
Training the agent¶
To train the agent, we will let the agent play one episode (one complete game is called an episode) at a time and then update it’s Q-values after each episode. The agent will have to experience a lot
of episodes to explore the environment sufficiently.
# hyperparameters
learning_rate = 0.01
n_episodes = 100_000
start_epsilon = 1.0
epsilon_decay = start_epsilon / (n_episodes / 2) # reduce the exploration over time
final_epsilon = 0.1
agent = BlackjackAgent(
Info: The current hyperparameters are set to quickly train a decent agent. If you want to converge to the optimal policy, try increasing the n_episodes by 10x and lower the learning_rate (e.g. to
from tqdm import tqdm
env = gym.make("Blackjack-v1", sab=False)
env = gym.wrappers.RecordEpisodeStatistics(env, deque_size=n_episodes)
for episode in tqdm(range(n_episodes)):
obs, info = env.reset()
done = False
# play one episode
while not done:
action = agent.get_action(obs)
next_obs, reward, terminated, truncated, info = env.step(action)
# update the agent
agent.update(obs, action, reward, terminated, next_obs)
# update if the environment is done and the current obs
done = terminated or truncated
obs = next_obs
Visualising the policy¶
Hopefully this tutorial helped you get a grip of how to interact with Gymnasium environments and sets you on a journey to solve many more RL challenges.
It is recommended that you solve this environment by yourself (project based learning is really effective!). You can apply your favorite discrete RL algorithm or give Monte Carlo ES a try (covered in
Sutton & Barto <http://incompleteideas.net/book/the-book-2nd.html>_, section 5.3) - this way you can compare your results directly to the book.
Best of luck! | {"url":"https://gymnasium.farama.org/introduction/train_agent/","timestamp":"2024-11-08T00:49:31Z","content_type":"text/html","content_length":"61755","record_id":"<urn:uuid:ca47f7df-ff92-46ed-ac9a-ca1df7ee057e>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00741.warc.gz"} |
CS2810 OOAIA: A4 solved
Create classes in C++ for representing Complex, Rational and Natural numbers. These classes should
support the following functionality:
1. Complex: add, subtract, multiply
2. Rational: add, subtract, multiply, reduce
3. Natural: check prime, calculate inverse modulo prime p
Create an inheritance structure between the classes to reuse code. The inheritance can be multilevel,
Complex —> Rational —> Natural, meaning that Rational inherits from base class Complex and
Natural inherits from Rational.
The reduce operation means, given a fraction p/q, obtain p’/q’, where GCD(p’, q’) = 1, or p’ and q’ are
coprime. Eg: reduce(6 / 9) = 2 / 3. In modulo n arithmetic, if the inverse of num is inv_num,then:
(num * inv_num) % n = 1. Refer to Modular arithmetic rules and Fermat’s Little Theorem for
theory related to modular arithmetic and calculation of inverse modulo p.
Some useful links for number-theoretic algorithms: Euclid’s GCD algorithm, Sieve algorithm, Modular
arithmetic rules, Fermat’s Little Theorem, Fast exponentiation
Input Format
For every test case, the first line contains n, the number of operations that follow. Each operation will
be of the form , where number type can be {“complex”,
“rational”, “natural”} and operation type depends on number type. The allowed operations for
complex are {“add”, “sub”, “mult”}, for rational they are {“add”, “sub”, “mult”, “reduce”} and for natural
they are {“isprime”, “inverse”}. This line will be followed by the actual inputs for each operation
Complex numbers in the input are represented as: with both
numbers as double. Rational numbers in input are represented as:
with both numbers as integers.
Arithmetic operations(add, sub, mult) over complex numbers and rationals will be followed by 2 lines
containing 1 number each represented in the above-specified format. The reduce operation on
rationals will be followed by 1 rational number represented as specified above. All operations on natural
numbers are followed by a single number. The prime p with respect to which inverse modulo p should
be evaluated is 1000000007.
Output Format
Print output for each operation in a separate line. Print complex numbers in the same format as the
input. Print both real as well as imaginary parts of the complex number even if any of them is 0. For
rationals, print the double representation of the rational for all operations except reduce. For the
reduce operation over rationals, print 2 integers , with the first
integer being negative if the result is negative. For the isprime operation, print 0/1 if the number is not
prime/ prime respectively and for the inverse mod p print a single natural number.
Number of operations: 1 <= n <= 10 5 . For complex numbers: -10 3 <= real, imaginary <= 10 3 , for rationals: -10 4 <= num, denom <= 10 4 , denom != 0, and for natural: 1 <= number <= 10 6 . Sample
Testcase Input: 9 complex add —> number type and operation type
1.2 2.3 —> 1st number: 1.2 + 2.3i
2.1 1.3 —> 2nd number: 2.1 + 1.3i
complex sub
1.2 3.1
2.2 1
complex mult
-1 2
2.3 1.2
rational add
1 2 —> 1/2
1 3 —> 1/3
rational sub
rational mult
rational reduce
natural isprime
natural inverse
3.300 3.600
-1.000 2.100
-4.700 3.400
Design Submission Format
For the design submission on Moodle, please submit a .tar.gz file named as your roll number.
All doubles have to be printed with a fixed precision of 3 decimal digits similar to the assignment A3.
Number Display
3.4 3.400
3.1415 3.142
2 2.000
This style of printing can be set by using the following statements before any “cout”. You only need to
write these statements once:
std::cout.precision(3); std::cout << std::fixed; | {"url":"https://codeshive.com/questions-and-answers/cs2810-ooaia-a4-solved/","timestamp":"2024-11-03T02:53:06Z","content_type":"text/html","content_length":"100742","record_id":"<urn:uuid:324552de-f8f3-43f9-839c-93602a04512d>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00230.warc.gz"} |
Optical Pumping | TeachSpin
top of page
Optical Pumping
• Optical Pumping of Rubidium Atoms, Rb85 and Rb87
• Explore Magnetic Hyperfine Interactions of Rubidium
• Observe Zero-Field Transitions
• Confirm Breit-Rabi Equation
• Observe Double Quantum Transitions
• Study Rabi Oscillations
• Measure Optical Pumping Times
• Study Temperature Dependence of Atomic Parameters Optical Pumping of Rubidium Gas
Optical Pumping is a widely used and powerful technique for exploring atomic energy states, atomic transitions, and atomic collisions using electromagnetism in the form of light, radio frequency, and
uniform constant magnetic fields. TeachSpin’s Optical Pumping apparatus explores the atomic physics of both isotopes of natural rubidium.
The rubidium atom is an ideal model system for students to study. Its energy states, in an externally applied uniform magnetic field, can be understood using a semi-classical model. This model
describes the coupling of a single electronic orbital and spin angular momentum with the nuclear spin angular momentum and of the coupled system to the external field. The experimental determination
of these atomic energy states can be compared to the theoretical predictions of the Briet-Rabi equation. The two isotopes of rubidium, Rb85 and Rb87, with different nuclear magnetic moments, make the
experimental data even richer.
TeachSpin's Optical Pumping Lab apparatus allows the student to explore a wealth of atomic physics, including temperature dependent cross-sections for photon absorption, zero magnetic field
transitions, spin-spin collision processes, field inversion measurements, Rabi oscillation of the atomic magnetic moment, optical pumping times, and other atomic physics experiments. It is only a
small exaggeration to claim these experiments constitute an atomic physics course.
The basic features of the experimental set up of TeachSpin's Optical Pumping apparatus
are shown in Figure 1. Rubidium resonance light from a heated rf discharge lamp is
collimated by a plano-convex lens and passes approximately parallel through an
interference filter, so that only the 795 nm line is transmitted. The light then passes
through a linear polarizer and a quarter wave plate to produce a circularly polarized
beam of light. This monochromatic, circularly polarized, light passes into the oven and
through the rubidium vapor absorption cell. The diverging light is focused by a second
plano-convex lens onto the photodiode detector.
The oven, with its absorption cell, resides inside two pairs of Helmholtz coils. The
studentmust align the instrument so that the absorption cell’s axis (the light path) lies
along thedirection of the local Earth’s magnetic field. One Helmholtz pair is used to
cancel the vertical component of the Earth’s magnetic field. The second pair is used to
create auniform horizontal magnetic field in opposition to the horizontal component of
the Earth’s field. Transitions are induced among the atomic energy levels by the radio frequency magnetic field which is applied transverse to the optic axis. These transitions are observed as
changes in the light intensity as measured by the photodiode optical detector.
The open construction of the Optical Pumping system allows students to manipulate the location of all the optical components. The can even insert the quarter wave plate and the linear polarizer in
the wrong order which becomes an important learning experience.
Students will explore the interaction of the alkali atom with a weak magnetic field, as described by the theoretical perturbation calculations known as the Briet-Rabi equation. This equation is:
W is the interaction energy, ΔW is the hyperfine splitting energy and B is the externally applied magnetic field
Figure 2 (below) shows the optical pumping signals for zero and very low magnetic fields. Both the Rb85and Rb87 isotopes produce only unresolved single lines in these low fields. However, when the
magnetic field is increased, each single line splits into resolved hyperfine lines, as shown in Figure 3 (below). This data can be compared to the Briet-Rabi predictions. Double quantum transitions
are also detected on this data. They can be studied as a function of rf power.
Both optical pumping times and the so-called Rabi oscillations can be studied by gating the rf power on and off at low magnetic fields.
Figure 4 (below) shows a measure of the optical pumping time when the rf power has been turned off and Figure 5 (below) is an exploded view of the signal after the rf has been gated on. It clearly
shows the oscillations of the transmitted light which are interpreted as precession of the atomic magnetization about the rf magnetic field. They can be studied as a function of rf power.
Other experiments include (but are not limited to) magnetic field reversal, photon absorption, spin exchange, effect of buffer gases on optical pumping and temperature dependence of all of the above
experiments. The instrument even includes a second mounted linear polarizer for students to study circularly polarized light.
Additional Resources
additional resources
Absorption Cell:
Natural Rb with 30 Torr Neon
RF Discharge of Isotopically Enriched Rb (63% Rb87)
PID Controller,
Range; Ambient – 100 °C
Res. 0.1°C, Reg. 0.05 °C/hr
50 mm Diameter
Interference Filter
2 Linear Polarizers and 1/4 Retarder Plate in 360° Rotation Mounts
2 Plano-Convex Lenses, f = 50 mm
Photodiode Detector:
Low-noise Current-to-Voltage Preamplifier
Bandwidth 0.1 Hz - 1 kHz
Noise: 20 µ Vp-p with Rgain = 1 M Ohm
Magnetic Field of Precision Helmholtz Coils:
0 – 1.4 x 10-4 T,
Stability, 2 x 10-7 T/hr,
8 x 10-4 T (internal supply)
22 x 10-4 T (external supply)
Stability, 4 x 10-7 T/hr
Homogeneity > 2 x 10-4 over cell
Horizontal Sweep: 0 – 10-4 T,
Time 1, 2, 5 . . . 1,000s
Stability 2 x 10-7 T/hr
External Modulation Input
Black Cloth Shroud
Mounted Linear Polarizer
Instruction Manual
High Current Supply
Different Absorption Cells
RF Signal Generator
Nonmagnetic Table
Circuit Diagrams
Extended Warranty
RF Amplifier:
10 kHz – 100 MHz
Input Impedance = 50 Ohms.
Output 150 mW, 100 mA Max.
Detector Amplifier:
Amplifier: Gain, 1, 2, 5,. . . 1000
Low-Pass, 12 db/oct
Time Constants, 1ms, 10 ms . . . 3s
Electronics 13’ x 15” x 10”
Exp. Table 28” x 15” x 16”
Two Years, parts and labor
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class lsst.ts.mthexapod.RangedPolynomial(coeffs: list[float], min_x: float, max_x: float)¶
Bases: object
A polynomial that is linear outside a specified range.
This model is suitable for systems which are mostly linear, with minor nonlinear corrections. The idea is to prevent unrealistic values outside the range of values over which the polynomial
coefficients were fit.
The equation is: ` y(x) = C0 + C1 x + C2 x^2 + ... for min_x <= x <= max_x y(x) = y(min_y) + C1 (x - min_x) for x < min_x y(x) = y(max_x) + C1 (x - max_x) for x > max_x `
coeffslist [float]
Polynomial coefficients C0, C1, … Must contain at least one element.
Minimum x below which the result is linear.
Maximum x above which the result is linear.
If coeffs has no elements or min_x >= max_x.
Methods Summary
__call__(x) Compute the value of the function.
Methods Documentation
__call__(x: float) float¶
Compute the value of the function.
Input value. | {"url":"https://ts-mthexapod.lsst.io/py-api/lsst.ts.mthexapod.RangedPolynomial.html","timestamp":"2024-11-06T14:33:03Z","content_type":"text/html","content_length":"13076","record_id":"<urn:uuid:dcec4d93-a2b1-46b7-affe-f2b1a080dc71>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00533.warc.gz"} |
Distance estimation and collision prediction for on-line robotic motion planning
An efficient method for computing the minimum distance and predicting collisions between moving objects is presented. This problem has been incorporated in the framework of an in-line motion planning
algorithm to satisfy collision avoidance between a robot and moving objects modelled as convex polyhedra. In the beginning, the deterministic problem where the information about the objects is
assumed to be certain is examined. If instead of the Euclidean norm, L[1] or L[∞] norms are used to represent distance, the problem becomes a linear programming problem. The stochastic problem is
formulated, where the uncertainty is induced by sensing and the unknown dynamics of the moving obstacles. Two problems are considered: First, filtering of the distance between the robot and the
moving object, at the present time. Second, prediction of the minimum distance in the future, in order to predict the collision time.
• Collision avoidance
• collision prediction
• distance functions
• random search
ASJC Scopus subject areas
• Control and Systems Engineering
• Electrical and Electronic Engineering
Dive into the research topics of 'Distance estimation and collision prediction for on-line robotic motion planning'. Together they form a unique fingerprint. | {"url":"https://nyuscholars.nyu.edu/en/publications/distance-estimation-and-collision-prediction-for-on-line-robotic-","timestamp":"2024-11-06T19:03:57Z","content_type":"text/html","content_length":"52509","record_id":"<urn:uuid:b0d4a847-3aa8-41de-98a6-77d6acac7bec>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00517.warc.gz"} |
Perfect square or cube - math word problem (82369)
Perfect square or cube
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
You need to know the following knowledge to solve this word math problem:
Units of physical quantities:
Grade of the word problem:
We encourage you to watch this tutorial video on this math problem:
Related math problems and questions: | {"url":"https://www.hackmath.net/en/math-problem/82369","timestamp":"2024-11-03T23:31:03Z","content_type":"text/html","content_length":"58579","record_id":"<urn:uuid:a066cd42-ce40-427c-93c6-e03da35752cc>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00021.warc.gz"} |
Yield Measures for Bonds with Embedded Options: Understanding Callable and Putable Bonds
4.2.3 Yield Measures for Bonds with Embedded Options
Bonds with embedded options present unique opportunities and challenges for investors. These financial instruments come with additional features that can significantly impact their yield calculations
and overall risk profile. In this section, we delve into the intricacies of bonds with embedded options, focusing on callable and putable bonds, and explore how these features affect yield measures
and investment decisions.
Understanding Embedded Options in Bonds
Embedded options are provisions within a bond contract that grant either the issuer or the bondholder certain rights. The two most common types of embedded options are callable and putable options.
Callable Bonds
Callable bonds give the issuer the right, but not the obligation, to redeem the bond before its maturity date. This feature is advantageous to the issuer in a declining interest rate environment, as
it allows them to refinance the debt at a lower cost. However, for investors, callable bonds introduce call risk, which is the risk that the bond will be called away when interest rates fall, forcing
them to reinvest at lower rates.
Consider a bond with a face value of $1,000, a coupon rate of 5%, and a maturity of 10 years. If the bond is callable after 5 years, the issuer may choose to call the bond if interest rates drop
below 5%, allowing them to issue new debt at a lower rate.
Putable Bonds
Putable bonds, on the other hand, provide the bondholder with the right to sell the bond back to the issuer at a predetermined price before maturity. This feature protects investors in a rising
interest rate environment, as they can reinvest the proceeds at higher rates.
Imagine a bond with a face value of $1,000, a coupon rate of 4%, and a maturity of 10 years, with a put option exercisable after 5 years. If interest rates rise above 4%, the bondholder can exercise
the put option and reinvest the proceeds at the higher prevailing rates.
Yield Measures for Bonds with Embedded Options
Yield measures for bonds with embedded options are more complex than those for plain vanilla bonds. The presence of options requires adjustments to cash flow projections and time horizons. The two
primary yield measures for bonds with embedded options are Yield to Call (YTC) and Yield to Put (YTP).
Yield to Call (YTC)
Yield to Call is the yield of a bond assuming it is called at the earliest possible date. It is calculated similarly to Yield to Maturity (YTM), but with the call date and call price replacing the
maturity date and face value.
$$ YTC = \frac{C + \frac{(Call\ Price - Current\ Price)}{n}}{\frac{(Call\ Price + Current\ Price)}{2}} $$
• \( C \) is the annual coupon payment.
