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Understanding R Square, F Square, and Q Square using SMART-PLSUnderstanding R Square, F Square, and Q Square using SMART-PLS
Understanding R Square, F Square, and Q Square using SMART-PLS
• R Square statistics explains the variance in the endogenous variable explained by the exogenous variable(s).
• For example, a variable Y influenced by X1, X2, and X3 has a R-Square value of 0.623. This would mean that 62.3% change in Y can be explained by X1, X2, X3.
• In order to make it easier to interpret, look for the arrows that are pointing towards the dependent (endogenous) variable.
• Falk and Miller (1992) recommended that R2 values should be equal to or greater than 0.10 in order for the variance explained of a particular endogenous construct to be deemed adequate.
• Cohen (1988) suggested R2 values for endogenous latent variables are assessed as follows: 0.26 (substantial), 0.13 (moderate), 0.02 (weak).
• Chin (1998) recommended R2 values for endogenous latent variables based on: 0.67 (substantial), 0.33 (moderate), 0.19 (weak).
• Hair et al. (2011) & Hair et al. (2013) suggested in scholarly research that focuses on marketing issues, R2 values of 0.75, 0.50, or 0.25 for endogenous latent variables can, as a rough rule of
thumb, be respectively described as substantial, moderate or weak.
• A variable in a structural model may be affected/influenced by a number of different variables.
• Removing an exogenous variable can affect the dependent variable.
• F-Square is the change in R-Square when an exogenous variable is removed from the model.
• f-square is effect size (>=0.02 is small; >= 0.15 is medium;>= 0.35 is large) (Cohen, 1988).
• Q-square is predictive relevance, measures whether a model has predictive relevance or not (> 0 is good).
• Further, Q2 establishes the predictive relevance of the endogenous constructs.
• Q-square values above zero indicate that your values are well reconstructed and that the model has predictive relevance.
• A Q2 above 0 shows that the model has predictive relevance.
• In order to find out the Q Square value, Run Blindfolding procedure in SMART-PLS.
• Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed.). New York: Routledge.
• Chin, W. W. (1998). The partial least squares approach to structural equation modeling. Modern methods for business research, 295(2), 295-336.
• Falk, R. F., & Miller, N. B. (1992). A primer for soft modeling. University of Akron Press.
• Hair, J. F., Ringle, C. M., & Sarstedt, M. (2011). PLS-SEM: Indeed a silver bullet. Journal of Marketing theory and Practice, 19(2), 139-152.
• Hair, J. F., Ringle, C. M., & Sarstedt, M. (2013). Partial least squares structural equation modeling: Rigorous applications, better results and higher acceptance. Long range planning, 46(1-2),
Watch the Video Tutorial for further Detail | {"url":"https://researchwithfawad.com/index.php/lp-courses/basic-and-advance-data-analysis-using-smart-pls/understanding-r-square-f-square-and-q-square-using-smart-pls/","timestamp":"2024-11-05T04:12:01Z","content_type":"text/html","content_length":"309572","record_id":"<urn:uuid:e4215c08-85bb-48e1-b36a-e9b190ca91fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00212.warc.gz"} |
turn off wall functions in Transition SST model?
That's a wrong notion that RANS or EVM models are introduced to get faster results or are expected to be used with coarse mesh. There is no such assumption behind development of these models. The
only assumption in EVM is that the turbulence is isotropic and non-EVM RANS, such as, RSM don't even have that assumption.
And when it comes to wall treatment, it is not directly linked with turbulence model; even LES requires wall treatment.
is a non-dimensional (Reynolds) number and for almost all industrial fluids, theoretically as well as experimentally, it is found that
up to
of 5. And if it is linear within this limit, it does not matter if you have 10 points or just 1 point, the line would be same. So,
being smaller than 1 is an overkill and does not help within anything.
Boundary conditions for both k and
at the wall is 0. | {"url":"https://www.cfd-online.com/Forums/fluent/102118-turn-off-wall-functions-transition-sst-model.html","timestamp":"2024-11-14T03:51:54Z","content_type":"application/xhtml+xml","content_length":"127865","record_id":"<urn:uuid:9b2c5a2d-25c7-4025-bd09-294ab1200f48>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00090.warc.gz"} |
Niels Möller nisse at lysator.liu.se
Sat Sep 7 22:17:13 UTC 2019
tg at gmplib.org (Torbjörn Granlund) writes:
> Should we rename div1 to put it into the GMP name space? How about
> mpn_div_11? (We could encode in its name that it is for use for small
> q, but I'd suggest that we don't do that.)
> That would allow for some assembly experiments.
If you think assembly will make a big difference (which seems likely),
that makes sense. And define it only when it's faster than a a div
instruction generated by the compiler?
> Taking out powers of two makes things complicated. Taking out a factor
> of two from one of the numbers corresponds to a matrix (1,0;0,2) with
> determinant 2. So the inverse is not an integer matrix, but an integer
> matrix + a shift count.
> Would a shift count hurt?
On second thought, I think there's a bigger problem: The way hgcd is
used, it's called on the most significant part of some numbers, but
applied to larger numbers. And a matrix based on trailing zeros in hgcd
won't be applicable to the larger numbers, since they have different
lowend bits.
> It would be interesting to try some left-to-right binary hgcd. Logic
> should be something like
> count_leading_zeros(a_bits, a);
> count_leading_zeros(b_bits, b);
> if (a_bits == b_bits)
> (a, b) <-- (min(a,b), |a-b|)
> else if a_bits > b_bits
> a <-- |a - b * 2^{a_bits - b_bits}|
> else
> b <-- |b - a * 2^{b_bits - a_bits}|
> OK! This is super interesting!
I think it may work nicely on machines with slow multiplication as well
as small-quotient division. But will be some overhead to make
branch-free. I haven't tried it seriously yet.
We'll sometimes get determinant == -1, so users of hgcd2 would need to
be updated to accept that. Or maybe one can just do a swap to ensure
that we have det == +1 in all cases. E.g.,
a <-- 2b - a
corresponds to the matrix (-1, 2; 0,1) with determinant -1. But if we
swap a and b, and set
(a, b) <-- (b, 2b - a)
that's the matrix (0,1; -1, 2) with determinant +1.
In the mean time, I've updated the replacement of div2. See attached
patch, using div1 except when HGCD2_METHOD == 2. Timing on my old laptop:
$ ./tune/speed -c -p1000000 -s1 mpn_hgcd2_1 mpn_hgcd2_2 mpn_hgcd2_3
overhead 6.00 cycles, precision 1000000 units of 8.33e-10 secs, CPU freq 1200.00 MHz
mpn_hgcd2_1 mpn_hgcd2_2 mpn_hgcd2_3
1 1492.62 2064.18 #1375.18
So this is 50% speedup over the old method.
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Niels Möller. PGP-encrypted email is preferred. Keyid 368C6677.
Internet email is subject to wholesale government surveillance.
More information about the gmp-devel mailing list | {"url":"https://gmplib.org/list-archives/gmp-devel/2019-September/005498.html","timestamp":"2024-11-15T04:29:02Z","content_type":"text/html","content_length":"5725","record_id":"<urn:uuid:b4d917bc-7e3f-484a-a1c3-63392a781d9e>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00201.warc.gz"} |
2012: The Higgs Is Found, Or Ruled Out
In two years the Higgs boson will be close to discovery, and its mass already known, or the particle will be already in the trash bin
. That is the single line which best summarizes the scenarios I depicted yesterday, in the concluding slides of a seminar I gave at IFIC, in beautiful Valencia (below, placa de la Virgen on a
pleasant evening, taken with my iphone).
that it takes one a substantial amount of work to prepare a well-constructed seminar, regardless of how well one knows the topic. And I ventured to hypothesize that among my readers there could be
somebody willing to invite me, to give an updated version of the same seminar at some other university or institute, preferably located in some interesting place to visit.
My wish materialized overnight when Martin Alonso, a PhD student at IFIC Valencia, took contact with me to organize a presentation there. As I write these notes, I am in the Valencia airport,
homebound after two very pleasant days. The people I met at IFIC were very kind and interesting, the city is quite beautiful to visit, and the food I ate there was spectacular. But the most
fulfilling part of the trip was, indeed, the seminar.
I hope my colleague Andrea Giammanco, who first invited me to Louvain, will not resent the fact that I am claiming my presentation in Valencia was appreciably better. Of course, unless you are really
dumb you learn from your mistakes: and although my talk in Louvain was reasonably good, after giving it I realized I might cut some material, reshape some other, cite a few more papers, and include
the latest results of searches that had not been published before February. The reshaped seminar received a very positive feedback at IFIC, where I got the attention of about thirty among staff
physicists and graduate students, despite the absence of many for Easter vacations.
I would be happy to advertise the package further, in the hope to collect other invitations; but I am traveling way too much these days, so for a while I will stop willfully overburdening my schedule
-unless invitations come from really appealing places! Instead, let me reproduce here, amended and simplified, the part of the seminar where I discussed the future of the Higgs boson searches, and
the scenarios that are taking shape in front of us.
2013: The Higgs is found, or excluded, or ...
One needs several ingredients in order to put together a meaningful prediction for when, how, and by whose hands this now over thirty-year-long saga of the Higgs boson will get to an end. But it so
happens that those ingredients are available today, and for the first time we have a rather credible way of extrapolating both Tevatron reach and LHC reach at least three years in the future. There
are, in fact, some factors making this particular moment of time a good pick to play the "seer".
First of all, the Tevatron: Higgs searches there have become stable in their output, in the sense that the technology is quite refined, and it is not foreseen to improve significantly in the future
(at least by yours truly). It might, but taking what now exists as a basis for extrapolation is a conservative, sound approach, and the one I will follow.
Second, the LHC has finally started running at 7 TeV energy in the center-of-mass. The first W bosons
are timidly popping up already!
This means that the ball is rolling, and I do not think there are significant caveats in trusting the integrated luminosity profile versus time that the LHC machinists have predicted for the next two
years, helium blowups allowing. And it also means that we can reliably take it for granted that CMS and ATLAS will be looking at roughly one inverse femtobarns of data by the end of 2011.
And third, much of the hard work has been already done by CDF some time ago. Basically the CDF predictions are provided in the form of two pairs of plots, which are shown below. The first pair
describes the 95% confidence-level limits on the ratio
between excluded Higgs cross section and standard model expectation, for two significant values (115 and 160 GeV) of the Higgs mass: if
is excluded, then the Higgs is disfavored to exist at the corresponding mass.
Above, expected 95% confidence-level limits on the ratio between production cross-section of a 115-GeV Higgs boson and its standard model prediction (on the vertical axis) are shown as a function of
available integrated luminosity (on the horizontal axis), in the hypothesis of (1) a combination CDF and DZERO searches, in the further hypothesis (2) that DZERO analyses at low mass become as strong
as the ones by CDF (they are significantly less stringent presently), and in the final hypothesis that (3) the data follow the expectation of the background (as opposed to producing a unlikely
fluctuation upward or downward, which may change the picture quite sizably).
The black like, which passes through the most recent CDF result (doubled in sensitivity to incorporate hypothesis 1) corresponds to a simple scaling of the R limit with the square root of the
integrated luminosity on which the analyses are based. The colored points and curves refer to earlier instantiations of the same analysis, which were less powerful and thus exploited less well the
available data.
Above you may instead see the twin plot corresponding to a 160 GeV Higgs boson search. As before, you may note how the limit has become more and more stringent with increases in luminosity, scaling
way more steeply than what the data increase predicted. The reason is that the time between the various results was used successfully by the experiments to improve the precision and care of their
results. But let us concentrate on the future, not on the past.
Instead, before I start discussing the power of Tevatron searches to actually
see a signal if one is there
, let me stress one point about these plots. What is shown is the power of the data and its analysis; even if future data should withstand a statistical fluctuation, the results shown would still
stand -because they are not actual results but averages, or rather
. A wild fluke might make the exclusion significantly stronger, or weaker. For instance, a 115-GeV Higgs boson might have already been excluded with 5 inverse femtobarns of CDF and D0 data, if
backgrounds had fluctuated down by about 2.5 standard deviations!
Once that is clear, let us examine the second pair of graphs: these are a different kind of sensitivity reach plots, again produced over a year ago. They still constitute our best predictions for the
chance of a Higgs boson sight; actually, the recent developments -Tevatron performance and added improvements in the analyses- make the predictions more trustable. These plots show, as a function of
the Higgs boson mass (on the horizontal axis), the probability (on the vertical axis) that CDF and DZERO, by combining their 5- (in red) or 10- (in blue) inverse femtobarns datasets, may obtain a
2-standard-deviation excess due to Higgs bosons,
if that particle exists at that mass
. Please ignore the dashed lines, which represent the case of further improvements in the analysis techniques which I doubt will ever be possible. Still, you can observe that the probability of a
2-sigma excess is quite sizable, and even a 3-sigma evidence is possible if the Tevatron gets lucky.
Above, the probabilities for a 2-sigma excess as a function of Higgs mass, with 5- and 10-inverse-femtobarns.
And above, the probability of a 3-sigma evidence as a function of Higgs mass.
Including Luminosity estimates and LHC predictions in three scenarios
None of what I showed above is new to you, if you have followed this blog in the course of the past year. Yet we can build something more interesting with that information. We need some more input to
make more precise predictions on what will happen in the next few years, on the west side of the Atlantic. We need to know what is the total integrated luminosity that CDF and DZERO will be able to
analyze, and when.
Please have a look at the graph below: it shows the integrated luminosity that the Tevatron collider delivered to the experiments as a function of time. The curve has followed an almost exponential
trend: it has fulfilled our wildest dreams of five years ago, delivering data in perfect accordance to a "design plan". Including the now customary yearly shutdowns of two months, the machine now
produces over 2 inverse femtobarns per year.
With that trend in mind, we can extrapolate to the end of 2011, which is the probable date when the Tevatron will stop running for good. It does not make much sense, in fact, for the US Department of
Energy to spend over 200 million dollars a year running a machine that is now outdated by the LHC startup: every additional year of running would add only 10% more reach for discoveries and
measurements (2 femtobarns over 10 is 20% luminosity, or 10% statistical power).
So by the end of 2011 it is a safe bet to claim that the Tevatron will have delivered about 11 inverse femtobarns of data to the experiments. This translates in a bit less than 10 inverse femtobarns
of collected collisions (the efficiency is never 100%, for several reasons which would get this post wildly off-topic).
In Europe, in the meantime, the LHC will deliver one inverse femtobarns of data to ATLAS and CMS. There exist predictions for the reach of these experiments in the search for the Higgs boson, but
they were computed assuming 10 or 14 TeV collisions. At 7 TeV, the Higgs boson has a cross section which is twice smaller; backgrounds are also smaller, but the net result is a significant hit in
sensitivity. It is however possible to make some extrapolations and compute the sensitivity that a combination of ATLAS and CMS data will have on the Higgs boson. I did the exercise, and have my own
Putting everything together, I can now post here a slide of the seminar, showing three different scenarios for what will happen by 2012.
Three Scenarios
As usual, my slides are quite descriptive, so there is little to comment on. It is important, however, to note that the existence of the "standard-model-like" Higgs boson is what makes a difference:
if the Higgs exists, it will most likely be found by the LHC experiments. If it does not exist, however, it will be the Tevatron to rule it out first.
Maybe one more thing to note remains to be made. The title of this post suggests that by 2012 the Higgs might be found and its mass known: this, for particle physicists, usually implies a 5-sigma
excess of signal events. But it needs not be so: in 1994, for instance, the top quark was found by CDF, and its mass was measured (with excellent accuracy, as demonstrated later). That was not a
discovery, since the effect amounted to only three standard deviations; the discovery came one year later, by both CDF and DZERO together. Nevertheless, by 1994 everybody was convinced that the top
quark was there. And those who doubted were later proven wrong... So if you are not a blind sceptic, 2012 may be the year for the Higgs as well!
And one final remark: however it goes, the next two years will be quite exciting! | {"url":"https://www.science20.com/quantum_diaries_survivor/2012_higgs_found_or_ruled_out","timestamp":"2024-11-05T19:40:24Z","content_type":"text/html","content_length":"45505","record_id":"<urn:uuid:7750a34e-27e0-4c36-bebd-46462456f43f>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00155.warc.gz"} |
Labeling schemes for tree representation
This paper deals with compact label-based representations for trees. Consider an n-node undirected connected graph G with a predefined numbering on the ports of each node. The all-ports tree labeling
ℒ[all] gives each node v of G a label containing the port numbers of all the tree edges incident to v. The upward tree labeling ℒ[up] labels each node v by the number of the port leading from v to
its parent in the tree. Our measure of interest is the worst case and total length of the labels used by the scheme, denoted M[up](T) and S[up](T) for ℒ[up] and M[all](T) and S[all](T) for ℒ[all].
The problem studied in this paper is the following: Given a graph G and a predefined port labeling for it, with the ports of each node v numbered by 0, . . . , deg(v) - 1, select a rooted spanning
tree for G minimizing (one of) these measures. We show that the problem is polynomial for M[up](T), S[up](T) and S[all](T) but NP-hard for M[all](T) (even for 3-regular planar graphs). We show that
for every graph G and port numbering there exists a spanning tree T for which S[up](T) = O(n log log n). We give a tight bound of O(n) in the cases of complete graphs with arbitrary labeling and
arbitrary graphs with symmetric port assignments. We conclude by discussing some applications for our tree representation schemes.
Original language English
Title of host publication Distributed Computing - IWDC 2005 - 7th International Workshop, Proceedings
Publisher Springer Verlag
Pages 13-24
Number of pages 12
ISBN (Print) 3540309594, 9783540309598
State Published - 2005
Externally published Yes
Event 7th International Workshop on Distributed Computing, IWDC 2005 - Kharagpur, India
Duration: 27 Dec 2005 → 30 Dec 2005
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 3741 LNCS
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 7th International Workshop on Distributed Computing, IWDC 2005
Country/Territory India
City Kharagpur
Period 27/12/05 → 30/12/05
Dive into the research topics of 'Labeling schemes for tree representation'. Together they form a unique fingerprint. | {"url":"https://cris.biu.ac.il/en/publications/labeling-schemes-for-tree-representation-3","timestamp":"2024-11-03T06:01:50Z","content_type":"text/html","content_length":"57853","record_id":"<urn:uuid:d472f5de-8d9b-49c0-b53f-b97ddf2cd968>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00331.warc.gz"} |
Arabic numerals
< 1 2 >
In 1202, the Italian mathematican Leonardo of Pisa (Fibonacci) introduced the arabic numerals in Europe.
The word cipher is derived from the Arabic صفر sifr, which means "zero" and is taken from the Sanskrit sunya, which means "empty".
There has been discussion for centuries about whether or not the digit 0 is a number. Nowadays we call it a number, and for writing decimal numbers we just need it.
In his book Liber Abaci (book of arithmetics) he described the Fibonacci numbers, which probably were known for hundreds of years. | {"url":"http://www.maeckes.nl/Arabische%20cijfers%20(historie)%20GB.html","timestamp":"2024-11-03T09:17:22Z","content_type":"application/xhtml+xml","content_length":"4542","record_id":"<urn:uuid:25811083-8d74-492f-8471-4ffbf8265424>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00683.warc.gz"} |
Consider the following simultaneous nonlinear equations: x^{2}+y+z=6
Consider the following simultaneous nonlinear equations: x^{2}+y+z=6 x y z+z=9 x^{2}+y^{2}+z^{2}=14 Starting from the initial guesses: xo = 1.5, yo = 2.5 and zo = 3.5, perform the following: a)
Determine the roots of the system of equations using the successive substitution method.Perform three complete iteration. Show your calculations. b) Determine the roots of the system of equations
using the Newton-Raphosn method. Perform the computations until & = 7%. Show your calculations. c) Write a MATLAB script that performs ten iterations of the Newton-Raphson method. d) Solve the system
of equations by calling the function newtmult, with an accuracy of 10 significant figures.Page 1 of 1
Fig: 1
Fig: 2
Fig: 3
Fig: 4
Fig: 5
Fig: 6
Fig: 7
Fig: 8
Fig: 9 | {"url":"https://tutorbin.com/questions-and-answers/consider-the-following-simultaneous-nonlinear-equations-x2yz6-x2y2z214","timestamp":"2024-11-03T02:50:46Z","content_type":"text/html","content_length":"77765","record_id":"<urn:uuid:cc7ca4f7-06f1-4843-aa63-f093047842f6>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00410.warc.gz"} |
How to Print Numbers with Ordinal Suffix in PHP
Ordinal suffixes are words that represent the rank of a number. In this tutorial, we will learn how to add ordinal numerals to the end of each number dynamically using PHP.
The first step is to create an array of ordinal suffixes and store them in a variable.
$ordinals = ['th','st','nd','rd','th','th','th','th','th','th'];
Now we can assign an appropriate number to a numeral.
for ($i=1; $i < 22; $i++) {
print_r($i . (($i%100) >= 11 && ($i%100) <= 13 ? 'th' : $ordinals[$i % 10]) . "<br>");
The only numbers that disrupt the ordinal suffix pattern are 11,12 and 13 which end in "th". The above example code looks for the 11th, 12th and 13th number in every 100 and uses a "th" suffix,
otherwise, the appropriate ordinal is picked from the array. | {"url":"https://www.skillsugar.com/how-to-print-numbers-with-ordinal-suffix-in-php","timestamp":"2024-11-03T03:11:57Z","content_type":"text/html","content_length":"37850","record_id":"<urn:uuid:01e63654-0371-4dc5-9360-a4cfe640f793>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00429.warc.gz"} |
Compound Annual Growth Rate (CAGR) Calculator
Compound Annual Growth Rate (CAGR) Calculator
This Compound Annual Growth Rate calculator will allow you to check the constant progression rate of return over a period of time. By using the CAGR calculator you can offset the periods of volatile
change between starting and ending periods to have a consistent growth reading. The CAGR measurement is most commonly used to analyze and standarise the change over time in directly quantifiable
To find CAGR
To find ending amount
To find # of years of required investment
Compound annual growth rate (CAGR) is a financial metric that is used to measure the rate at which an investment or business has grown over a specific period. It is a valuable tool for evaluating the
performance of an investment or business because it takes into account the impact of compounding, which is the reinvestment of earnings back into the investment or business. CAGR is commonly used to
analyze and standardize the change over time in directly quantifiable data such as revenue, profit, or sales.
CAGR is an effective way to measure the growth rate of an investment or business because it accounts for the compounding effect. For example, suppose that an investment has grown by 10% in the first
year, 20% in the second year, and 30% in the third year. The simple average of the growth rates is (10% + 20% + 30%) / 3 = 20%. However, the compound growth rate is calculated by multiplying the
growth rates together and taking the nth root, where n is the number of years. In this example, the compound growth rate is (1 + 10%) x (1 + 20%) x (1 + 30%)^(1/3) - 1 = 21.44%.
CAGR is an important tool for investors because it allows them to compare the performance of different investments over the same period. For example, suppose that an investor is considering two
different stocks, A and B. Stock A has grown by 10% per year for the past three years, while stock B has grown by 5% per year for the past five years. To compare the performance of these two stocks,
the investor can calculate the CAGR for each stock. The CAGR for stock A is (1 + 10%)^3 - 1 = 33.10%, while the CAGR for stock B is (1 + 5%)^5 - 1 = 28.34%. Based on these calculations, the investor
may decide that stock A is a better investment because it has a higher CAGR.
CAGR is also a useful tool for businesses because it allows them to evaluate their performance over time. For example, suppose that a company has had revenue of $100,000 in the first year, $120,000
in the second year, and $150,000 in the third year. The simple average growth rate is (20% + 25%) / 2 = 22.5%. However, the CAGR is (1 + 20%) x (1 + 25%)^(1/2) - 1 = 22.04%. This calculation shows
that the company has grown at an average rate of 22.04% per year over the three-year period.
CAGR is particularly useful for businesses that are looking to expand into new markets or launch new products. For example, suppose that a company is considering launching a new product line that is
expected to generate $1 million in revenue in the first year, $2 million in the second year, and $3 million in the third year. The CAGR for this product line is (1 + 100%) x (1 + 50%) x (1 + 33.33%)^
(1/3) - 1 = 57.09%. This calculation shows that the product line is expected to grow at an average rate of 57.09% per year over the three-year period. By using CAGR, the company can evaluate whether
this level of growth is sufficient to justify the investment in the new product line.
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Inappropriate data-types - SQL Smells - Duhallow Grey Geek
Inappropriate data-types – SQL Smells
Inappropriate SQL Data-types cause SQL Smells!
People choose inappropriate data-types. This isn’t surprising. There are lots of SQL data-types, so people make inappropriate choices. Phil Factor names “Using inappropriate data-types” as a smell in
his article on SQL Smells.
I’m going to concentrate on dates and numbers in this post. I will explain why people choose inappropriate data-types. I will also describe an approach which will encourage you to choose the right
This will be a superficial treatment. I’m going to look at the problem from a high level. Dates and numbers can suffer from detailed technical problems as well.
Why do people choose inappropriate data-types?
SQL: Storage, Interchange and Presentation forms of data
Phil Factor identifies the main reason for this problem as:
“Confusing how data will be presented with how it will be stored”.
I agree. Here are some reasons Analysts choose inappropriate data-types:
• We approach problems from the outside and take the users’ point of view. We should consider presentation. Inside the database, data should be stored in an appropriate form.
• The same argument applies for interfaces and interchange formats. Interface requirements should not determine the way data is stored internally. Interfaces are still important. Where possible,
standard interchange formats should be used.