• \( n \) is the number of years until the call date.
• \( Call\ Price \) is the price at which the bond can be called.
Example Calculation:
Consider a callable bond with a face value of $1,000, a coupon rate of 5%, a current price of $1,050, and a call price of $1,020 callable in 5 years.
$$ YTC = \frac{50 + \frac{(1020 - 1050)}{5}}{\frac{(1020 + 1050)}{2}} $$
$$ YTC = \frac{50 - 6}{1035} $$
$$ YTC = \frac{44}{1035} $$
$$ YTC \approx 4.25\% $$
This calculation shows that the yield to call is lower than the coupon rate, reflecting the potential for the bond to be called early.
Yield to Put (YTP)
Yield to Put is the yield of a bond assuming it is put back to the issuer at the earliest possible date. It is calculated similarly to YTM, but with the put date and put price replacing the maturity
date and face value.
$$ YTP = \frac{C + \frac{(Put\ Price - Current\ Price)}{n}}{\frac{(Put\ Price + Current\ Price)}{2}} $$
• \( Put\ Price \) is the price at which the bond can be put.
Example Calculation:
Consider a putable bond with a face value of $1,000, a coupon rate of 4%, a current price of $950, and a put price of $980 putable in 3 years.
$$ YTP = \frac{40 + \frac{(980 - 950)}{3}}{\frac{(980 + 950)}{2}} $$
$$ YTP = \frac{40 + 10}{965} $$
$$ YTP = \frac{50}{965} $$
$$ YTP \approx 5.18\% $$
This calculation indicates that the yield to put is higher than the coupon rate, providing an incentive for the bondholder to exercise the put option if interest rates rise.
Impact of Embedded Options on Yields and Risk
Embedded options significantly affect the risk and return profile of bonds. Understanding these impacts is crucial for making informed investment decisions.
Call Risk and Reinvestment Risk
Callable bonds expose investors to call risk, which is the risk that the bond will be redeemed before maturity, typically when interest rates fall. This can lead to reinvestment risk, as investors
may have to reinvest the proceeds at lower rates, reducing their overall returns.
Yield Calculations and Adjustments
Yield calculations for bonds with embedded options require adjustments to account for the potential exercise of the options. Investors must consider multiple yield measures, such as YTC and YTP, to
assess the most relevant yield based on their expectations of interest rate movements.
Comparing YTM, YTC, and YTP
Investors should compare Yield to Maturity (YTM), Yield to Call (YTC), and Yield to Put (YTP) to determine which yield measure is most relevant for their investment strategy. In a declining interest
rate environment, YTC may be more relevant for callable bonds, while YTP may be more relevant for putable bonds in a rising rate environment.
Illustrating Yield to Call and Yield to Put Calculations
To further illustrate the concepts of YTC and YTP, let’s consider a detailed example of a callable bond and a putable bond.
Example: Callable Bond
A company issues a 10-year callable bond with a face value of $1,000, a coupon rate of 6%, and a call price of $1,050 callable after 5 years. The current market price is $1,080.
1. Calculate YTM:
$$ YTM = \frac{60 + \frac{(1000 - 1080)}{10}}{\frac{(1000 + 1080)}{2}} $$
$$ YTM = \frac{60 - 8}{1040} $$
$$ YTM \approx 4.96\% $$
2. Calculate YTC:
$$ YTC = \frac{60 + \frac{(1050 - 1080)}{5}}{\frac{(1050 + 1080)}{2}} $$
$$ YTC = \frac{60 - 6}{1065} $$
$$ YTC \approx 5.07\% $$
In this scenario, the YTC is slightly higher than the YTM, indicating that if the bond is called, the yield will be slightly better than holding the bond to maturity.
Example: Putable Bond
A company issues a 10-year putable bond with a face value of $1,000, a coupon rate of 5%, and a put price of $970 putable after 4 years. The current market price is $960.
1. Calculate YTM:
$$ YTM = \frac{50 + \frac{(1000 - 960)}{10}}{\frac{(1000 + 960)}{2}} $$
$$ YTM = \frac{50 + 4}{980} $$
$$ YTM \approx 5.51\% $$
2. Calculate YTP:
$$ YTP = \frac{50 + \frac{(970 - 960)}{4}}{\frac{(970 + 960)}{2}} $$
$$ YTP = \frac{50 + 2.5}{965} $$
$$ YTP \approx 5.43\% $$
In this case, the YTP is slightly lower than the YTM, suggesting that if the bond is put, the yield will be slightly less favorable than holding the bond to maturity.
Considerations When Investing in Bonds with Embedded Options
Investing in bonds with embedded options requires careful consideration of various factors:
1. Interest Rate Environment: Understanding the current and expected interest rate environment is crucial for assessing the likelihood of options being exercised.
2. Issuer’s Financial Health: The issuer’s ability to refinance or meet put obligations affects the risk associated with callable and putable bonds.
3. Yield Comparisons: Evaluating YTM, YTC, and YTP helps investors make informed decisions based on their investment goals and risk tolerance.
4. Market Conditions: Market conditions, such as liquidity and demand for bonds, can influence the pricing and attractiveness of bonds with embedded options.
5. Investment Strategy: Aligning bond investments with overall investment strategy and objectives is essential for maximizing returns and managing risk.
By understanding the yield measures and risks associated with bonds with embedded options, investors can make more informed decisions and optimize their bond portfolios.
Quiz Time!
📚✨ Quiz Time! ✨📚
### What is a callable bond? - [x] A bond that can be redeemed by the issuer before maturity. - [ ] A bond that can be sold back to the issuer by the holder before maturity. - [ ] A bond that cannot
be redeemed before maturity. - [ ] A bond with no embedded options. > **Explanation:** Callable bonds allow the issuer to redeem the bond before its maturity date, typically in a declining interest
rate environment. ### What is the primary risk associated with callable bonds? - [x] Call risk - [ ] Default risk - [ ] Inflation risk - [ ] Liquidity risk > **Explanation:** Call risk is the risk
that the bond will be called away when interest rates fall, forcing investors to reinvest at lower rates. ### How is Yield to Call (YTC) calculated? - [x] Using the call date and call price instead
of the maturity date and face value. - [ ] Using the maturity date and face value. - [ ] Using the put date and put price. - [ ] Using the current market price only. > **Explanation:** YTC is
calculated using the call date and call price, reflecting the yield if the bond is called at the earliest possible date. ### What does Yield to Put (YTP) assume? - [x] The bond is put back to the
issuer at the earliest possible date. - [ ] The bond is held to maturity. - [ ] The bond is called at the earliest possible date. - [ ] The bond is sold in the secondary market. > **Explanation:**
YTP assumes the bond is put back to the issuer at the earliest possible date, reflecting the yield if the put option is exercised. ### In a rising interest rate environment, which bond feature is
more beneficial to investors? - [x] Putable bond - [ ] Callable bond - [ ] Convertible bond - [ ] Zero-coupon bond > **Explanation:** Putable bonds are more beneficial in a rising interest rate
environment, as they allow investors to sell the bond back and reinvest at higher rates. ### What is the impact of a callable bond being called? - [x] Investors may have to reinvest at lower rates. -
[ ] Investors will receive a higher yield. - [ ] The bond will increase in value. - [ ] The bond will decrease in value. > **Explanation:** If a callable bond is called, investors may have to
reinvest the proceeds at lower rates, leading to reinvestment risk. ### How does a putable bond protect investors? - [x] By allowing them to sell the bond back to the issuer if interest rates rise. -
[ ] By providing a fixed interest rate. - [ ] By offering a higher coupon rate. - [ ] By guaranteeing a minimum yield. > **Explanation:** Putable bonds protect investors by allowing them to sell the
bond back to the issuer if interest rates rise, enabling reinvestment at higher rates. ### What should investors compare when evaluating bonds with embedded options? - [x] YTM, YTC, and YTP - [ ]
Only YTM - [ ] Only YTC - [ ] Only YTP > **Explanation:** Investors should compare YTM, YTC, and YTP to determine which yield measure is most relevant for their investment strategy. ### Which yield
measure is most relevant for callable bonds in a declining interest rate environment? - [x] Yield to Call (YTC) - [ ] Yield to Maturity (YTM) - [ ] Yield to Put (YTP) - [ ] Current Yield >
**Explanation:** In a declining interest rate environment, YTC is more relevant for callable bonds, as it reflects the potential for the bond to be called early. ### True or False: Embedded options
in bonds can significantly impact their yield calculations and risk profile. - [x] True - [ ] False > **Explanation:** Embedded options can significantly impact yield calculations and the risk
profile of bonds, influencing investment decisions. | {"url":"https://csccourse.ca/4/2/3/","timestamp":"2024-11-15T03:20:56Z","content_type":"text/html","content_length":"95534","record_id":"<urn:uuid:62e69ff4-6ace-4552-8164-2d0a765a1e55>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00485.warc.gz"} |
Dr. Jenny Bay-Williams, Productive Ways to Build Fluency with Basic Facts
ROUNDING UP: SEASON 1 | EPISODE 15
Ensuring students master their basic facts remains a shared goal among parents and educators. That said, many educators wonder what should replace the memorization drills that cause so much harm to
their students’ math identities. Today on the podcast, Dr. Jenny Bay-Williams talks about how to meet that goal and shares a set of practices that also support student reasoning and sensemaking.
Eight Unproductive Practices in Developing Fact Fluency
Mike Wallus: Ensuring students master their basic facts remains a shared goal among parents and educators. That said, many educators wonder what should replace the memorization drills that cause so
much harm to their students' math identities. Today on the podcast, Jenny Bay-Williams talks about how to meet that goal and shares a set of productive practices that also support student reasoning
and sensemaking.
Mike: Welcome to the podcast, Jenny. We are excited to have you.
Jennifer Bay-Williams: Well, thank you for inviting me. I'm thrilled to be here and excited to be talking about basic facts.
Mike: Awesome. Let's jump in. So, your recommendations start with an emphasis on reasoning. I wonder if we could start by just having you talk about the “why” behind your recommendation and a little
bit about what an emphasis on reasoning looks like in an elementary classroom when you're thinking about basic facts.
Jenny: All right, well, I'm going to start with a little bit of a snarky response: that the non-reasoning approach doesn't work.
Mike and Jenny: (laugh)
Jenny: OK. So, one reason to move to reasoning is that memorization doesn't work. Drill doesn't work for most people. But the reason to focus on reasoning with basic facts, beyond that fact, is that
the reasoning strategies grow to strategies that can be used beyond basic facts. So, if you take something like the making 10 idea—that nine plus six, you can move one over and you have 10 plus
five—is a beautiful strategy for a 99 plus 35. So, you teach the reasoning up front from the beginning, and it sets students up for success later on.
Mike: That absolutely makes sense. So, you talk about the difference between telling a strategy and explicit instruction. And I raised this because I suspect that some people might struggle to think
about how those are different. Could you describe what explicit instruction looks like and maybe share an example with listeners?
Jenny: Absolutely. First of all, I like to use the whole phrase: “explicit strategy instruction.” So, what you're trying to do is have that strategy be explicit, noticeable, visible. So, for example,
if you're going to do the making 10 strategy we just talked about, you might have two 10-frames. One of them is filled with nine counters, and one of them is filled with six counters. And students
can see that moving one counter over is the same quantity. So, they're seeing this flexibility that you can move numbers around, and you end up with the same sum. So, you're just making that idea
explicit and then helping them generalize. You change the problems up and then they come back, and they're like, “Oh, hey, we can always move some over to make a 10 or a 20 or a 30” or whatever
you're working on. And so, I feel like, in using the counters—or they could be stacking unifix cubes or things like that—that's the explicit instruction.
Jenny: It's concrete. And then, if you need to be even more explicit, you ask students in the end to summarize the pattern that they noticed across the three or four problems that they solved. “Oh,
that you take the bigger number, and then you go ahead and complete a 10 to make it easier to add.” And then, that's how you're really bringing those ideas out into the community to talk about. For
multiplication, I'm just going to contrast. Let's say we're doing add a group strategy with multiplication. If you were going to do direct instruction, and you're doing six times eight, you might
say, “All right, so when you see a six,” then a direct instruction would be like, “Take that first number and just assume it's a five.” So then, “Five eights is how much? Write that down.” That's
direct instruction. You're like, “Here, do this step here, do this step here, do this step.”
Jenny: The explicit strategy instruction would have, for example—I like eight boxes of crowns because they oftentimes come in eight. So, but, they'd have five boxes of crowns and then one more box of
crowns. So, they could see you've got five boxes of crowns. They know that fact is 40, they—if they're working on their sixes, they should know their fives. And so, then what would one more group be
about? So, just helping them see that with multiplication through visuals, you're adding on one group, not one more, but one group. So, they see that through the visuals that they're doing or through
arrays or things like that. So, it's about them seeing the number of relationships and not being told what the steps are.
Mike: And it strikes me, too, Jenny, that the role of the teacher in those two scenarios is pretty different.
Jenny: Very different. Because the teacher is working very hard (chuckles) with the explicit strategy instruction to have the visuals that really highlight the strategy. Maybe it's the colors of the
dots or the exact 10-frames they've picked, and have they filled them or whether they choose to use the unifix cubes, and how they're going to color them, and things like that. So, they're doing a
lot of thinking to make that pattern noticeable, visible. As opposed to just saying, “Do this first, do that second, do that third.”
Mike: I love the way that you said that you're doing a lot of thinking and work as a teacher to make a pattern noticeable. That's powerful, and it really is a stark contrast to, “Let me just tell you
what to do.” I'd love to shift a little bit and ask you about another piece of your work. So, you advocate for teaching facts in an order that stresses relationships rather than simply teaching them
in order. I'm wondering if you can tell me a little bit more about how relationships-based instruction has an impact on student thinking.
Jenny: So, we want every student to enact the reasoning strategies. So, I'm going to go back to addition, for example. And I'm going to switch over to the strategy that I call pretend-to-10, also
called use 10 or compensation. But if you're going to set them up for using that strategy, [there are] a lot of steps to think through. So, if you're doing nine plus five, then in the pretend-to-10
strategy, you just pretend that nine is a 10. So now you've got 10 plus five and then you've got to compensate in the end. You’ve got to fix your answer because it's one too much. And so, you've got
to come back one. That's some thinking. Those are some steps. So, what you want is to have the students automatic with certain things so that they're set up for that task. So, for that strategy, they
need to be able to add a number onto 10 without much thought.
Jenny: Otherwise, the strategy is not useful. The strategy is useful when they already know 10 plus five. So, you teach them this, you teach them that relationship, you know 10 and some more, and
then they know that nine’s one less than 10. That relationship is hugely important, knowing nine is one less than 10—um, and so then they know their answer has to be one less. Nine’s one less than
10. So, nine plus a number is one less than 10 plus the number. Huge idea. And there's been a lot of research done in kindergarten on students understanding things like seven’s one more than six,
seven’s one less than eight. And they're predictive studies looking at student achievement in first grade, second grade, third grade. And students—it turns out that one of the biggest predictors of
success is students understanding those number relationships. That one more, one less, um, two more, two less—hugely important in doing the number sense. So that's what the relationship piece is, is
sequencing facts so that what is going to be needed for the next thing they're going to do, the thinking that's going to be needed, is there for them. And then build on those relationships to learn
the next strategy.
Mike: I mean, it strikes me that there's a little bit of a twofer in that one. The first is this idea that what you're doing is purposely setting up a future idea, right? It's kind of like saying,
“I'm going to build this prior knowledge about 10-ness, and then I'm going to have kids think about the relationship between 10 and nine.” So, like, the care in this work is actually really
understanding those relationships and how you're going to leverage them. The other thing that really jumps out from what you said—this has long term implications for students’ thinking—it's not just
fact acquisition, it's what you said, research shows that this has implications for how kids are thinking further down the road. Am I understanding that right?
Jenny: That's absolutely correct. So just that strategy alone. Let's say they're adding 29 plus 39. And they're like, “Oh hey, both of those numbers are right next to the next benchmark. So instead
of 29 plus 39, I'm going to add 30 plus 40, 70. And I got, I went up two, so I'm going to come back down two. And I know that two less than a benchmark's going to land on an eight to that.” Again,
it's coming back to this relationship of how far apart numbers are—what's right there within a set of 10—helps then to generalize within 10s or within 100s. And by the way, how about fractions?
Mike: Hmm. Talk about that.
Jenny: (laughs) It generalizes to fractions. So, let's take that same idea of adding. Let's just say it's like, two and seven-eighths plus two and seven-eighths. So, if we just pretended those were
both threes because they're both super close to three, then you'd have six, and then you added on two-eighths too much. So, you come back two-eighths, or a fourth, and you have your answer. You don't
have to do the regrouping with fractions and all the mess that really gets bogged down. And it's a much more efficient method that, again, you set students up for when they understand these number
relationships. When you get into fractions, you're thinking about, like, how close are you to the next whole number maybe, instead of to the next 10s number.
Mike: It strikes me that if you have a group of teachers who have a common understanding of this approach to facts, and everyone's kind of playing the long game and thinking about how what they're
doing is going to support what's next, it just creates a system that's much more intentional in helping kids not only acquire the facts, but build a set of ways of thinking.
Jenny: Mike, that's exactly it. I mean, here we are, we're trying to make up for lost time. We never have enough time in the classroom. We want an efficient way to make sure our kids get the most
learning in. And so, to me that is about investing early in the fact strategies. Because then actually when you get up to those other things that you're adding or subtracting or multiplying or
whatever you're doing, you benefit from the fact that you took time early to learn those strategies. Because those strategies are now very useful for all this other math that you're doing. And then
students are more successful in making good choices about how they're going to solve those problems that are, oftentimes—especially when, I like to mention fractions and decimals at least once in a
basic facts talk because we get back, by the time we get into fractions and decimals, we're back to just sometimes only showing one way, the sort of standard algorithm way. When, in fact, those basic
facts strategies absolutely apply to almost-always more-efficient strategies for working with fractions and decimals.
Mike: I want to shift a little bit. One of the things that was really helpful for me in growing my understanding is, the way that you talk about a set of facts that you would describe as
“foundational” facts and another set of facts that you would describe as “derived” facts. And I'm wondering if you can unpack what those two subsets are and how they're related to one another.
Jenny: Yeah. So, the foundational facts are ones where automaticity is needed in order to enact a strategy. So, to me, the foundational fact strategies are, they're names. Like the doubling strategy
or double and double again, some people call it. Or add a group for multiplication, and the addition ones of making 10s and pretend-to-10 strategies. And in those strategies, you can solve lots of
different facts. But there's too much going on (laughs) in your brain if you don't have automaticity with the facts you need. So, for example, if you have your six facts, and you're trying to get
your six facts down. And you already know your fives, like, automaticity with your fives, then that becomes a useful way to get your sixes. So, if you have six times eight, and you know five times
eight is 40, then you're like, “I got one more 8, 48.”
Jenny: That's an added group strategy. But if you're not automatic with your fives, this is how this sounds when you're interviewing a child. They're going to use add a group strategy, but they don't
know their fives. So, then they're like, “Let's see, five times eight is 5, 10, 15, 20, 25, 30, 40. Now, what was I doing?” Like, they can't finish it because they were skip-counting with their
fives. They lose track of what they're doing, is my point. So, the key is that they just know those facts that they need in order to use a strategy. And that, going back to, like, the pretend-to-10,
they got to know 10-and-some-more facts to be successful. They have to know nine’s one less than 10 to be successful. So, that's the idea is, if they reach automaticity with the foundational fact
sets, then their brain is freed up to go through those reasoning strategies.
Mike: That totally makes sense. I want to shift a little bit now. One of the things that I really appreciated about the article was that you made what I think is a very strong, unambiguous case for
ending many of the past practices used for fact acquisition—worksheets and timed tests, in particular. This can be a tough sell because this is often what is associated with elementary mathematics,
and families kind of expect this kind of practice. How would you help an educator explain the shift away from these practices to folks who are out in the larger community? What is it that we might
help say to folks to help them understand this shift?
Jenny: That's a great question, and the real answer is, it depends, again, on audience. So, who is your audience? Even if the audience is parents, what do those parents prioritize and want for their
children? So, I feel like [there are] lots of reasons to do it, but to really speak to what matters to them. So, I'm going to give a very generic answer here. But for everyone, they want their child
to be successful. So, I feel that that opportunity to show, to give a problem like 29 plus 29, and ask how parents might add that problem. And if they think 30 plus 30 and subtract two to get to the
answer, whatever, then that gives this case to say, “Well this is how we're going to work on basic facts. We're building up so that your child is ready to use these strategies. We're going to start
right with the basic facts, learning these strategies. These really matter.”
Jenny: And the example I gave could be whatever fits with the level of their kid. So, it could be like 302 minus 299. It's a classic one where you don't want your child to implement an algorithm
there, you want them to notice those numbers are three apart. And so, there's this work that begins early. So, I think that's part of it. I think another part of it is helping people just reflect on
their own learning experiences. What were your learning experiences with basic facts? And even if they liked the speed drills, they oftentimes recognize that it was not well-liked by most people. And
also, then, they really didn't learn strategies. So, I feel like we have to be showing that we're not taking something away, we're adding something in. They are going to become automatic with their
facts. They're not going to forget them because we're not doing this memorizing that leads to a lot of forgetting. And bonus, they're going to have these strategies that are super useful going
forward. So, to me, those are some of the really strong speaking points. I like to play a game and then just stop and pause for a minute and just say, “Did you see how hard it was for me to get you
quiet? Do you see how much fun you were having?” And then I just hold up a worksheet (laughs). I'm like, “And how about this?” You know, again, that emotional connection to the experience and the
Mike: That is wonderful. Since you brought it up, let's talk about replacements for worksheets and timed tests.
Jenny: Um-hm.
Mike: So, you advocate for games as you said, and for an activity-based approach. I think that what I want to try to do is get really specific so that if I'm a classroom teacher, and I can't see a
picture of that yet, can you help paint a picture? Like what might that look like?
Jenny: I love that question because [there are] lots of good games and lots of places. But again, like I said earlier, this thinking really deeply about what game I'm choosing and for what. What do
my students need to practice? And then being very intentional about game choice is really important. So, for example, if students are working on their 10-and-some-more facts, then you want to play a
game where all the facts are 10-and-some-more facts. That's what they're working on. And then maybe you mix in some that aren't. Or you play a game with that and then they sort cards and find all the
solve the 10 and more, or [there are] lots of things they can do. They can play concentration, where the fact is hidden and the answer is hidden and things like that. So, you can be very focused. And
then when you get to the strategies, you want to have a game that allows for students to say, allow their strategies.
Jenny: So, I'm a big fan of, like, sentence frames, for example. So, [there are] games that we have in our Math Fact Fluency book that are in other places that specifically work on a strategy. So,
for example, if I'm working on the pretend-to-10 strategy, I like to play the game fixed-addend war, which is the classic game of war, except, there's an addend in the middle, and it's a nine, to
start. And then each of the two players turns up a card. So, Mike, if you turn up a seven, then you're going to explain how you're going to use the pretend-to-10 strategy to add it. And I turned up a
six, so I'm going to, I'm going to do this, then, I'll—you can do it. So, I turned up a six. So, I'm going to say, “Well, 10 and six is 16, so nine and six is one less, 15.” I've just explained the
pretend-to-10 strategy. And then you get your turn.
Mike: And I'd say, “Well seven and 10—I know seven and 10 is 17, so seven and nine has to be one less, and that's 16.”
Jenny: Yeah. So, your total's higher than mine, you win those two cards, you put them in your deck, and we move on. So, that's a way to just practice thinking through that strategy. Notice there's no
time factor in that. You have a different card than I have. You have as much time, and we're doing think-aloud. These are all high-leverage practices. Then we get to the games where it's like, you
might turn up a six and a five where you're not going to use the pretend-to-10 strategy for that. You've got to think, “Oh, that doesn't really fit that strategy because neither one of those numbers
is really close to 10. Oh, hey, it's near a double, I'm going to use my double.” So, you sequence these games to—if you start with one of those open-ended games, it might be too big of a jump because
students aren't ready to choose between their strategies. They have to first be adept at using their strategies. And once they're adept at using them, then they're ready to play games where they get
to choose among the strategies.
Mike: So, you're making me think a couple things, Jenny. One is, it's not just that we're shifting to using games as a venue to practice to get to automaticity. You're actually saying that when we
think about the games, we really need to think about, “What are the strategies that we're after for kids?” And then make sure that the way that the game is structured, like, when you're talking about
the pretend-to-10, with the fixed addend. That's designed to elicit that strategy and have kids work on developing their language and their thinking around that particularly. So, there's a level of
intent around the game choice and the connection to the strategies that kids are thinking about. Am I understanding that right?
Jenny: That's it. That's exactly right. That's exactly right. And a huge, a lot of intentionality so that they have that opportunity and a no-pressure, a low-stress, think through the strategy. If
they make a mistake, their peer or themselves usually correct it in the moment, and they get so much practice in. I mean, imagine going through half a deck of cards playing that game.
Mike: Yeah.
Jenny: That's 26 facts. And then picture those 26 facts on a page of paper. And then, and again, in the game that you've got the added benefit of think-aloud, and then you're hearing what your peer
has said.
Mike: You know, one of the things that strikes me is, if I'm a teacher, I might be thinking like, “This is awesome, I'm super excited about it. Holy mackerel, do I have to figure these games out
myself?” And I think the good news is, there's a lot of work that's been done on this. I know you've done some. Do you have any recommendations for folks? There's of course curriculum. But do you
have recommendations for resources that you think help a teacher think about this or help a teacher see some of the games that we're talking about?
Jenny: Well, I'm going to start with my Math Fact Fluency book because that is where we go through each of these strategies, each of the foundational facts sets and the strategies, and for each one
supply a game. And then from those games, they're easily adaptable to other settings. And some of the games are classic games. So, there's a game, for example, called “Square Deal.” And the idea is
that you're covering a game board, and you're trying to make a square. So, you get a two-by-two grid taken, and you score a point or five points or whatever you want to score. Well, we have that game
housed under the 10-and some-more facts. So, all the answers are like 19, 16, 15, and the students turn over a 10 card and another card, and if it's a 10 and a five, they get to claim a 15 spot on
the game board.
Jenny: Well, that game board can be easily adapted to any multiplication fact sets, any other addition. I like to do a Square Deal with 10 and some more, and then I like to do Square Deal with nine
and some more. There's my effort, again, to come back to either pretend-to-10 or making 10. Where they're like, “Oh, I just played 10 and some more. Now we're doing the same game, but it's nine and
some more.” So, I feel like there's a lot of games there. And there is a free companion website that has about half of the games ready to download in English and in Spanish.
Mike: Any chance you'd be willing to share it?
Jenny: Yeah, absolutely. So, you can just Google it. The Kentucky Center for Mathematics created it during COVID, actually, as a gift to the math community. And so, if you type in “Kentucky Center
for Math” or “KCM math fact fluency companion website,” it will pop up.
Mike: That's awesome. I want to ask you about one more thing before we close because we've really talked about the replacement for worksheets, the replacements for timed tests. But there is a piece
of this where people think about, “How do I know?” right? “How can I tell that kids have started to build this automaticity?” And you make a pretty strong case for interviewing students to understand
their thinking. I'm wondering if you could just talk again about the “why” behind it and a little bit about what it might look like.
Jenny: So, first of all, timed tests are definitely a mistake for many reasons. And one of the reasons—beyond the anxiety they cause—they're just very poor assessment tools. So, you can't see if the
student is skip-counting or not, for example, for multiplication facts. You can't see if they're counting by ones for the addition facts. You can't see that when they're doing the test, and you can't
assume that they're working at a constant rate; that they're just solving one every, you know, couple of seconds, which is the way those tests are designed. Because I can spend a lot of time on one
and less time on the other. So, they're just not, they're just not effective as an assessment tool. So, if you flip that—let's say they're playing the game we were talking about earlier, and you just
want to know, can they use the pretend-to 10 strategy?
Jenny: That's your assessment question of the day. Well, you just wander around with a little checklist (chuckles), you know? Yes, they can. No, they can't. And so, a checklist can get at the
strategies, and a checklist can also get at the facts like how well are they doing with their facts? So, once they do some of those games that are more open-ended, you can just observe and listen to
them and get a feel for that. If they're playing Square Deal with whatever fact, you know. So, what happens is you're, like, “I wonder how they're doing with their fours. We've really been working
with their fours a lot.” Well, you can play Square Deal or a number of other games where that day you're working on fours. The fixed-addend war can become fixed-factor war, and you put a four in the
middle. So adaptable games and then you're just listening and watching.
Jenny: And if you're not comfortable with that approach, then they can be playing those games, and you can have students channeling through where you do a little mini-interview. It only takes a few
questions to get a feel for whether a student knows their facts. And you can really see who's automatic and who's still thinking. So, for example, a student who's working on their fours, if you give
them four times seven, they might say, “Twenty-eight.” I call that automatic. Or they might, they might do four times seven, and they pause, and they're like, “Twenty-eight.” Then I'm like, “How did
you think about that?” And they're like, “Well, I doubled and doubled again.” “Great.” So, I can mark off that they are using a strategy, but they're not automatic yet. So that to me is a check, not
a star. And if I ask, “How did you do it?” And they say, “Well, I skip-counted.” Well then, I'm marking down the skip-counted. Because that means they need a strategy to help them move toward
Mike: I think what strikes me about that, too, is, when you understand where they're at on their journey to automaticity, you can actually do something about it as opposed to just looking at the
quantity that you might see on a timed test. What's actionable about that? I'm not sure, but I think what you're suggesting really makes the case that I can do something with data that I observe or
data that I hear in an interview or see in an interview.
Jenny: Absolutely. I mean this whole different positioning of the teacher as coaching the student toward their growth, helping them grow in their math proficiency, their math fluency. You see where
they're at and then you're monitoring that in order to move them forward instead of just marking them right or wrong on a timed test. I think that's a great way to synthesize that.
Mike: Well, I have to say, it has been a pleasure talking with you. Thank you so much for joining us today.
Jenny: Thank you so much. I am again thrilled to be invited and always happy to talk about this topic.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence
and ability. | {"url":"https://www.mathlearningcenter.org/blog/dr-jenny-bay-williams-productive-ways-build-fluency-basic-facts","timestamp":"2024-11-09T00:14:48Z","content_type":"text/html","content_length":"55018","record_id":"<urn:uuid:fcc50b83-f6ec-45c8-b2c4-2b773026553e>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00335.warc.gz"} |
Multiphase flow simulation of rich gas pipelines with traces of liquid glycol - Flerfasesimulering av rikgassrørledninger med små mengder glykol væske
pipelines with traces of liquid glycol
Flerfasesimulering av rikgassrørledninger med små mengder glykol væske
Jørgen Børsum Haugen Petter Holm Reiso
Master of Science in Mechanical Engineering Supervisor: Even Solbraa, EPT
Department of Energy and Process Engineering Submission date: June 2016
Norwegian University of Science and Technology
This Master’s thesis was written at the Department of Energy and Process Engineering at the Norwegian University of Science and Technology in Trondheim in the spring of 2016. The ex- perimental work
was conducted at Statoil Research and Development center at Rotvoll, Trond- heim. Statoil provided field data from the Åsgard transport pipeline which have been used in the OLGA simulations. The
report is a continuation of a project thesis written in the fall of 2015.
This assignment represents the workload of 30 ECTS.
Trondheim, 2016-06-13
Petter Holm Reiso Jørgen Børsum Haugen
We would like to thank our supervisor, professor II Even Solbraa, for his guidance and help throughout the assignment. We encountered several complex issues where he was essential in finding the
We would like to thank Ole Johan Berg at Statoil for guidance and support with the experimental work. We would also like to thank Knud Lunde who provided field data and OLGA files for the Åsgard
transport pipeline. Finally, we would like to thank Marie Vikre Danielsen at Statoil for measuring the liquid mixtures utilized for the experimental work.
P.H.R J.B.H
Long distance transport of multiphase flow is an important technology in the development of oil and gas fields. Predicting phase behaviour in long pipelines is a demanding and complicated process. To
realistically simulate these situations, the industry is reliant on software that can cal- culate accurate fluid properties. The most used simulation tool today, is a computer program named OLGA (OiL
and GAs simulator). OLGA requires input in the form of a fluid property table to conduct these simulations. These property tables are generated by tools like PVTsim (Pressure, Volume and Temperature
simulator) and NeqSim (Non-Equilibrium Simulator). The purpose of this Master’s thesis was to further develop and improve NeqSim as a fluid property table generator. This task was specifically aimed
towards liquid viscosity and interfacial tensions of aqueous TEG (TriEthylene Glycol). These properties are regarded among the most influential parameters for fluid behaviour.
Experimental work was conducted to measure the interfacial tensions of high pressure aqueous TEG and methane. There is low availability of such data in the literature. The measurement method utilized
was the pendant drop method. The interfacial tensions were measured with an uncertainty of less than 2%. Relevant experimental data for liquid viscosities and interfacial tensions were also
The measured data and the collected experimental data were compared to calculated values from NeqSim and PVTsim. These tools utilize similar empirical methods to calculate liquid viscosities.
Interfacial tensions are calculated by the Firoozabadi Ramey Method in PVTsim.
NeqSim offers several calculation methods for interfacial tension. These include the Firooz- abadi Ramey Method, the Parachor Method, Linear Gradient Theory, Gradient Theory Simple and Gradient
Theory. NeqSim proved to predict the most accurate liquid viscosity values. In regard to interfacial tensions, Gradient Theory in NeqSim provided the most accurate results.