• Spreadsheets have made us lazy. You don’t have to think about the “data-type” when you key something into a cell. Format and validation are often added afterwards.
• There are “folk memories” about problems with data in old file-based systems. These systems did not have the rich range of data-types of modern databases and languages.
Consequences of using inappropriate data-types
Inappropriate data types can have serious consequences for a system we are building. Some of these problems are not obvious. Many of these problems apply to all systems. Some of these problems become
more important with larger databases.
• Having the wrong format makes validation harder. It prevents the database engine from checking the content and increases the risk of “garbage data” getting into the system.
• The possible presence of garbage data makes error handling throughout the system harder.
• Storing data in “display format” imbeds that format deep inside the system.
• “Dates” are associated with useful functions which simplify program design.
• Inappropriate data-types can change how data is sorted. This influences how indexes work and cause performance issues.
• The “correct” data-types are usually very space efficient. Using alternatives can waste space in the database for no benefit.
Let’s look at some specific examples:
Inappropriate data-types for Numbers, especially currency
SQL: Storing numbers as character strings
It is possible to present numbers as strings, even including the decimal and thousands separators and any related currency symbols.
Interchange files can contain numbers as text, because it is convenient.
Numbers stored as strings are harder to validate.
Numbers stored as strings are sorted differently to numbers stored as numbers. If you doubt me, then try the experiment of illustrated in the Figure: “Number versus Character sorting” in your
favourite spreadsheet.
SQL: Number versus Character sorting
Inappropriate data-types for Dates
Some early databases did not handle dates very well. This encouraged designers to do-it-themselves with varying degrees of success.
It is possible to represent a date as an integer. Such a “date” will sort as you expect, but needs its own validation and will not help you with date arithmetic.
It is also possible (and unfortunately common) to store dates in character fields. In most cases this is simply “an accident waiting to happen”!
All these do-it-yourself options are vulnerable to the problem that Americans tend to specify dates “mm-dd-yy” and Europeans (including the British) tend to specify dates “dd-mm-yy”. There is nothing
we as analysts can do about this except to make sure that the test data for any system always includes a date with “13th of the Month”!
Benefits of using the appropriate data-types
The benefits of using the appropriate data-types far outweigh any perceived costs. Most of the “cost” is simply being aware that there are options and then not choosing the inappropriate data-types!
Using the appropriate data-types will:
• Help protect you from “garbage data” (The database will reject an incorrect leap year 29th of February!)
• Sort as you (and the business) expect without the need to work out the details.
• Allow you to specify the presentation separately from the storage. Many languages and presentation frameworks have these facilities built in.
• Take up less storage space.
• Make your system perform better!
How to prevent appropriate data-types
Choosing the Appropriate Data-type – Problem Prevention
People choose inappropriate data-types in the transition from the “Conceptual Model” to the “Logical Model” (or possibly the “Physical Model”). We have selected the entities and attributes the system
needs, but we have chosen inappropriate data-types.
The solution is to separate different aspects of the data and the decisions we need to make in our minds.
Here is the approach I recommend:
In the Conceptual Model:
• Decide what data you need (the “attribute”) and decide what kind of data it is. Do this from a “non-computer” point of view.
• If it is a number, say what it counts or measures and what the units are.
• Treat “money” as a unit.
• For dates and times, label them loosely as “date”, “time” or “date-time”.
• Record any “limits”. The database designer may use them in detailed design.
• Say how the users of the system will expect to see it presented. You will use this information in the user-interface design.
In the Logical Model:
• Decide what kind of “bucket” the database should put it into. A database professional may help you with this.
• If it is a number, and a precise value, say it is an Integer (of some kind). For decimals, say how many decimal places you need.
• Look hard at dates and times. Do you mean “date” or “time of day”? Do you need an “elapsed time” or a “point in time”?
In the Physical Model:
• Decide exactly what SQL Data-type you are going to use. Many of the basic data-types have alternatives. There are several types of “Integer” and quite a lot of “Date and Time” types.
• This is a good time to talk to a database professional.
There are two main reasons for choosing inappropriate data-types in SQL:
• Concentrating too much on how data will be presented, rather than how it will be stored
• Making decisions about the Physical Model prematurely
Using inappropriate data-types can have wide-ranging harmful effects on your database and system.
Avoid the problems by following a simple process:
1. Concentrate on what data the system needs in the Conceptual Model.
2. Outline how that data should be stored in the Logical Model.
3. Confirm the exact SQL data-type in the Physical Model.
This does not have to be difficult or time consuming. It fits perfectly well with “Agile” development.
Where next?
“Inappropriate data-types” was a problem with converting a Conceptual Model into a Logical Model. In the next article I’m going to look at the SQL Smells and Requirements Smells around – “Using
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Nuclei Class 12 Important Extra Questions Physics Chapter 13
Here we are providing Class 12 Physics Important Extra Questions and Answers Chapter 13 Nuclei. Important Questions for Class 12 Physics with Answers are the best resource for students which helps in
Class 12 board exams.
Class 12 Physics Chapter 13 Important Extra Questions Nuclei
Nuclei Important Extra Questions Very Short Answer Type
Question 1.
What will be the ratio of the radii of two nuclei of mass numbers A[1] and A[2]?
The ratio is \(\frac{R_{1}}{R_{2}}=\left(\frac{A_{1}}{A_{2}}\right)^{1 / 3}\)
Question 2.
Two nuclei have mass numbers in the ratio 1: 2. What is the ratio of their nuclear densities?
The densities of both nuclei are equal as they do not depend upon mass number.
Question 3.
A nucleus of mass number A has a mass defect Δm. Give the formula, for the binding energy per nucleon of this nucleus.
The formula is E = \(\frac{\Delta m \times c^{2}}{A}\)
Question 4.
Write the relation between half-life and decay constant of a radioactive sample.
The relation is T[1/2] = \(\frac{0.693}{\lambda}\)
Question 5.
Write the nuclear decay process for β-decay of 15^32P.
The process is
Question 6.
State the relation between the mean life (τ) of a radioactive element and its decay constant λ.
The two are related as τ = 1 / λ.
Question 7.
Write any two characteristic properties of nuclear force. (CBSE AI 2011)
1. Do not obey inverse square law and
2. Spin-dependent.
Question 8.
How is the radius of a nucleus related to its mass number? (CBSE AI 2011C, AI 2013C)
The radius R of the nucleus and mass number A is related as R = R[o]A^1/3, where R[o] is a constant.
Question 9.
A nucleus undergoes β – decay. How does
(i) the mass number,
(ii) atomic number change? (CBSE Delhi 2011C)
During β – decay
(i) the mass number remains the same,
(ii) atomic number increases by one.
Question 10.
Define the activity of a given radioactive substance. Write its SI unit. (CBSE AI 2013)
The rate of disintegration in a radioactive substance is known as its activity. SI unit is becquerel (Bq).
Question 11.
Why is it found experimentally difficult to detect neutrinos in nuclear β-decay? (CBSE AI 2014)
They are very difficult to detect since they can penetrate a large quantity of matter (even earth) without any interaction.
Question 12.
Four nuclei of an element undergo fusion to form a heavier nucleus, with the release of energy. Which of the two — the parent or the daughter nucleus – would have higher binding energy per nucleon?
(CBSE AI 2018, Delhi 2018)
Daughter nucleus.
Question 13.
Why is nuclear fusion not possible In the laboratory?
Because temperature as high as 10^7 K cannot be sustained in the laboratory.
Question 14.
Why is the penetrating power of gamma rays very large?
Because they have high energy and are neutral.
Question 15.
Can it be concluded from beta decay that electrons exist inside the nucleus?
No, the beta particle although an electron is actually created at the instant of beta decay and ejected at once. It cannot exist inside the nucleus as its de-Broglie wavelength is much larger than
the dimensions of the nucleus.
Question 16.
Why is the ionizing power of α – parties greater than that of γ-rays?
Because α – particles are heavy particles and their speed is comparatively small, so they collide more frequently with atoms of the medium and ionize them.
Question 17.
When a nucleus undergoes alpha decay, is the product atom electrically neutral in beta decay?
No, in alpha decay, the atomic number decreases by 2 hence the atom is left with 2 extra orbital electrons. It, therefore, has a double negative charge. In beta decay, the atom is left with a net
positive charge.
Question 18.
You are given two nuclides [3]^7X and [3]^4Y. Are they the isotopes of the same element? Why?
Yes, because an atomic number of both nuclides is 3.
Question 19.
The variation of the decay rate of two radioactive samples A and B with time is shown in the figure. Which of the two has a greater decay constant? (NCERT Exemplar)
The decay constant of A is greater than that of B but it does not always decay faster than B.
Question 20.
Does the ratio of neutrons to protons in a nucleus increase, decrease, or remain the same after the emission of an alpha particle? (NCERT Exemplar)
The ratio of neutrons to protons in a nucleus increases after the emission of an alpha particle.
Question 21.
Which property of nuclear force explains the approximate constancy of binding energy per nucleon with mass number A for nuclei in the range 30 < A < 170? (CBSE2019C)
The short-range nature of the nuclear force explains the approximate constancy of binding energy per nucleon with mass number A in the range 30 < A < 170.
Question 22.
Draw a graph showing the variation of decay rate with a number of active nuclei. (NCERT Exemplar)
The graph is as shown.
Question 23.
Which sample, A or B, shown in the figure has a shorter mean-life? (NCERT Exemplar)
B has shorter mean life as λ is greater forB.
Question 24.
Why do stable nuclei never have more protons than neutrons? (NCERT Exemplar)
Protons are positively charged and repel one another electrically. This repulsion becomes so great in nuclei with more than 10 protons or so that an excess of neutrons, which produce only attractive
forces, is required for stability.
Question 25.
Why does the process of spontaneous nuclear fission occur in heavy nuclei? (CBSE 2019C)
Because heavy nuclei contain a large number of protons that exert strong repulsive forces on one another.
Nuclei Important Extra Questions Short Answer Type
Question 1.
Draw the curve showing the binding energy/nucleon with a mass number of different nuclei. Briefly state, how nuclear fusion and nuclear fission can be explained on the basis of this graph.
The diagram is as shown.
Light nuclei have a small value of binding energy per nucleon, therefore to become more stable they fuse to increase their binding energy per nucleon.
A very heavy nucleus, say A 240, has Lower binding energy per nucLeon compared to that of a nucleus with A = 120. Thus if a nucleus A = 240 breaks into two A = 120 nuclei, nucleons get more tightLy
bound. This implies energy would be released in the process.
Question 2.
Define decay constant for a radioactive sample. Which of the following radiations α, β, and γ rays
(i) are similar to X-rays,
(ii) are easily absorbed by matter, and
(iii) are similar in nature to cathode rays?
The decay constant is defined as the reciprocal of that time duration for which the number of nuclei of the radioactive sample decays to 1 / e or 37 % of its original value.
(i) Gamma
(ii) Alpha
(iii) Beta
Question 3.
State the law of radioactive decay.
Plot a graph showing the number of undecayed nuclei as a function of time (t) for a given radioactive sample having a half-life T[1/2].
Depict In the plot the number of undecayed nuclei at (i) t = 3T[1/2] and (ii) t = 5 T[1/2] (CBSE Delhi 2011)
The number of nuclei disintegrating per second is proportional to the number of nuclei present at the time of disintegration and is independent of alt physical conditions like temperature, pressure,
humidity, chemical composition, etc.
The plot is as shown.
Question 4.
Draw a plot of the potential energy of a pair of nucleons as a function of their separations. Mark the regions where the nuclear force is (i) attractive and (ii) repulsive. Write any two
characteristic features of nuclear forces. (CBSE AI 2012)
For r > r[0] (attraction), For r < r[o] (repulsion)
1. Strong attractive force (stronger than the repulsive electric force between the protons)
2. Are short-range forces.
Question 5.
(a) Write the relation for binding energy (BE) (in MeV) of a nucleus of mass [Z]^AM atomic number (Z) and mass number (A) in terms of the masses of its constituents – neutrons and protons.
The required expression is
ΔE = (Zm[p] + (A – Z)m[n] – M) × 931 MeV
(b) Draw a plot of BE/A versus mass number A for 2 ≤ A ≤ 170. Use this graph to explain the release of energy in the process of nuclear fusion of two light nuclei. (CBSE Delhi 2014C)
Since the binding energy of the smaller nuclei like hydrogen is less, therefore they fuse together to form helium in order to increase their binding energy per nucleon and become stable. This means
that the final system is more tightly bound than the initial system. Again energy would be released in such a process of fusion.
Question 6.
If both the number of neutrons and the number of protons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain. (CBSE
We know that the binding energy of a nucleus gives a negative contribution to the mass of the nucleus (mass defect). Now, since proton number and neutron number are conserved in a nuclear reaction
the total rest mass of neutrons and protons is the same on either side of a reaction. But the total binding energy of nuclei on the left side need not be the same as that on the right-hand side.
The difference in these binding energies appears as the energy released or absorbed in a nuclear reaction. Since binding energy contributes to mass, we say that the difference in the total mass of
nuclei on the two sides gets converted into energy or vice-versa.
Question 7.
State two properties of nuclear forces. Write the relation between half-life and decay constant of a radioactive nucleus. (CBSE AI 2017C)
1. They are saturated forces.
2. They are charge-independent.
The required relation is
T = \(\frac{\ln 2}{\lambda}=\frac{2.303 \log 2}{\lambda}=\frac{0.693}{\lambda}\)
Question 8.
(a) Draw a graph showing the variation of binding energy per nucleon (BE/A) vs mass number A for the nuclei in 20 ≤ A ≤ 170.
Since the binding energy of the smaller nuclei like hydrogen is less, therefore they fuse together to form helium in order to increase their binding energy per nucleon and become stable. This means
that the final system is more tightly bound than the initial system. Again energy would be released in such a process of fusion.
(b) A nucleus of mass number 240 and having binding energy/nucleon 7.6 MeV splits into two fragments Y, 1 of mass numbers 110 and 130 respectively. If the binding energy/ nucleon of Y, 1 is equal to
8.5 MeV each, calculate the energy released in the nuclear reaction. (CBSE Al 2017C)
Energy released per fission
= (110 + 130) × 8.5 – 240 × 7.6
= 240 × (8.5 – 7.6) MeV
= 240 × 0.9
= 216.0 MeV
Question 9.
Explain with the help of an example, whether the neutron-proton ratio in a nucleus increases or decreases due to beta decay.
Consider the following decay
Number of neutrons before beta decay
= 234-90 = 144
Number of neutrons after beta decay
= 234-91 =143
Number of protons before beta decay
= 90
Number of protons after beta decay
= 91
Neutron-proton ratio before beta decay
= \(\frac{144}{90}\) = 1.6
Neutron-proton ratio after beta decay
= \(\frac{143}{91}\) = 1.57
Thus neutron-proton ratio decreases during beta decay.
Question 10.
How is the size of a nucleus experimentally determined? Write the relation between the radius and mass number of the nucleus. Show that the density of the nucleus is independent of its mass number.
(CBSE Delhi 2011C)
The size of the nucleus can be determined by the Rutherford experiments on alpha particles scattering. The distance of the nearest approach is approximately the size of the nucleus. Here it is
assumed that only coulomb repulsive force caused scattering. With alpha rays of 5.5 MeV, the size of the nucleus was found to be less than 4 × 10^-14 m. By doing scattering experiments with fast
electrons bombarding targets of different elements, the size of the nuclei of various elements determined accurately.
The required relation is
R = R[o]A[1]/3, where R[o] = 1.2 × 10^-15 m
The density of a nucleus of mass number A and radius R is given by
which is independent of the mass number A.
Question 11.
(a) What characteristic property of nuclear force explains the constancy of binding energy per nucleon (BE/A) in the range of mass number ‘A’ lying 30 < A < 170?
The nuclear force between two nucleons falls rapidly to zero as their distance is more than a few femtometres. This leads to the saturation of forces in a medium or a large-sized nucleus, i.e. nuclei
for which A is 30 < A < 170, which is the reason for the constancy of the binding energy per nucleon.
(b) Show that the density of nucleus over a wide range of nuclei is constant- independent of mass number A. (CBSE AI 2012)
The size of the nucleus can be determined by the Rutherford experiments on alpha particles scattering. The distance of the nearest approach is approximately the size of the nucleus. Here it is
assumed that only coulomb repulsive force caused scattering. With alpha rays of 5.5 MeV, the size of the nucleus was found to be less than 4 x 10^-14 m. By doing scattering experiments with fast
electrons bombarding targets of different elements, the size of the nuclei of various elements determined accurately.
The required relation is
R = R[o]A[1]/3, where R[o] = 1.2 × 10^-15 m
The density of a nucleus of mass number A and radius R is given by
which is independent of the mass number A.
Question 12.
A radioactive nucleus ‘A’ decays as given below:
If the mass number and atomic number of A[1] are 180 and 73 respectively, find the mass number and an atomic number of A and A[2].
For A : Z = 72 and A = 180
For A[2]: Z = 71 and A = 176
Question 13.
The sequence of stepwise decay of a radioactive nucleus is[2] are 176 and 71 respectively, what are the corresponding values of D and D[3]? Justify your answer in each case.
For D: A = 180, Z = 72
For D[3]: A = 172, Z = 69
During alpha decay mass number decreases by 4 and the atomic number decreases by 2, while during beta decay the mass number remains the same, and the atomic number increases by 1.
Question 14.
Write symbolically the nuclear β^+ decay process of [6]^11C. Is the decayed product X an isotope or isobar of[6]^11C?
Given the mass values m ([6]^11C) = 11.011434 u and m (X) = 11.009305 u. (CBSE AI 2015)
Estimate the Q – value in this process.
The required equation is
X is an isobar
Mass defect = m(C) – m(X)
= (11.011434- 11.009305) u = 0.002129 u
Therefore Q = Δm × 931.5 MeV
= 0.002129 × 931.5 = 1.98 MeV
Question 15.
Two radioactive samples, X, Y have the same number of atoms at t = 0. Their half¬lives are 3 h and 4 h respectively. Compare the rates of disintegration of the two nuclei after 12 hours. (CBSE AI
Let N0 be the nuclei present in X and Y at t = 0. Given T[x] = 3 h and T[y] = 4 h, t = 12 h.
The number of nuclei present in X and Y after 12 hours is
Question 16.
A radioactive sample has the activity of 10,000 disintegrations per second after 20 hours. After the next 10 hours, its activity reduces to 5,000 dis. sec^-1. Find out its half-life and initial
activity. (CBSE Delhi 2017C)
Since activity reduces to half in 10 hours from 10000 dis. sec^-1 to 5000 dis. sec^-1, therefore half-life of the sample will be 10 years.
Question 17.
Why is the energy of the beta particles emitted during beta decay continuous?
The phenomenon of beta decay arises due to the conversion of a neutron in the nucleus into a proton, electron, and an anti-neutrino. Because the energy available in beta decay is shared by the
electron and the anti-neutrino in all possible ratios as they come out of the nucleus, therefore the beta ray energy spectrum is continuous in nature.
Question 18.
Explain, how radioactive nuclei can emit β-particles even though atomic nuclei do not contain these particles. Hence explain why the mass number of a radioactive nuclide does not change during
Beta-particles (or electrons) as such are not present inside a nucleus. However, in the case of a radioactive nuclide, sometimes a neutron decays into a proton, an electron, and an antineutrino as
given by the following equation:
where mass and charge of antineutrino particle is zero. Out of the particles formed, the proton remains within the nucleus itself but electron along with antineutrino comes out of the nucleus. It is
this electron that is being emitted as a beta-particle.
As in the process of β-emission, one proton is produced in the nucleus at the expense of a neutron and the mass number of both is the same, hence the mass number of the nuclide remains unchanged
during p-decay.
Question 19.
Consider a radioactive nucleus A which decays to a stable nucleus C through the following sequence: A → B → C. Here B is an intermediate nucleus that is also radioactive. Considering that there are N
[o] atoms of A initially, plot the graph showing the variation of the number of atoms of A and B versus time. (NCERT Exemplar)
At t = 0, N[A] = N[o] while N[B] = 0. As time increases, N[A] falls off exponentially, while the number of atoms of B increases, becomes maximum, and finally decays to zero ∞ (following exponential
decay law).
Hence the graph is as shown.
Nuclei Important Extra Questions Long Answer Type
Question 1.
Define the terms: half-life period and decay constant of a radioactive sample. Derive the relation between these terms.
The half-life is the time required for the number of radioactive nuclei to decrease to one-half the original number.
The decay constant is defined as the reciprocal of that time duration for which the number of nuclei of the radioactive sample decays to 1 / e or 37% of its original value.
To get the relation for half life T and decay constant λ we set N = \(\frac{N_{0}}{2}\) and t = T in the equation N = No e^-λt, obtaining \(\frac{1}{2}\) = e^-λt
Taking the logarithm of both sides and solving for T we have
T = \(\frac{\ln 2}{\lambda}=\frac{2.303 \log 2}{\lambda}=\frac{0.693}{\lambda}\)
Question 2.
(a) Draw a graph showing the variation of the potential energy of a pair of nucleons as a function of their separation. Indicate the regions in which nuclear force is (i) attractive, and (ii)
Graph showing the variation of potential energy U (in MeV) of a pair of nucleons as a function of their separation r (in fm) is shown here.
1. In the graph region AD (r > r[o]) shows the region where nuclear force is strongly attractive.
2. The region DE (r < r[o]) shows the region where nuclear force is strongly repulsive.
(b) Write two characteristic features of nuclear force which distinguish it from the Coulomb force.
Two characteristics of nuclear forces which distinguish it from Coulomb’s force are
• It is charge Independent.
• It is an extremely short-range force and does not obey the inverse square law.
Question 3.
Prove that the Instantaneous rate of change of the activity of a radioactive substance is Inversely proportional to the square of Its half-life.
The activity of a radioactive substance is
A = \(\frac{dN}{dt}\).
We know that the number of nuclei of a radioactive substance Left behind after time t is given by N = N[o]e^-λt
Differentiating the above relation with respect to time we have
A = \(\frac{d N}{d t}=\frac{d}{d t}\)N[o]e^-λt = – N[o]λe^-λt
Differentiating the above equation with respect to time we have
Therefore the Instantaneous rate of change of the activity of a radioactive substance is inversely proportional to the square of Its half Life.
Question 4.
(a) Deduce the expression N = N[o]e^-λt the law of radioactive decay.
(b) (i) Write symbolically the process expressing the β^+ decay of, [11]^22Na Also write the basic nuclear process underlying this decay.
(ii) Is the nucleus formed in the decay of the nucleus [11]^22Na an Isotope or isobar? (CBSE Delhi 2014)
(a) Let N0 be the number of nuclei present in a freshly separated sample of a radioactive substance. Let after time t the number of nuclei left behind be N. Let dN number of nuclei disintegrate in a
small time interval dt. Then by the – decay law,
where λ is a constant of proportionality.
Question 5.
(a) Complete the following nuclear reactions:
(b) Write the basic process Involved in nuclei responsible for (i) β^– and (ii) β^+ decay.
The basic nuclear process underlying β^– decay is the conversion of the neutron to proton
n → p + e^– + v^–
while for v+ decay, it is the conversion of a proton into a neutron
p → n + e^+ + v
(c) Why is it found experimentally difficult to detect neutrinos? (CBSE AI 2015 C)
Neutrinos are neutral particles with very small (possibly, even zero) mass compared to electrons. They have only weak interaction with other particles. They are, therefore, very difficult to detect,
since they can penetrate a large quantity of matter (even earth) without any interaction.
Question 6.
(a) Explain the processes of nuclear fission and nuclear fusion by using the plot of binding energy per nucleon (B.E./A) versus the mass number A.
For the graph
From the plot, we note that
• During nuclear fission: A heavy nucleus in the larger mass region (A > 200) breaks into two middle-level nuclei, resulting in an increase in B.E./ nucleon. This results in the release of energy,
• During nuclear fusion: Light nuclei in the lower mass region (A < 20) fuse to form a nucleus having higher B.E. / nucleon. Hence Energy gets released.
(b) A radioactive Isotope has a half-life of 10 years. How long will It take for the activity to reduce to 3.125%? (CBSE AI2018)
3.125% means that the number of nuclei decays to 1/32 of its original value.
Now we know that
N = N0\(\left(\frac{1}{2}\right)^{t / T}\)
Therefore we have
\(\left(\frac{1}{2}\right)^{5}=\left(\frac{1}{2}\right)^{t / T}\)
t = 5T = 5 × 10 = 50 years
Question 7.
Group the following six nuclides into three pairs of (?) isotones, (ii) isotopes, and (iii) isobars: [6]^12C, [2]^3He, [80]^198Hg, [1]^3H, [79]^197Au [6]^14C. How does the size of the nucleus depend
on its mass number? Hence explain why the density of nuclear matter should be independent of the size of the nucleus.
(a) Isotones: [80]^198Hg, [79]^197Au
(6) Isotopes: [6]^12C , [6]^14C
(c) Isobars: [2]^3He, [1]^3H
The size of a nucleus depends upon its mass number as R = R[0] A^1/3
The nuclear density is given by the expression
The calculations show that the nuclear density is independent of the mass number.
Question 8.
Define the term ‘decay constant’ of a radioactive sample. The rate of disintegration of a given radioactive nucleus is 10,000 disintegrations/s and 5,000 disintegration/s after 20 hr and 30 hr
respectively from start. Calculate the half-life and an initial number of nuclei at t = 0. (CBSE Delhi 2019)
The decay constant of a radioactive element is the reciprocal of the time in which the number of its nuclei reduces to 1 /e of its original number.