A parameter study was conducted in OLGA to establish how liquid viscosity and interfacial ten- sions affects the simulations of multiphase flow. The simulations were conducted using field data
provided by Statoil of the Åsgard transport pipeline. The simulations were conducted us- ing both the standard OLGA module and the OLGA HD module. It was shown that alterations in liquid viscosity
and interfacial tensions have small impacts on the simulations. Increasing the mass flow of TEG did impact the simulations. The impact was negligible with the standard OLGA module, but significant
with the OLGA HD module. It is concluded that the OLGA HD module results in more accurate simulations of the Åsgard transport pipeline. It is also con- cluded that both NeqSim and PVTsim calculates
sufficiently accurate property values to be used in the current OLGA version 7.3.5.
The development of NeqSim as a property generator is a continuous process. There are still areas where NeqSim can be improved. These are discussed in the final chapter, and include further
improvements in the calculations of liquid viscosity and interfacial tension and the gen- eration of wax property tables.
Langdistansetransport av flerfasestrømning er en viktig teknologi i utviklingen av olje- og gassfelt.
Å forutsi faseoppførsel i lange rørledninger er en krevende og komplisert prosess. For å realistisk simulere disse situasjonene er industrien avhengig av programvare som kan kalkulere presise
fluidegenskaper. Det mest brukte simuleringsverktøyet idag er et dataprogram kalt OLGA (OiL and GAs simulator). OLGA behøver innmating i form av en tabell med fluidegenskaper for å utføre
simuleringene. Disse tabellene er genererert av verktøy som PVTsim (Pressure, Volume and Temperature simulator) og NeqSim (Non-Equilibrium Simulator). Hensikten med denne masteroppgaven var å
videreutvikle og forbedre programvaren NeqSim som en tabellgenera- tor. Oppgaven var spesifikt rettet mot væskesviskositet og overflatespenning av vannholdig TEG (TriEthylene Glycol). Disse
egenskapene er ansett blant de mest innflytelsesrike parametrene for fluidoppførsel.
Eksperimentelt arbeid ble utført for å måle overflatespenning av TEG og metan under høyt trykk.
Det er liten tilgjengelighet på slike data i litteraturen. Målingsmetoden som ble benyttet var
"pendant drop"-metoden. Overflatespenningene ble målt med en usikkerhet på mindre enn 2%. Relevant eksperimentell data for væskeviskositet og overflatespenning har ble også innhen- tet.
De målte verdiene og de innhentede eksperimentelle data ble sammelignet med beregnede verdier fra NeqSim og PVTsim. Programvarene benytter lignende empiriske metoder for å beregne væskeviskositet.
Overflatespenning blir beregnet med Firoozabadi Ramey-metoden i PVTsim.
NeqSim tilbyr flere beregningsmetoder for overflatespenning. Dette inkluderer Firoozabadi Ramey- metoden, Parachor-metoden, "Linear Gradient Theory", "Gradient Theory Simple" og "Gradi- ent Theory".
NeqSim beregnet mest nøyaktige verdier for væskeviskositet. Med hensyn til over- flatespenning beregnet "Gradient Theory" i NeqSim de mest nøyaktige verdiene.
En parameterstudie ble gjennomført i OLGA for å etablere hvordan væskeviskositet og over- flatespenning påvirker simuleringer av flerfasestrømning. Simuleringene ble utført ved bruk av feltdata fra
Statoil av Åsgard transport rørledningen. Simuleringene ble utført både med stan- dard OLGA-modulen og med OLGA HD-modulen. Det ble vist at endringer i væskeviskositet og overflatespenning har liten
påvirkning på simuleringene. Økning av massestrømmen av TEG påvirket simuleringene. Effekten var neglisjerbar med standard OLGA-modulen, men betydelig med OLGA HD-modulen. Det er konkludert at OLGA
HD-modulen resulterer i mer nøyaktige simuleringer av Åsgard transport rørledningen. Det er også konkludert at både NeqSim og PVT- sim beregner tilstrekkelig nøyaktige egenskapsverdier til å brukes i
den nåverende OLGA versjo- nen 7.3.5.
Utviklingen av NeqSim som en tabellgenerator er en kontinuerlig prosess. Det finnes fremdeles områder hvor NeqSim kan forbedres. Disse er diskutert i det siste kapittelet, og omhandler yt- terligere
forbedringer i beregningene av væskeviskositet og overflatespenning, samt generering av egenskapstabeller for voks.
Preface . . . iv
Acknowledgement . . . v
Abstract. . . vi
Sammendrag . . . viii
1 Introduction 1 1.1 Multiphase Flow. . . 1
1.2 Thesis Specification . . . 2
1.3 Report Structure. . . 2
2 Dehydration 4 2.1 Hydrates . . . 4
2.2 Absorption . . . 4
2.3 Glycols . . . 5
2.3.1 TEG . . . 5
2.3.2 TEG Content in Gas Phase . . . 6
3 Properties Needed for Multiphase Flow Simulation 7 3.1 Conservation Laws . . . 8
3.1.1 Mass Conservation . . . 9
3.1.2 Momentum Conservation . . . 10
3.1.3 Energy Conservation . . . 10
3.2 Stratified Two Phase Flow . . . 10
3.2.1 Simplified Conservation Laws . . . 11
3.2.2 Closure Relations - Interfacial Shear . . . 12
3.3 Other Properties for Closure Relations. . . 13
4 Theoretical Concepts 14 4.1 Liquid Viscosity . . . 14
4.2 Models to Calculate Liquid Viscosity. . . 15
4.2.1 Empirical Models . . . 15
4.2.2 Friction Theory . . . 17 x
4.3 Surface Tension . . . 18
4.3.1 Attractive Intermolecular Forces . . . 18
4.3.2 Excess Surface Free Energy . . . 19
4.3.3 Forces on a Curved Surface . . . 20
4.4 Models for Calculating Surface Tension . . . 22
4.4.1 Parachor Method . . . 22
4.4.2 Firoozabadi and Ramey . . . 22
4.4.3 Gradient Theory . . . 22
5 Software 25 5.1 OLGA . . . 25
5.1.1 OLGA HD . . . 26
5.2 NeqSim: A General Non-Equilibrium Simulator . . . 27
5.2.1 Property Simulation . . . 27
5.2.2 Viscosity . . . 27
5.2.3 Interfacial Tension . . . 28
5.3 PVTsim . . . 28
5.3.1 Viscosity . . . 29
5.3.2 Interfacial Tension . . . 29
6 Collected Experimental Data 30 6.1 Viscosity . . . 30
6.1.1 Viscosity of TEG . . . 30
6.1.2 Viscosity of Aqueous TEG . . . 31
6.1.3 Viscosity of Aqueous High Pressure TEG and Methane . . . 32
6.2 Interfacial Tension . . . 33
6.2.1 Measurement Methods . . . 33
6.2.2 Interfacial Tension of Aqueous TEG and Air . . . 34
6.2.3 Interfacial Tension of High Pressure Water and Methane . . . 34
6.2.4 Interfacial Tension of High Pressure Aqueous TEG and Methane . . . 35
7 Experimental Setup 37 7.1 Experimental Apparatus . . . 37
7.1.1 Temperature Test Chamber . . . 37
7.1.2 Schematic Overview . . . 38
7.1.3 Pendant Drop Cell . . . 38
7.1.4 Densitometer . . . 39
7.1.5 Pressure Sensors . . . 39
7.1.6 Circulation Pump . . . 39
7.2 Experimental Procedure . . . 41
7.2.1 Filling the Rig . . . 41
7.2.2 Mixing Process . . . 41
7.2.3 Measurements . . . 42
7.3 Fluid Specification . . . 43
8 Experimental Results 44 8.1 Experimental Conditions . . . 44
8.2 Experimental Results . . . 45
8.2.1 100 wt% TEG . . . 45
8.2.2 90 wt% TEG . . . 46
8.2.3 Comparing Values of Liquid Mixtures . . . 47
8.3 Uncertainty Analysis . . . 48
8.3.1 Drop Shape Analysis . . . 49
8.3.2 Measured Conditions . . . 49
8.3.3 Overall Uncertainty. . . 50
9 Comparison of Experimental and Simulated Data 52 9.1 Collected Viscosity Data Compared to Simulated Data . . . 52
9.1.1 Viscosity of TEG . . . 52
9.1.2 Viscosity of Aqueous TEG . . . 53
9.1.3 Viscosity of High Pressure Aqueous TEG and Methane . . . 58
9.2 Collected Interfacial Tension Data Compared to Simulated Data . . . 60
9.2.1 Interfacial Tension of Aqueous TEG and Air . . . 60
9.2.2 Interfacial Tension of High Pressure Water and Methane . . . 62
9.2.3 Interfacial Tension of High Pressure Aqueous TEG and Methane . . . 63
9.3 Measured Values of this Study Compared to Simulated Data. . . 67
9.3.1 Interfacial Tensions of 100 wt% TEG . . . 68
9.3.2 Interfacial Tensions of 90 wt% TEG . . . 71
9.3.3 Interfacial Tensions of both Liquid Mixtures . . . 73
9.3.4 Densities . . . 76
9.4 Measured Values of this Study Compared to Ng et al. (2009) . . . 79
9.5 Evaluation of Simulation Tools . . . 82
9.5.1 Viscosity . . . 83
9.5.2 Interfacial Tension . . . 84
10 Simulations in OLGA 86 10.1 Input Structure . . . 86
10.1.1 Åsgard Transport . . . 87
10.1.2 NeqSim . . . 87
10.1.3 PVTsim . . . 88
10.1.4 OLGA . . . 88
10.2 Simulations with the Standard OLGA Module . . . 90
10.2.1 Scenario One - Carryover of TEG . . . 90
10.2.2 Scenario Two - Initial Dump of TEG . . . 94
10.3 Simulations with the OLGA HD Module . . . 97
10.3.1 Scenario One - Carryover of TEG . . . 97
10.3.2 Scenario Two - Initial Dump of TEG . . . 101
10.4 Evaluation of OLGA Simulations . . . 104
10.4.1 The Standard OLGA Module . . . 104
10.4.2 The OLGA HD Module . . . 105
10.4.3 Comparing OLGA Modules and Scenarios . . . 106
11 Discussion 109 11.1 Laboratory Work . . . 109
11.2 Comparison of Experimental and Simulated Data . . . 110
11.3 Simulations in OLGA . . . 112
12 Conclusion 116 12.1 Further work . . . 118
12.1.1 Liquid Viscosity Calculations in NeqSim . . . 118
12.1.2 Interfacial Tension in NeqSim . . . 118
12.1.3 Other Properties in NeqSim . . . 118
12.1.4 Wax tables . . . 119
Bibliography 120 A The OLGA HD Module 124 B Additional Viscosity Information 125 B.1 Viscosity of Aqueous TEG, Begum et al. (2012) . . . 125
B.2 Viscosity of Aqueous TEG, Sun and Teja (2003). . . 127
B.3 Viscosity of High Pressure Aqueous TEG and Methane, Ng et al. (2009) . . . 128
B.4 Viscosity Data from Statoil. . . 129
C Additional Interfacial Tension Information 130 C.1 Interfacial Tension of High Pressure Water and Methane, Kashefi (2012). . . 130
C.2 Interfacial Tension of High Pressure Aqueous TEG and Methane, Ng et al. (2009) . 132 C.3 Measured Values of this Study Compared to NeqSim, 100 wt% TEG . . . 133
C.4 Measured Values of this Study Compared to NeqSim, 90 wt% TEG . . . 134
C.5 Interfacial Tension of High Pressure Aqueous MEG and Methane, Norgaard and Nygaard (2014) . . . 135
D Parameter Study Results 137
D.1 Simulated values with carryover TEG, OLGA . . . 137
D.2 Simulated values with TEG dump, OLGA . . . 139
D.3 Simulated values with carryover TEG, OLGA HD. . . 140
D.4 Simulated values with TEG, OLGA HD . . . 141
3.1 Illustration of symbols used in the conservation equations of the two fluid model . 8
4.1 Intermolecular forces. . . 19
4.2 Forces on a curved surface. . . 20
4.3 A Pendant drop . . . 21
5.1 Illustration of the generic layer model utilized in OLGA HD . . . 26
6.1 Experimental data of the viscosity of TEG at atmospheric pressure and tempera- tures 25 - 190^◦C. . . 31
6.2 Experimental data of the viscosity of aqueous TEG at atmospheric pressure and temperatures 20 - 180^◦C . . . 31
6.3 Experimental data of the viscosity of aqueous TEG at atmospheric pressure and weight fractions 0 to 1 . . . 32
6.4 Experimental data for the viscosity of TEG containing methane in equilibrium at pressures 34 - 138 bar. . . 33
6.5 Experimental data of interfacial tension of aqueous TEG and air at 30^◦C . . . 34
6.6 Experimental data of the interfacial tension of water and methane in equilibrium at pressures 12 - 1064 bar . . . 35
6.7 Experimental data of the interfacial tension of aqueous TEG and methane in equi- librium at pressures 34 - 138 bar . . . 36
6.8 Experimental data of the interfacial tension of aqueous TEG and methane in equi- librium at temperatures 25-60^◦C . . . 36
7.1 Climate Test Chamber VC^34034 . . . 37
7.2 Flow sheet of High Pressure Interfacial Tension Rig . . . 38
7.3 Pressure stabilization process of the liquid phase for 100 wt% TEG and methane at 20^◦C. . . 42
7.4 Pressure stabilization process of the vapor phase for 100 wt% TEG and methane at 20^◦C. . . 42
7.5 Picture of droplet from DropImage Advanced. 20^◦C, 90 wt% TEG and 196.6 bar.. . 43
8.1 Experimental data of the interfacial tension of methane and TEG in equilibrium at pressures 50-225 bar, 100 wt% TEG. . . 46 8.2 Experimental data of the interfacial tension of methane and aqueous
TEG in equi-
librium at pressures 50-210 bar, 90 wt% TEG.. . . 46 8.3 Experimental data of the interfacial tension of methane and aqueous TEG in equi-
librium at pressures 50-220 bar and 4.3^◦C. . . 47 8.4 Experimental data of the interfacial tension of methane and aqueous TEG in equi-
librium at pressures 50-220 bar and 20^◦C.. . . 47 8.5 Experimental data of the interfacial tension of methane and aqueous TEG in equi-
librium at pressures 50-200 bar and 41.5^◦C.. . . 48 8.6 Experimental interfacial tensions over time for 100 wt% TEG at 4.3^◦C . . . 49 9.1 Calculated liquid viscosities of TEG compared to
experimental data at atmospheric
pressure and temperatures 25 - 160^◦C. . . 53 9.2 Calculated liquid viscosities of 98.5 wt% TEG in aqueous solution compared to
experimental data at atmospheric pressure and temperatures 25 - 190^◦C . . . 53 9.3 Calculated liquid viscosities of weight fractions 0 to 1 of TEG in aqueous solution
compared to experimental data at atmospheric pressure and 30^◦C . . . 54 9.4 Calculated liquid viscosities in NeqSim of weight fractions 0 to 1 of TEG in aqueous
solution as percentage of experimental data at atmospheric pressure . . . 55 9.5 Calculated liquid viscosities in PVTsim of weight fractions 0 to 1 of TEG in aqueous
solution as percentage of experimental data at atmospheric pressure . . . 55 9.6 Calculated liquid viscosities of 0.74 weight fraction of TEG in aqueous solution
compared to experimental data at atmospheric pressure and temperatures 20 - 175^◦C . . . 56 9.7 Calculated liquid viscosities in NeqSim of aqueous TEG as percentage of experi-
mental data at atmospheric pressure and temperatures 20 - 175^◦C . . . 57 9.8 Calculated liquid viscosities in PVTsim of aqueous TEG as percentage of experi-
mental data at atmospheric pressure and temperatures 20 - 175^◦C . . . 57 9.9 Calculated liquid viscosities of aqueous TEG and methane compared to experi-
mental data at 43.3^◦C and pressures 34.5 - 138 bar . . . 58 9.10 Calculated liquid viscosities in NeqSim of aqueous TEG and methane as percent-
age of experimental data at pressures 34.5 - 138 bar . . . 59 9.11 Calculated liquid viscosities in PVTsim of aqueous TEG and methane as percent-
age of experimental data at pressures 34.5 - 138 bar . . . 60 9.12 Calculated interfacial tensions of methane and aqueous TEG compared to experi-
mental data at atmospheric pressure and 30^◦C. . . 61 9.13 Calculated interfacial tensions of methane and aqueous TEG as a percentage of
experimental data at atmospheric pressure and 30^◦C . . . 62
9.14 Calculated interfacial tensions of high pressure methane and water compared to experimental data at 100^◦C and pressures 12 - 305 bar . . . 63 9.15 Calculated interfacial tensions of methane and
aqueous TEG compared to experi-
mental data at 43.3^◦C and pressures 34.5 - 138 bar . . . 64 9.16 Calculated interfacial tensions of methane and aqueous TEG compared to experi-
mental data at 43.3^◦C and pressures 34.5 - 138 bar, omitting the Firoozabdi Ramey methods . . . 64 9.17 Calculated interfacial tensions of methane and aqueous TEG compared to experi-
mental data at 138 bar and temperatures 26.7 - 60^◦ . . . 65 9.18 Average deviations of calculated and experimental interfacial tensions by pressure 66 9.19 Average deviations of calculated and
experimental interfacial tensions by temper-
ature . . . 66 9.20 Average deviations of calculated and experimental interfacial tensions . . . 67 9.21 Calculated interfacial tensions of methane and TEG compared to the experimental
results of this study at 20^◦C and pressures 57 - 219 bar. . . 68 9.22 Average deviations of calculated and experimental interfacial tensions of this study
by pressure groups . . . 69 9.23 Average deviations of calculated and experimental interfacial tensions of this study
by temperature . . . 70 9.24 Average deviations of calculated and experimental interfacial tensions of this study 70 9.25 Calculated interfacial tensions of methane and aqueous TEG compared to the
perimental results of this study at 4.3^◦C and pressures 55 - 210 bar . . . 71 9.26 Average deviations of calculated and experimental interfacial tensions of this study
by pressure groups . . . 72 9.27 Average deviations of calculated and experimental interfacial tensions of this study
by temperature . . . 72 9.28 Average deviations of calculated and experimental interfacial tensions of this study 73 9.29 Average deviations of calculated and experimental interfacial tensions of
this study
by liquid mixtures. . . 74 9.30 Average deviations of calculated and experimental interfacial tensions of this study
by pressure groups . . . 74 9.31 Average deviations of calculated and experimental interfacial tensions of this study
by temperature . . . 75 9.32 Average deviations of calculated and experimental interfacial tensions of this study 76 9.33 Calculated liquid density in NeqSim compared to experimental results of this
at 4.3^◦C, 100 wt% TEG and pressures 54 - 193 bar. . . 77 9.34 Calculated liquid density in NeqSim compared to experimental results of this study
at 41.5^◦C, 90 wt% TEG and pressures 53 - 189 bar. . . 77
9.35 Calculated vapor density in NeqSim compared to experimental results of this study at 4.3^◦C, 100 wt% TEG and pressures 54 - 193 bar. . . 78 9.36 Calculated vapor density in NeqSim compared to
experimental results of this study
at 41.5^◦C, 90 wt% TEG and pressures 53 - 189 bar. . . 78 9.37 Comparison between the experimental results of this study and Ng et al. (2009).. . 79 9.38 Comparison between the experimental results
of this study at 20^◦C and Ng et al.
(2009) at 26.7^◦C. . . 80 9.39 Comparison between the experimental results of this study at 41.5^◦C and Ng et al.
(2009) at 43.3^◦C . . . 80 9.40 Average deviations of calculated results in NeqSim and experimental results of this
study and Ng et al. (2009). . . 81 10.1 Simulated pressure along pipeline with the standard OLGA module and TEG car-
ryover and TEG mass flow parameter factor 10. . . 91 10.2 Simulated total pressure drop with the standard OLGA module and TEG carryover
with parameter factors 0.1 to 10 . . . 91 10.3 Simulated accumulated TEG along pipeline with the standard OLGA module and
TEG carryover with parameter factors 1 . . . 92 10.4 Simulated accumulated TEG volume along pipeline with the standard OLGA mod-
ule and TEG carryover for liquid viscosity and interfacial tension parameter factors 0.1 to 10. . . 93 10.5 Simulated accumulated TEG volume along pipeline with the standard OLGA mod-
ule and TEG carryover for TEG mass flow parameter factors 0.1 to 10 . . . 93 10.6 Simulated total pressure drop with the standard OLGA module and initial dump
of TEG for parameter factors 0.1 to 10 . . . 94 10.7 Simulated accumulated TEG volume along pipeline with the standard OLGA mod-
ule and initial dump of 30 m^3TEG for parameter factors 0.1 to 10 . . . 95 10.8 Simulated entrainment percentage of TEG in gas with the standard OLGA module
after 6 minutes and liquid viscosity parameter factor 0.2 . . . 96 10.9 Simulated entrainment percentage of TEG in gas with the standard OLGA module
after 22.5 hours and liquid viscosity parameter factor 0.2. . . 96 10.10Simulated total pressure drop with the OLGA HD module and TEG carryover for
liquid viscosity and interfacial tension parameter factors 0.1 to 10 . . . 98 10.11Simulated total pressure drop with the OLGA HD module and TEG carryover for
mass flow parameter factors 0.1 to 10 . . . 99 10.12Simulated accumulated TEG volume along pipeline with the OLGA HD module
and TEG carryover for liquid viscosity and interfacial tension parameter factors 0.1 to 10. . . 99
10.13Simulated accumulated TEG volume along pipeline with the OLGA HD module and TEG carryover for mass flow parameter factors 0.1 to 10 . . . 100 10.14Simulated pressure drop with OLGA HD module
with initial dump of 30 m^3TEG
and parameter factors 0.1 to 10 . . . 101 10.15Simulated accumulated TEG with OLGA HD module and initial dump of 30 m^3
TEG for parameter factors 0.1 to 10 . . . 102 10.16Simulated percentage entrainment of TEG in the gas after 65 hours with the OLGA
HD module and parameter factors 1 . . . 102 10.17Simulated percentage entrainment of TEG in the gas after 65 hours with the OLGA
HD module and liquid viscosity parameter factor 10 . . . 103 B.1 Calculated liquid viscosities of weight fractions 0 to 1 of TEG in aqueous solution
compared to experimental data at atmospheric pressure and 35^◦C . . . 125 B.2 Calculated liquid viscosities of weight fractions 0 to 1 of TEG in aqueous solution
compared to experimental data at atmospheric pressure and 40^◦C . . . 126 B.3 Calculated liquid viscosities of weight fractions 0 to 1 of TEG in aqueous solution
compared to experimental data at atmospheric pressure and 45^◦C . . . 126 B.4 Calculated liquid viscosities of weight fractions 0 to 1 of TEG in aqueous solution
compared to experimental data at atmospheric pressure and 50^◦C . . . 126 B.5 Calculated liquid viscosities of 0.89 weight fraction of TEG in aqueous solution
compared to experimental data at atmospheric pressure and temperatures 21 - 174^◦C . . . 127 B.6 Calculated liquid viscosities of 0.96 weight fraction of TEG in aqueous solution
compared to experimental data at atmospheric pressure and temperatures 21 - 175^◦C . . . 127 B.7 Calculated liquid viscosities of aqueous TEG compared to experimental data at
26.7^◦C and pressures 69 - 138 bar. . . 128 B.8 Calculated liquid viscosities of aqueous TEG compared to experimental data at
60^◦C and pressures 69 - 138 bar . . . 128 B.9 Calculated liquid viscosities of TEG compared to experimental data at atmospheric
pressure and temperatures 0 - 50^◦C.. . . 129 C.1 Calculated interfacial tensions of methane and aqueous TEG compared to experi-
mental data at 37^◦C and pressures 34 - 282 bar. . . 130 C.2 Calculated interfacial tensions of methane and aqueous TEG compared to experi-
mental data at 150^◦C and pressures 27 - 278 bar . . . 131 C.3 Calculated interfacial tensions of methane and aqueous TEG compared to experi-
mental data at 200^◦C and pressures 24 - 212 bar . . . 131 C.4 Calculated interfacial tensions of methane and aqueous TEG compared to experi-
mental data at 26.7^◦C and pressures 69 - 138 bar, . . . 132
C.5 Calculated interfacial tensions of methane and aqueous TEG compared to experi- mental data at 60^◦C and pressures 69 - 138 bar. . . 132 C.6 Calculated interfacial tensions of methane and aqueous
TEG compared to experi-
mental data at 69 bar and temperatures 26.7 - 60^◦ . . . 133 C.7 Calculated interfacial tensions of methane and TEG compared to the experimental
results of this study at 4.3^◦C and pressures 54 - 193 bar.. . . 133 C.8 Calculated interfacial tensions of methane and TEG compared to the experimental
results of this study at 41.5^◦C and pressures 53 - 198 bar. . . 134 C.9 Calculated interfacial tensions of methane and aqueous TEG compared to the ex-
perimental results of this study at 20^◦C and pressures 55 - 197 bar . . . 134 C.10 Calculated interfacial tensions of methane and aqueous TEG compared to the ex-
perimental results of this study at 41.5^◦C and pressures 53 - 189 bar . . . 135 C.11 Experimental data of the interfacial tension of methane and MEG in equilibrium
at pressures 22 - 133 bar, 100 wt% MEG . . . 135 C.12 Experimental data of the interfacial tension of methane and aqueous MEG in equi-
librium at pressures 19 - 154 bar, 80 wt% MEG . . . 136 C.13 Experimental data of the interfacial tension of methane and aqueous MEG in equi-
librium at pressures 26 - 145 bar, 50 wt% MEG . . . 136
2.1 Physical and chemical properties of TEG . . . 5 3.1 Table of symbols and units used in the conservation equations of the two phase
model . . . 9 3.2 Table of properties produced in NeqSim . . . 13 6.1 Presentation of collected experimental data . . . 30 7.1 Technical Equipment Information . . . 40 7.2 Compostition of TEG/water
mixtures . . . 43 8.1 Experimental matrix . . . 45 8.2 Uncertainty of measured conditions . . . 48 8.3 Input variables to Equation 8.2 . . . 50 8.4 Uncertainties for 100wt% TEG at 4.3^◦C . . . 51 8.5
Experimental uncertainty . . . 51 9.1 Summary of main deviations of liquid viscosity of the property generation tools. . 83 9.2 Summary of main interfacial tension deviations of the calculation
methods . . . . 84 10.1 Property tables generated in NeqSim by parameter factors. . . 88 10.2 OLGA input structure. . . 88 10.3 Mass flow of TEG parameter factors . . . 89 10.4 Maximum time and
distance of entrainment of TEG in gas, standard OLGA . . . . 97 10.5 Distance of entrainment of TEG in gas at 65 hours of simulated time, OLGA HD . . 103 10.6 Sensitivity of the standard OLGA module
to the parameter factors, both scenarios 104 10.7 Sensitivity of the OLGA HD module to the parameter factors, both scenarios . . . . 105 10.8 Simulated pressure drop after 10 days in OLGA and OLGA HD
with parameter fac-
tor 1 . . . 106 10.9 Simulated accumulated TEG after 10 days in OLGA and OLGA HD with parameter
factor 1 . . . 107 10.10Simulated maximum time and distance of entrainment in OLGA and OLGA HD
with parameter factor 1 . . . 107 xxi
D.1 Simulated pressure drop from parameter studies with NeqSim property tables.
Scenario 1. . . 137 D.2 Simulated accumulated TEG from parameter studies with NeqSim property tables.
Scenario 1. . . 137 D.3 Simulated pressure drop from parameter studies with PVTsim property tables. Sce-
nario 1. . . 138 D.4 Simulated accumulated TEG from parameter studies with PVTsim property tables.
Scenario 1. . . 138 D.5 Simulated pressure drop from parameter studies with NeqSim property tables.
Scenario 2. . . 139 D.6 Simulated accumulated TEG from parameter studies with NeqSim property tables.
Scenario 2. . . 139 D.7 Results from parameter studies with PVTsim property tables. Scenario 2. . . 139 D.8 Simulated pressure drop from parameter studies with NeqSim property tables.
Scenario 1 with the OLGA HD module . . . 140 D.9 Simulated accumulated TEG from parameter studies with NeqSim property tables.
Scenario 1 with the OLGA HD module. . . 140 D.10 Simulated pressure drop from parameter studies with PVTsim property tables. Sce-
nario 1 with the OLGA HD module . . . 140 D.11 Simulated accumulated TEG from parameter studies with PVTsim property tables.
Scenario 1 with the OLGA HD module. . . 141 D.12 Simulated pressure drop from parameter studies with NeqSim property tables.
Scenario 2 with the OLGA HD module. . . 141 D.13 Simulated accumulated TEG from parameter studies with NeqSim property tables.
Scenario 2. . . 141 D.14 Results from parameter studies with PVTsim property tables. Scenario 2 with the
OLGA HD module. . . 142
CPA - Cubic Plus Associating
DEG - Diethylene glycol
EG - Ethylene glycol
EoS - Equation of State
GC - Gas chromatograph
LBVC - Lohrenz-Bray-Clark
MEG - Monoethylene glycol
Mole% - Mole percentage
MSm^3 - Million standard cubic meter
NeqSim - Non-Equilibrium simulator
OH - Hydroxyl
OLGA - Oil and gas simulator
PFCT - Pedersen-Fredenslund-Christensen-Thomassen
PR - Peng-Robinson
PVTsim - Pressure, Volume and Temperature simulator
RK - Redlich-Kwong
SAFT - Statistical Associating Fluid Theory
SRK - Soave-Redlich-Kwong
TEG - Triethylene glycol
T[4]EG - Tetraethylene glycol
VDW - van der Waals
wt% - Weight percentage
Symbol Description Unit
α Phase fraction -
A Area m^2
β Bond number -
C[p] Heat capacity J
D Pipe diameter m
e Empirical interaction parameter -
E[η] Viscosity coefficient -
ε Surface roughness m
f Friction factor -
g gravity m/s^2
G Excess free energy J
H Enthalpy J/kg
k Thermal conductivity W/m·K
κ Friction coefficient -
l Length m
m Mass transfer kg/m·s
M Molecular weight kg/mol
η Dynamic viscosity mPa·s
ρ Density kg/m^3
P Pressure bar
Pi Parachor value -
q Interfacial heat flux J/m·s
Q Heat flux J/m·s
R Universal gas constant J/mol·K
R Radius m
Re Reynolds number -
Rs Gas mass fraction -
s Cross section m
S Entropy J/K
τ Shear stress N/m^2
t Thickness m
T Temperature K
u Velocity m/s
U Internal energy J
µ Chemical potenial J/mol
υ Kinematic viscosity m^2/s
W Work J
x Mole fraction -
ξ Viscosity reducing parameter -
y Mole fraction -
γ Interfacial tension mN/m
Φ Grand potential J
θ Angle ^◦
1.1 Multiphase Flow
Multiphase flow is present in many processes surrounding us, both environmental and indus- trial. In fluid mechanics, multiphase flow is simultaneous flow of various phases in contact. It is usually
gases and liquids appearing together, but there can also be solids present. The sub- ject has been of growing interest in the Norwegian petroleum industry the last decades. Long distance transport of
unprocessed natural gas in the same pipeline means less transport ex- penditures and the possibility of processing the mixture onshore. This constitutes a significant economical advantage.
Fluids transported in rich gas transport pipelines, such as Åsgard transport, will eventually reach seabed temperature of close to 0^◦C. At low temperatures and high pressures, hydrates can form if
liquid water is present. Hydrates are solid particles which can cause severe operating prob- lems in downstream equipment. Because hydrocarbons in reservoirs contain water, one option is to remove
the water from the hydrocarbons before transportation. TEG (TriEthylene Glycol) is among the most used liquid solvents for absorption of water.
The consequence of using TEG as a solvent is that small amounts of it will be transported within the rich gas. This influences properties like liquid viscosity and interfacial tension, which are
among the most influential parameters for fluid behaviour. These properties have considerable effects on fluid flow characteristics and consequently capacity and processing aspects. To de- termine
size and design of the transport pipelines and downstream process equipment accurate simulation models are needed to describe the behaviour of the multiphase flow. The computer program OLGA (OiL and
GAs simulator) is a modelling tool for multiphase flow. It was commer- cialized by the Schlumberger company SPT Group, and is the most used simulation tool today.
Multiphase flow technology is highly advanced and challenging, and is a process in develop- ment.
Multiphase flow simulators comprise advanced fluid mechanical and numerical models. As in- put it needs a property table comprising various thermodynamic and physical properties for the fluid. The
property table is generated by a property generation tool which calculates properties for a set amount of temperature and pressure points. An example of a property table genera- tor is PVTsim
(Pressure, Volume and Temperature simulator), which calculates fluid properties using a classical equation of state. These equations calculate accurate property values for flu- ids containing
hydrocarbons, but struggle when applied to polar components like water and TEG. Another option is to generate property tables using NeqSim (Non-Equilibrium Simulator), which is developed at the
Department of Refrigeration and Air Conditioning, at the Norwegian University of Science and Technology. NeqSim calculates fluid properties using the Cubic Plus Association Equation of State, which
is more compatible with polar components.
1.2 Thesis Specification
NeqSim comprise several mathematical models for predicting viscosity and interfacial tensions.
The accuracy of these models when applied to TEG are however uncertain. The aim of this mas- ter’s thesis is to evaluate the models used for property generation and to compare generated data with
experimental data for aqueous TEG and methane.