We have R = λN
R(20hrs) = 100o0 = λN[20]
R(30hrs) = 5000 = λN[30]
\(\frac{N_{20}}{N_{30}}\) = 2
This means that the number of nuclei, of the given radioactive nucleus, gets halved in a time of (30 – 20) hours = 10 hours
Half-life = 10 hours
This means that in 20 hours (= 2 half-Lives), the original number of nuclei must have gone down by a factor of 4.
Hence rate of decay at t = 0
λ N[0] = 4 λ N[20]
R[0] = 4 × 10,000 = 40,000 disintegration per second
Question 9.
(a) Write the relation between half-life and an average life of a radioactive nucleus.
The relation is τ = 1 .44T^1/2
(b) In a given sample two isotopes A and B are initially present in the ratio of 1:2. Their half-lives are 60 years and 30 years respectively. How long will it take so that the sample has these
isotopes in the ratio of 2:1? (CBSE Delhi 2019)
Question 10.
Distinguish between nuclear fission and fusion. Show how in both these processes energy is released. Calculate the energy release in MeV in the deuterium-tritium fusion reaction:
Using the data m(21H) = 2.014102 u,
m([1]^3H) = 3.016949u, m([2]^4He) = 4.002603 u,
m[n] = 1.008665 u,1 u = 931.5 MeV/c^2 (CBSE Delhi 2015)
The distinction is shown in the table below.
┃Nuclear Fission │Nuclear Fusion ┃
┃1. It is the splitting of a heavy nucleus into two or tighter unstable nuclei.│1. It is the combining of two light nuclei into a heavier nucleus. ┃
┃2. It may or may not be a chain reaction. │2. It is always a chain reaction. ┃
┃3. It is independent of temperature. │3. It is temperature-dependent. ┃
┃4. It can be controlled. │4. It can’t be controlled. ┃
┃5. Tremendous amount of energy is released. │5. Energy released per unit mass is seven times the energy released during fission.┃
┃6. By-products are harmful. │6. By-products are not harmful. ┃
┃7. Example of reaction – The atom bomb. │7. Example of reaction – Reaction in stars, hydrogen born ┃
In both reactions, there is a mass defect that is converted into energy.
Now energy released in the reaction
Question 11.
(a) Draw a plot showing the variation of the potential energy of a pair of nucleons as a function of their separation. Mark the regions where the nuclear force is (a) attractive and (b) repulsive.
For r > r[o], the force is attractive For r < r[o], the force is repulsive
(b) In the nuclear reaction
determine the values of a and b. (CBSE Delhi 2018 C)
We have,
1 + 235 = a + 94 + 2 × 1
∴ a = 236 – 96 = 140
0 + 92 = 54+ 6 + 2 × 0
∴ b = 92 – 54 = 38
Question 12.
Binding energy per nucleon versus mass number curve is as shown. [Z]^AS, [Z1]^A1w, [Z2]^A2X, and [Z3]^A3Y, are four nuclei indicated on the curve.
Based on the graph:
(а) Arrange X, W, and S in the increasing order of stability.
(a) S, W, and X
(b) Write the relation between the relevant A and Z values for the following nuclear reaction. S → X + W
The equation is
Z = Z[1] + Z[2] and A = A[1] + A[2]
(c) Explain why binding energy for heavy nuclei is low. (CBSE Sample Paper 2018-19)
Reason for low binding energy: For heavier nuclei, the Coulomb repulsive force between protons increases considerably and offsets the attractive effects of the nuclear forces. This can result in such
nuclei being unstable.
Question 13.
(a) Derive the law of radioactive decay,
viz. N = Noe-λt
Let N0 be the number of nuclei present in a freshly separated sample of a radioactive substance. Let after time t the number of nuclei left behind be N. Let dN number of nuclei disintegrate in a
small time interval dt. Then by the – decay law,
where λ is a constant of proportionality.
(b) Explain, giving necessary reactions, how energy is released during
(i) fission
Nuclear Fission: It is a process in which a heavy nucleus splits up into two Lighter nucLei of nearly equal masses. It is found that the sum of the masses of the product nuclei and particles is less
than the sum of the masses of the reactants, i.e. there is some mass defect. This mass defect appears as energy. One such fission reaction is given below:
The Q value of the above reaction is about 200 MeV. The sum of the masses of Ba, Kr, and 3 neutrons is less than the sum of the masses of U and one neutron.
(ii) fusion
Nuclear Fusion: It is the process in which two light nuclei combine together to form a heavy nucleus. For fusion very high temperature of l is required. One such fusion reaction is given below:
The Q value of this nuclear reaction is 24 MeV. It is the energy equivalent of the mass defect in the above reaction. The energy released in fusion is much less than in fission but the energy
released per unit mass infusion is much greater than that released in fission.
Question 14.
(a) Distinguish between isotopes and isobars, giving one example for each.
(b) Why is the mass of a nucleus always less than the sum of the masses of its constituents? Write one example to justify your answer.
(a) Classify the following six nuclides into (i) isotones, (ii) Isotopes, and (iii) isobars: (CBSEAI2019)
(b) How does the size of a nucleus depend on its mass number? Hence explain why the density of nuclear matter should be independent of the size of the nucleus.
(a) Isotopes have the same atomic number while isobars have the same mass number
Examples of isotopes [6]^12C, [6]^14C
Examples of isobars [2]^3He, [1]^3H
(b) Mass of a nucleus is less than its constituents because in the bound state some mass is converted into binding energy which is energy equivalent of mass defect e.g., the mass of 1860 nucleus is
less than the sum of masses of 8 protons and 8 neutrons
(a) Isotones: [80]^198Hg, [79]^197Au
(6) Isotopes: [6]^12C , [6]^14C
(c) Isobars: [2]^3He, [1]^3H
(b) The radius of the nucleus is given by
R = R[o]A^1/3
Volume of the nucleus \(\frac{4}{3}\)πR^3 = \(\frac{4}{3}\)πR[o]^3 A
If m is the average mass of the nucleon then the mass of the nucleus M = mA
Hence nuclear density
Which is independent of the A i.e., the size of the nucleus.
Numerical Problems:
• Radius of the nucleus R = R[o]A^1/3
• Mass defect, Δm = Z m[p] + (A – Z) m[n]– M
• Energy released ΔE = Δm × 931 MeV
ΔE = [Z m[p] + (A – Z) m[n] – M] × 931 MeV
• Binding energy per nucleon BE/N = ΔE/A
• Relation between original nuclei (N) and nuclei left (N[o]) after time t N = N[o]e^-λt
• Relation between decay constant (λ) and half-life (T) = \(\frac{\ln 2}{\lambda}=\frac{2.303 \log 2}{\lambda}=\frac{0.693}{\lambda}\)
• Half-life is also given by the expression N = N[o]\(\left(\frac{1}{2}\right)^{n}\) where n = t/T
• The average life is given by
τ = \(\frac{1}{\lambda}=\frac{T}{\ln 2}=\frac{T}{0.693}\) = 1.44 T
• Activity is given by A = -λN = \(\frac{0.693N}{T}\)
Question 1.
Calculate the binding energy per nucleon of Fe^56[26] Given m[Fe] = 55.934939 u, m[n] = 1.008665 u and m[p] = 1.007825 u
Number of protons Z = 26
Number of neutrons (A – Z) = 30
Now mass defect is given by
Δm = Z m[p] + (A – Z)m[n] – M
Δm = 26 × 1.007825 + 30 × 1.008665 – 55.934939
= 0.528461 u
Therefore binding energy
BE = Δm × 931 MeV = 0.528461 × 931
= 491.99 MeV
BE/nucleon = 491.99/56 = 8.785 MeV
Question 2.
The activity of a radioactive element drops to one-sixteenth of its initial value in 32 years. Find the mean life of the sample.
32/T = 4 or 7 = 32 / 4 = 8 years.
Therefore mean life of the sample is τ = 1.44 7 = 1.44 × 8 = 11.52 years.
Question 3.
A radioactive sample contains 2.2 mg of pure 116C which has a half-life period of 1224 seconds. Calculate (i) the number of atoms present initially and (ii) the activity when 5 pg of the sample will
be left.
Mass of sample = 2.2 pg
Now 11 g of the sample contains 6.023 × 10^23 nuclei, therefore the number of nuclei in 2.2 mg = 2.2 × 10^-3 g are
Question 4.
The half-life of 238 92U is 4.5 × 10^9 years. Calculate the activity of 1 g sample of [92]^238U.
Given T = 4.5 × 10^9 years.
Number of nuclei of U in 1 g
= N = \(\frac{6.023 \times 10^{23}}{238}\) = 2.5 × 10^21
Therefore activity
Question 5.
The decay constant for a given radioactive sample is 0.3456 per day. What percentage of this sample will get decayed in a period of 4 days?
Given λ = 0.3456 day^-1
T[1/2] = 0.693/λ = 0. 693/ 0.3456 = 2.894 days, t = 4 days.
Let N be the mass left behind, then N = N[o]e^-λt
N = N[o] e^-0 3456 × 4
N = N[0] e^-1 3824 = N[o] × 0.25
Therefore the percentage of undecayed is
Question 6.
It is observed that only 6.25 % of a given radioactive sample is left undecayed after a period of 16 days. What is the decay constant of this sample per day?
Given N/N[o] = 6.25 %, t = 16 days, λ = ?
16/ T = 4 or T = 4 days.
Therefore λ = 1/T = 1/4 = 0.25 day^-1
Question 7.
A radioactive substance decays to 1/32^th of its initial value in 25 days. Calculate its half-life.
Given t = 25 days, N = N[o] / 32, using
25/7= 5 or T= 25 / 5 = 5 days.
Question 8.
The half-life of a radioactive sample is 30 s.
(i) the decay constant, and
Given T[1]/2 = 30 s, N = 3N[o] / 4, λ = ?, t = ?
(i) Decay constant
λ = \(\frac{0.693}{T_{1 / 2}}=\frac{0.693}{30}\) = 0.0231 s^-1
(ii) time taken for the sample to decay to 3/4 th of its initial value.
Using N = N[o]e^-λt we have
Question 9.
The half-life of 14 6C is 5700 years. What does it mean?
Two radioactive nuclei X and Y initially contain an equal number of atoms. Their half-lives are 1 hour and 2 hours respectively. Calculate the ratio of their rates of disintegration after 2 hours.
It means that in 5700 years the number of nuclei of carbon decay to half their original value.
Given N[ox] = N[oY], T[X] = 1 h, T[Y] = 2 h, therefore
\(\frac{\lambda_{X}}{\lambda_{Y}}=\frac{2}{1}\) = 2
Now after 2 hours X will reduce to one- fourth and Y will reduce to half their original value.
If activities at t = 2 h are R[x] and R[y] respectively, then
Thus their rate of disintegration after 2 hours is the same.
Question 10.
A star converts all its hydrogen to helium achieving 100% helium composition. It then converts helium to carbon via the reaction.
The mass of the star is 5 × 10^32 kg and it generates energy at the rate of 5 × 10^30 watt. How long will it take to convert all the helium to carbon at this rate?
As 4 × 10^-3 kg of He consists of 6.023 × 10^23 He nuclei so 5 × 10^32 kg He will contain
\(\frac{6.023 \times 10^{23} \times 5 \times 10^{32}}{4 \times 10^{-3}}\) = 7.5 × 10^58 nuclei
Now three nuclei of helium produce 7.27 × 1.6 × 10^-13 J of energy
So all nuclei in the star will produce
E = \(\frac{7.27 \times 1.6 \times 10^{-13}}{3}\) × 7.5 × 10^58
= 2.9 × 10^46 J
As power generated is P = 5 × 10^30 W, therefore time taken to convert all He nuclei into carbon is
t = \(\frac{E}{P}=\frac{2.9 \times 10^{46}}{5 \times 10^{30}}\) = 5.84 × 10^15 s
1.85 × 10^8 years
Question 11.
Radioactive material is reduced to (1/16)^th of its original amount in 4 days. How much material should one begin with so that 4 × 10^-3 kg of the material is left after 6 days?
N = N[o] / 16, t = 4 days,
N = 4 × 10^-3 kg,
t = 6 days
To calculate half-life of the material we have
Now using the expression 4 × 10^-3 = No\(\left(\frac{1}{2}\right)^{6 / 1}\)
Solving we have N[o] = 0.256 kg
Question 12.
Two different radioactive elements with half-lives T[1] and T[2] have N[1] and N[2] (undecayed) atoms respectively present at a given instant. Determine the ratio of their activities at this instant.
The activity of a radioactive sample is given by the relation
A = – λN
Therefore the ratio of activity of these two radioactive elements is
\(\frac{A_{1}}{A_{2}}=\frac{-\lambda_{1} N_{1}}{-\lambda_{2} N_{2}}=\frac{T_{2} N_{1}}{T_{1} N_{2}}\)
Question 13.
Given the mass of the iron nucleus as 55.85 u and A = 56. Find the nuclear density? (NCERT)
Given m[Fe] = 55.85 u = 9.27 × 10^-26kg
The density of matter in neutron stars (an astrophysical object) is comparable to this density. This shows that matter in these Neutron stars has been compressed to such an extent that they resemble
a big nucleus.
Question 14.
We are given the following atomic masses: [92]^238U = 238.05079 u, [2]^4He = 4.00260 u, [90]^234Th = 234.04363 u [1]^1H = 1.00783 u, [91]^237Pa =237.05121 u Here the symbol Pa is for the element
protactinium (Z = 91). (a) Calculate the energy released during the alpha decay of [92]^238U. (b) Show that cannot spontaneously emit a proton. (NCERT)
(i) The alpha decay of [92]^238Uis given by
The energy released in this process is given by
Q= (M[u] – M[Th] – M[He]) × 931.5 MeV
Substituting the atomic masses as given in the data we find that
Q = (238.05079 – 234.04363 – 4.00260) × 931.5 MeV ⇒ Q = 4.25 MeV.
(ii) If 29®U spontaneously emits a proton, the decay process would be
The Q for this process to happen is Q = (M[u] – M[pa] – M[H]) × 931.5 MeV
Q = (238.05079 – 237.05121 – 1.00783) × 931.5 MeV ⇒ Q = – 7.68 MeV
Thus the Q of the process is negative and therefore it cannot proceed spontaneously. We will have to supply energy of 7.68 MeV to the [92]^238U nucleus to make it emit a proton.
Question 15.
The half-life of 90Sr is 28 years. What is the disintegration rate of 15 mg of this isotope? (NCERT)
Given T[1/2] = 28 years, m = 15 mg
Now the rate of disintegration is given by | {"url":"https://www.learninsta.com/class-12-physics-important-questions-chapter-13/","timestamp":"2024-11-04T14:46:46Z","content_type":"text/html","content_length":"117139","record_id":"<urn:uuid:650262fd-9d07-489c-805d-f94364484a2b>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00787.warc.gz"} |
Consider a real-life system havi
No. 1. 2. 3. 4. 5. 6. 7. 8. 9. Tasks Consider a real-life system having at least a 3rd order, type-0 and over-damped open-loop response; get the mathematical model of the system explaining its
dynamics. G(s) = 1/(s+A)(s+B)(s+C) A 9, B 8, C = 3 so, G(s) = 1/(s+9)(s+8)(s+3) Check the stability of this system. Considering a step input and certain initial conditions on input and output,
compute and plot the open-loop natural and forced components, and the total output. Compute and plot the closed-loop unit step response (unity feedback) without initial conditions; analyze this
closed-loop response by finding out the closed-loop poles (stability), and computing the unity feedback error co-efficient and corresponding steady-state error. Identify the issues (problems) with
the closed-loop performance considering the analyses done in (4.) above. Give the different alternative designs for PID compensation. Design a suitable PID compensations using root locus technique
following the detailed procedures Compare the performance of these both designs with the one found in (4.) above; comment and propose the possible corrections in any one of these designs. Produce the
MATLAB® codes and plots for steps (3.), (4.) and (7.) above. And submit the full report with the dedicated tasks. | {"url":"https://tutorbin.com/questions-and-answers/no-1-2-3-4-5-6-7-8-9-tasks-consider-a-real-life-system-having-at-least-a-3rd-order-type-0-and-over-damped-open-loop","timestamp":"2024-11-05T22:20:24Z","content_type":"text/html","content_length":"62184","record_id":"<urn:uuid:d6612389-79ac-4e6d-8729-84e6704c4fe1>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00701.warc.gz"} |
What is the diameter of a square if it is 15cm x 15cm?
The 엠카지노 https://safearea79.com to write out the difference of 9 and a number is n-9.
The way to write out the difference of 9 and a number is n-9.
tuffy rhodes
The simplest form of the improper fraction 5/4 is 1-1/4.
The way to write out the difference of 9 and a number is n-9.
An inequality.
10 more than y
The way to write out the difference of 9 and a number is n-9.
The size of the koi will depend on many factors such as what type of koi it is, how much and what you feed it and the size of the aquarium or pond you keep it in.
There is no simple answer to the question because the children’s genders are not independent events. They depend on the parents’ ages and their genes. Unfortunately there is no readily available
research into the genders of seven or more children to establish the experimental probability for such an outcome.
The way to write out the difference of 9 and a number is n-9.
The GCF is 3.
The way to write out the difference of 9 and a number is n-9.
The New Deal.
George Best.
WikiAnswers does not divulge private or personal information, such as telephone numbers, email addresses or home addresses, for individuals.
because they contain large amount of iron oxides
1/4 times 2/3 = (1 times 2)/(4 times 3) = 4/6 = 2/3
10 more than y
Take initiatives, take actions or call for taking an action.
There is no simple answer to the question because the children’s genders are not independent events. They depend on the parents’ ages and their genes. Unfortunately there is no readily available
research into the genders of seven or more children to establish the experimental probability for such an outcome.
His favorite color was blue,.
The way to write out the difference of 9 and a number is n-9.
The way to write out the difference of 9 and a number is n-9.
No it’s spelled soldier
‘The industry is up for the challenge but we need all stakeholders, including government, charge point providers and energy companies, to match manufacturers’ commitment by providing the competitive
incentives and infrastructure that assures a zero-emission future.’
10 more than y
The simplest form of the improper fraction 5/4 is 1-1/4.
The way to write out the difference of 9 and a number is n-9.
two ,4 and 6
no not that i am aware of because i think they related to coper heads ( from us )
How about 22 + 32
The way to write out the difference of 9 and a number is n-9.
10 more than y
2 Varsity years
The way to write out the difference of 9 and a number is n-9.
The way to write out the difference of 9 and a number is n-9.
It is length * width * height.
9.2 = 92/10 = 46/5
Does Lisa Kelly on ice road trucker smoke
The way to write out the difference of 9 and a number is n-9.
The way to write out the difference of 9 and a number is n-9.
10 more than y
Before meeting the heir to the Spanish throne, Letizia Ortiz Rocasolano whose father Jesús José Ortiz Álvarez and stepmother Ana Togores are both journalists, enjoyed a lengthy career in TV and
*pregnant? No.
10 more than y
That is decidedly dangerous as the bleach can cause serious skin damage to this sensitive area
The way to write out the difference of 9 and a number is n-9.
she attended Indian springs high school in los angeles
In my opion its soccer.All sports are cool.Maybe try Basketball or vollyball.
27 hours is 1 day and 3 hours.
The way to write out the difference of 9 and a number is n-9.
No. The sum of any odd number of odd numbers will be an odd number.
They suggest this number won’t be enough to match demand, are worried that there will be a serious lack of fast chargers and raised issue with the postcode lottery of charging solutions around the
UK, with areas such as Northern Ireland and the North West of England having far fewer devices than other parts of the country.
The way to write out the difference of 9 and a number is n-9.
10 more than y
There are no diameters for squares; let us just make that clear. Diameter only applies for circles (Line that goes through centre of circle and touches edges). However, diameters in squares would be
simply length(but went through centre). Therefore it could be anywhere between 15-~21.2132. (Straight length/width or diagonal or anywhere else.)
Yes, when they are both 90o and the parallelogram is usually referred to as a rectangle.
The very best are, but on average no
The way to write out the difference of 9 and a number is n-9.
The way to write out the difference of 9 and a number is n-9.
10 more than y
These figures will be music to the ears of Boris Johnson, who continues to press ahead with plans to ban the sale of all new petrol and diesel cars by 2030 – and hybrids from 2035 – in a bid to
achieve ambitious net-zero targets.
YES!!! Because vineger is a solution of ethanoic(acetic) acid. | {"url":"https://pitfmb2024.membership-afismi.org/2024/03/19/what-is-the-diameter-of-a-square-if-it-is-15cm-x-15cm/","timestamp":"2024-11-05T13:31:58Z","content_type":"text/html","content_length":"44458","record_id":"<urn:uuid:dd81fcd8-a7a6-446f-a7e2-81cb88d82c3e>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00003.warc.gz"} |
Top 4 Repositories on GitHub to Learn Pandas
Top 4 Repositories on GitHub to Learn Pandas
Some of the most popular repositories to brush up on Pandas for beginners and experts alike.
You can find lots of Pandas in Github!
Everyone knows what Github is.
If you’re a newbie like me, you might still be afraid of touching it. While I haven’t really progressed past git commit + git push, I do know that you can use Github as more than a version control
tool for your projects.
In addition to the open-source projects that anyone can commit to, Github also has countless resources you can use as learning materials.
While following an online course can be great, sometimes having extra practice can help you better retain what you previously learned. The popular sites “Codewars” and “Codekata” are one way to get
extra practice every day, as you can select a language of your choice and solve as many problems as you’d like.
For those of you specifically searching for Pandas practice, you can benefit from this list of the Top 4 Repositories on GitHub for Pandas! There’s a repository for every level, whether you’ve just
gotten started with Pandas or if you’re already looking to bring your skills from basic to advanced. I’ve included the ones with the most forks as a measure of popularity.
Pandas Exercises — All Topics (4k Forks)
Pandas Github repository from guipsamora
This repository has 11 different sections with exercises from getting data into a DataFrame to creating advanced visualizations. Each folder has multiple data sets, which all have different
You can download the IPYNB files to open up the Jupyter notebooks and try out the exercises for yourself. There are empty cells below each question, so you can input your code and then check your
answers by looking at the “Exercise_with_Solution.ipynb” file.
There are a total of 27 notebooks for you to go through, so this is definitely a comprehensive resource. Even if you’re already familiar with Pandas, it’s worth going through the “Getting and
knowing” section, because you may find functions like .describe(include=all) and .nunique() that you haven’t seen before.
There’s also a link to videos of data scientists going through all the notebooks, so if you’d prefer to watch a walkthrough of the solutions instead of just reading them, you can check that out here.
Pandas Videos — All Topics w/ Videos (1.2k Forks)
Pandas GitHub repository from justmarkham
This repository contains Jupyter notebooks with code from a video series that goes through a lot of different Pandas functionality. The author goes through how to solve a question using a real
dataset (has been posted online by the author and is included in the notebook).
Ideally, you would have a Jupyter notebook open and follow along with the video. Then, once you’ve finished with the video and gone through all the code, you can use the notebooks included in the
repository as an answer sheet. There are also some additional footnotes in the notebooks that may help clarify the output of certain cells.
This list of videos and associated notebooks is very comprehensive, so odds are if you have a Pandas related question you will find a walkthrough here. There are simple, niche questions like “How do
I sort a Pandas DataFrame or Series” and broad, complex ones like “How do I use pandas with sci-kit learn to create Kaggle submissions”.
100 Pandas Puzzles (1k forks)
Pandas GitHub repository from ajcr
This repository has just one Jupyter notebook for you to download with all the exercises. Each question has a cell below where you can fill in your code, which you can check against the relevant cell
in the solutions notebook.
The notebook is divided into different sections like “Importing Pandas”, “DataFrame basics”, “Series and DatetimeIndex” and so on. You’ll find that most questions can be solved with just a couple
lines, so ideally you won’t have giant blocks of code for a single question.
There’s also a cool “Minesweeper” section, where:
we’ll make a DataFrame that contains the necessary data for a game of Minesweeper: coordinates of the squares, whether the square contains a mine and the number of mines found on adjacent
It’s categorized as “medium to hard” in difficulty, but if you’ve gone through the previous exercises, you should be able to get through it. I thought it was a fun break from traditional data
analysis, as it forces you to think of how to manipulate a DataFrame in a unique situation.
The author also notes that the list of puzzles isn’t complete, so if you also want to contribute to the list of puzzles, you can submit requests for additional exercises, corrections, and
Pycon 2019 Tutorial — Intermediate Level (180 forks)
Pandas GitHub repository from justmarkham
This repository includes a (very long) notebook with the code discussed in the “Data Science Best Practices with Pandas” video produce by the author. It’s best for intermediate Pandas users, as it
doesn’t include a walkthrough of Pandas basics.
There are eight main sections, which don’t really follow a “tutorial” type format. Instead, the notebook reads like an actual data analysis project, from examining the data to cleaning it to creating
preliminary visualizations to answering specific questions like “Which occupations deliver the funniest TED talks on average?”.
If you’re new to data analysis projects with Python and Pandas, it may be worth going through the whole video to see how someone would approach the different steps of cleaning, exploration, and
analysis. Then, you could apply those best practices on your own projects.
I hope you found this compilation of popular repositories useful! There are plenty of different ways to learn, so definitely give one of these resources on GitHub a go if they fit with your level of
Pandas and style of learning.
If you’re interested in looking at a data analysis type project where I analyzed Medium’s popular page for what kinds of stories are trending, you can check this out:
Have fun with your Pandas learning! | {"url":"https://readmedium.com/top-4-repositories-on-github-to-learn-pandas-1008cb769f77","timestamp":"2024-11-14T23:48:57Z","content_type":"text/html","content_length":"74465","record_id":"<urn:uuid:9abfa045-86cb-4db3-9c60-76f213243161>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00642.warc.gz"} |
Program for Friday, December 8th
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View: session overviewtalk overview
10:30-12:00 Session 18: Modeling languages
A Categorical Approach to Synthetic Chemistry
10:30 ABSTRACT. We introduce a mathematical framework for retrosynthetic analysis, an important research method in synthetic chemistry. Our approach represents molecules and their interaction using
string diagrams in layered props - a recently introduced categorical model for partial explanations in scientific reasoning. Such principled approach allows one to model features currently not
available in automated retrosynthesis tools, such as chirality, reaction environment and protection-deprotection steps.
Closure and Decision Properties for Higher-Dimensional Automata
ABSTRACT. We develop the language theory of higher-dimensional automata (HDAs). We show a pumping lemma which allows us to expose a class of non-regular ipomset languages. We also give an
11:00 example of a regular language with unbounded ambiguity. Then we pass to decision and closure properties of regular languages. We show that inclusion of regular languages is decidable (hence is
emptiness), and that intersections of regular languages are again regular. On the other hand, complements of regular languages are not regular. We introduce a width-bounded complement and show
that width-bounded complements of regular languages are again regular.
Robustness in Metric Spaces over Continuous Quantales and the Hausdorff-Smyth Monad
Eugenio Moggi
ABSTRACT. Generalized metric spaces are obtained by weakening the requirements (e.g., symmetry) on the distance function and by allowing it to take values in structures (e.g., quantales) that
are more general than the set of non-negative real numbers. Quantale-valued metric spaces have gained prominence due to their use in quantitative reasoning on programs/systems, and for defining
various notions of behavioral metrics. We investigate imprecision and robustness in the framework of quantale-valued metric spaces, when the quantale is continuous. In particular, we study the
relation between the robust topology, which captures robustness of analyses, and the Hausdorff-Smyth hemi-metric. To this end, we define a preorder-enriched monad P_S, called the
Hausdorff-Smyth monad, and when Q is a continuous quantale and X is a Q-metric space, we relate the topology induced by the metric on P_S(X) with the robust topology on the powerset P(X)
defined in terms of the metric on X.
14:00-15:30 Session 19: Verification I
Synchronous Agents, Verification, and Blame — A Deontic View
ABSTRACT. A question we can ask of multi-agent systems is whether the agents’ collective interaction satisfies a particular set of objectives or specifications. This goal can be individual or
14:00 collective. When a collaborative goal is not reached, or a specification is violated, a pertinent question is whether any agent is to blame. We give trace semantics to our logic and use it to
define blame assignments for violations. We also provide quantitative semantics to compare different interactions in terms of the required reparations. Finally, we give an automaton
construction for the logic, which we use as the base for model checking and blame analysis.
Store Locally, Prove Globally
ABSTRACT. The use of message-passing process calculi for the verification of distributed algorithms requires support for state-based reasoning that goes beyond their traditional action-based
14:30 style: knowledge about (local) states is at best provided implicitly. Therefore, we propose a dis- tributed process calculus with locations, the units of distribution, which we equip with
explicit state information in the form of memories. On top, we provide a simple formal model for location failure and failure detection such that we can deal with the verification of
fault-tolerant distributed algorithms. We exhibit the use of our calculus by formalizing a simple distributed consensus algorithm and prove its correctness. The proof exploits global invariants
by direct access to the local memories.
Denotational Semantics for Symbolic Execution
ABSTRACT. Symbolic execution is a technique to systematically explore all possible paths through a program. This technique can be formally explained by means of small-step transition systems
15:00 that update symbolic states and compute a precondition corresponding to the taken execution path (called the path condition). To enable swift and robust compositional reasoning, this paper
defines a denotational semantics for symbolic execution. We prove the correspondence between the denotational semantics and both the small-step execution semantics and a concrete semantics. The
denotational semantics is a function defined piecewise using a partitioning of the input space. Each part of the input space is a path condition obtained from symbolic execution, and the
semantics of this part is the corresponding symbolic substitution interpreted as a function on the initial state space. Correctness and completeness of symbolic execution is encapsulated in a
graceful identity of functions. We provide mechanizations in Coq for our main results.
16:00-17:30 Session 20: Verification II
TOOL PAPER: Tessla-ROS-Bridge - Runtime Verification of Robotic Systems
ABSTRACT. Runtime Verification is a formal method to check a run of a system against a specification. To this end, a monitor is generated from the specification checking the system under
16:00 scrutiny. Typically, runtime verification is used for checking exceutions of programs. However, it may equally be well suited for runs of robotic systems; often these are build and controlled
on top the Robot Operating System (ROS). In stream run- time verification the specifications are given as stream transformations. This approach has become popular recently and several stream
runtime verification systems starting from LOLA have emerged. In this paper the TeSSLa-ROS-Bridge is introduced allowing to interact with robotic systems based on ROS via the temporal stream
specification language TeSSLa.
Simplifying process parameters by unfolding algebraic data types
ABSTRACT. Complex abstract data types are often used to facilitate creating concise models of the behavior of realistic systems. However, static analysis techniques that aim to optimize such
16:30 models often consider variables of complex types as a single indivisible unit. The use of complex data types thus negatively affects the optimizations that can be performed. In this paper we
revisit and extend a technique by Groote and Lisser that can be used to replace a single, complex variable by multiple variables of simpler data types, improving the effectiveness of other
static analyzes. We describe the technique in the context of the process algebraic specification language mCRL2, and establish its correctness. We demonstrate using an implementation in the
mCRL2 toolset that it sometimes reduces the size of the underlying state spaces, and it typically reduces the verification times when using symbolic model checking.
Modular Soundness Checking of Feature Model Evolution Plans
ABSTRACT. A software product line (SPL) is a family of closely related software systems which capitalizes on the variability and reusability of the software products and can be formalized by a
feature model. Feature model evolution plans (FMEP) capture the current SPL as well as the planned evolution of the SPL to ensure successful long-term development. As business requirements
17:00 often change, FMEPs should support intermediate update. This modification may cause paradoxes in an FMEP, e.g. a node left without a parent, making the plan impossible to realise. Current tools
exist to validate FMEPs, but require analysis of the entire plan even when a modification affects only small parts of it. There is a need for a method which detects such paradoxes in a more
efficient way.
In this paper, we present a representation for FMEPs, called an interval based feature model (IBFM). This representation enables local validation, which validates only the parts of the plan
affected by the change. We define operations for updating the FMEPs and the preconditions under which they preserve soundness. We show the correctness of the method. | {"url":"https://easychair.org/smart-program/ICTAC2023/2023-12-08.html","timestamp":"2024-11-13T20:09:56Z","content_type":"application/xhtml+xml","content_length":"17262","record_id":"<urn:uuid:2a7b59d3-e248-41cd-83e5-5469ccf450f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00766.warc.gz"} |
Revision history of "Talk:Mathematical analysis"
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How to Cite This Entry:
Mathematical analysis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mathematical_analysis&oldid=31476 | {"url":"https://encyclopediaofmath.org/index.php?title=Talk:Mathematical_analysis&action=history","timestamp":"2024-11-14T17:22:19Z","content_type":"text/html","content_length":"20957","record_id":"<urn:uuid:a3f59aba-b0a6-4a05-aee2-462020ec3c3f>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00584.warc.gz"} |
Calculates Profit and Loss (PnL) per share (or unit of the
ratioPutSpread {bullishTrader} R Documentation
Calculates Profit and Loss (PnL) per share (or unit of the underlying) and Breakeven point at expiration for Ratio Put Spread and draws its graph in the Plots tab.
This strategy consists of a short position in NS close to ATM put options with a strike price X1L, and a long position in NL ITM put options with a strike price X2H, where NL is less than NS.
Typically, NL is equal to 1 and NS is equal to 2, or NL is 2 and NS is 3. This is an income strategy if it is structured as a net credit trade. The trader’s outlook is neutral to bullish
(Kakushadze & Serur, 2018).
hl = 0,
hu = 1.7,
xlab = "Spot Price ($) on Expiration",
ylab = "Profit / Loss [ PnL ] at Expiration ($)",
main = "Ratio Put Spread ",
sub = "bullishTrader / MaheshP Kumar"
ST Spot Price at time T.
X2H Higher Strike Price or eXercise price.
X1L Lower Strike Price or eXercise price.
PX2H Premium received for the sold s at higher Strike.
PX1L Premium paid for the bought s at Lower Strike.
hl lower bound value for setting lower limit of X axis displaying spot price.
hu upper bound value for setting upper limit of X axis displaying spot price.
xlab X axis label.
ylab Y axis label.
main Title of the Graph.
sub Subtitle of the Graph.
According to conceptual details given by Cohen (2015), and a closed form solution provided by Kakushadze and Serur (2018), this method is developed, and the given examples are created, to compute per
share Profit and Loss at expiration and also the Breakeven (BE) point for Ratio Put Spread and draws its graph in the Plots tab.
returns a profit and loss graph of Ratio Put Spread.
MaheshP Kumar, maheshparamjitkumar@gmail.com
Cohen, G. (2015). The Bible of Options Strategies (2nd ed.). Pearson Technology Group.
Kakushadze, Z., & Serur, J. A. (2018, August 17). 151 Trading Strategies. Palgrave Macmillan. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3247865
version 1.0.1 | {"url":"https://search.r-project.org/CRAN/refmans/bullishTrader/html/ratioPutSpread.html","timestamp":"2024-11-14T12:18:24Z","content_type":"text/html","content_length":"4537","record_id":"<urn:uuid:0b3aa93e-9b94-41c6-add2-0b61deeac5d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00013.warc.gz"} |
Ring source/force excitation of axisymmetric shell structure
The formulation of the near field pressures, far field pressures and acoustic particle displacements of a ring source and a ring axial force is presented. These quantities are necessary ingredients
for calculations of far field radiation in the presence of the scattering action of an axisymmetric shell structure. Numerical results for specific spherical and prolate spheroidal shells show that
the former amplifies the radiation, at resonance, much more than does the latter. The programs can be used to predict far field sound radiation from branched shells with or without rib stiffening.
Pub Date:
November 1988
□ Particle Motion;
□ Pressure Distribution;
□ Resonance;
□ Sound Waves;
□ Spherical Shells;
□ Underwater Acoustics;
□ Circles (Geometry);
□ Scatter Propagation;
□ Scattering Coefficients;
□ Spherical Coordinates;
□ Symmetry;
□ Atomic and Molecular Physics | {"url":"https://ui.adsabs.harvard.edu/abs/1988rsfe.rept.....J/abstract","timestamp":"2024-11-12T17:41:48Z","content_type":"text/html","content_length":"34227","record_id":"<urn:uuid:cc0f6015-5dbd-40cd-9112-05d54f60679b>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00728.warc.gz"} |
Évariste Galois – Wikipedia
2023-11-27 14:19:21
French mathematician (1811–1832)
Évariste Galois (;^[1] French: [evaʁist ɡalwa]; 25 October 1811 – 31 Might 1832) was a French mathematician and political activist. Whereas nonetheless in his teenagers, he was capable of decide a
necessary and sufficient condition for a polynomial to be solvable by radicals, thereby fixing an issue that had been open for 350 years. His work laid the foundations for Galois theory and group
theory,^[2] two main branches of abstract algebra.
Galois was a staunch republican and was closely concerned within the political turmoil that surrounded the French Revolution of 1830. On account of his political activism, he was arrested repeatedly,
serving one jail sentence of a number of months. For causes that stay obscure, shortly after his launch from jail, Galois fought in a duel and died of the injuries he suffered.^[3]
Formative years[edit]
Galois was born on 25 October 1811 to Nicolas-Gabriel Galois and Adélaïde-Marie (née Demante).^[2]^[4] His father was a Republican and was head of Bourg-la-Reine’s liberal party. His father grew to
become mayor of the village^[2] after Louis XVIII returned to the throne in 1814. His mom, the daughter of a jurist, was a fluent reader of Latin and classical literature and was answerable for her
son’s training for his first twelve years.
The Cour d’honneur of the Lycée Louis-le-Grand, which Galois attended as a boy.
In October 1823, he entered the Lycée Louis-le-Grand the place his trainer Louis Paul Émile Richard acknowledged his brilliance.^[5] On the age of 14, he started to take a severe curiosity in
Galois discovered a replica of Adrien-Marie Legendre‘s Éléments de Géométrie, which, it’s mentioned, he learn “like a novel” and mastered on the first studying. At 15, he was studying the unique
papers of Joseph-Louis Lagrange, such because the Réflexions sur la résolution algébrique des équations which doubtless motivated his later work on equation concept,^[6] and Leçons sur le calcul des
fonctions, work supposed for skilled mathematicians, but his classwork remained uninspired and his academics accused him of placing on the airs of a genius.^[4]
Budding mathematician[edit]
In 1828, Galois tried the doorway examination for the École Polytechnique, essentially the most prestigious establishment for arithmetic in France on the time, with out the same old preparation in
arithmetic, and failed for lack of explanations on the oral examination. In that very same yr, he entered the École Normale (then often called l’École préparatoire), a far inferior establishment for
mathematical research at the moment, the place he discovered some professors sympathetic to him.^[citation needed]
Augustin-Louis Cauchy reviewed Galois’s early mathematical papers.
Within the following yr Galois’s first paper, on continued fractions,^[7]
was revealed. It was at across the identical time that he started making elementary discoveries within the concept of polynomial equations. He submitted two papers on this matter to the Academy of
Sciences. Augustin-Louis Cauchy refereed these papers, however refused to simply accept them for publication for causes that also stay unclear. Nonetheless, despite many claims on the contrary, it’s
extensively held that Cauchy acknowledged the significance of Galois’s work, and that he merely instructed combining the 2 papers into one with the intention to enter it within the competitors for
the Academy’s Grand Prize in Arithmetic. Cauchy, an eminent mathematician of the time although with political beliefs that had been diametrically against these of Galois, thought-about Galois’s work
to be a possible winner.^[8]
On 28 July 1829, Galois’s father died by suicide after a bitter political dispute with the village priest.^[9] A few days later, Galois made his second and final try and enter the Polytechnique and
failed but once more.^[9] It’s undisputed that Galois was greater than certified; accounts differ on why he failed. Extra believable accounts state that Galois made too many logical leaps and baffled
the incompetent examiner, which enraged Galois. The latest loss of life of his father might have additionally influenced his habits.^[4]
Having been denied admission to the École polytechnique, Galois took the Baccalaureate examinations with the intention to enter the École normale.^[9] He handed, receiving his diploma on 29 December
1829.^[9] His examiner in arithmetic reported, “This pupil is usually obscure in expressing his concepts, however he’s clever and exhibits a outstanding spirit of analysis.”
Galois submitted his memoir on equation concept a number of instances, nevertheless it was by no means revealed in his lifetime. Although his first try was refused by Cauchy, in February 1830
following Cauchy’s suggestion he submitted it to the Academy’s secretary Joseph Fourier,^[9] to be thought-about for the Grand Prix of the Academy. Sadly, Fourier died quickly after,^[9] and the
memoir was misplaced.^[9] The prize can be awarded that yr to Niels Henrik Abel posthumously and in addition to Carl Gustav Jacob Jacobi. Regardless of the misplaced memoir, Galois revealed three
papers that yr. One laid the foundations for Galois theory.^[10] The second was in regards to the numerical decision of equations (root finding in trendy terminology).^[11] The third was an necessary
one in number theory, wherein the idea of a finite field was first articulated.^[12]
Political firebrand[edit]
Battle for the City Corridor by Jean-Victor Schnetz. Galois, as a staunch republican, would have needed to take part within the July Revolution of 1830 however was prevented by the director of the
École Normale.
Galois lived throughout a time of political turmoil in France. Charles X had succeeded Louis XVIII in 1824, however in 1827 his party suffered a major electoral setback and by 1830 the opposition
liberal party became the majority. Charles, confronted with political opposition from the chambers, staged a coup d’état, and issued his infamous July Ordinances, touching off the July Revolution^[9]
which ended with Louis Philippe changing into king. Whereas their counterparts on the Polytechnique had been making historical past within the streets, Galois, on the École Normale, was locked in by
the varsity’s director. Galois was incensed and wrote a blistering letter criticizing the director, which he submitted to the Gazette des Écoles, signing the letter along with his full identify.
Though the Gazette‘s editor omitted the signature for publication, Galois was expelled.^[13]
Though his expulsion would have formally taken impact on 4 January 1831, Galois give up faculty instantly and joined the staunchly Republican artillery unit of the National Guard. He divided his time
between his mathematical work and his political affiliations. Attributable to controversy surrounding the unit, quickly after Galois grew to become a member, on 31 December 1830, the artillery of the
Nationwide Guard was disbanded out of worry that they may destabilize the federal government. At across the identical time, nineteen officers of Galois’s former unit had been arrested and charged
with conspiracy to overthrow the federal government.
In April 1831, the officers had been acquitted of all fees, and on 9 Might 1831, a banquet was held of their honor, with many illustrious individuals current, akin to Alexandre Dumas. The proceedings
grew riotous. In some unspecified time in the future, Galois stood and proposed a toast wherein he mentioned, “To Louis Philippe,” with a dagger above his cup. The republicans on the banquet
interpreted Galois’s toast as a risk towards the king’s life and cheered. He was arrested the next day at his mom’s home and held in detention at Sainte-Pélagie prison till 15 June 1831, when he had
his trial.^[8] Galois’s protection lawyer cleverly claimed that Galois really mentioned, “To Louis-Philippe, if he betrays,” however that the qualifier was drowned out within the cheers. The
prosecutor requested a couple of extra questions, and maybe influenced by Galois’s youth, the jury acquitted him that very same day.^[8]^[9]^[13]^[14]
On the next Bastille Day (14 July 1831), Galois was on the head of a protest, sporting the uniform of the disbanded artillery, and got here closely armed with a number of pistols, a loaded rifle, and
a dagger. He was once more arrested.^[9] Throughout his keep in jail, Galois at one level drank alcohol for the primary time on the goading of his fellow inmates. One in every of these inmates,
François-Vincent Raspail, recorded what Galois mentioned whereas drunk in a letter from 25 July. Excerpted from the letter:^[8]
And I let you know, I’ll die in a duel on the event of some coquette de bas étage. Why? As a result of she’s going to invite me to avenge her honor which one other has compromised.
Have you learnt what I lack, my pal? I can confide it solely to you: it’s somebody whom I can love and love solely in spirit. I’ve misplaced my father and nobody has ever changed him, do you hear
Raspail continues that Galois, nonetheless in a delirium, tried suicide, and that he would have succeeded if his fellow inmates hadn’t forcibly stopped him.^[8] Months later, when Galois’s trial
occurred on 23 October, he was sentenced to 6 months in jail for illegally sporting a uniform.^[9]^[15]^[16] Whereas in jail, he continued to develop his mathematical concepts. He was launched on 29
April 1832.
Ultimate days[edit]
Siméon Denis Poisson reviewed Galois’s paper on equation concept and declared it “incomprehensible”.
Galois returned to arithmetic after his expulsion from the École Normale, though he continued to spend time in political actions. After his expulsion grew to become official in January 1831, he tried
to begin a non-public class in superior algebra which attracted some curiosity, however this waned, because it appeared that his political activism had precedence.^[4]^[8] Siméon Denis Poisson
requested him to submit his work on the theory of equations, which he did on 17 January 1831. Round 4 July 1831, Poisson declared Galois’s work “incomprehensible”, declaring that “[Galois’s] argument
is neither sufficiently clear nor sufficiently developed to permit us to guage its rigor”; nevertheless, the rejection report ends on an encouraging notice: “We might then recommend that the writer
ought to publish the entire of his work with the intention to type a definitive opinion.”^[17]
Whereas Poisson’s report was made earlier than Galois’s 14 July arrest, it took till October to succeed in Galois in jail. It’s unsurprising, within the gentle of his character and state of affairs
on the time, that Galois reacted violently to the rejection letter, and determined to desert publishing his papers by way of the Academy and as a substitute publish them privately by way of his pal
Auguste Chevalier. Apparently, nevertheless, Galois didn’t ignore Poisson’s recommendation, as he started amassing all his mathematical manuscripts whereas nonetheless in jail, and continued
sharpening his concepts till his launch on 29 April 1832,^[13] after which he was someway talked right into a duel.^[9]
Galois’s deadly duel occurred on 30 Might.^[18] The true motives behind the duel are obscure. There was a lot hypothesis about them. What is thought is that, 5 days earlier than his loss of life, he
wrote a letter to Chevalier which clearly alludes to a damaged love affair.^[8]
Some archival investigation on the unique letters means that the girl of romantic curiosity was Stéphanie-Félicie Poterin du Motel,^[19] the daughter of the doctor on the hostel the place Galois
stayed over the past months of his life. Fragments of letters from her, copied by Galois himself (with many parts, akin to her identify, both obliterated or intentionally omitted), can be found.^[20]
The letters trace that du Motel had confided a few of her troubles to Galois, and this may need prompted him to impress the duel himself on her behalf. This conjecture can also be supported by
different letters Galois later wrote to his associates the night time earlier than he died. Galois’s cousin, Gabriel Demante, when requested if he knew the reason for the duel, talked about that
Galois “discovered himself within the presence of a supposed uncle and a supposed fiancé, every of whom provoked the duel.” Galois himself exclaimed: “I’m the sufferer of an notorious coquette and
her two dupes.”^[13]
Far more detailed hypothesis primarily based on these scant historic particulars has been interpolated by lots of Galois’s biographers, such because the steadily repeated hypothesis that your entire
incident was stage-managed by the police and royalist factions to get rid of a political enemy.^[citation needed]
As to his opponent within the duel, Alexandre Dumas names Pescheux d’Herbinville,^[14] who was really one of many nineteen artillery officers whose acquittal was celebrated on the banquet that
occasioned Galois’s first arrest.^[21] Nonetheless, Dumas is alone on this assertion, and if he had been appropriate it’s unclear why d’Herbinville would have been concerned. It has been speculated
that he was du Motel’s “supposed fiancé” on the time (she finally married another person), however no clear proof has been discovered supporting this conjecture. Then again, extant newspaper
clippings from only some days after the duel give an outline of his opponent (recognized by the initials “L.D.”) that seem to extra precisely apply to one in all Galois’s Republican associates,
likely Ernest Duchatelet, who was imprisoned with Galois on the identical fees.^[22] Given the conflicting info obtainable, the true id of his killer could be misplaced to historical past.
Regardless of the causes behind the duel, Galois was so satisfied of his impending loss of life that he stayed up all night time writing letters to his Republican associates and composing what would
develop into his mathematical testomony, the well-known letter to Auguste Chevalier outlining his concepts, and three hooked up manuscripts.^[23] Mathematician Hermann Weyl mentioned of this
testomony, “This letter, if judged by the novelty and profundity of concepts it accommodates, is maybe essentially the most substantial piece of writing in the entire literature of mankind.”
Nonetheless, the legend of Galois pouring his mathematical ideas onto paper the night time earlier than he died appears to have been exaggerated.^[8] In these last papers, he outlined the tough edges
of some work he had been doing in evaluation and annotated a replica of the manuscript submitted to the Academy and different papers.
The Galois memorial within the cemetery of Bourg-la-Reine. Évariste Galois was buried in a standard grave and the precise location is unknown.
Early within the morning of 30 Might 1832, he was shot within the abdomen,^[18] was deserted by his opponents and his personal seconds, and was discovered by a passing farmer. He died the next
morning^[18] at ten o’clock within the Hôpital Cochin (in all probability of peritonitis), after refusing the places of work of a priest. His funeral resulted in riots.^[18] There have been plans to
provoke an rebellion throughout his funeral, however throughout the identical time the leaders heard of Basic Jean Maximilien Lamarque‘s loss of life and the rising was postponed with none rebellion
occurring till 5 June. Solely Galois’s youthful brother was notified of the occasions previous to Galois’s loss of life.^[24] Galois was 20 years previous. His last words to his youthful brother
Alfred had been:
“Ne pleure pas, Alfred ! J’ai besoin de tout mon braveness pour mourir à vingt ans !”
(Do not weep, Alfred! I would like all my braveness to die at twenty!)
On 2 June, Évariste Galois was buried in a standard grave of the Montparnasse Cemetery whose actual location is unknown.^[18]^[16] Within the cemetery of his native city – Bourg-la-Reine – a cenotaph
in his honour was erected beside the graves of his family.^[25]
Évariste Galois died in 1832. Joseph Liouville started learning Galois’s unpublished papers in 1842 and acknowledged their worth in 1843. It isn’t clear what occurred within the 10 years between 1832
and 1842 nor what finally impressed Joseph Liouville to start studying Galois’s papers. Jesper Lützen explores this topic at some size in Chapter XIV Galois Idea of his e book about Joseph Liouville
with out reaching any definitive conclusions.^[26]
It’s actually attainable that mathematicians (together with Liouville) didn’t need to publicize Galois’s papers as a result of Galois was a republican political activist who died 5 days earlier than
the June Rebellion, an unsuccessful anti-monarchist revolt of Parisian republicans. In Galois’s obituary, his pal Auguste Chevalier virtually accused academicians on the École Polytechnique of
getting killed Galois since, if that they had not rejected his work, he would have develop into a mathematician and wouldn’t have devoted himself to the republican political activism for which some
believed he was killed.^[26]
Provided that France was nonetheless residing within the shadow of the Reign of Terror and the Napoleonic era, Liouville may need waited till the June Rebellion‘s political turmoil subsided earlier
than turning his consideration to Galois’s papers.^[26]
Liouville lastly revealed Galois’s manuscripts within the October–November 1846 problem of the Journal de Mathématiques Pures et Appliquées.^[27]^[28] Galois’s most well-known contribution was a
novel proof that there isn’t any quintic formula – that’s, that fifth and better diploma equations will not be usually solvable by radicals. Though Niels Henrik Abel had already proved the
impossibility of a “quintic formula” by radicals in 1824 and Paolo Ruffini had revealed an answer in 1799 that turned out to be flawed, Galois’s strategies led to deeper analysis into what’s now
referred to as Galois Theory, which can be utilized to find out, for any polynomial equation, whether or not it has an answer by radicals.