The following tasks are to be considered:
1. Review of experimental data of interfacial tension and viscosity of TEG/water solutions.
2. Experimental measurement of interfacial tension of high pressure TEG and natural gas.
3. Status and further development of NeqSim as a property generation tool for multiphase flow simulators.
4. Simulation of a rich gas pipeline (Åsgard transport) with a free liquid glycol phase. Com- parison of the effect of various property generation models and tools.
1.3 Report Structure
Chapter 2, 3 and 4 in the study are theoretical chapters. Chapter 2 includes aspects of dehy- dration, hydrates and glycols. Chapter 3 focuses on conservation laws and closure relations for
multiphase flow. Chapter 4 describes the properties liquid viscosity and interfacial tension along with the models for calculating them. Chapter 5 gives an introduction to the software OLGA, NeqSim
and PVTsim. Chapter 6 presents a review of existing experimental data on vis- cosity and interfacial tension of TEG, water and methane. Methods for measuring interfacial tension are also presented in
this chapter. Chapter 7 describes the experimental set up, ap- paratus and procedure of the laboratory work conducted in this study. Chapter 8 presents the
experimental results obtained in this study along with an uncertainty analysis. Chapter 9 com- pares experimental data and generated properties from NeqSim and PVTsim. It also compares the obtained
measured interfacial tension values to the measured values obtained in a simi- lar study. Chapter 10 presents a parameter study simulated in OLGA of the Åsgard transport pipeline. Chapter 11 is the
discussion of our work. Chapter 12 presents the conclusion to our thesis and recommendations for further work. The thesis ends with bibliography and appen- dices.
2.1 Hydrates
The formation of hydrates is one of the most common problems in multiphase flow pipelines.
Hydrates are ice-like structures where the water molecules form crystalline structures which are held together by hydrogen bonding. The structures are stabilized by light natural gas com- pounds such
as methane. The formation can occur if liquid water is present in the flow. Accord- ing toAnyadiegwu et al.(2014) the temperature must be below the gas dew point temperature and the pressure above
the gas dew point pressure, meaning they can form at higher temper- atures than ice. The high pressure and low temperatures in rich gas transport pipelines make them susceptible to hydrate formation.
Hydrates can cause several operating problems, the most severe being partial or complete block- ages of pipes and downstream equipment. Hydrate plugs large enough to block a pipeline com- pletely can
form within minutes (Kidnay et al.,2011). Other problems are erosion of expanders and fouling and plugging of heat exchangers (Rojey,1997). To prevent the formation of hydrates the usual solution is
to strip the water content from the flow. In order to remove and control the water content of the natural gas, a dehydration process is utilized.
2.2 Absorption
There are several ways to achieve dehydration. Examples are absorption, adsorption, gas per- meation and refrigeration. The most widely used method in oil and gas processing is absorption.
Absorption is a process where a gas (or liquid) is contained within a liquid solvent to remove spe- cific compounds. Physical absorption is the transfer of mass from one phase to another.
In oil and gas processing, water is usually removed from the natural gas by absorption dehydra- tion, using a liquid solvent. The contact is normally accomplished in tray or packed towers. To
achieve viable absorption, a low cost solvent with strong affinity for water is favoured.
2.3 Glycols
Glycols are the common name for dihydric alcohols, which are alcohols containing two hydroxyl (–OH) groups. They are considered among the more effective liquid solvents and are commonly used in oil
and gas processing. Dehydration by absorption using glycol is usually economically more attractive than dehydration by a solid desiccant (Anyadiegwu et al.,2014). The OH-bonds have a strong affinity
for water molecules and will extract water from the natural gas when the gas is exposed to the liquid glycol. Glycols used for dehydrating natural gas are ethylene glycol (EG), diethylene glycol
(DEG), triethylene glycol (TEG), and tetraethylene glycol(T[4]EG). Nor- mally a single type of pure glycol is used in a dehydrator, but sometimes a glycol blend is eco- nomically attractive (Guo and
2.3.1 TEG
TEG (TriEthylene Glycol) is the most used glycol for natural gas dehydration. It provides the best combination of dew point depression, operating cost and reliability (Guo and Ghalambo, 2005). It is
an odourless viscous liquid. The advantages of TEG is the ease of regeneration and operation, minimal losses of drying agent during operation, high affinity for water and chemical stability. TEG has
been successfully utilized to dehydrate natural gases over wide ranges of oper- ating conditions (Anyadiegwu et al.,2014). The physical and chemical properties of triethylene glycol are given in
Parameter Unit Properties
Empirical formula - C[6]H[14]O[4] Molecular weight g/mol 150.17 Density at 25^◦C g/cm^3 1.120
Flash point ^◦C 176
Ignition point ^◦C 371
Boiling point at 1 atm ^◦C 287.7
Freezing point ^◦C -4.3
Critical Temperature ^◦C 440 Critical Pressure kPa 3313.3 Viscosity at 20^◦C mPa·s 49.0 Vapor pressure at 20^◦C kPa <0.001
Table 2.1: Physical and chemical properties of TEG,Company(2007)
2.3.2 TEG Content in Gas Phase
The loss of TEG from the dehydration unit to the transport pipeline (carryover of TEG) occurs in two ways:
• Mechanical TEG carryover:Small droplets carried over into the pipeline mechanically.
• TEG vaporization: TEG vaporizes and is solved in the gas. Normally, this amount is con- siderably higher than the mechanical carryover. The amount is dependant on the opera- tion conditions of the
TEG contactor.
Over time, liquid TEG can accumulate within the pipeline. Condensed liquid may cause slug formation in a pipeline with transient conditions, or cause an undesirably high pressure drop along the
pipeline. The condensed TEG and water may cause long-term corrosion on the inside of the wall of the pipeline if the water content is high enough (Kordabadi and Dinon,2013).
Properties Needed for Multiphase Flow Simulation
In fluid mechanics, multiphase flow is simultaneous flow of various phases in contact. It is usu- ally gases and liquids appearing together, but there can also be solids present. In oil and gas
production, it is crucial to account for the occurring multiphase flow. The wells produce gas, water and oil at the same time, which leads to three phase flow. In addition, methanol and gly- col is
often injected in the well stream to avoid hydrate formation in the pipelines. Flow models play a major part in predicting accurate production rates and flow assurance, but calculation models have
historically been inaccurate. Even the best equations of state have their limitations.
However, technology for transporting multiphase flow have advanced rapidly in recent decades.
New calculation models like SAFT (Huang and Radosz,1990) and CPA-EoS (Kontogeorgis et al., 1996) provide significantly better simulation models. This has already had an enormous eco- nomical impact
on several offshore developments. Multiphase flow pipelines have in some places replaced topside offshore installations. In the development of future oil and gas fields, long-distance multiphase
transport of gas, water, oil and chemicals will be an important fea- ture. This chapter is based on material written bySolbraa(2002),Bratland(2010) andBjortuft (2014). It will be limited to a two
phase flow scenario consisting of one gas phase and one liquid phase.
Two phase flow can generally be treated as separated flow or dispersed flow. Separated flow regimes, such as stratified or annular flow, has a well defined interface. This may not be the case when
dealing with the more complex interface of dispersed flow regimes, like bubble/droplet or slug flow. However, simulation of two phase pipe flow can be done using the same mathemati- cal models for
both flow regimes. The respective closure relations on the other hand, will have to differ. The next sections will discuss the conservation laws and closure relations for the men- tioned two phase
system. It will describe the thermodynamic and physical properties needed in
multiphase flow simulators.
In Section3.1, the basic equations for the two fluid model are described. Section3.2limits the model to a simple situation of stratified flow for simplicity. Section 3.3includes some general comments
on closure relations.
3.1 Conservation Laws
The model presented in this section uses a transient and one dimensional basis for all conser- vation laws. Only the x-axis is applied. An introduction to one dimensional modelling of two phase flow
was given byWallis(1969).Ishii(1975) presented the basic theory and equations for the two fluid model.
In transient single phase flow, three conservation equations are sufficient to describe the main conservation principles - mass conservation, momentum conservation, and energy conserva- tion. The
same equations apply for multiphase flow, one set of equations for each phase. The conservation equations for mass, momentum and energy will be given in the following sections, both for gas phase and
for liquid phase. In Figure3.1bySolbraa(2002) some of the character- istic parameters used in the two fluid model are presented. Table3.1presents the symbols and units used in the conservation
Figure 3.1: Illustration of symbols used in the conservation equations of the two fluid model, Solbraa(2002)
Variable Description Unit
m Mass transfer kg / m·sec
τi Interfacial liquid-gas shear stress N/m^2
τw g Wall-gas shear stress N/m^2
τw l Wall-liquid shear stress N/m^2 αl Liquid phase fraction (holdup)³
´ - αg Gas phase fraction³
Q Heat flux from surroundings J/m·sec
q[l g] Interfacial heat flux J/m·sec
D Pipe diameter m
ε Surface roughness m
g Gravity m/sec^2
Table 3.1: Table of symbols and units used in the conservation equations of the two phase model
3.1.1 Mass Conservation
The conservation equations of mass for the gas and the liquid are given as
∂t +∂(αgρgugA)
∂x =m[l g]−m[g w] (3.1)
∂t +∂(αlρlu[l]A)
∂x = −m[l g]−m[l w] (3.2)
whereρis the density,Ais the Area, anduis the velocity.αis the phase fraction defined as αk= A[k]
A (3.3)
m[l g]is the mass transfer between the phases, andm[kw]is the mass transfer between phasekand other sources, such as inflow perforations in the pipe wall. The equations assume mass transfer from the
liquid phase to the gas phase, which makesm[l g] negative in the liquid equation. Also, the gain of one phase, must be the loss of the other, as phase change cannot result in altered total mass
m[ki]=0 (3.4)
Another useful relation follows from the definition of what a fraction is. The sum of all phase fractions must equal 1 to fill the cross section of the pipe
αk=1 (3.5)
3.1.2 Momentum Conservation
The conservation equations of momentum for the gas and the liquid are given as
∂t +∂(αgρgu^2[g]A)
∂x =m[l g]u[i]−m[g w]u[g]−αgA∂P[g]
∂x −αgρgAg si nθ−s[g w]τg w−s[i]τi (3.6)
∂t +∂(αlρlu^2[l]A)
∂x = −m[l g]u[i]−m[l w]u[l]−αlA∂P[l]
∂x −αlρlAg si nθ−s[l w]τl w+s[i]τi (3.7) wheregis the gravity, andsis the cross sectional contact length between the phases or the wall.
τkw is a frictional term for the wall, andτi is the interfacial friction. In the same manner as for the mass conservation equations, the interfacial friction term appears with opposite signs in the
two equations.
3.1.3 Energy Conservation
The conservation equations of energy for the gas and the liquid are given as
∂αgρgA³ U[g]+^u
2 g z[g]´
∂t +
∂αgρgu[g]A³ H[g]+^u
2 +g z[g]´
∂x =q[l g]+Q[g] (3.8)
∂αlρlA³ U[l]+^u
2 l
2 g z[l]´
∂t +
∂αlρlu[l]A³ H[l]+^u
2 l
2 +g z[l]´
∂x = −q[l g]+Q[l] (3.9)
wherezis a vertical coordinate,U is internal energy,His enthalpy, andQis heat transfer from the surroundings.q[l g]is heat transfer from the liquid to the gas, and is therefore negative for the
liquid equation. Also, the heat gain of one phase, must be the equal heat loss of the other.
q[ki]=0 (3.10)
3.2 Stratified Two Phase Flow
All equations in the previous section are general in their presented form. There are not brought in any fluid specific properties, such as how viscosity, density, surface tension and specific en-
thalpy varies with pressure and temperature. The equations presented are valid for any fluid, but they are not complete in this sense. To describe friction or heat, other correlations must be added.
They are often referred to as closure relations, as they are needed to fulfil the equation set. With the help of these correlations, we are able to match the number of unknowns with the
number of equations.
To avoid getting lost in details a relatively simple situation is considered. There are only two fluids, one gas and one liquid. The pressures and temperatures are such that there is no evapo- ration
or condensation. The gas does not dissolve in the liquid (which in real life is never quite true), and there are no perforations in the pipe. The flow is strictly stratified. To make the energy
conservation equation redundant, the flow is assumed to be isothermal. These assumptions might seem unreasonable, but the mass transfer terms are often insignificant in cases where the rate of mass
transfer is small compared to the flow rate of the phases. For long multiphase pipelines containing oil and gas this is usually the case, as pressures and temperatures change gradually.
3.2.1 Simplified Conservation Laws
In the given situation there are zero mass transfer between the phases, and also no mass trans- fer through the walls of the pipe. The mass conservation equations for the gas and the liquid,
Equations3.1and3.2, reduce to
∂t +∂(αgρgu[g])
∂x =0 (3.11)
∂t +∂(αlρlu[l])
∂x =0 (3.12)
with Equation3.5reduced to
αg+αl=1 (3.13)
As pressure variations due to different elevation of the two phases are neglected, the pressure on the interface can be defined asP. This reduces the momentum conservation equation for the gas and
the liquid, Equations3.6and3.7, to
∂t +∂(αgρgu^2[g]A)
∂x =αgA∂P
∂x−αgρgAg si nθ−s[g w]τg w−s[i]τi (3.14)
∂t +∂(αlρlu^2[l]A)
∂x =αlA∂P
∂x −αlρlAg si nθ−s[l w]τl w+s[i]τi (3.15) As already mentioned, the energy conservation equations, Equations3.8and3.9, are not needed in this case.
3.2.2 Closure Relations - Interfacial Shear
The closure relations which needs to be established for this particular situation is the interfacial shear, and the shear between the phases and the wall. The wall-gas shear can be calculated from
τw g=f[w g]ρg
8 (3.16)
where f[w g] is the Darcy friction factor. Similarly the wall-liquid shear can be calculated from τw l=f[w l]ρl
8 (3.17)
The interfacial shear can be calculated from τi=f[i]ρg
8 (3.18)
whereu[i] is the interfacial velocity. For simplicity the interfacial velocity can be assumed to be the same as the liquid velocity. f[i] is the Darcy friction factor on the interface. To estimate
the friction factor between a fluid and the wall the correlation presented byHaaland(1983) can be used
pf[w] = −1.8 log[10]h6.9 Re +
³ ε 3.7D[h]
(3.19) whereRe is the Reynolds number,D[h] is the hydraulic diameter, andεis the roughness of the pipe wall. The interfacial friction factor can be estimated from an empirical correlation given by
fw g =1+0.75αl (3.20)
whereαl is the liquid phase fraction (holdup). The Reynolds number is defined as Re=uD[h]ρ
η (3.21)
whereηis the viscosity. The hydraulic diameter is calculated from D[h]=4A
O (3.22)
where Ais the cross section area, andO is the wetted perimeter. TheHaaland(1983) equation is an approximation of the implicit Colebrook equation. As described above the frictional fac- tor in fully
developed turbulent pipe flow depends on the Reynolds number and the relative pipe roughness [D]^ε. A functional form of this dependence cannot be obtained from a theoreti- cal analysis, and the
available results are obtained from experiments. The results are presented in tabular, graphical and functional form obtained by curve-fitting experimental data. Cyril F.
Colebrook combined the available data for transition and turbulent flow in smooth and rough pipes into the Colebrook equation given by
pf = −2.0l og
3.7 + 2.51 Rep
(3.23) The graphical visualization of the Darcy friction factor led to the forming of the famousMoody chartwhich is one of the most widely accepted and used charts in engineering today (Cengel and
3.3 Other Properties for Closure Relations
Generally, more closure relations than shear stress are needed to complete a set of equations for multiphase flow. Examples are heat capacity, entropy and thermal conductivity. Simulation models rely
on these properties to generate results. Table3.2presents the properties generated in NeqSim to use in simulation tools.
Variable Description
ρ Density
R[s] Gas mass fraction
η Viscosity
C[p] Heat capacity
H Enthalpy
k Thermal conductivity γ Surface tension
S Entropy
Table 3.2: Table of properties produced in NeqSim
Theoretical Concepts
The critical concepts in this study are liquid viscosity and surface tension/interfacial tension.
These are influential properties for fluid behaviour. This chapter will describe the properties as well as the models used to calculate them. There are numerous calculation models. We will focus on
the models that are available in NeqSim and PVTsim, which are the software we will use to generate properties. NeqSim and PVTsim will be described in the next chapter.
In Section4.1the concept of liquid viscosity is described. Section4.2describes the models to calculate liquid viscosity. Section4.3describes the concept of surface tension, and section4.4 describes
the models used to calculate surface tension.
4.1 Liquid Viscosity
Liquid viscosity is the measure of a liquids internal resistance to flow. It is a function of tem- perature and pressure, and can be termed a drag force. Viscosity determines the hydrodynamic
characteristics of fluids, such as flow rate and pressure drop. Because of this, liquid viscosities are critical for simulating pipeline flow. Pipeline design, pump characteristics, injection and
transportation design heavily depend on liquid viscosity. The overall pressure drop is also es- sential to the hydraulic capacity of the pipeline.
There are two distinct forms of viscosity: Dynamic viscosity and kinematic viscosity. Dynamic viscosity is often simply referred to as viscosity in the literature and is the focus of this study.