Contributions to arithmetic[edit]
The ultimate web page of Galois’s mathematical testomony, in his personal hand. The phrase “to decipher all this mess” (“déchiffrer tout ce gâchis”) is on the second to the final line.
From the closing traces of a letter from Galois to his pal Auguste Chevalier, dated 29 Might 1832, two days earlier than Galois’s loss of life:^[23]
Tu prieras publiquement Jacobi ou Gauss de donner leur avis, non sur la vérité, mais sur l’significance des théorèmes.
Après cela, il y aura, j’espère, des gens qui trouveront leur revenue à déchiffrer tout ce gâchis.
(Ask Jacobi or Gauss publicly to provide their opinion, not as to the reality, however as to the significance of those theorems. Later there might be, I hope, some individuals who will discover
it to their benefit to decipher all this mess.)
Throughout the 60 or so pages of Galois’s collected works are many necessary concepts which have had far-reaching penalties for almost all branches of arithmetic.^[29]^[30]
His work has been in comparison with that of Niels Henrik Abel (1802–1829), a recent mathematician who additionally died at a really younger age, and far of their work had important overlap.
Whereas many mathematicians earlier than Galois gave consideration to what are actually often called groups, it was Galois who was the primary to make use of the phrase group (in French groupe) in a
way near the technical sense that’s understood at the moment, making him among the many founders of the department of algebra often called group theory. He referred to as the decomposition of a
gaggle into its left and proper cosets a correct decomposition if the left and proper cosets coincide, which is what at the moment is named a standard subgroup.^[23] He additionally launched the idea
of a finite field (also referred to as a Galois field in his honor) in primarily the identical type as it’s understood at the moment.^[12]
In his final letter to Chevalier^[23] and hooked up manuscripts, the second of three, he made primary research of linear teams over finite fields:
Galois concept[edit]
Galois’s most important contribution to arithmetic is his improvement of Galois concept. He realized that the algebraic answer to a polynomial equation is expounded to the construction of a gaggle of
permutations related to the roots of the polynomial, the Galois group of the polynomial. He discovered that an equation might be solved in radicals if one can discover a sequence of subgroups of its
Galois group, each regular in its successor with abelian quotient, that’s, its Galois group is solvable. This proved to be a fertile strategy, which later mathematicians tailored to many different
fields of arithmetic apart from the theory of equations to which Galois initially utilized it.^[29]
Galois additionally made some contributions to the idea of Abelian integrals and continued fractions.
As written in his final letter,^[23] Galois handed from the examine of elliptic capabilities to consideration of the integrals of essentially the most normal algebraic differentials, at the moment
referred to as Abelian integrals. He labeled these integrals into three classes.
Continued fractions[edit]
In his first paper in 1828,^[7] Galois proved that the common continued fraction which represents a quadratic surd ζ is solely periodic if and provided that ζ is a reduced surd, that’s, $
{displaystyle zeta >1}$ and its conjugate ${displaystyle eta }$
satisfies ${displaystyle -1<eta <0}$.
In reality, Galois confirmed greater than this. He additionally proved that if ζ is a decreased quadratic surd and η is its conjugate, then the continued fractions for ζ and for (−1/η) are each
purely periodic, and the repeating block in a type of continued fractions is the mirror picture of the repeating block within the different. In symbols now we have
{displaystyle {start{aligned}zeta &=[,{overline {a_{0};a_{1},a_{2},dots ,a_{m-1}}},][3pt]{frac {-1}{eta }}&=[,{overline {a_{m-1};a_{m-2},a_{m-3},dots ,a_{0}}},],finish{aligned}}}
the place ζ is any decreased quadratic surd, and η is its conjugate.
From these two theorems of Galois a end result already identified to Lagrange will be deduced. If r > 1 is a rational quantity that isn’t an ideal sq., then
${displaystyle {sqrt {r}}=left[,a_{0};{overline {a_{1},a_{2},dots ,a_{2},a_{1},2a_{0}}},right].}$
Particularly, if n is any non-square optimistic integer, the common continued fraction enlargement of √n accommodates a repeating block of size m, wherein the primary m − 1 partial denominators type
a palindromic string.
See additionally[edit]
Exterior hyperlinks[edit]
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Poincaré, Einstein and Picasso: children of time
A great thanks to Marco Fulvio Barozzi: his post^(1) about Miller's book is the main inspiration of my post.
on the Guardian
Arthur I. Miller
, the author of the book
Einstein, Picasso: Space, Time and the Beauty that Causes Havoc
, wrote a briefly article in which he resumed his thesis about the connections between
Poincaré and Einstein
, between
Poincaré and Picasso
and, for translation, between
Einstein and Picasso
Henri Poincaré
was one of the most important mathematician of the early XX century: his most important contributions, that have a great impact also in physics, are in group theory and representation theory. His
work was indeed important for the birth of the ray representations (the theory was developed in particular by
Valentine Bargmann
starting from
's works) and basic for special relativity and in particular for general relativity. Poincaré was the first to propose the symmetrical form of the Lorentz transformations, and his work was important
for the creation of the Poincaré group, the symmetry group of the general relativity. In particular about the relativity, Poincaré written on his book
Science and Hypothesis
Our Euclidean geometry is itself a sort of linguistic convention; we may state the facts of mechanics in relation to a non-Euclidean space, but this would be a less convenient reference, although
legitimate like our ordinary space.^(1)
He also defined the
principle of relative motion
the physical impossibility of observing absolute motion.^(1)
Two years later he named it
Principle of Relativity
At the other hand, Einstein did not cite Poincaré's works in his paper published in 1905 by
Annalen der Physik
and only in a conference in 1921 Einstein confirmed his debt to the french mathematician, but only about general relativity and non-euclidean geometry. And this is the only documented connection
between Einstein and Poincaré: we must suppose that the two scientists worked indipendetly and also after his first paper Einstein used Poincaré's discoveries in order to develop the mathematical
formalism of the general relativity.
Some years later the first Einstein's paper, the cubism was born in France:
A circle of poets and critics, and followers of the philosopher Bergson, stood up for cubism in the visual arts. This group became known as the Cubists. The poet and publicist G. Apollinaire
became the undisputed leader of this movement.^(2)
It seems that relativity played a relevant role in the phylosophy of the artistic movement
Like the scientists, the artists has come to recognize thatclassic conceptions of space and volume are limited and one-sided. (...) The presentation of objects from several point of view
introduces a principle which is intimately bound up with modern life - simultaenity. It is a temporal coincidence that Einstein should havebegan his famous work (...) with a careful definition of
In this quotation by
Sigfried Giedion
, the connection was simply casual, only a
temporal coincidence
, but a lot of art historians think that the connection is not so casual. One of this is
Paul M. Laporte
, who published two paper about cubism and relativity, and submitted them to Albert Einstein. The great physicist reply with a long letter, in which he concludes:
This new artistic "language" has nothing in common with the Theory of Relativity.^(5)
And probably it is so. Indeed in 1903 the
Introduction to Metaphysics
Henri Bergson
was published. In the book Bergson argued that
human consciousness experiences space and time as ever-changing and heterogeneous. With the passage of time, an observer accumulates in his memory a store of perceptual information about a given
object in the external visible world, and this accumulated experience becomes the basis for the observer’s conceptual knowledge of that object. By contrast, the intellect or reasoning faculty
always represents time and space as homogenous. Bergson argued that intellectual perception led to a fundamentally false representation of the nature of things, that in nature nothing is ever
absolutely still. Instead the universe is in a constant state of change or flux. An observer views an object and its surrounding environment as a continuum, fusing into one another. The task of
metaphysics, according to Bergson, is to find ways to capture this flux, especially as it is expressed in consciousness. To represent this flux of reality, Picasso began to make references to the
fourth dimension by "sticking together" several three-dimensional spaces in a row.^(4)
It seems a good inspiration for Picasso, better than Poincaré's
Science and Hypothesis
, like Miller thinks. But Picasso would not necessarily know the work of Poincaré to receive inspiration. This source could be arrived from another protagonist of the birth of the cubism. Indeed,
Maurice Vlaminck
, the secret origins of the
new artistic "language"
had three father: a painter, a poet, and a mathematichian.
I witnessed the birth of cubism, its growth, its decline. Picasso was the obstetrician, Guillaume Apollinaire the midwife, Princet the godfather.^(1, 3)
Jean Metzinger
seemed confirm the importance of Princet in the first steps of the cubism:
Maurice Princet joined us often. Although quite young, thanks to his knowledge of mathematics he had an important job in an insurance company. But, beyond his profession, it was as an artist that
he conceptualized mathematics, as an aesthetician that he invoked n-dimensional continuums. He loved to get the artists interested in the new views on space that had been opened up by Schlegel
and some others. He succeeded at that.^(1, 3)
And Princet was interested in advanced mathematics, in particular in Poincaré's work and, in general, in non-Euclidean geometries.
It also seems that Princet has made known to the Spanish painter Elémentaire Traité de géométrie à quatre dimensions (1902) by Esprit Jouffret, which were described the hypercubes and other
complex polyhedra in four dimensions and it was shown how to play on a two-dimensional plane objects with more than three dimensions.^(1)
Listening to Princet, Picasso realised that geometry offered the language to express the deep meaning of primitive Iberian art, which he was working on at the time. In Les Demoiselles d'Avignon,
he depicts one of the demoiselles simultaneously full face and in profile, two perspectives at once, a projection from the fourth dimension. He had gone beyond Poincaré.^(6)
And the art critic?
Jorge Romero Brest
emphasize the importance of Princet in the birth of cubism:
The oscillation of planes suggests, but does not represent, a psace that trascends the threedimensional; in other words the forms appear to be space-time symbols. The link between the new
painting and concurrent scientific developments was emphasized by the mathematician Maurice Princet.^(5)
At the other hand,
Louis Vauxcelles
describes the creation of the cubism like a Goldberg machine!
M. Princet has studied at length non-Euclidean geometry and the theorems of Riemann, of which Gleizes and Metzinger speak rather carelessly. Now then, M. Princet one day met M. Max Jacob and
confided him one or two of his discoveries relating to the fourth dimension. M. Jacob informed the ingenious M. Picasso of it, and M. Picasso saw there a possibility of new ornamental schemes. M.
Picasso explained his intentions to M. Apollinaire, who hastened to write them up in formularies and codify them. The thing spread and propagated. Cubism, the child of M. Princet, was born.^(1,
Probably Einstein and Poincaré didn't have a direct importance in the birth of cubism. The Einstein's revolution is simply a natural development of science, starting from the observation that the
Maxwell's equations aren't invariant under the action of the
Galilei's group
(the symmetry group of Schroedinger equation).
Poincaré's work is the natural consequences of Reimann's work, and Jouffret's paper is a
of this research subject.
Princet could have been the role of the
: when the friends met in some
, Princet simply started to tell about mathematics, the non-euclidean geometry, and the space with four and more dimensions. And this conversations may have sown the cubism in the mind of the
In this sense Miller could be right after all, but the most probably solution of the quest is that Poincaré, Einstein and Picasso (and also Princet!) are simply children of time. (1)
Marco Fulvio Barozzi
Einstein e Picasso, con qualche dubbio
Einstein and Picasso, with some doubts
(2) Photography school:
The birth of cubism
(3) Wikipedia:
Maurice Princet
Cubism: A New Vision - The Birth of Cubism
Paul M. Laporte
Cubism and Relativity
) via
Stagesof discovery
Arthur I. Miller Henri Poincaré: the unlikely link between Einstein and Picasso
3 comments:
1. Art and science, the perfect combination ;)
2. Amazing post, Gian! Between Art and Science, there are more connections than you can imagine.
3. Wow! Such an amazing and helpful post this is. I really really love it. It's so good and so awesome. I am just amazed. I hope that you continue to do your work like this in the future also
Markup Key:
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Lesson Getting Started 1
Getting Started
What are the three things you need to get started using a whiteboard?
1. Whiteboard
Whiteboards have different surfaces such as enameled metal, glass, or melamine. There are electronic smartboards that also save your work. You can get different sizes of whiteboards with or without a
2. Markers
Be sure to use only dry erase markers. They often are chisel pointed.
Avoid markers not made to use on a whiteboard such as permanent markers, transparency markers, or washable markers.
3. Erasers
Erasers are more important than you might think. You will need to keep a marker in one hand and an eraser in the other so you can remove misspellings or marks that are out of line as you sketch.
Use a felt bottom marker to remove most of the dry erase ink from the white board when you are done sketching. Next, use a cleaning wipe (tub of Clorox wipes from cleaning supplies store) to remove
any traces of the ink. Then, dry the whiteboard with a paper towel. Dry erase markers won’t write on a wet surface.
Note that the whiteboard needs to be as clean as possible so it is easy to see your work. In order to have a great photograph, the whiteboard needs to have a white surface. You may be recording your
work for use later in PowerPoint decks, or in videos.
If there is a stain or ink in a scratch on the whiteboard, you can use a magic eraser (see cleaning supplies in the grocery store) to remove the marks.
Click the NEXT red arrow to learn how to draw the cleaning wipe container. Click the NEXT red arrow again to learn how to draw the roll of paper towels.
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ZPEnergy.com - Andrei Sakharov 1967 Emergent Gravity from ZPF & Nonlocality of Gravity Energy
Andrei Sakharov 1967 Emergent Gravity from ZPF & Nonlocality of Gravity Energy
Date: Monday, March 07, 2005 @ 22:55:26 UTC
Topic: Science
From Dr. Jack Sarfatti: The foundations of metric engineering the fabric of space-time geometry for interstellar space travel are implicit in this paper.
The pdf version is at: http://qedcorp.com/APS/EmergentGravityGauge.pdf
I attach a much smaller WORD file that you can edit. Windows machines should be able to read it from MAC.
Comments, corrections, suggestions welcomed.
Emergent Gravity Gauge Force/Geometrodynamic Duality - Abstract:
Andrei Sakharov’s 1967 conjecture for the emergence of gravity from zero point vacuum fluctuations is brought to completion. The missing idea was the cohering of these fluctuations by spontaneous
breakdown of U(1) symmetry inducing the inflationary vacuum phase transition to the Big Bang. The warped anholonomic part of the Einstein-Cartan tetrad is related to the Goldstone phase of the
post-inflationary Higgs field. The paradox of nonlocality of the pure gravity field stress-energy in geometrodynamics is solved using the equivalence principle that imposes a gauge force/
geometrodynamic duality. | {"url":"http://zpenergy.com/modules.php?name=News&file=print&sid=1210","timestamp":"2024-11-06T04:41:19Z","content_type":"text/html","content_length":"2807","record_id":"<urn:uuid:2206d794-656a-44df-a780-ddceb83410ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00413.warc.gz"} |
T- score used in the solution is for alpa, not divided by 2. Why – Q&A Hub – 365 Data Science
Resolved: T- score used in the solution is for alpa, not divided by 2. Why
Please am confused with how to check the t-score table. The video shows that you divide the alpha by 2 e.g alpha is 0.05, divide by 2 which gives 0.025.
We are supposed to check 0.025 against the degree of freedom, to get the intersect.
But solution keeps using the alpha figure(without dividing it by 2).
Please I need clarification.
2 answers ( 1 marked as helpful)
Super learner
This user is a Super Learner. To become a Super Learner, you need to reach Level 8.
Which solution?
in the exercise solution file, t-statistics for alpha (0.05/2) or (alpha 0.025) is correct -> 2.26.
Hi there, a 95% CI suggests that we need to capture 95% of data inbetween 2 points. Therefor we need 2.5% of the data to the left of the lower point, 95% of data within the two points and 2.5% of
data above the higher point. This is why for a 2 tail test we need to consider alpha/2. If this was a 1-tail test then this suggests there is a strict known value for a lower bound like 0 of +/-
infinite, therefor we would want to have 95% data below the upper value and 5% data above this value for a 95% CI. | {"url":"https://365datascience.com/question/t-score-used-in-the-solution-is-for-alpa-not-divided-by-2-why/","timestamp":"2024-11-14T05:03:33Z","content_type":"text/html","content_length":"113148","record_id":"<urn:uuid:5be9dd22-a153-461a-b12d-00363d9a9563>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00815.warc.gz"} |
Problem B
Biologists have discovered a strange DNA molecule, best described as a sequence of $N$ characters from the set $\{ A, B\} $. An unlikely sequence of mutations has resulted in a DNA strand consisting
only of $A$’s. Biologists found that very odd, so they began studying the mutations in greater detail.
They discovered two types of mutations. One type results in changing any single character of the sequence ($A \rightarrow B$ or $B \rightarrow A$). The second type changes a whole prefix of the
sequence, specifically replacing all characters in positions from $1$ to $K$ (for some $K$ between $1$ and $N$, inclusive) with the other character ($A$ with $B$, $B$ with $A$).
Compute the least possible number of mutations that could convert the starting molecule to its end state (containing only $A$ characters). Mutations can occur in any order.
The first line of input contains the positive integer $N$ ($1 \le N \le 1\, 000\, 000$), the length of the molecule. The second line of input contains a string with $N$ characters, with each
character being either $A$ or $B$. This string represents the starting state of the molecule.
The first and only line of output must contain the required minimum number of mutations.
Sample Input 1 Sample Output 1
Sample Input 2 Sample Output 2
Sample Input 3 Sample Output 3 | {"url":"https://nus.kattis.com/courses/CS3233/CS3233_S2_AY2223/assignments/w4j2ec/problems/dna","timestamp":"2024-11-08T18:56:46Z","content_type":"text/html","content_length":"29009","record_id":"<urn:uuid:cec11335-be83-4220-a2a6-8d58894e118a>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00876.warc.gz"} |
Data Structures - Trees
Written by Mike James
Thursday, 02 November 2017
For example, if you already have the perfectly balanced tree in Figure 4a and the value 2 has to be added to it then the result is the perfectly balanced tree in Figure 4b.
Figure 4a: A perfectly balanced tree
Figure 4b: Adding (2) keeps the tree in balance.
However it isn't always possible to insert a new data value and keep the tree in perfect balance. For example, there is no way to add 9 to the tree in Figure 4a and keep it in perfect balance (Figure
4c) without reordering chunks of the tree. It turns out the effort expended in reorganising the tree to maintain perfect balance just isn't worth it.
Figure 4c: Adding (9) makes the tree unbalanced
AVL Trees
So it looks as though using trees to store data such that searching is efficient is problematic. Well there might be a remedy if a less restricted form of balance were used.
One such approach is to insist that the depths of each sub-tree differ by at most one. A tree that conforms to this definition is called an AVL tree, or sometimes just a "balanced" as opposed to
"perfectly balanced" tree.
Many programmers have puzzled what AVL might stand for - Averagely Very Long tree?
The answer is that AVL trees were invented by Adelson-Velskii and Landis in 1962. Notice that every perfectly balanced tree is an AVL tree but the reverse isn't true as shown in Figure 5.
Figure 5: An AVL tree that isn't perfectly balanced
It turns out that an AVL tree will never be more than 45% deeper than the equivalent perfectly balanced tree. This means that searching an AVL tree is likely to be as quick as searching the tree in
perfect balance and the payoff is that adding a node to an existing AVL tree so that it stays an AVL tree isn't difficult.
In short re-balancing an AVL is easy, as can be seen in Figure 6a & b.
Figure 6a: A simple reorganisation converts a not quite AVL tree into an AVL tree
Figure 6b: A slightly more complicated example of a reorganisation converting a not quite AVL tree into an AVL tree
At this point the story could come to an end and we could all happily use AVL trees to store data that needs to be found again in double quick time. Unfortunately AVL trees and binary trees in
general aren't ideal when it comes to implementing an index stored on disk. The trouble is that disks are accessed one large chunk - i.e. a sector - at a time and mapping a search tree onto a disk so
that there is one node per sector would produce a very slow search.
One of the nicest ideas to get around this problem is the B-Tree. A B-Tree is constructed using a `page' of storage at every node. The B-Tree is constrained to have no fewer than n items and no more
than 2n items on every page. In other words, the page size is fixed at 2n and we insist that at least 50% of the page is used if possible.
The organisation of each page, and its links to the next page, is more complicated than for a binary tree but not that difficult.
The m items on each page are entered in order and either a page is terminal or it has m+1 pointers to pages at the next level. The first pointer is to a page that contains items that are smaller than
the first item on the page, the second is to a page that contains items that lie between the first and second item and so on to the last pointer, which points to pages that contain items larger than
the last item. It sounds confusing but the example of a B-Tree in Figure 7 should make everything look simpler and make it possible to understand an exact definition of a B-Tree.
A B-Tree of order n satisfies the following:
1. Every page contains at most 2n items
2. Every page, except the root page, contains at least n items
3. Every page with m items is either a leaf page or has m+1 descendants
4. All leaf pages are at the same level
You can spend a few happy hours working out how this rather abstract definition results in the friendly B-Tree in Figure 7.
Figure 7: A B-Tree
You should also find it easy to work out the algorithm for searching a B-Tree. Basically it comes down to starting at the root page and if the key isn't found moving to the page that contains items
in the same range as the one you are looking for. If there isn't a suitable descendant page then the item you are looking for isn't in the B-Tree and so you very likely have to add it. Notice that
the number of page accesses required to find or not find an item is very reasonable.
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Using Vectors
"Vector" redirects here. For a type of graphic, see Vector Graphics.
Vectors are ordered sets of numbers, used in algebra to represent many different things. Most of the time, they are used to represent various physical quantities in physics simulations or games. This
article will explain how to use vectors with Scratch to produce interesting effects.
What Vectors Are
Vectors are like lists, except you cannot add or remove items. Their "items" (components) are numbers. For this article, we will only consider 2-dimensional vectors with 2 components, however,
vectors can have any number of dimensions or components, even 4 dimensions. For example, raytracing requires 3-dimensional vectors. Vectors by definition are quantities consisting of one magnitude
and one direction. They can be graphically defined with an arrow.
2D vectors can be represented as points on the Cartesian plane be graphing their X component on the X axis and Y component on the Y axis. So we can represent the positions of sprites as vectors.
Vectors are used in graphics to represent many things such as positions, velocities, forces, etc. . Using vector math, you can simulate complex phenomena such as collisions.
Due to the limitation of mathematical notation, the following notations are used throughout:
• (x, y) or <x, y> is the vector with x and y.
• a.x and a.y are the x and y components respectively.
• a +,-,*,/ b are the respective operations with a and b as operands.
• a • b is a dot product.
• a × b is a cross product.
• a! is the unit vector a.
• |a| is the length of a.
Vector Operations
A lot of the standard operations of natural numbers can be performed on vectors. Vector addition consists of simply adding the respective components of two vectors: (1, 2) + (3, 4) = (4, 6). Imagine
drawing the second vector at the tip of the first.
Subtraction can be derived from addition: just subtract the respective components: (5, 4) - (3, 2) = (2, 2)
For our purposes, vector multiplication is between a scalar and a vector, where the scalar is multiplied by each of the components —the same holds for division, with the reciprocal of the scalar
operand: 5 * (2, 3) = (10, 6)
A vector can be divided by a scalar but a scalar cannot be divided by a vector:
• (15, 25) / 5 = (3, 5)
• 4 / (16, 24) = NaN
The magnitude of a vector can be calculated using the Pythagorean theorem: |a| = √[(a.x)^2+(a.y)^2]
The direction of a n-dimensional vector is defined by (n-1) angles; for a 2D vector that means just one angle defined by atan2(a.y, a.x).
One common quantity we need is the unit vector, which is a vector in the same direction as another, but of unit length. It can be calculated by dividing each component by the length of the vector: v!
= (v.x/|v|, v.y/|v|).
The dot product and cross product are the two main methods of multiplying two vectors.
• The dot product: a • b of two vectors is a scalar defined by two equations:
□ a • b = (a.x * b.x) + (a.y * b.y). To generalize, the dot product is the sum of the products of the respective components of two vectors.
□ |a|*|b|*cos(theta). theta is the angle between the two vectors.
• The cross product: a × b of two vectors is only defined in 3 or 7 dimensional space and produces a vector. The cross product is non-commutative, meaning that a × b ≠ b × a.
Let a × b = c.
□ The Three-Dimensional cross-product is defined by these two equations:
☆ c = |a|*|b|*sin(theta)*n. theta is the angle between the two vectors and n is the unit vector perpendicular two the vectors a and b.
☆ The individual components of c are defined by these equations.
○ c.x = (a.y * b.z) - (a.z * b.y)
○ c.y = (a.z * b.x) - (a.x * b.z)
○ c.z = (a.x * b.y) - (a.y * b.x)
□ The Seven-Dimensional cross-product defined by these two equations:
☆ The individual components of c are defined by these equations.
○ c.x = (a.y * b.z) - (a.z * b.y) + (a.w * b.v) - (a.v * b.w) + (a.u * b.t) - (a.t * b.u)
○ c.y = (a.z * b.x) - (a.x * b.z) + (a.w * b.u) - (a.u * b.w) + (a.v * b.t) - (a.t * b.v)
○ c.z = (a.x * b.y) - (a.y * b.x) + (a.w * b.t) - (a.t * b.w) + (a.u * b.v) - (a.v * b.u)
○ c.w = (a.v * b.x) - (a.x * b.v) + (a.u * b.y) - (a.y * b.u) + (a.t * b.z) - (a.z * b.t)
○ c.v = (a.x * b.w) - (a.w * b.x) + (a.t * b.y) - (a.y * b.t) + (a.z * b.u) - (a.u * b.z)
○ c.u = (a.x * b.t) - (a.t * b.x) + (a.y * b.w) - (a.w * b.y) + (a.v * b.z) - (a.z * b.v)
○ c.t = (a.u * b.x) - (a.x * b.u) + (a.y * b.v) - (a.v * b.y) + (a.z * b.w) - (a.w * b.z)
Note: Normally, this unit vector is represented with a hat (e.g. â), not an exclamation point, but this is hard to typeset in Wiki syntax.
Scratch Representation
To represent a vector in Scratch, we use a pair of variables, normally called <name>.x and <name>.y. We can then perform addition, subtraction, multiplication, and division on each of those variables
Create a pair of variables position.x and position.y, and write a script to make a sprite continually go to the coordinates given by the position vector. Now, if you set those variable watchers to
sliders, you can change the position with sliders.
when gf clicked
set x to (position.x)
set y to (position.y)
For something more interesting, create another variable pair called "velocity" (i.e. create velocity.x and velocity.y). Add a new script which changes the respective components of the position vector
by the components of the velocity vector. Now, by changing the value of velocity, you should have a much smoother motion.
when gf clicked
change [position.x v] by (velocity.x)
change [position.y v] by (velocity.y)
set x to (position.x)
set y to (position.y)
Finally, we can create an effect of gravity by also changing the velocity by some vector:
when gf clicked
set [position.x v] to (0)
set [position.y v] to (0)
set [velocity.x v] to (10)
set [velocity.y v] to (10)
change [velocity.y v] by (-1)
change [position.x v] by (velocity.x)
change [position.y v] by (velocity.y)
set x to (position.x)
set y to (position.y)
A common use: bouncing
A common use of vectors is to simulate a body bouncing off an arbitrarily angled surface. To make a bouncing script, we need to calculate the perpendicular vector to the surface, then project the
velocity vector of the body on that to find the component of the vector that is reflected.
To find the perpendicular, we switch the X and Y components, and negate any one. Projection is a bit tougher, and requires the dot product. The dot product (a • b). | {"url":"https://test.scratch-wiki.info/wiki/Eng:Using_Vectors","timestamp":"2024-11-02T08:30:54Z","content_type":"text/html","content_length":"40512","record_id":"<urn:uuid:b69570ba-a10a-40ec-9bb2-6b3ecd10131d>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00812.warc.gz"} |
Numbering In Google Sheets
What Is Numbering In Google Sheets?
Numbering in Google Sheets is a technique that enables us to automatically populate cells with sequential numbers to organize rows of data. We can use options such as the ROW function and fill
handle to number cells in Google Sheets.
Users can utilize the numbering technique in Google Sheets to number key cells in massive datasets containing financial and accounting data. It helps save time to manage such standardized
For example, the source dataset lists employee names and their entry times.
The aim is to update the entry number for each employee in column A, with the entry number starting with 1 and incrementing by one in the subsequent cells till cell A11.
Then, we can insert numbering in Google Sheets using the ROW(), which works similar to Excel ROW function, in each target cell.
In this automatic numbering in Google Sheets example, we enter the ROW() in a target cell, with the reference to the cell above the target cell as its argument value.
The reason for supplying such an argument value to the ROW() is that the top-most cell, which we must number as 1, is the second cell in column A, cell A2. So, supplying the reference to cell A1 as
the input to the ROW() in cell A2 ensures the function output is 1.
So, we can then continue numbering in Google Sheets by dragging the fill handle till cell A11 to number the cells from A3:A11 in the required sequence.
Key Takeaways
• The numbering in Google Sheets is a method that helps us to update cells with sequential numbers automatically.
• The more straightforward methods to number cells in Google Sheets are to use the fill handle option, ‘+’ arithmetic operator-based or ROW function-based customized formulas.
• We can dynamically number cells in Google Sheets using inbuilt functions such as the ARRAYFORMULA and SEQUENCE. Also, we can create customized formulas using other inbuilt functions, such as
OFFSET and COUNTA, for sequentially numbering the desired cells in Google Sheets.
• Numbering cells in Google Sheets helps organize and manage massive financial and statistical datasets, making the data more readable.
Numbering() In Google Sheets Formula
The Numbering() in Google Sheets is the ROW(), which has the same logic as the Excel ROW function. Its syntax in Google Sheets is the following:
• cell_reference: The cell whose row number the ROW() must return. The argument value is optional. However, when we omit it, the ROW() returns the number of the row where the cell in which we
entered the formula is located.
Furthermore, assume the cell_reference argument value is a cell range containing more than a single cell, and the expression is not an array formula. Then, in such a case, the ROW() output is the
number denoting the first row in the range supplied as the cell_reference argument value.
How To Add Numbering In Google Sheets?
We shall see three techniques of adding automatic numbering in Google Sheets
#1 – Using The Fill Handle Option
1. Please choose the top-most cell in the set of cells, which we aim to number automatically.
2. Please enter the number we wish to view in the chosen cell and press Enter.
3. Enter the second number in the required sequence we want to view in the next cell. Press Enter.
4. Choose the top two cells. Next, place the mouse cursor at the right corner at the bottom of the chosen cell range. Next, by pressing the mouse’s left key, drag the cursor up to the cell we want
to number.
Thus, in this way we can add numbering in Google Sheets in the required set of cells, with the numbers in the sequence we set in the top two cells.
#2 – Using The Addition Operator-based Customized Formula
1. Please select the top-most cell in the set of cells, which we aim to number automatically.
2. Update the number we want to view in the chosen cell and press Enter.
3. Enter the following formula in the next cell.
=previous_cell_reference + number
The number term in the above formula is the numeric value to add to the value in the cell above the current cell. We do so to set the required sequence of numbers in the target cells.
• Select the second cell. Next, place the mouse cursor at the right corner at the bottom of the chosen cell. After that, by pressing the mouse’s left key, drag the cursor to the cell we want to
#3 – Using The ROW Function-based Formulas
The steps to use the ROW()-based insert numbering in Google Sheets technique are as follows:
1. Please choose the top-most cell in the set of cells, which we aim to number automatically.
2. Type the appropriate ROW() formula to populate the chosen cell with the required number.
3. Utilize the fill handle to continue numbering in Google Sheets target cells using the ROW()-based formula.
The examples below show the different ways to add numbering in Google Sheets.
Example #1
The source dataset contains a list of the top 10 laptops in the US.
We must update their ranks, starting from 1 to 10, in column A.
Then, the steps to number the required cells using the fill handle are as follows:
Step 1: Select cell A2, enter the value of 1, and press Enter.
Step 2: Select cell A3, enter the value of 2, and press Enter.
Step 3: Select the cell range A2:A3.
Next, place the mouse cursor at the right corner at the bottom of the cell range A2:A3. Next, by pressing the mouse’s left key, drag the cursor till cell A11.
Thus, the required cells get numbered automatically, following the sequence we updated in cells A2:A3.
Example #2
The source dataset contains the date-wise inventory level data of various product categories at a store.
We must update the order numbers, from 1 to 11, in column A to complete the dataset.
Then, the steps to use the ‘+’ operator to number the targeted column A cells are as follows:
Step 1: Select cell A2, enter the value of 1, and press Enter.
Step 2: Choose cell A3, enter the following expression, and press Enter.
Step 3: Utilize the fill handle option to implement the formula in the cell range A4:A12.
The formula we entered in cells A3:A11 keeps adding the value of 1 to the previous cell value. Thus, in this way, the target cells get numbered, with each number incrementing by 1.
Example #3
We have a list of stationery items and their units sold data.
We must update their invoice numbers, from 1 to 10, in column A cells A2:A11 of the source dataset. However, the invoice number must be preceded by the text “INV_”.
Then, the steps to achieve the desired outcome using the ROW() are as follows:
Step 1: Select cell A2, enter the following ROW()-basedformula, and press Enter.
Step 2: With cell A2 chosen, utilize the fill handle option to copy the formula in cells A3:A11.
The ROW() does not have an argument value supplied to it. So, it returns the current row number in each target cell. For instance, the ROW() in the cell A11 formula returns the value of 11.
Next, the formula deducts the value of 1 from the ROW() output. So, in cell A11, the value now is 10. Finally, the expression appends the text “INV_” before the determined value to return the
required invoice number in the desired format. For example, the cell A11 formula adds the specified text before the value of 10 to return INV_10 as the required invoice number for the corresponding
stationery item.
Important Things To Note
• Ensure that the ROW()-based customized formula used for numbering in Google Sheets is correct. Otherwise, the cells may get populated with incorrect numbers or numbers, not in the desired
• Ensure that the number we update in the top-most cell of the set of target cells we aim to number in Google Sheets is correct. Otherwise, continuing the sequence in the rest of the cells with the
fill handle or the addition arithmetic operator-based formula will lead to incorrect numbering.
Frequently Asked Questions
How to do dynamic auto serial numbering in Google Sheets?
We can do dynamic auto-serial numbering in Google Sheets using the ARRAYFORMULA or SEQUENCE function.
For example, we have a list of employees.
The task is to update the car park slot for each employee, with the allotted car park slot being consecutive, even, and odd in columns B, C, and D, respectively.
Then, the steps are as follows:
Step 1: Select cell B2, enter the following ARRAYFORMULA(), and press Enter.
The formula executes as an array formula. The ROW(A2:A8) returns the row number 2 in cell B2, 3 in cell B3, and so on till 8 in cell B8. However, the expression inside the ARRAYFORMULA() deducts the
value of 1 from the respective ROW() output in each target cell. Thus, the cells B2:B8 get numbered from 1 to 7.
Step 2: Choose cell C2, enter the following SEQUENCE(), and press Enter.
The SEQUENCE() accepts four input values. The first input is the number of rows we aim to number, which is 7 (rows 2 to 8). The second input is the number of columns we want to number, which is 1
(column C). The third input is the value that must appear in the top-most cell of the concerned set of cells, which is 2 since we must display the even car park slots in column B. Finally, the last
input is the value by which we want the numbers in the subsequent cells to increment. In this case, the last argument value is 2 since we require the numbers to be even.
Thus, for the input values, the SEQUENCE() populates cells B2:B8 with the required even car park slots.
Step 3: Choose cell D2, enter the following SEQUENCE(), and press Enter.
The logic of the SEQUENCE() in column D is the same as explained in the previous step. However, while the first, second, and fourth input values remain the same, the third is different. The number to
show in the top-most cell of the concerned set of cells is 1 instead of 2. The reason is that we must display odd car park slots.
Can we customize the starting number when numbering cells in Google Sheets?
We can customize the starting number when numbering cells in Google Sheets by entering the required number in the top-most cell. After that, we can use the fill handle to continue the sequential
numbering of the required cells.
How to remove numbering in Google Sheets?
We can remove numbering in Google Sheets by first selecting the cells where we must remove the numbering. Next, right-click and use the applicable Delete option to remove the numbers from the chosen
Download Template
This article must be helpful to understand the Numbering In Google Sheets, with its formula and examples. You can download the template here to use it instantly.
Recommended Articles
Guide to What Is Numbering In Google Sheets. Here we explain how to add Numbering in Google Sheets with examples (automatically with examples). You can learn more from the following articles. – | {"url":"https://www.excelmojo.com/numbering-in-google-sheets/","timestamp":"2024-11-10T12:37:21Z","content_type":"text/html","content_length":"211586","record_id":"<urn:uuid:1dfa8f9d-73d6-4fbe-87a4-993a691d6f1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00686.warc.gz"} |
PODS 2024: Tutorials
Tutorial 1: Closing the Gap Between Theory and Practice in Query Optimization
Speaker: Thomas Neumann (TU Munich)
Query optimization, and in particular the problem of join ordering, has a huge impact on the performance of database systems. Accordingly, it has been widely studied in the literature, but there is
a, perhaps surprising, gap between techniques that have been proposed in venues like PODS and the techniques that are used in typical systems. There are several reasons for that, but one of them is
that many theoretical approaches look at asymptotic complexity, while systems tends to primarily care about the performance of a query for a given database instance in absolute terms. This tutorial
looks at the differences and tries to bring both worlds closer together.
Tutorial 2: Approximation Algorithms on Matrices - with some Database Applications!
Speaker: David Woodruff (CMU)
In this talk I will cover dimensionality reduction techniques, or sketching, for solving optimization problems on large matrices. Classical polynomial time solutions to optimization problems
involving matrices are often no longer efficient given the large size of matrices arising on big data sets. Instead, one seeks linear or sometimes even sublinear time solutions to such optimization
problems. These often involve randomized algorithms and come at the price of a small approximation error. The idea behind sketching techniques is to reduce a large instance of a problem to a much
smaller instance of the same problem in such a way that solving the small instance gives an approximate solution to the large instance. Oftentimes the smaller problem is now so small that one can
directly apply a classical polynomial time algorithm to solve it directly. Thus, the emphasis is on how to perform the reduction itself to the small problem very quickly, i.e., in linear or
sub-linear time. I will survey a wide range of technique for this, which often have the form of multiplying the input matrix by a random matrix, or have the form of sampling or recursive sampling. | {"url":"https://sigmodconf.hosting.acm.org/2024/pods_tutorial.shtml","timestamp":"2024-11-01T20:36:39Z","content_type":"application/xhtml+xml","content_length":"22398","record_id":"<urn:uuid:e0be5e72-7ebb-43fb-885e-03ff4b9c86d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00899.warc.gz"} |
Beyoncé and Guns N' Roses had the two top-grossing concert tours for 2016 , together generating \(\$ 300\) million in ticket sales. Guns \(\mathrm{N}\) ' Roses took in \(\$ 38\) million less than
Beyoncé. How much did each tour generate? (Data from Pollstar.)
Short Answer
Expert verified
Beyoncé: 169 million, Guns N' Roses: 131 million
Step by step solution
Define Variables
Let \( B \) be the amount of money generated by Beyoncé's tour and \( G \) be the amount of money generated by Guns N' Roses' tour.
Formulate Equations
Based on the given information, we can write two equations:1. \( B + G = 300 \text{ million} \)2. \( G = B - 38 \text{ million} \)
Substitute and Solve
Substitute the second equation into the first equation: \( B + (B - 38) = 300 \)Combine like terms to solve for \( B \):\( 2B - 38 = 300 \)Add 38 to both sides:\( 2B = 338 \)Divide both sides by 2:\(
B = 169 \text{ million} \)
Find Amount for Guns N' Roses
Using the value of \( B \), find \( G \) using the second equation:\( G = 169 - 38 \) \( G = 131 \text{ million} \)
Verify the Solution
Check the sum of both amounts to ensure it equals 300:\( 169 \text{ million} + 131 \text{ million} = 300 \text{ million} \)The solution is verified.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Algebra is a branch of mathematics that helps us solve problems involving unknown values, or variables. In this exercise, algebra is used to determine the amount of money generated by two concert
tours. By forming equations based on given data, we can use algebraic techniques to find the values of unknown quantities.
Understanding algebra helps us translate real-world problems into mathematical expressions. In this case, we created equations based on the total earnings and the difference in earnings. These
equations allow us to systematically solve for the unknowns.
Variable Definition
Variables are symbols used to represent unknown numbers. Here, we defined variables as follows:
• Let \(B\) be the amount of money generated by Beyoncé's tour.
• Let \(G\) be the amount of money generated by Guns N' Roses' tour.
Defining variables is an essential step in solving any algebraic problem. It helps us to clearly identify and work with the unknown values.
In this exercise, defining the variables \(B\) and \(G\) allowed us to write equations that represent the relationships between the amounts generated by the two tours.
Equation Solving
Once we have defined our variables and formulated the equations, the next step is to solve them. Here’s how we solved the equations:
1. The given equations were:
• \(B + G = 300\text { million}\)
• \(G = B - 38\text{ million}\)
2. We substituted the second equation into the first:\(B + (B - 38) = 300\)
3. By combining like terms, we simplified the equation:\(2B - 38 = 300\)
4. We added 38 to both sides, resulting in:\(2B = 338\)
5. Finally, we divided by 2:\(B = 169\text{ million}\)
These steps illustrate how we can transform and manipulate equations to find the values of our variables.
After finding the solution, it's crucial to verify it to ensure accuracy. Verification includes checking if the obtained values satisfy the original equations.
1. We calculated \(G\) using the second equation:\(G = 169 - 38\)
Which gave us:\(G = 131\text { million}\).
2. We then verified by summing up both amounts to ensure they match the total revenues:
\(169 \text { million} + 131 \text{ million} = 300 \text{ million}\).
Verification reassures us that the solution is correct and consistent with the problem's conditions. Always double-check your results. | {"url":"https://www.vaia.com/en-us/textbooks/math/beginning-and-intermediate-algebra-7-edition/chapter-2/problem-27-beyonce-and-guns-n-roses-had-the-two-top-grossing/","timestamp":"2024-11-14T05:00:44Z","content_type":"text/html","content_length":"252882","record_id":"<urn:uuid:b44c618e-e329-4002-82ce-45fae5828d72>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00205.warc.gz"} |
Is Trigonometry Hard? - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day
However, an exact answer to this question depends upon a number of factors as some people find trigonometry hard while others think it to be relatively easy. In many cases, students don’t comprehend
the problem properly, which creates all the difficulties if the problem itself is quite easy and straightforward.
In this article, we will discuss the features or course outlines which make trigonometry hard for some students and share some tips on how to overcome these difficulties.
Is Trigonometry Hard?
Trigonometry is hard for some students while others find it easy. Science students learn trigonometry at the school level, while complex or advanced trigonometry is taught in high school. High-level
trigonometry is unfortunately hard for students as it contains many formulas and becomes complex, especially when we have to find the unknown angles and values of multiple connected triangles.
Students often ask questions such as,: “Is trigonometry harder than statistics?” “Is trigonometry geometry?” “Is trigonometry harder than geometry?” “Why is trigonometry so confusing?” “Is
trigonometry important?” etc.
Let us first discuss what trigonometry means and its significance, and then we will discuss the reasons which make trigonometry hard. Hopefully, our explanation will clear out most of the questions
we mentioned above.
Trigonometry is the branch of mathematics that deals with the calculation of unknown angles and sides of right-angled triangles. The Greek mathematician Hipparchus introduced the concept of
trigonometry and it evolved over time.
Trigonometry defines six different ratios for a right-angle triangle. Using these ratios, we can find out the unknown values of the angle and sides in a right-angle triangle. The names of these six
ratios are:
1. Sine
2. Cosine
3. Tangent
4. Secant
5. Cosecant
6. Cot
The definitions of these ratios are given in the table below. We can use these definitions to determine the sides and angles of a right-angle triangle. For example, if the angle between the base and
hypotenuse is “x,” then it can be determined by using the ratio $tan(x) = \dfrac{perpedicular}{base}$ or $cos(x) = \dfrac{base}{hypotenuse}$.
Let us now discuss the reasons which make trigonometry difficult.
Difficulty of Trigonometry
Trigonometry is considered hard by students due to the following reasons:
1. The memorization of formulas and values
2. Non-linear functions
3. Angle measurement in Radians/Degree
4. Polar and Cartesian coordinates
5. Unit Circle Calculations
6. Lengthy and Complex Calculations
7. Domain and Range of trigonometric functions
8. Visualization
The Memorization of Formulas and Values
In order to be efficient in solving trigonometric problems, it is essential to memorize many formulas along with formulas and values of the trigonometric ratios. For example, you will have to learn
the values of sin, cos, tan, cot, cosec and sec at angles of $0^{o}$, $30^{o}$ ,$60^{o}$,$90^{o}$ along with other formulas.
After learning the basic formulas, students then have to memorize long and complex formulae such as the law of cosines and the law of sines, etc., and you cannot solve most of the problems in exams
unless you have learned the formulas by heart.
Learning all these formulas is a bit tedious, but instead of cramming them, a simple workaround is to do lots of practice. If you regularly solve trigonometric questions, you will realize that you
remember all the formulas effortlessly.
Non-linear Functions
As already discussed, trigonometry defines six different ratios. If we plot these ratios as a function of the angle $\theta$, we get non-linear functions, and non-linear functions are more
challenging to work with as opposed to linear functions, making it hard for students to solve questions related to trigonometry.
Also, unlike simple algebra where you use similar formulas to solve most of the problems, in trigonometry, we have varied formulas and each question requires a unique application of these formulas to
arrive at the solution. This can be confusing to the students when they first approach trigonometry. However, again, with practice, these difficulties appear to melt away, and you start to enjoy the
fact that each question has its own flavor.
Angle Measurement in Radians/Degrees
It is already tough for students to solve trigonometric equations involving angles with degrees but when they have to convert answers to radians or radians to degrees, it just makes the problem more
complex. To convert to degrees from radians, you have to multiply your answer with 180 and then divide it by $\pi$ and conversely, when you convert from degree to radians, you multiply the value with
$\pi$ and then divide it by 180.
A simple mistake or confusion in the conversion of angles can alter the values of all trigonometric functions resulting in incorrect solutions.
In some questions, you are allowed to use a calculator. You have to be mindful if the mode of the calculator is set to radians or degrees and you would have to re-adjust the mode based on the
question that you are solving. It is a common mistake for students to use the incorrect calculator mode while solving trigonometric questions, resulting in incorrect answers.
Note that the conversion between radians to degrees is not hard in itself. The difficulty lies in attention to detail. So when solving questions, keep on asking yourself if you are working with
radians or degrees and if you encounter calculations with very large or very small numbers, it is better to check if you are working with the correct units of angle.
Polar and Cartesian Coordinates
The formulas and non-linear functions alone are tough enough for students, but to make the matter more complex, the students must have a solid background in the polar and cartesian systems. For
example, students must know what is an ordered pair and what is meant by the coordinate points. If a point $(-3,2)$ is given, the student should know the value of “$x$” and “$y$” coordinates, and
furthermore, he should know in which coordinate this point lies in the cartesian system.
Trigonometric questions use the Cartesian system coordinates to solve the problems, so if you are not familiar with the cartesian system and even if you know the trigonometric functions, you won’t be
able to solve the problems.
Initial or beginner-level problems related to trigonometric equations require an understanding of the Cartesian system, but as you go further and study advance level trigonometric systems, you will
also have to deal with a polar coordinate system. The polar coordinate system has its alternative for $x$ and $y$ coordinates as “$r$” and “$\theta$”.
The polar coordinate system uses radians or degrees while plotting a function, so students not only have to deal with the conversion from cartesian coordinate to polar coordinate, but they also have
to deal with radian to degree and the degree to radian conversion when dealing with polar coordinates. This conversion, along with the trigonometric functions, makes trigonometry complex.
Unit Circle and Triangles
Trigonometry makes a lot of use of the unit circle. A unit circle is a circle having a radius of 1. Trigonometry uses the unit circle in many of its problems, and then you have to solve for the
triangles inside the unit circle.
The problem becomes complex when you start dealing with a circle having a radius greater than 1. In Trigonometry, many assumptions are made while dealing with problems involving a unit circle so such
problems become complex, and if students don’t remember the basic function of a unit circle, then they will find it very hard to solve trigonometric problems involving a unit circle.
Lengthy and Complex Calculations
Trigonometry hard questions involve lengthy and complex calculations. Some of the calculations in trigonometry can become quite long and students who like it short and easy will find it difficult to
solve such problems.
The problems become lengthy because of the calculations of all sides and angles of a given function or triangle, and to make matters worse, you might also have to deal with the conversion from radian
to degree or cartesian to polar coordinates. Some students just get confused by the sheer length of the problems in Trigonometry. It should be remembered that while the questions might be long, they
involve the same calculations over and over and a little practice and patience from the students will definitely help them overcome the difficulty.
Domain and Range of Trigonometric Functions
The domain and range of any function are the input and expected output values of the function, and the same is the case with trigonometric functions. The domain of the trigonometric function is the
value of the angles used in any of the six trigonometric functions, while the resultant value will be the range. Note that the trigonometric ratios become the trigonometric functions if we view them
as a function of the angle $\theta$.
The values of angle can have a variety of range values as they can be positive or negative, so the range changes according to that, and to make the matter more difficult, students not only have to
deal with domain and range of normal functions, they also have to find out the domain and range of the inverse of six trigonometric functions. For example, the domain and range of $tan(\theta)$ is $R
– (2n+1) \dfrac{\pi}{2}$ and $(-\infty,\infty)$ respectively while the domain and range of $tan^{-1}(\theta)$ is $(-\infty,\infty)$ and $( -\dfrac{\pi}{2}, \dfrac{\pi}{2})$.
We have mentioned only the domain and range of a general $tan(\theta)$ and its inverse function, and when we put in the value of $\theta$ and we have to convert it from radians to degree or vice
versa, things will surely get complicated. There will be open-end and close-end domains and ranges so students need to know the difference between them as well while solving problems related to
finding domains and range of trigonometric functions. So, in short, the more you go deeper into Trigonometry, the harder it becomes.