The SI-system utilizes the unit Pa·s for dynamic viscosity, where 1 Pa·s = 1 N·s/m^2= 1 kg/(m·s).
For practical use it is common to specify viscosity in milliPascal second or centipoise (1 mPa·s = 1 cP) as this gives more practical numerical values. Dynamic viscosity is defined byViswanath et al.
(2007) as "the tangential force per unit area required to slide one layer against another layer when the two layers are maintained at a unit distance". This means that dynamic viscosity is the ratio
of the shear stress to the strain rate.
v (4.1)
whereηis viscosity,τis shear stress,xis length andvis velocity. Kinematic viscosity is the ratio of the dynamic viscosity of a fluid to its density.
ρ (4.2)
whereυis kinematic viscosity andρis density. Polling et al.(2001) establishes that liquid vis- cosity decreases with increasing temperatures and increases with increasing pressures.
4.2 Models to Calculate Liquid Viscosity
Historically the theoretical methods to calculate liquid viscosity have been inaccurate. This is due to the high complexity of liquid molecular structures and interactions. This has led the studies
of viscosity to mainly focus on experimental measurements, and the establishment of empirical and semi-empirical formulas.
Theoretical models based on the corresponding state principle, the absolute rate theory of Eyring and the free volume theory have been developing in parallel. The book "Viscosity" byTouloukian et al.
(1975) published several theories and models. Accurate theoretical models have emerged the last decades by combining these models with the cubic equations of state (EoS). The newly emerged friction
theory, which is partially based on empirical formulas, is able to give good pre- dictions for liquid viscosity. This section will give a brief introduction the mathematical models available in
NeqSim and PVTsim, starting with the empirical models.
4.2.1 Empirical Models
Pure Compounds
For simplicity, it is often desirable to determine liquid viscosity from experimental data. There have been published numerous compilations of compound parameters that correlate these data. Arrhenius
proposed in 1899 an equation which acts as a formula for the temperature de- pendence of reaction rates. In the case of liquid viscosity it is given by
η=η0e^−E^η^/(RT^) (4.3)
whereη0is the viscosity at some reference temperature,T is the temperature,E[η]is the temper- ature coefficient for viscosity andRis the universal gas constant (Laidler,1984). Equation4.3 is an
empirical relationship with numerous modifications and alterations developed over the years. Different datasets utilize empirical equations often based on the Arrhenius equation to
correlate their data. NeqSim utilizes a compound parameter list including empirical equations developed by Statoil, which is described in Section5.2.2.
Grunberg and Nissan
Grunberg and Nissan(1949) established the one-parameter equation for correlating the liquid viscosity of nonpolar mixtures. It is based on the Arrhenius equation, Equation4.3, but includes an
additional term. It is given by
l n(η1,2)=x[1]l n(η1)+x[2]l n(η2)+ex[1]x[2] (4.4) wherexis the mole fraction of the respective compounds ande is an empirical interaction pa- rameter.
LBC Correlation
The Lohrenz-Bray-Clark (LBC) correlation byLohrenz et al.(1964) proposed an empirical corre- lation for the prediction of liquid viscosity of hydrocarbon mixtures based on their composition.
According toYoung et al.(2007) it is the most widely used viscosity model in reservoir engineer- ing. This is due to its simplicity, consistency and flexibility. It is based on the empirical residual
concept and the general structure of the LBC correlation is given by
+a[4]ρr4 (4.5)
where η0 is the dilute gas limit viscosity, ξ is the viscosity reducing parameter and ρr is the reduced density of the fluid. The model predicts reasonable gas viscosities, but the oil viscosities
are not accurate. Because of this it is necessary to tune the calculated viscosities.
PFCT Correlation
A popular correlation based on north sea oil is the Pedersen-Fredenslund-Christensen-Thomassen (PFCT) correlation by Pedersen et al. (1984). The model uses a parameter α to account for molecular size
and density effects. It is given by
ηmi x(P,T)=
³T[c,mi x] T[c,o]
´−1/6³P[c,mi x] P[c,o]
´2/3³M[mi x] M[o]
´1/2³αmi x
´ηo(P[o],T[o]) (4.6) where o refers to the reference component, T is temperature, P is pressure, M is molecular weight andαis given by
αmi x=1+7.747x10^−^5ρr4.265M[mi x]^0.8579 (4.7) | {"url":"https://9pdf.net/document/q2n9j64e-multiphase-simulation-pipelines-traces-liquid-flerfasesimulering-rikgassr%C3%B8rledninger-mengder.html","timestamp":"2024-11-09T09:59:27Z","content_type":"text/html","content_length":"183739","record_id":"<urn:uuid:a13bfe11-92ce-4476-ba99-377960d7e082>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00664.warc.gz"} |
Theory stronger, incrementally!! (Exercise 8)
The key lesson of all these exercises is that you can push yourself to be better and more confident in theory by tackling simple, paradigmatic problems in an incremental way. You must put pencil to
paper! You must do it regularly. But once you do, the benefits come quickly. Each mini-realization builds into knowledge. Each solved simple problem builds your intuition for understanding complex
Below are solutions to the problems from last time. Looking for more? Scroll down for some additional problems.
Here are the answers to the questions from last time, which explain the connection between continuous and discrete-time kinetic descriptions.
1. Give the full definition of
1. This is the conditional to probability to be in state
2. If
1. At time
3. Using the exact solutions to the continuous-time two-state probabilities, previously, calculate the discrete-time transition probabilities into state A. This problem is trickier than it sounds
(though the math is easy), but don’t worry the derivation is there for you in the notes. The key trick is to write
1. For this, please just look at the notes. It’s written out carefully there.
Going forward, here are some key problems you should solve.
1. Demonstrate the correctness of the (differential) continuity equation in one dimension by integrating the probability over a small increment.
2. Derive the diffusion equation via a Taylor expansion of the time-varying probability distribution. See my online notes.
3. Derive the Smoluchowski equation via a Taylor expansion of the time-varying probability distribution. See my online notes.
4. Show that the stationary distribution of the Smoluchowski equation is the Boltzmann distribution.
5. Show that overdamped dynamics lead to the Boltzmann distribution. For hints, see Sec. IIB of this paper by Chandler’s group.
6. Derive the replica-exchange Metropolis acceptance criterion. For hints, see Ch. 12 of my textbook.
7. Explain why one takes a linear average of MD or MC snapshots (equally spaced in time) to obtain a Boltzmann-weighted average. See Ch. 2 of my textbook.
8. Derive the ideal gas partition function from scratch. See Ch. 5 of my textbook.
Good luck! Work hard, but work slow … | {"url":"https://statisticalbiophysicsblog.org/?p=446","timestamp":"2024-11-09T18:52:38Z","content_type":"text/html","content_length":"47436","record_id":"<urn:uuid:c97b544a-0327-43ee-a0ea-d95e49dd8d2d>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00439.warc.gz"} |
Source code for networkx.generators.classic
"""Generators for some classic graphs.
The typical graph builder function is called as follows:
>>> G = nx.complete_graph(100)
returning the complete graph on n nodes labeled 0, .., 99
as a simple graph. Except for `empty_graph`, all the functions
in this module return a Graph class (i.e. a simple, undirected graph).
import itertools
import numbers
import networkx as nx
from networkx.classes import Graph
from networkx.exception import NetworkXError
from networkx.utils import nodes_or_number, pairwise
__all__ = [
# -------------------------------------------------------------------
# Some Classic Graphs
# -------------------------------------------------------------------
def _tree_edges(n, r):
if n == 0:
# helper function for trees
# yields edges in rooted tree at 0 with n nodes and branching ratio r
nodes = iter(range(n))
parents = [next(nodes)] # stack of max length r
while parents:
source = parents.pop(0)
for i in range(r):
target = next(nodes)
yield source, target
except StopIteration:
@nx._dispatchable(graphs=None, returns_graph=True)
def full_rary_tree(r, n, create_using=None):
"""Creates a full r-ary tree of `n` nodes.
Sometimes called a k-ary, n-ary, or m-ary tree.
"... all non-leaf nodes have exactly r children and all levels
are full except for some rightmost position of the bottom level
(if a leaf at the bottom level is missing, then so are all of the
leaves to its right." [1]_
.. plot::
>>> nx.draw(nx.full_rary_tree(2, 10))
r : int
branching factor of the tree
n : int
Number of nodes in the tree
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
G : networkx Graph
An r-ary tree with n nodes
.. [1] An introduction to data structures and algorithms,
James Andrew Storer, Birkhauser Boston 2001, (page 225).
G = empty_graph(n, create_using)
G.add_edges_from(_tree_edges(n, r))
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def kneser_graph(n, k):
"""Returns the Kneser Graph with parameters `n` and `k`.
The Kneser Graph has nodes that are k-tuples (subsets) of the integers
between 0 and ``n-1``. Nodes are adjacent if their corresponding sets are disjoint.
n: int
Number of integers from which to make node subsets.
Subsets are drawn from ``set(range(n))``.
k: int
Size of the subsets.
G : NetworkX Graph
>>> G = nx.kneser_graph(5, 2)
>>> G.number_of_nodes()
>>> G.number_of_edges()
>>> nx.is_isomorphic(G, nx.petersen_graph())
if n <= 0:
raise NetworkXError("n should be greater than zero")
if k <= 0 or k > n:
raise NetworkXError("k should be greater than zero and smaller than n")
G = nx.Graph()
# Create all k-subsets of [0, 1, ..., n-1]
subsets = list(itertools.combinations(range(n), k))
if 2 * k > n:
universe = set(range(n))
comb = itertools.combinations # only to make it all fit on one line
G.add_edges_from((s, t) for s in subsets for t in comb(universe - set(s), k))
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def balanced_tree(r, h, create_using=None):
"""Returns the perfectly balanced `r`-ary tree of height `h`.
.. plot::
>>> nx.draw(nx.balanced_tree(2, 3))
r : int
Branching factor of the tree; each node will have `r`
h : int
Height of the tree.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
G : NetworkX graph
A balanced `r`-ary tree of height `h`.
This is the rooted tree where all leaves are at distance `h` from
the root. The root has degree `r` and all other internal nodes
have degree `r + 1`.
Node labels are integers, starting from zero.
A balanced tree is also known as a *complete r-ary tree*.
# The number of nodes in the balanced tree is `1 + r + ... + r^h`,
# which is computed by using the closed-form formula for a geometric
# sum with ratio `r`. In the special case that `r` is 1, the number
# of nodes is simply `h + 1` (since the tree is actually a path
# graph).
if r == 1:
n = h + 1
# This must be an integer if both `r` and `h` are integers. If
# they are not, we force integer division anyway.
n = (1 - r ** (h + 1)) // (1 - r)
return full_rary_tree(r, n, create_using=create_using)
@nx._dispatchable(graphs=None, returns_graph=True)
def barbell_graph(m1, m2, create_using=None):
"""Returns the Barbell Graph: two complete graphs connected by a path.
.. plot::
>>> nx.draw(nx.barbell_graph(4, 2))
m1 : int
Size of the left and right barbells, must be greater than 2.
m2 : int
Length of the path connecting the barbells.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Only undirected Graphs are supported.
G : NetworkX graph
A barbell graph.
Two identical complete graphs $K_{m1}$ form the left and right bells,
and are connected by a path $P_{m2}$.
The `2*m1+m2` nodes are numbered
`0, ..., m1-1` for the left barbell,
`m1, ..., m1+m2-1` for the path,
and `m1+m2, ..., 2*m1+m2-1` for the right barbell.
The 3 subgraphs are joined via the edges `(m1-1, m1)` and
`(m1+m2-1, m1+m2)`. If `m2=0`, this is merely two complete
graphs joined together.
This graph is an extremal example in David Aldous
and Jim Fill's e-text on Random Walks on Graphs.
if m1 < 2:
raise NetworkXError("Invalid graph description, m1 should be >=2")
if m2 < 0:
raise NetworkXError("Invalid graph description, m2 should be >=0")
# left barbell
G = complete_graph(m1, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
# connecting path
G.add_nodes_from(range(m1, m1 + m2 - 1))
if m2 > 1:
G.add_edges_from(pairwise(range(m1, m1 + m2)))
# right barbell
(u, v) for u in range(m1 + m2, 2 * m1 + m2) for v in range(u + 1, 2 * m1 + m2)
# connect it up
G.add_edge(m1 - 1, m1)
if m2 > 0:
G.add_edge(m1 + m2 - 1, m1 + m2)
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def binomial_tree(n, create_using=None):
"""Returns the Binomial Tree of order n.
The binomial tree of order 0 consists of a single node. A binomial tree of order k
is defined recursively by linking two binomial trees of order k-1: the root of one is
the leftmost child of the root of the other.
.. plot::
>>> nx.draw(nx.binomial_tree(3))
n : int
Order of the binomial tree.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
G : NetworkX graph
A binomial tree of $2^n$ nodes and $2^n - 1$ edges.
G = nx.empty_graph(1, create_using)
N = 1
for i in range(n):
# Use G.edges() to ensure 2-tuples. G.edges is 3-tuple for MultiGraph
edges = [(u + N, v + N) for (u, v) in G.edges()]
G.add_edge(0, N)
N *= 2
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def complete_graph(n, create_using=None):
"""Return the complete graph `K_n` with n nodes.
A complete graph on `n` nodes means that all pairs
of distinct nodes have an edge connecting them.
.. plot::
>>> nx.draw(nx.complete_graph(5))
n : int or iterable container of nodes
If n is an integer, nodes are from range(n).
If n is a container of nodes, those nodes appear in the graph.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
>>> G = nx.complete_graph(9)
>>> len(G)
>>> G.size()
>>> G = nx.complete_graph(range(11, 14))
>>> list(G.nodes())
[11, 12, 13]
>>> G = nx.complete_graph(4, nx.DiGraph())
>>> G.is_directed()
_, nodes = n
G = empty_graph(nodes, create_using)
if len(nodes) > 1:
if G.is_directed():
edges = itertools.permutations(nodes, 2)
edges = itertools.combinations(nodes, 2)
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def circular_ladder_graph(n, create_using=None):
"""Returns the circular ladder graph $CL_n$ of length n.
$CL_n$ consists of two concentric n-cycles in which
each of the n pairs of concentric nodes are joined by an edge.
Node labels are the integers 0 to n-1
.. plot::
>>> nx.draw(nx.circular_ladder_graph(5))
G = ladder_graph(n, create_using)
G.add_edge(0, n - 1)
G.add_edge(n, 2 * n - 1)
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def circulant_graph(n, offsets, create_using=None):
r"""Returns the circulant graph $Ci_n(x_1, x_2, ..., x_m)$ with $n$ nodes.
The circulant graph $Ci_n(x_1, ..., x_m)$ consists of $n$ nodes $0, ..., n-1$
such that node $i$ is connected to nodes $(i + x) \mod n$ and $(i - x) \mod n$
for all $x$ in $x_1, ..., x_m$. Thus $Ci_n(1)$ is a cycle graph.
.. plot::
>>> nx.draw(nx.circulant_graph(10, [1]))
n : integer
The number of nodes in the graph.
offsets : list of integers
A list of node offsets, $x_1$ up to $x_m$, as described above.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
NetworkX Graph of type create_using
Many well-known graph families are subfamilies of the circulant graphs;
for example, to create the cycle graph on n points, we connect every
node to nodes on either side (with offset plus or minus one). For n = 10,
>>> G = nx.circulant_graph(10, [1])
>>> edges = [
... (0, 9),
... (0, 1),
... (1, 2),
... (2, 3),
... (3, 4),
... (4, 5),
... (5, 6),
... (6, 7),
... (7, 8),
... (8, 9),
... ]
>>> sorted(edges) == sorted(G.edges())
Similarly, we can create the complete graph
on 5 points with the set of offsets [1, 2]:
>>> G = nx.circulant_graph(5, [1, 2])
>>> edges = [
... (0, 1),
... (0, 2),
... (0, 3),
... (0, 4),
... (1, 2),
... (1, 3),
... (1, 4),
... (2, 3),
... (2, 4),
... (3, 4),
... ]
>>> sorted(edges) == sorted(G.edges())
G = empty_graph(n, create_using)
for i in range(n):
for j in offsets:
G.add_edge(i, (i - j) % n)
G.add_edge(i, (i + j) % n)
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def cycle_graph(n, create_using=None):
"""Returns the cycle graph $C_n$ of cyclically connected nodes.
$C_n$ is a path with its two end-nodes connected.
.. plot::
>>> nx.draw(nx.cycle_graph(5))
n : int or iterable container of nodes
If n is an integer, nodes are from `range(n)`.
If n is a container of nodes, those nodes appear in the graph.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
If create_using is directed, the direction is in increasing order.
_, nodes = n
G = empty_graph(nodes, create_using)
G.add_edges_from(pairwise(nodes, cyclic=True))
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def dorogovtsev_goltsev_mendes_graph(n, create_using=None):
"""Returns the hierarchically constructed Dorogovtsev--Goltsev--Mendes graph.