The last and final reason for trigonometry to be confusing and difficult is the concept of visualization. The branch of Trigonometry heavily relies on visualization and visual analysis. As most of
the graphs are non-linear and students are required to deduce the properties, domain and range of a given function by looking at the available graph, it becomes a difficult process and it requires
good visual analysis skills.
The students with good visual analysis skills will find it easier to comprehend a given graph or to draw the graph by using the calculated values, while students who don’t have good visual analysis
skills will find it hard to relate a given problem to a circle, triangles and other non-linear bell shaped graphs.
These are some of the reasons which make Trigonometry so confusing for students, but in general, it is easier than statistics but harder than algebra and geometry.
Let us conclude this topic by revisiting what we have learned so far.
• Trigonometry is a branch of mathematics that uses trigonometric functions to find angles and sides of right-angle triangles.
• Remembrance of various formulas, conversion from radians to degrees, the degree to radians, Cartesian to polar coordinates, along with lengthy calculations, make Trigonometry difficult for some
• Beginner-level Trigonometry is not difficult if you memorize the formulas and understand the basics of Trigonometry.
After going through the article, it will be clear to you why trigonometry is considered hard by most students. Having said that, if you are good at remembering formulas and values, you may not find
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From NNs to Transformers
The development of Transformer models in AI research is built upon a rich scientific heritage spanning several decades. Key milestones and contributions from basic neural networks (NNs) to modern
Transformer architectures include:
1. Perceptrons (1950s-1960s): The perceptron, introduced by Frank Rosenblatt in 1957, is a type of linear binary classifier that laid the foundation for neural networks. It uses a simple algorithm to
learn the weights of input features for making binary decisions.
2. Multi-layer perceptrons (MLPs) and backpropagation (1980s): Multi-layer perceptrons extend the perceptron concept to multiple layers of interconnected neurons. The backpropagation algorithm,
introduced by Rumelhart, Hinton, and Williams in 1986, allows the efficient training of MLPs by computing gradients of the error with respect to the model's parameters using the chain rule of
3. Recurrent Neural Networks (RNNs) (1980s): RNNs, introduced by John Hopfield in 1982, are a class of neural networks designed to handle sequential data. They maintain a hidden state that can
capture information from previous time steps, making them suitable for tasks involving sequences, such as time series analysis and natural language processing.
4. Long Short-Term Memory (LSTM) networks (1997): LSTMs, introduced by Hochreiter and Schmidhuber, address the vanishing gradient problem in RNNs by using a gating mechanism that allows for the
controlled flow of information through the network. This enables LSTMs to learn and model long-range dependencies in sequence data more effectively than traditional RNNs.
5. Word Embeddings (2000s): Word embeddings, such as Word2Vec (2013) by Mikolov et al. and GloVe (2014) by Pennington et al., represent words as dense vectors in a high-dimensional space. These
continuous representations capture semantic and syntactic relationships between words, making them useful for many natural language processing tasks.
6. Encoder-Decoder architecture and Attention mechanism (2014): The encoder-decoder architecture, popularized by Sutskever et al. and Cho et al. in 2014, is a two-part neural network used for
sequence-to-sequence tasks like machine translation. The same year, Bahdanau et al. introduced the attention mechanism to improve encoder-decoder performance by allowing the model to weigh different
parts of the input sequence when generating the output sequence.
7. Convolutional Neural Networks (CNNs) for text (2014): While CNNs have been widely used in computer vision since the 1980s, they were applied to natural language processing tasks by researchers
like Kim (2014), who demonstrated that CNNs could be used for sentence classification and other text-related tasks.
8. Transformer architecture (2017): Vaswani et al. introduced the Transformer architecture, which replaces the sequential processing of RNNs and LSTMs with self-attention mechanisms and positional
encodings. Transformers can process input sequences in parallel, enabling faster training and improved handling of long-range dependencies.
9. Pre-trained Language Models (2018): Models like ELMo by Peters et al., OpenAI's GPT by Radford et al., and BERT by Devlin et al. demonstrated the power of pre-training large-scale language models
on massive text corpora. These models can then be fine-tuned for specific tasks, often achieving state-of-the-art performance with relatively small amounts of labeled data.
10. GPT-3 and beyond (2020-present): OpenAI's GPT-3 is one of the largest and most powerful language models, with 175 billion parameters. It is capable of performing many tasks through few-shot
learning and prompt engineering without extensive fine-tuning. GPT-3 has demonstrated impressive performance on a wide range of natural language processing tasks, including text generation,
translation, summarization, question-answering, and more.
The scientific heritage of Transformer models in AI research is built upon decades of progress in neural networks, natural language processing, and machine learning. From basic neural networks to
advanced pre-trained models like GPT-3, the field has evolved significantly, incorporating new techniques and architectures to create more powerful and versatile models. As research continues, we can
expect to see even more advanced models and applications that push the boundaries of AI and natural language understanding.
Perceptrons and MLPs
Multi-layer Perceptrons (MLPs) are a class of feedforward artificial neural networks that consist of multiple layers of interconnected nodes or neurons. They are a foundational model in machine
learning and serve as a building block for more advanced neural network architectures. MLPs can be used for a wide range of tasks, including regression, classification, and feature extraction.
An MLP typically consists of three types of layers: an input layer, one or more hidden layers, and an output layer. Each layer contains a certain number of nodes, also known as neurons or units, that
are interconnected with nodes in the subsequent layer through weighted connections. The layers can be described as follows:
1. Input layer: The input layer takes in the features of the input data and passes them to the first hidden layer. The number of nodes in this layer corresponds to the dimensionality of the input
2. Hidden layers: Hidden layers are responsible for transforming the input data into a more abstract representation that can be used to solve the given problem. Each node in a hidden layer computes a
weighted sum of its inputs from the previous layer, applies a bias term, and then passes the result through an activation function. The activation function introduces non-linearity into the model,
allowing MLPs to learn complex, non-linear relationships between inputs and outputs. Common activation functions include the sigmoid, hyperbolic tangent (tanh), and Rectified Linear Unit (ReLU).
3. Output layer: The output layer generates the final predictions or outputs of the MLP. It is similar to the hidden layers in terms of computation, but its activation function depends on the
specific task. For regression tasks, a linear activation function can be used, while for classification tasks, a softmax function can be used to produce probabilities for each class.
Training an MLP involves optimizing the weights and biases of the connections between nodes to minimize the error between the predicted outputs and the ground truth. This is typically achieved using
the backpropagation algorithm, which computes gradients of the error with respect to the model's parameters and updates the weights and biases using an optimization method, such as stochastic
gradient descent (SGD) or an adaptive optimization algorithm like Adam.
In summary, Multi-layer Perceptrons are a foundational neural network architecture that consists of an input layer, one or more hidden layers, and an output layer. The nodes in each layer are
interconnected through weighted connections, and non-linear activation functions are used to allow the model to learn complex relationships between inputs and outputs. Training an MLP involves
optimizing the weights and biases using the backpropagation algorithm and an optimization method.
Optimization: SGD and Adam
Stochastic Gradient Descent (SGD) and Adam are optimization algorithms widely used in training deep learning models. They are iterative methods that aim to minimize a loss function by updating the
model's parameters based on gradients computed from the data.
1. Stochastic Gradient Descent (SGD):
SGD is an optimization algorithm used to minimize an objective function iteratively by updating the model's parameters using the gradient of the loss function with respect to the parameters.
a. Gradient computation: The gradient of the loss function indicates the direction of the steepest increase in the loss. It is computed using backpropagation, which calculates the gradients with
respect to each parameter by applying the chain rule of calculus.
b. Parameter update: The parameters are updated by taking a step in the opposite direction of the gradient, scaled by a learning rate (η). This step aims to minimize the loss function:
θ = θ - η * ∇L(θ)
Here, θ represents the model's parameters, η is the learning rate, and ∇L(θ) is the gradient of the loss function with respect to the parameters.
c. Mini-batch processing: In practice, SGD operates on mini-batches of data instead of individual data points or the entire dataset. This approach provides a balance between computational efficiency
and gradient estimation accuracy.
d. Learning rate scheduling: The learning rate is a crucial hyperparameter in SGD. Often, a learning rate schedule is used to decrease the learning rate over time, allowing for more aggressive steps
early in training and finer adjustments later.
2. Adam (Adaptive Moment Estimation):
Adam is an optimization algorithm that extends SGD by incorporating adaptive learning rates for individual parameters and momentum. It combines the ideas of RMSProp and momentum-based optimization
a. First moment estimation (momentum): Adam computes the exponential moving average of the gradients, which is an estimate of the first moment (mean) of the gradients:
m_t = β1 * m_(t-1) + (1 - β1) * g_t
Here, m_t is the first moment estimate at time step t, β1 is the exponential decay rate for the first moment estimate, and g_t is the gradient at time step t.
b. Second moment estimation (RMSProp): Adam also computes the exponential moving average of the squared gradients, which is an estimate of the second moment (uncentered variance) of the gradients:
v_t = β2 * v_(t-1) + (1 - β2) * g_t^2
Here, v_t is the second moment estimate at time step t, β2 is the exponential decay rate for the second moment estimate, and g_t^2 is the squared gradient at time step t.
c. Bias correction: To account for the initialization of the first and second moment estimates with zeros, bias-corrected estimates are computed:
m_t_hat = m_t / (1 - β1^t)
v_t_hat = v_t / (1 - β2^t)
d. Parameter update: The parameters are updated using the bias-corrected first and second moment estimates, scaled by an adaptive learning rate:
θ = θ - η * m_t_hat / (sqrt(v_t_hat) + ε)
Here, η is the learning rate, and ε is a small constant to prevent division by zero (typically 1e-8).
In summary, SGD and Adam are standard optimization algorithms used to train deep learning models by minimizing a loss function. SGD operates on mini-batches and updates parameters using the gradient
of the loss function, while Adam extends SGD by incorporating adaptive learning rates and momentum to improve convergence and stability.
Other Optimization Algorithms
In addition to Stochastic Gradient Descent (SGD) and Adam, there are several other optimization algorithms commonly used in deep learning. This technical overview covers some of the popular ones,
including Momentum, Nesterov Accelerated Gradient (NAG), AdaGrad, RMSProp, and AdaDelta.
1. Momentum:
Momentum is an extension of SGD that accelerates convergence by considering the past gradients. It introduces a velocity term, which accumulates past gradients with an exponential decay rate, helping
the optimization process to overcome local minima and converge faster.
Velocity update:
v_t = γ * v_(t-1) + η * ∇L(θ)
Parameter update:
θ = θ - v_t
Here, θ represents the model's parameters, η is the learning rate, ∇L(θ) is the gradient of the loss function with respect to the parameters, γ is the momentum coefficient (typically 0.9), and v_t is
the velocity at time step t.
2. Nesterov Accelerated Gradient (NAG):
NAG is a modification of the momentum algorithm that incorporates a lookahead step to improve convergence. It computes the gradient not at the current parameter values but at the approximate future
position, resulting in more accurate updates.
Approximate future position:
θ_future = θ - γ * v_(t-1)
Gradient computation:
Velocity and parameter updates are the same as in the momentum algorithm.
3. AdaGrad (Adaptive Gradient):
AdaGrad is an optimization algorithm that adapts the learning rate for each parameter based on the historical gradients. It accumulates the squared gradients element-wise in a diagonal matrix and
scales the learning rate inversely proportional to the square root of this accumulated sum.
Squared gradient accumulation:
G_t = G_(t-1) + ∇L(θ) ⊙ ∇L(θ)
Parameter update:
θ = θ - (η / sqrt(G_t + ε)) ⊙ ∇L(θ)
Here, G_t is the accumulated squared gradients at time step t, ⊙ denotes element-wise multiplication, and ε is a small constant to prevent division by zero (typically 1e-8).
4. RMSProp (Root Mean Square Propagation):
RMSProp is an optimization algorithm that addresses AdaGrad's aggressive learning rate decay for non-convex optimization problems. It computes an exponential moving average of the squared gradients
instead of accumulating them, leading to more suitable learning rate updates.
Squared gradient moving average:
E[g^2]_t = β * E[g^2]_(t-1) + (1 - β) * (∇L(θ))^2
Parameter update:
θ = θ - (η / sqrt(E[g^2]_t + ε)) ⊙ ∇L(θ)
Here, β is the exponential decay rate (typically 0.9).
5. AdaDelta
AdaDelta is an extension of RMSProp that eliminates the need for a manually set learning rate. It computes the exponential moving averages of both squared gradients and parameter updates and uses
their ratio for parameter updates.
Squared gradient moving average:
E[g^2]_t = β * E[g^2]_(t-1) + (1 - β) * (∇L(θ))^2
Parameter update:
Δθ_t = - (sqrt(E[Δθ^2]_(t-1) + ε) / sqrt(E[g^2]_t + ε)) ⊙ ∇L(θ)
θ = θ + Δθ_t
Squared update moving average:
E[Δθ^2]_t = β * E[Δθ^2]_(t-1)
6. AdaMax
AdaMax is an extension of Adam that replaces the L2 norm of the second moment estimate with an L∞ norm. This change leads to a more stable update rule, particularly for sparse gradients.
First moment estimation (same as Adam):
m_t = β1 * m_(t-1) + (1 - β1) * g_t
Second moment estimation (using L∞ norm):
u_t = max(β2 * u_(t-1), abs(g_t))
Bias correction (same as Adam):
m_t_hat = m_t / (1 - β1^t)
Parameter update:
θ = θ - η * m_t_hat / (u_t + ε)
7. AMSGrad
AMSGrad is a modification of Adam that addresses the potential lack of convergence in certain cases. It uses the maximum of all second moment estimates up to the current time step, ensuring the
learning rate remains non-increasing throughout the optimization process.
First and second moment estimations (same as Adam):
m_t = β1 * m_(t-1) + (1 - β1) * g_t
v_t = β2 * v_(t-1) + (1 - β2) * g_t^2
Max second moment estimation:
v_t_max = max(v_t_max, v_t)
Bias correction (same as Adam):
m_t_hat = m_t / (1 - β1^t)
Parameter update:
θ = θ - η * m_t_hat / (sqrt(v_t_max) + ε)
In summary, various optimization algorithms have been developed to improve the training of deep learning models, each with its strengths and limitations. These algorithms, including Momentum,
Nesterov Accelerated Gradient, AdaGrad, RMSProp, AdaDelta, AdaMax, and AMSGrad, build upon the foundation of Stochastic Gradient Descent and introduce adaptive learning rates, momentum, and other
techniques to address specific challenges in optimization, such as faster convergence, robustness to noisy gradients, and handling sparse gradients.
Backpropagation is an essential algorithm for training Multi-layer Perceptrons (MLPs) and other feedforward neural networks. It computes the gradients of the error (loss) with respect to the model's
parameters (weights and biases) and updates these parameters to minimize the error. The algorithm leverages the chain rule of calculus to efficiently compute gradients in a reverse pass through the
network, hence the name "backpropagation."
Here is a technical description of the backpropagation algorithm and the training process for an MLP:
1. Forward pass: The input data is passed through the network to generate predictions. In each layer, the neurons compute a weighted sum of their inputs, add a bias term, and pass the result through
an activation function. This process is repeated until the output layer produces the final predictions.
2. Compute loss: The loss function measures the difference between the predictions and the ground truth (target values). Common loss functions include mean squared error (MSE) for regression tasks
and cross-entropy loss for classification tasks.
3. Backward pass (Backpropagation): The backpropagation algorithm computes the gradients of the loss with respect to the model's parameters (weights and biases). It starts at the output layer and
moves backward through the network, calculating the gradients layer by layer using the chain rule of calculus.
- Compute the gradient of the loss with respect to the output layer's pre-activation values. This is obtained by taking the derivative of the loss function with respect to the output layer's
pre-activation values and depends on the specific loss function used.
- For each layer (starting from the last hidden layer to the first hidden layer):
a. Compute the gradient of the loss with respect to the layer's output values by multiplying the gradient of the loss with respect to the layer's pre-activation values by the derivative of the
activation function with respect to the pre-activation values.
b. Compute the gradient of the loss with respect to the layer's weights and biases using the gradient of the loss with respect to the layer's output values and the output values of the previous
c. Compute the gradient of the loss with respect to the layer's pre-activation values of the previous layer by multiplying the gradient of the loss with respect to the layer's output values by the
layer's weights.
4. Update parameters: Use the computed gradients to update the model's parameters (weights and biases) using an optimization algorithm. The most basic optimization algorithm is Stochastic Gradient
Descent (SGD), which updates the parameters by subtracting the gradient multiplied by a learning rate. More advanced optimization algorithms, like Adam, RMSProp, and Adagrad, adapt the learning rate
for each parameter based on their past gradients.
The process of forward pass, computing loss, backpropagation, and updating parameters is performed iteratively for a given number of epochs or until the model's performance converges. The model is
typically trained using mini-batches of input data to improve computational efficiency and make better use of available hardware.
In summary, backpropagation is a crucial algorithm for training MLPs, involving a forward pass to generate predictions, loss computation, a backward pass to compute gradients, and updating parameters
using an optimization algorithm. This process is iterated until the model's performance converges or a stopping criterion is met.
RNNs and LSTMs
Recurrent Neural Networks (RNNs) are a class of neural networks designed to process and model sequential data. Unlike feedforward neural networks, such as Multi-layer Perceptrons (MLPs), RNNs have
internal loops that allow them to maintain a hidden state over time, making them suitable for tasks involving sequences, like time series analysis, natural language processing, and speech
1. RNN architecture: The core idea behind RNNs is the recurrent connection, which allows the network to maintain a hidden state that can capture information from previous time steps. An RNN can be
thought of as a chain of repeating modules, where each module takes an input at the current time step and the hidden state from the previous time step, and produces an output and an updated hidden
2. RNN equations: Let's denote the input at time step 't' as 'x_t', the hidden state at time step 't' as 'h_t', and the output at time step 't' as 'y_t'. The RNN computes the hidden state and output
at each time step using the following equations:
- h_t = activation_function(W_hh * h_(t-1) + W_xh * x_t + b_h)
- y_t = W_hy * h_t + b_y
Here, W_hh, W_xh, and W_hy are weight matrices, b_h and b_y are bias vectors, and 'activation_function' is a non-linear function, such as tanh or ReLU.
3. Forward pass: To compute the outputs of an RNN, perform the following steps for each time step in the input sequence:
- Update the hidden state using the current input and the previous hidden state.
- Compute the output using the updated hidden state.
4. Loss computation: The loss function measures the difference between the RNN's outputs and the target values. For sequence-to-sequence tasks, the loss is typically computed at each time step and
then averaged over the entire sequence. Common loss functions include mean squared error (MSE) for regression tasks and cross-entropy loss for classification tasks.
5. Backpropagation through time (BPTT): RNNs are trained using a variant of the backpropagation algorithm called backpropagation through time (BPTT). BPTT computes the gradients of the loss with
respect to the model's parameters by unfolding the RNN through time and applying the chain rule of calculus to calculate gradients at each time step. The gradients are then used to update the RNN's
parameters using an optimization algorithm, such as SGD or Adam.
6. Vanishing and exploding gradients: RNNs can suffer from the vanishing and exploding gradient problem, which makes it difficult to learn long-range dependencies in the input sequence. The gradients
can either become too small (vanish) or too large (explode) when propagated through many time steps, causing slow convergence or unstable training.
7. Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) cells: LSTMs and GRUs are specialized RNN cells designed to mitigate the vanishing gradient problem. They use gating mechanisms to
control the flow of information through the network, allowing the model to learn long-range dependencies more effectively.
Recurrent Neural Networks are a class of neural networks for processing sequential data. They maintain a hidden state over time, allowing them to capture temporal relationships in input sequences.
RNNs are trained using a variant of backpropagation called backpropagation through time (BPTT). However, they can suffer from vanishing and exploding gradients, which can be mitigated using
specialized cells like Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU).
8. Bidirectional RNNs: Bidirectional RNNs are a variation of RNNs that process the input sequence in both forward and backward directions. They consist of two separate RNNs, one processing the input
from the start to the end and the other from the end to the start. The hidden states from both RNNs are combined at each time step to produce the output. Bidirectional RNNs can capture both past and
future context, making them more effective at tasks like sequence labeling and machine translation.
9. Sequence-to-sequence (seq2seq) models: Seq2seq models are a popular application of RNNs for tasks that require mapping an input sequence to an output sequence, such as machine translation and
speech recognition. A seq2seq model typically consists of an encoder RNN, which processes the input sequence and generates a context vector, and a decoder RNN, which uses the context vector to
generate the output sequence.
10. Attention mechanisms: Attention mechanisms are a powerful extension to RNNs, particularly for seq2seq models. They allow the model to weigh different parts of the input sequence when generating
the output sequence, effectively enabling the model to focus on relevant information. Attention mechanisms can improve the performance of RNN-based models on tasks with long sequences and complex
dependencies, such as machine translation and summarization.
To implement and train an RNN in practice, programmers can use popular deep learning frameworks like TensorFlow, PyTorch, or Keras. These frameworks provide built-in support for RNNs, LSTMs, GRUs,
and attention mechanisms, as well as tools for gradient computation, parameter optimization, and GPU acceleration.
Recurrent Neural Networks are a powerful tool for modeling and processing sequential data. They can capture temporal dependencies and have been successfully applied to various tasks, including
natural language processing, speech recognition, and time series analysis. RNNs can be extended with specialized cells like LSTMs and GRUs, bidirectional processing, and attention mechanisms to
improve their performance and overcome limitations such as the vanishing gradient problem.
Word Embeddings
Word embeddings are dense vector representations of words that capture their semantic and syntactic meaning in a continuous vector space. They are widely used in natural language processing (NLP)
tasks, as they enable models to efficiently process textual data and capture the relationships between words. Word embeddings can be learned using unsupervised or supervised techniques, with popular
methods including Word2Vec, GloVe, and FastText.
1. Motivation: Traditional text representation techniques, such as one-hot encoding and bag-of-words, suffer from high dimensionality and sparsity. They also fail to capture the semantic
relationships between words. Word embeddings address these issues by representing words as continuous, dense vectors with fixed dimensions. These dense vectors can capture semantic and syntactic
relationships, allowing models to generalize better and perform more complex reasoning.
2. Word2Vec: Word2Vec is a popular unsupervised technique for learning word embeddings. It consists of two main architectures: Continuous Bag-of-Words (CBOW) and Skip-Gram. Both architectures learn
word embeddings by predicting a target word based on its context (surrounding words) or vice versa. The main difference is that CBOW predicts the target word using the context words' average, while
Skip-Gram predicts context words using the target word.
3. GloVe: Global Vectors for Word Representation (GloVe) is another unsupervised technique for learning word embeddings. GloVe builds on the idea of co-occurrence matrices and factorizes a matrix of
word co-occurrence probabilities to learn word embeddings. This approach allows GloVe to capture both global and local semantic relationships between words.
4. FastText: FastText is an extension of the Word2Vec approach that learns embeddings for subword units (n-grams) instead of entire words. This allows FastText to generate embeddings for
out-of-vocabulary words and capture morphological information, making it suitable for languages with rich morphology and large vocabularies.
5. Preprocessing and training: To learn word embeddings, text data must be preprocessed, typically including tokenization, lowercasing, and removal of stopwords and rare words. The preprocessed text
is then used to generate training examples based on a sliding window approach. For example, with a window size of 2, the context words for the word "cat" in the sentence "The quick brown cat jumped
over the lazy dog" would be ["quick", "brown", "jumped", "over"]. The embeddings are learned by optimizing an objective function (e.g., negative log-likelihood) using stochastic gradient descent or
other optimization algorithms.
6. Dimensionality and similarity: Word embeddings typically have a fixed dimensionality, ranging from 50 to 300 dimensions. The choice of dimensionality depends on the task and dataset size, with
larger dimensions capturing more information at the cost of increased computational complexity. The similarity between word embeddings can be measured using cosine similarity, Euclidean distance, or
other distance metrics.
7. Transfer learning and pre-trained embeddings: Pre-trained word embeddings, such as Word2Vec, GloVe, and FastText, have been trained on large text corpora and can be used as a starting point for
downstream NLP tasks. Transfer learning with pre-trained embeddings can lead to faster convergence and improved performance, especially when training data is limited.
In summary, word embeddings are dense vector representations of words that capture semantic and syntactic relationships in a continuous vector space. They are a powerful tool for natural language
processing tasks, enabling models to efficiently process text and generalize better. Popular methods for learning word embeddings include Word2Vec, GloVe, and FastText. Pre-trained embeddings can be
used for transfer learning to improve performance on downstream tasks.
Encoder-Decoder Architecture and Attention mechanism
Encoder-decoder architectures and attention mechanisms are essential components in modern neural network-based systems for tasks that involve mapping one sequence to another, such as machine
translation, summarization, and speech recognition. The encoder-decoder architecture is a two-part neural network that encodes the input sequence into a fixed-size vector and then decodes it into an
output sequence. Attention mechanisms improve this process by allowing the decoder to focus on relevant parts of the input sequence.
1. Encoder-decoder architecture: The encoder-decoder architecture consists of two main components:
a. Encoder: The encoder is typically a Recurrent Neural Network (RNN), such as an LSTM or GRU, or a Transformer-based model that processes the input sequence and generates a context vector. This
context vector is a fixed-size representation of the input sequence, which captures its essential information.
b. Decoder: The decoder is also usually an RNN or a Transformer-based model that takes the context vector generated by the encoder and produces the output sequence. The decoder generates the output
sequence one element at a time, conditioning its predictions on the context vector and the previously generated elements.