The Dorogovtsev--Goltsev--Mendes [1]_ procedure deterministically produces a
scale-free graph with ``3/2 * (3**(n-1) + 1)`` nodes
and ``3**n`` edges for a given `n`.
Note that `n` denotes the number of times the state transition is applied,
starting from the base graph with ``n = 0`` (no transitions), as in [2]_.
This is different from the parameter ``t = n - 1`` in [1]_.
.. plot::
>>> nx.draw(nx.dorogovtsev_goltsev_mendes_graph(3))
n : integer
The generation number.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. Directed graphs and multigraphs are not supported.
G : NetworkX `Graph`
If `n` is less than zero.
If `create_using` is a directed graph or multigraph.
>>> G = nx.dorogovtsev_goltsev_mendes_graph(3)
>>> G.number_of_nodes()
>>> G.number_of_edges()
>>> nx.is_planar(G)
.. [1] S. N. Dorogovtsev, A. V. Goltsev and J. F. F. Mendes,
"Pseudofractal scale-free web", Physical Review E 65, 066122, 2002.
.. [2] Weisstein, Eric W. "Dorogovtsev--Goltsev--Mendes Graph".
From MathWorld--A Wolfram Web Resource.
if n < 0:
raise NetworkXError("n must be greater than or equal to 0")
G = empty_graph(0, create_using)
if G.is_directed():
raise NetworkXError("directed graph not supported")
if G.is_multigraph():
raise NetworkXError("multigraph not supported")
G.add_edge(0, 1)
new_node = 2 # next node to be added
for _ in range(n): # iterate over number of generations.
new_edges = []
for u, v in G.edges():
new_edges.append((u, new_node))
new_edges.append((v, new_node))
new_node += 1
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def empty_graph(n=0, create_using=None, default=Graph):
"""Returns the empty graph with n nodes and zero edges.
.. plot::
>>> nx.draw(nx.empty_graph(5))
n : int or iterable container of nodes (default = 0)
If n is an integer, nodes are from `range(n)`.
If n is a container of nodes, those nodes appear in the graph.
create_using : Graph Instance, Constructor or None
Indicator of type of graph to return.
If a Graph-type instance, then clear and use it.
If None, use the `default` constructor.
If a constructor, call it to create an empty graph.
default : Graph constructor (optional, default = nx.Graph)
The constructor to use if create_using is None.
If None, then nx.Graph is used.
This is used when passing an unknown `create_using` value
through your home-grown function to `empty_graph` and
you want a default constructor other than nx.Graph.
>>> G = nx.empty_graph(10)
>>> G.number_of_nodes()
>>> G.number_of_edges()
>>> G = nx.empty_graph("ABC")
>>> G.number_of_nodes()
>>> sorted(G)
['A', 'B', 'C']
The variable create_using should be a Graph Constructor or a
"graph"-like object. Constructors, e.g. `nx.Graph` or `nx.MultiGraph`
will be used to create the returned graph. "graph"-like objects
will be cleared (nodes and edges will be removed) and refitted as
an empty "graph" with nodes specified in n. This capability
is useful for specifying the class-nature of the resulting empty
"graph" (i.e. Graph, DiGraph, MyWeirdGraphClass, etc.).
The variable create_using has three main uses:
Firstly, the variable create_using can be used to create an
empty digraph, multigraph, etc. For example,
>>> n = 10
>>> G = nx.empty_graph(n, create_using=nx.DiGraph)
will create an empty digraph on n nodes.
Secondly, one can pass an existing graph (digraph, multigraph,
etc.) via create_using. For example, if G is an existing graph
(resp. digraph, multigraph, etc.), then empty_graph(n, create_using=G)
will empty G (i.e. delete all nodes and edges using G.clear())
and then add n nodes and zero edges, and return the modified graph.
Thirdly, when constructing your home-grown graph creation function
you can use empty_graph to construct the graph by passing a user
defined create_using to empty_graph. In this case, if you want the
default constructor to be other than nx.Graph, specify `default`.
>>> def mygraph(n, create_using=None):
... G = nx.empty_graph(n, create_using, nx.MultiGraph)
... G.add_edges_from([(0, 1), (0, 1)])
... return G
>>> G = mygraph(3)
>>> G.is_multigraph()
>>> G = mygraph(3, nx.Graph)
>>> G.is_multigraph()
See also create_empty_copy(G).
if create_using is None:
G = default()
elif isinstance(create_using, type):
G = create_using()
elif not hasattr(create_using, "adj"):
raise TypeError("create_using is not a valid NetworkX graph type or instance")
# create_using is a NetworkX style Graph
G = create_using
_, nodes = n
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def ladder_graph(n, create_using=None):
"""Returns the Ladder graph of length n.
This is two paths of n nodes, with
each pair connected by a single edge.
Node labels are the integers 0 to 2*n - 1.
.. plot::
>>> nx.draw(nx.ladder_graph(5))
G = empty_graph(2 * n, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
G.add_edges_from(pairwise(range(n, 2 * n)))
G.add_edges_from((v, v + n) for v in range(n))
return G
@nx._dispatchable(graphs=None, returns_graph=True)
@nodes_or_number([0, 1])
def lollipop_graph(m, n, create_using=None):
"""Returns the Lollipop Graph; ``K_m`` connected to ``P_n``.
This is the Barbell Graph without the right barbell.
.. plot::
>>> nx.draw(nx.lollipop_graph(3, 4))
m, n : int or iterable container of nodes
If an integer, nodes are from ``range(m)`` and ``range(m, m+n)``.
If a container of nodes, those nodes appear in the graph.
Warning: `m` and `n` are not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
The nodes for `m` appear in the complete graph $K_m$ and the nodes
for `n` appear in the path $P_n$
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Networkx graph
A complete graph with `m` nodes connected to a path of length `n`.
The 2 subgraphs are joined via an edge ``(m-1, m)``.
If ``n=0``, this is merely a complete graph.
(This graph is an extremal example in David Aldous and Jim
Fill's etext on Random Walks on Graphs.)
m, m_nodes = m
M = len(m_nodes)
if M < 2:
raise NetworkXError("Invalid description: m should indicate at least 2 nodes")
n, n_nodes = n
if isinstance(m, numbers.Integral) and isinstance(n, numbers.Integral):
n_nodes = list(range(M, M + n))
N = len(n_nodes)
# the ball
G = complete_graph(m_nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
# the stick
if N > 1:
if len(G) != M + N:
raise NetworkXError("Nodes must be distinct in containers m and n")
# connect ball to stick
if M > 0 and N > 0:
G.add_edge(m_nodes[-1], n_nodes[0])
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def null_graph(create_using=None):
"""Returns the Null graph with no nodes or edges.
See empty_graph for the use of create_using.
G = empty_graph(0, create_using)
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def path_graph(n, create_using=None):
"""Returns the Path graph `P_n` of linearly connected nodes.
.. plot::
>>> nx.draw(nx.path_graph(5))
n : int or iterable
If an integer, nodes are 0 to n - 1.
If an iterable of nodes, in the order they appear in the path.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
_, nodes = n
G = empty_graph(nodes, create_using)
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def star_graph(n, create_using=None):
"""Return the star graph
The star graph consists of one center node connected to n outer nodes.
.. plot::
>>> nx.draw(nx.star_graph(6))
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
The graph has n+1 nodes for integer n.
So star_graph(3) is the same as star_graph(range(4)).
n, nodes = n
if isinstance(n, numbers.Integral):
nodes.append(int(n)) # there should be n+1 nodes
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
if len(nodes) > 1:
hub, *spokes = nodes
G.add_edges_from((hub, node) for node in spokes)
return G
@nx._dispatchable(graphs=None, returns_graph=True)
@nodes_or_number([0, 1])
def tadpole_graph(m, n, create_using=None):
"""Returns the (m,n)-tadpole graph; ``C_m`` connected to ``P_n``.
This graph on m+n nodes connects a cycle of size `m` to a path of length `n`.
It looks like a tadpole. It is also called a kite graph or a dragon graph.
.. plot::
>>> nx.draw(nx.tadpole_graph(3, 5))
m, n : int or iterable container of nodes
If an integer, nodes are from ``range(m)`` and ``range(m,m+n)``.
If a container of nodes, those nodes appear in the graph.
Warning: `m` and `n` are not checked for duplicates and if present the
resulting graph may not be as desired.
The nodes for `m` appear in the cycle graph $C_m$ and the nodes
for `n` appear in the path $P_n$.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Networkx graph
A cycle of size `m` connected to a path of length `n`.
If ``m < 2``. The tadpole graph is undefined for ``m<2``.
The 2 subgraphs are joined via an edge ``(m-1, m)``.
If ``n=0``, this is a cycle graph.
`m` and/or `n` can be a container of nodes instead of an integer.
m, m_nodes = m
M = len(m_nodes)
if M < 2:
raise NetworkXError("Invalid description: m should indicate at least 2 nodes")
n, n_nodes = n
if isinstance(m, numbers.Integral) and isinstance(n, numbers.Integral):
n_nodes = list(range(M, M + n))
# the circle
G = cycle_graph(m_nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
# the stick
nx.add_path(G, [m_nodes[-1]] + list(n_nodes))
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def trivial_graph(create_using=None):
"""Return the Trivial graph with one node (with label 0) and no edges.
.. plot::
>>> nx.draw(nx.trivial_graph(), with_labels=True)
G = empty_graph(1, create_using)
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def turan_graph(n, r):
r"""Return the Turan Graph
The Turan Graph is a complete multipartite graph on $n$ nodes
with $r$ disjoint subsets. That is, edges connect each node to
every node not in its subset.
Given $n$ and $r$, we create a complete multipartite graph with
$r-(n \mod r)$ partitions of size $n/r$, rounded down, and
$n \mod r$ partitions of size $n/r+1$, rounded down.
.. plot::
>>> nx.draw(nx.turan_graph(6, 2))
n : int
The number of nodes.
r : int
The number of partitions.
Must be less than or equal to n.
Must satisfy $1 <= r <= n$.
The graph has $(r-1)(n^2)/(2r)$ edges, rounded down.
if not 1 <= r <= n:
raise NetworkXError("Must satisfy 1 <= r <= n")
partitions = [n // r] * (r - (n % r)) + [n // r + 1] * (n % r)
G = complete_multipartite_graph(*partitions)
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def wheel_graph(n, create_using=None):
"""Return the wheel graph
The wheel graph consists of a hub node connected to a cycle of (n-1) nodes.
.. plot::
>>> nx.draw(nx.wheel_graph(5))
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Node labels are the integers 0 to n - 1.
_, nodes = n
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
if len(nodes) > 1:
hub, *rim = nodes
G.add_edges_from((hub, node) for node in rim)
if len(rim) > 1:
G.add_edges_from(pairwise(rim, cyclic=True))
return G
@nx._dispatchable(graphs=None, returns_graph=True)
def complete_multipartite_graph(*subset_sizes):
"""Returns the complete multipartite graph with the specified subset sizes.
.. plot::
>>> nx.draw(nx.complete_multipartite_graph(1, 2, 3))
subset_sizes : tuple of integers or tuple of node iterables
The arguments can either all be integer number of nodes or they
can all be iterables of nodes. If integers, they represent the
number of nodes in each subset of the multipartite graph.
If iterables, each is used to create the nodes for that subset.
The length of subset_sizes is the number of subsets.
G : NetworkX Graph
Returns the complete multipartite graph with the specified subsets.
For each node, the node attribute 'subset' is an integer
indicating which subset contains the node.
Creating a complete tripartite graph, with subsets of one, two, and three
nodes, respectively.
>>> G = nx.complete_multipartite_graph(1, 2, 3)
>>> [G.nodes[u]["subset"] for u in G]
[0, 1, 1, 2, 2, 2]
>>> list(G.edges(0))
[(0, 1), (0, 2), (0, 3), (0, 4), (0, 5)]
>>> list(G.edges(2))
[(2, 0), (2, 3), (2, 4), (2, 5)]
>>> list(G.edges(4))
[(4, 0), (4, 1), (4, 2)]
>>> G = nx.complete_multipartite_graph("a", "bc", "def")
>>> [G.nodes[u]["subset"] for u in sorted(G)]
[0, 1, 1, 2, 2, 2]
This function generalizes several other graph builder functions.
- If no subset sizes are given, this returns the null graph.
- If a single subset size `n` is given, this returns the empty graph on
`n` nodes.
- If two subset sizes `m` and `n` are given, this returns the complete
bipartite graph on `m + n` nodes.
- If subset sizes `1` and `n` are given, this returns the star graph on
`n + 1` nodes.
See also
# The complete multipartite graph is an undirected simple graph.
G = Graph()
if len(subset_sizes) == 0:
return G
# set up subsets of nodes
extents = pairwise(itertools.accumulate((0,) + subset_sizes))
subsets = [range(start, end) for start, end in extents]
except TypeError:
subsets = subset_sizes
if any(size < 0 for size in subset_sizes):
raise NetworkXError(f"Negative number of nodes not valid: {subset_sizes}")
# add nodes with subset attribute
# while checking that ints are not mixed with iterables
for i, subset in enumerate(subsets):
G.add_nodes_from(subset, subset=i)
except TypeError as err:
raise NetworkXError("Arguments must be all ints or all iterables") from err
# Across subsets, all nodes should be adjacent.
# We can use itertools.combinations() because undirected.
for subset1, subset2 in itertools.combinations(subsets, 2):
G.add_edges_from(itertools.product(subset1, subset2))
return G | {"url":"https://networkx.org/documentation/stable/_modules/networkx/generators/classic.html","timestamp":"2024-11-09T16:06:01Z","content_type":"text/html","content_length":"116925","record_id":"<urn:uuid:0e6795a4-d481-467c-a2e0-dbd666591383>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00112.warc.gz"} |
Duplications and deletions in genomes: theory and applications
Montana State University - Bozeman, College of Engineering
In computational biology, duplications and deletions in genome rearrangements are important to understand an evolutionary process. In cancer genomics research, intra-tumor genetic heterogeneity is
one of the central problems. Gene duplications and deletions are observed occurring rapidly in cancer during tumour formation. Hence, they are recognized as critical mutations of cancer evolution.
Understanding these mutations are important to understand the origins of cancer cell diversity which could help with cancer prognostics as well as drug resistance explanation. In this dissertation,
first, we prove that the tandem duplication distance problem is NP-complete, even if |sigma| > or = 4, settling a 16-year old open problem. And we obtain some positive results by showing that if one
of the input sequences, S, is exemplar, then one can decide if S can be transformed into T using at most k tandem duplications in time 2 O (k 2) + poly(n). Motivated by computing duplication patterns
in sequences, a new fundamental problem called the longest letter-duplicated subsequence (LLDS) is investigated. We investigate several variants of this problem. Due to fast mutations in cancer,
genome rearrangements on copy number profiles are used more often than genome themselves. We explore the Minimum Copy Number Generation problem. We prove that it is NP-hard to even obtain a constant
factor approximation. We also show that the corresponding parameterized version is W[1]-hard. These either improve the previous hardness result or solve an open problem. And we then give a polynomial
algorithm for the Copy Number Profile Conforming problem. Finally, we investigate the pattern matching with 1-reversal distance problem. With the known results on Longest Common Extension queries,
one can design an O(n+m) time algorithm for this problem. However, we find empirically that this algorithm is very slow for small m. We then design an algorithm based on the Karp-Rabin fingerprints
which runs in an expected O(nm) time. The algorithms are implemented and tested on real bacterial sequence dataset. The empirical results shows that the shorter the pattern length is (i.e., when m <
200), the more substrings with 1-reversal distance the bacterial sequences have. | {"url":"https://scholarworks.montana.edu/items/8d02035c-48d6-4ee6-bc1e-db25aab46530","timestamp":"2024-11-04T18:48:24Z","content_type":"text/html","content_length":"451596","record_id":"<urn:uuid:c48abdce-d29f-4a86-9fa0-f073a3b4edcd>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00529.warc.gz"} |
Advanced Activity
From WikiEducator
The goals of this activity is to improve your multiplication speed on the medium numbers and become comfortable with multiplying larger numbers.
1. Try to complete the multiplication table in the simulation.
2. Easy peezy? Try using the clock.
3. Record your times. Do you get faster each time you practice?
4. Easy peezy, lemon squeezy? Choose your level if you dare!.
Special thanks to the Physics Education Technology PhET Team
Arithmetic Workout
When you have filled in all the squares, check out the diagonal line of numbers that runs from the × to the lower right corner of grid. Do you notice any patterns?
Hopefully you noticed that the same numbers occur on each side of the diagonal. Think about why this happens and talk to your mates about it. | {"url":"https://wikieducator.org/Arithmetic_Workout/Multiplication_Workout/Activities/Advanced_Activity","timestamp":"2024-11-08T07:41:57Z","content_type":"text/html","content_length":"29915","record_id":"<urn:uuid:8a476772-d5fe-482c-a3f7-014af73374ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00252.warc.gz"} |