2. Limitations of fixed-size context vectors: One limitation of the basic encoder-decoder architecture is that it relies on a fixed-size context vector to represent the entire input sequence. For
long sequences or sequences with complex dependencies, the context vector may not capture all the necessary information, leading to poor performance.
3. Attention mechanisms: Attention mechanisms address the limitations of fixed-size context vectors by allowing the decoder to dynamically focus on different parts of the input sequence when
generating the output sequence. Instead of using a single context vector, the attention mechanism computes a weighted sum of the encoder's hidden states at each decoding step, with the weights
determined by an attention score function.
4. Types of attention mechanisms:
a. Dot-product attention: This attention mechanism computes the attention scores by taking the dot product of the decoder's hidden state and the encoder's hidden states. The dot product measures the
similarity between the decoder's hidden state and each encoder's hidden state, giving higher weights to more similar states.
b. Scaled dot-product attention: This is a variant of dot-product attention used in the Transformer architecture, where the dot product is scaled by the square root of the hidden state dimension.
This scaling helps stabilize gradients during training.
c. Additive attention (Bahdanau attention): This attention mechanism computes the attention scores using a trainable feedforward neural network that takes the decoder's hidden state and the encoder's
hidden states as inputs. The neural network learns to compute the attention scores that result in the best performance on the target task.
5. Incorporating attention into the encoder-decoder architecture: To use an attention mechanism in an encoder-decoder architecture, modify the decoder to compute the attention scores and the weighted
sum of the encoder's hidden states at each decoding step. The weighted sum, also known as the context vector, is then used in combination with the decoder's hidden state to generate the output
6. Benefits of attention mechanisms: Attention mechanisms have several benefits for sequence-to-sequence tasks:
a. Improved performance on long sequences and complex dependencies.
b. Faster convergence during training, as attention allows the model to focus on relevant parts of the input sequence.
c. Interpretability, as the attention scores can be visualized to understand which parts of the input sequence the model focuses on when generating the output sequence.
7. Transformer architecture: The Transformer architecture, introduced by Vaswani et al. (2017), is a powerful alternative to RNN-based encoder-decoder models that relies solely on attention
mechanisms. Transformers use self-attention in both the encoder and decoder, allowing them to process input and output sequences in parallel, which can result in faster training and improved
performance on long sequences. The encoder and decoder in a Transformer consist of multiple layers of multi-head self-attention, position-wise feedforward networks, and layer normalization.
8. Multi-head attention: Multi-head attention is a technique used in the Transformer architecture to capture different aspects of the relationships between words in a sequence. Instead of computing a
single attention score, the model computes multiple attention scores using different learned linear projections of the input vectors. The resulting context vectors from each head are then
concatenated and projected to generate the final output. Multi-head attention allows the model to capture various types of dependencies and relationships between words in a sequence.
9. Positional encoding: Since the Transformer architecture does not have any inherent notion of the order of elements in a sequence, positional encoding is used to inject positional information into
the input embeddings. Positional encoding can be done using sinusoidal functions or learned positional embeddings. The positional encodings are added to the input word embeddings before they are fed
into the model, allowing the Transformer to capture both content and positional information.
10. Applications of encoder-decoder architectures with attention: Encoder-decoder architectures with attention mechanisms have been successfully applied to a wide range of sequence-to-sequence tasks,
a. Machine translation: Translating text from one language to another.
b. Summarization: Generating a concise summary of a given text.
c. Speech recognition: Converting spoken language into written text.
d. Image captioning: Generating textual descriptions of images.
e. Conversational AI: Building chatbots and dialogue systems that can carry on a conversation with humans.
In conclusion, encoder-decoder architectures and attention mechanisms are essential components in modern neural network-based systems for sequence-to-sequence tasks. Attention mechanisms allow the
model to focus on relevant parts of the input sequence, resulting in improved performance on long sequences and complex dependencies. The Transformer architecture is a powerful alternative to
RNN-based models that relies on attention mechanisms, enabling faster training and improved performance on a wide range of tasks.
Convolutional Neural Networks (CNNs) are a class of deep learning models designed to efficiently process grid-like data, such as images, audio spectrograms, and time series. They are particularly
effective at capturing local patterns and hierarchies within data, making them suitable for tasks like image recognition, object detection, and natural language processing. CNNs consist of several
layers, including convolutional, pooling, and fully connected layers, which work together to extract meaningful features and make predictions.
1. Convolutional layers: Convolutional layers are the core building blocks of CNNs. They consist of multiple filters (also known as kernels) that are applied to the input data through a convolution
operation. This operation involves sliding the filter over the input data and computing the element-wise product and sum between the filter and the input at each location. Convolutional layers learn
to detect local patterns, such as edges, corners, and textures, by adjusting the filter weights during training.
2. Stride and padding: The stride is the step size by which the filter moves across the input data during convolution. A larger stride results in a smaller output size, reducing the computational
complexity at the cost of potentially losing some information. Padding involves adding extra pixels or data points around the input to control the output size. There are two common types of padding:
"valid" padding, which does not add any padding, and "same" padding, which adds padding such that the output size remains the same as the input size.
3. Activation functions: After the convolution operation, an activation function is applied to introduce non-linearity into the model. Common activation functions include ReLU (Rectified Linear
Unit), sigmoid, and tanh. ReLU is particularly popular in CNNs due to its simplicity and effectiveness at mitigating the vanishing gradient problem.
4. Pooling layers: Pooling layers are used to downsample the input data, reducing its spatial dimensions and computational complexity. They aggregate local information by applying a pooling
operation, such as max pooling or average pooling, over non-overlapping regions of the input data. Max pooling, for example, takes the maximum value within each region, effectively preserving the
most important features while discarding redundant information.
5. Fully connected layers: Fully connected layers are used in the final stages of a CNN to combine the extracted features and produce the output. These layers are similar to those used in Multilayer
Perceptrons (MLPs) and are often followed by a softmax activation function to generate class probabilities for classification tasks.
6. Dropout: Dropout is a regularization technique used to prevent overfitting in neural networks, including CNNs. During training, dropout randomly sets a fraction of the input units to zero at each
update, effectively forcing the model to learn redundant representations and improving its generalization capabilities.
7. CNN architectures: Several popular CNN architectures have been developed over the years, such as LeNet, AlexNet, VGGNet, ResNet, and Inception. These architectures differ in their layer
configurations, depth, and design principles but share the common goal of efficiently processing grid-like data and capturing hierarchical features.
8. Training CNNs: CNNs are typically trained using stochastic gradient descent (SGD) or its variants, such as Adam and RMSProp. The model learns by minimizing a loss function, such as cross-entropy
for classification tasks, which measures the discrepancy between the predicted and true labels. Backpropagation is used to compute gradients with respect to the model's parameters, which are then
updated using the chosen optimization algorithm.
9. Batch normalization: Batch normalization is a technique used to improve the training of CNNs by normalizing the activations of each layer. By ensuring that the input to each layer has a mean of
zero and a standard deviation of one, batch normalization helps mitigate the internal covariate shift problem, which occurs when the distribution of inputs to a layer changes during training. This
leads to faster convergence, improved generalization, and allows the use of higher learning rates.
10. Residual connections: Residual connections, introduced in the ResNet architecture, are a technique to address the degradation problem that occurs when training very deep CNNs. Degradation refers
to the decrease in performance as the network depth increases. Residual connections involve adding the input of a layer (or a group of layers) to its output, effectively allowing the model to learn
residual functions that capture the difference between the input and output. This makes it easier for the network to learn identity functions when necessary, enabling the training of much deeper
models without performance degradation.
11. Dilated convolutions: Dilated convolutions, also known as atrous convolutions, are a variant of the standard convolution operation that incorporates a dilation factor. The dilation factor
determines the spacing between the values in the filter, effectively allowing the filter to cover a larger receptive field without increasing the number of parameters. Dilated convolutions are
particularly useful for tasks that require capturing information from larger contexts, such as semantic segmentation and image synthesis.
12. Applications of CNNs: CNNs have been successfully applied to a wide range of tasks, including:
a. Image classification: Assigning a label to an image based on its content.
b. Object detection: Identifying and localizing objects within an image.
c. Semantic segmentation: Labeling each pixel in an image with the class of the object it belongs to.
d. Style transfer: Combining the content of one image with the style of another image.
e. Natural language processing: Processing and understanding text data using 1D CNNs or character-level CNNs.
f. Speech recognition: Converting spoken language into written text using 1D CNNs on audio spectrograms.
In conclusion, Convolutional Neural Networks are a versatile and powerful class of deep learning models, capable of processing grid-like data and capturing local patterns and hierarchies. Key
components and techniques used in CNNs include filters, activation functions, stride, padding, dropout, batch normalization, residual connections, and dilated convolutions. CNNs have been applied to
a wide range of tasks across various domains, including image classification, object detection, semantic segmentation, style transfer, natural language processing, and speech recognition.
Transformers are a class of deep learning models that have revolutionized the field of natural language processing (NLP) and sequence-to-sequence tasks. Introduced by Vaswani et al. in 2017,
Transformers rely on self-attention mechanisms to process input sequences in parallel, resulting in faster training and better performance on long sequences compared to traditional recurrent neural
networks (RNNs) and convolutional neural networks (CNNs). Transformers have become the foundation of many state-of-the-art models, such as BERT, GPT, and T5.
1. Architecture: The Transformer architecture is composed of an encoder and a decoder, both of which consist of multiple identical layers. Each layer in the encoder and decoder contains a multi-head
self-attention mechanism, a position-wise feedforward network, and layer normalization.
2. Self-attention mechanism: Self-attention is the core component of the Transformer model. It allows the model to weigh the importance of different tokens in the input sequence relative to a
specific token. The self-attention mechanism computes three linear projections of the input embeddings: the query (Q), key (K), and value (V) matrices. The attention scores are calculated as the dot
product of the query and key matrices, scaled by the square root of the key dimension, and followed by a softmax activation to produce the attention weights. These weights are then applied to the
value matrix to generate the attention output.
3. Multi-head attention: Multi-head attention is an extension of the self-attention mechanism, which computes multiple attention outputs using different learned linear projections of the input
embeddings. The resulting attention outputs from each head are concatenated and linearly transformed to produce the final output. Multi-head attention allows the model to capture various aspects of
the relationships between tokens in a sequence.
4. Position-wise feedforward networks: These are fully connected feedforward networks that are applied to each token's output from the self-attention mechanism independently. They consist of two
linear layers with a non-linear activation function (e.g., ReLU) in between. The purpose of position-wise feedforward networks is to introduce non-linearity and model complex interactions between
5. Layer normalization: Layer normalization is a technique used to stabilize the training of deep neural networks by normalizing the activations of each layer. It computes the mean and standard
deviation of the activations across the feature dimension and normalizes them to have zero mean and unit variance. In Transformers, layer normalization is applied after the self-attention mechanism
and the position-wise feedforward networks.
6. Positional encoding: Transformers do not have an inherent notion of the order of tokens in a sequence. Therefore, positional encoding is used to inject positional information into the input
embeddings. Positional encoding can be done using sinusoidal functions or learned positional embeddings. The positional encodings are added to the input word embeddings before they are fed into the
model, allowing the Transformer to capture both content and positional information.
7. Training: Transformers are trained using standard optimization algorithms like stochastic gradient descent (SGD) or Adam. The model learns by minimizing a loss function, such as cross-entropy for
classification or sequence generation tasks. Gradients are computed with respect to the model's parameters using backpropagation and updated using the chosen optimization algorithm.
8. Applications: Transformers have been successfully applied to a wide range of NLP and sequence-to-sequence tasks, including machine translation, text summarization, question answering, text
generation, and more. Pre-trained Transformer models, such as BERT and GPT, have been fine-tuned for specific tasks, achieving state-of-the-art performance across various benchmarks.
Pre-trained Language Models and GPT
Pre-trained Language Models (PLMs) are a class of deep learning models that have been trained on large amounts of text data to learn representations and patterns in natural language. These models can
then be fine-tuned for specific tasks, such as text classification, machine translation, or question answering, by training them on smaller, task-specific labeled datasets. The idea behind
pre-training is to leverage the vast knowledge encoded in the model from pre-training and adapt it to a wide range of tasks with relatively small amounts of labeled data.
GPT (Generative Pre-trained Transformer) is one such pre-trained language model, based on the Transformer architecture, that has achieved state-of-the-art performance across various natural language
processing tasks.
Here is a technical introduction to Pre-trained Language Models and GPT, suitable for programmers:
1. Pre-training: Pre-training involves training a language model on a large corpus of unlabeled text data. The objective during pre-training is to predict the next word in a sequence given the
previous words, also known as the language modeling task. The model learns to generate contextually appropriate words and, in the process, captures rich linguistic information about syntax,
semantics, and world knowledge.
2. GPT architecture: GPT is based on the Transformer architecture, specifically the decoder part of the original Transformer. It consists of a stack of identical layers, each containing multi-head
self-attention and position-wise feedforward networks, along with layer normalization and residual connections. GPT also incorporates positional encoding to capture the order of tokens in a sequence.
3. Masked self-attention: Unlike the original Transformer, GPT uses masked self-attention to ensure that the model cannot access future tokens during the pre-training and fine-tuning phases. This
masking ensures that the model learns to generate text in an autoregressive manner, predicting one token at a time based on the previous tokens.
4. Fine-tuning: After pre-training, GPT can be fine-tuned on a specific task using task-specific labeled data. During fine-tuning, the input sequence is formatted according to the task, and the
output layer is adapted to produce task-specific predictions. For example, for a text classification task, a special classification token is added to the input sequence, and the final hidden state
corresponding to this token is used to produce a probability distribution over the classes using a linear layer followed by a softmax activation.
5. Transfer learning: The process of adapting a pre-trained model to a specific task is called transfer learning. The idea is that the knowledge captured in the pre-trained model can be effectively
transferred to the target task, often leading to better performance compared to training a model from scratch on the task-specific data.
6. Versions of GPT: There have been several versions of GPT, with each subsequent version featuring a larger architecture and trained on more data. For example, GPT-3, the third version of GPT, has
175 billion parameters and has been trained on hundreds of gigabytes of text data, making it one of the largest and most powerful language models to date.
Pre-trained Language Models like GPT leverage large-scale unsupervised learning on vast text corpora to capture rich linguistic information. GPT, based on the Transformer architecture, is pre-trained
using a masked language modeling task and can be fine-tuned for various specific tasks using smaller labeled datasets. The process of transferring knowledge from a pre-trained model to a target task
is called transfer learning, and it has proven to be highly effective in achieving state-of-the-art performance across a wide range of natural language processing tasks. | {"url":"https://netrunner.one/0022","timestamp":"2024-11-10T08:31:37Z","content_type":"text/html","content_length":"66071","record_id":"<urn:uuid:372d1539-f341-4c35-af53-7627175fdb5f>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00450.warc.gz"} |
wall angle calculator online
Beam angle online calculator. Use the number boxes to enter the value in feet, inches, and fraction, inches and fraction. Calculate unknown angles or lengths by entering ANY TWO (2) known variables
into the text boxes. From Figure 30.3-1, page 335 Kzt = Kzt default = 1.00 Refer to section 26.8 and figure 26.8-1 to determine the wind speed-up effect. An important factor here is also the distance
between the light source and the illuminated object. You can find roof pitch degrees manually and it is not very difficult either. All you need to do is just enter the Calculate button: There is more
than one way to find a roof pitch. Especially due to the influence of the distance between lamp and floor or wall, an estimation is often difficult. From Wikipedia, the free encyclopedia – Roof pitch
– pitch of a roof is its vertical ‘rise’ over its horizontal ‘span’, From the source of hunker – By MICHAEL LOGAN – Use the roof pitch to find the height of a roof peak – How to Figure the Center
Height of Rafters, The source of carpentry-pro-framer provided – how to calculate roof pitch – Find Existing Roof Pitch, By DELAURIER ROOFING – What Kind of Roof Pitch Is Best For Your Home – The
Importance of roof pitch. Feel Free to Contact. ; Step 2 Use SOHCAHTOA to decide which one of Sine, Cosine or Tangent to use in this question. Decorative lighting is often used to highlight certain
objects in a room. Roof pitch chart is based on the rise, span and slope. The full luminous intensity (100%) is only achieved exactly in the center of the light circle. Use wall quotes to display,
room names, walls of fame, children’s names, company names and slogans, emergency exits, fire extinguisher designators, and building directories. Accent lighting is used to highlight certain areas in
a room. - 1 right angle (90°) - The opposite side to the right angle is called the hypotenuse. Two boxes will appear. They emit their entire luminous flux in only one specific direction. As far as
any shape and size of the views are concerned, there is no difference between these two systems. Determine the desired length for one side of the angle. It gives you the required angle, length of the
rafter and rise to span ratio. Wind Load Calculator. Now you have to click the calculate button of the above roof angle calculator. This roof pitch calculator is an easily accessible tool that will
assist you to measure the pitch and length of the rafters you are going to need for construction. The beam angle determines how large the generated light circle appears on the illuminated object. It
can be either m, ft., in, or yd. Calculating how fast I was going when I hit the ground after free falling off of a rock wall. Right after you have to add the value of run in the given space. ...
Heat Conduction through a Wall Equation and Calculator. The beam angle of an LED spot determines the diameter of the generated light circle on the illuminated surface or object. This simple,
easy-to-use moment of inertia calculator will find moment of inertia for a circle, rectangle, hollow rectangular section (HSS), hollow circular section, triangle, I-Beam, T-Beam, L-Sections (angles)
and channel sections, as well as centroid, section modulus and many more results. A 4/12 roof pitch is referred to as the roof rises 4 inches in height for every 12 inches, as it measured
horizontally from the edge of the roof to the centerline. Calculation of roofs: pitched, gable, gambrel . Click on the “Calculate” button to solve for all unknown variables. When the Co-Browse window
opens, give the session ID that is located in the toolbar to the representative. in the case of shingles, the minimum roof pitch is 1:6. Ø: Diameter of the object or surface to be illuminated, d:
Distance between lamp and object or surface, arctan: Inverse function of the tangent for angle calculation. All you need to do is just to rearrange the equations. In case of a Circular Pitch you need
to apply following formula: Center Distance C = Np (mG + 1) / 2P C = ( Dp + DG ) / 2 C = ( NG + Np ) / 2P C = (NG + Np) p / 2P C = (NG + Np) p / 6.2832, P = π / p P = N / D P = [ Np ( mG + 1) ] / 2C.
All types of online civil engineering design surveying calculations for highways, roadways and concrete calculations are made easier here. The tools in this website are provided "as is" without any
warranty of any kind. 2) Now click the end of the first Wall and start moving your mouse to start placing the second Wall. Example 1: Suppose that a 10 meter ladder is leaning against a building such
that the angle of elevation from ground to the building is 62 degrees. The angle should be selected according to the application as described above. How Do Light Emitting Diodes And LED Lights Work?
The gentle slope of a 4/12 roof pitch falls on the cusp between moderate-pitch and low-pitch. We can use it as a roof pitch finder as well. Tip: You will need a scientific calculator or tangent chart
to calculate the proper length of each line to match your desired angle. In this magazine you will find all about modern LED lighting.More about LampHQ.com, Copyright © LampHQ.com | About | Privacy
Policy | Contact. The Calculator automatically determines the number of correct digits in the operation result, and returns its precise result. ASCE 7-16 design pressure calculator. This can be
determined very well with the beam angle calculator. Using the Roof Pitch Degree Calculator program, you can make sure that your house or shed or whatever it is safe and balanced. It is named
according to the quadrant in this which the object is imagined no place for purposes of line projection. Also, you can use the online gravel calculator that helps you to calculate the volume (cubic
yards, cubic ft) and weight (tons, lbs) of gravel required to complete your project. The most common stud spacing with wall framing are 16", 19.2" and 24" on centers. Isosceles triangle rectangle: -
1 right angle - 2 equal sides. You will have a value of rise to run ratio as a third output. To ward off drainage problems, homeowners should have turn to roofing systems that can be sealed. Got a
School Code? Also try cos and cos-1.And tan and tan-1. For the water to run off there is always going to be little slop. It will give you a roof pitch angle and a ratio between rise and run in the
form of x:12. This can be a sitting area or a colored wall. After they're up, you can snap a line to mark and trim the tails in a nice straight line, to make up for any ridge warp or marking, cutting
and fixing errors. Informative Calculators ▶ Roof Pitch Calculator, For further assistance, please contact us. Now you have chosen the Rise and Run option, all you need to do is to fill in these two
different blanks. The percentage ratio between the rise and the run of the roof in the form of x:12. Here you can calculate the approximate illuminance or light output using the given beam angle and
the distance between luminaire and illuminated plane (floor, table, work surface, wall). If you can write it out, you can print a wall quote for it. The ideal beam angle usually depends very
individually on the place of use of the lamp and on your own use-case. In the living area there are mainly these three lighting applications: A beam angle of 120° is a good choice for the basic
lighting of a room. A: By taking three measurements for the adjoining Walls, Solid can calculate the angle for you. The smallest pitch for a standing seam metal roof is 1:4. This Roof Pitch
Calculator is a miraculous tool that will quickly guess the precise roof slope from every angle. This tool uses formulas from IPC-2221 to calculate the width of a copper printed circuit board
conductor or "trace" required to carry a given current while keeping the resulting increase in trace temperature below a specified limit. How high does the ladder reach? In the light circle, the beam
angle defines the area where the lamp radiates at least half (50%) of its maximum luminous intensity. SoilStructure RAPID RETAIN, version 4 is a tool for designing Abutment Wall, Wing Wall,
Cantilever Retaining Wall & Basement Wall. (Make sure to include shoe height) The beam angle of LED luminaires and illuminants determines how large the light cone appears in the room. Here you can
find out what the bream angle in degrees means and what you should pay attention to when choosing a directional LED light. 3) Before you click the mouse to place the second Wall, click the F3 button
on your keyboard. As a second output, you will have a Roof pitch angel. A 5/12 roof pitch angle is equal to 22.62 degrees. GoodCalculators.com A collection of really good online calculators for … In
the Roof pitch calculator application, you can also use Angle in Degrees feature. Click here Calculators for Education & Industry Graphing | Scientific | Financial SHOP NOW Calculator Engraving
Available, Extra $10 SHOP NOW TI-Nspire™ CX II GraphingCalculators Faster Performance | Improved Graphics | Added Functions SHOP NOW Calculators VIEW ALL TI-Innovator VIEW ALL ARKON VIEW ALL
Calculated Industries VIEW ALL … Examples of the angle of a slope include such things as the angle of the driveway, the pitch of a roof, the angle of a hill and so on. I have more than 7 years of
experience in Internet Marketing. It can also be calculated in terms of degree. A pitch 1:12 represents that per every 36 feet of construction length the rise will be equivalent to 3 feet. GU10 LED
illuminants are also available with different beam angles. It is recognized as a standard roof pitch. Basically, a very simple calculation, convert the wall length in feet to wall length in inches
and divide this amount by the on centre stud spacing, then add one more to this amount. The state of the luminaire or the lamp glass where the GU10 lamp is used is also important. Besides the beam
angle there is also the so-called field angle. This is valuable because sometimes it is essential to cut the bevel via spherical saw and settings on the saw are standardized in degrees as well.
HomeAdvisor's Tile Calculator helps you figure out how much floor, shower, bathroom or wall tile you need. α is the surface inclination angle. Typically, the slate shingles or standard asphalt
shingles roof are used in moderate to high-pitched roofs cannot sustain the health of a low-roof-pitch! Also, find the distance from the ground to the top of the ladder. First, of fall you have to
measure the run length. It is also known as pitch.Pitch: it is an incline of the roof. Angle: It will be signified in degrees as an output. The angle made by a beam with the horizontal surface; The
percentage ratio between the rise and the run of the roof in the form of x:12. This will help you to carry your project in all the depth and detail and make sure that it will maintain an effective
and sturdy nature throughout. Use on interactive whiteboards, angles can be automatically shown or measured with a protractor. In order for a structure to be sound and secure, the foundation, roof,
and walls must be strong and wind resistant. High precision calculator (Calculator) allows you to specify the number of operation digits (from 6 to 130) in the calculation of formula. LampHQ.com is a
participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to products on
Amazon.com and affiliated sites. This can be constructed only with the help of built-up roofing or some particular artificial roofing. You can assess it in dual ways: Also, you can use the online
gravel calculator that helps you to calculate the volume (cubic yards, cubic ft) and weight (tons, lbs) of gravel required to complete your project. Thank you for your questionnaire. Please select
your height from the drop down listed above. 4:12 to 9:12 range of the pitch represents conventional roofs. Calculation of foundations, walls, concrete, rebar. To get an idea about different roof
pitch angles, you can take a look at roof pitch chart! The beam angle calculator helps you to determine the correct angle. The following formula and the calculator will help you to determine the best
fitting beam angle. Add Roof Pitch Calculator to your website through which the user of the website will get the ease of utilizing calculator directly. This free triangle calculator computes the
edges, angles, area, height, perimeter, median, as well as other values of a triangle. LED Beam Angle Guide – With Beam Angle Calculator, Meaning of a beam angle of 15°, 60° or 120° degrees,
Conclusion: The optical impression is important, Convert Lumens to Watts and Watts to Lumens the Right Way. Roof pitch can be efficiently calculated with the rise and run using the simple formula:
Once you know the rise and run, you can apply Pythagorean Theorem to determine rafter, roof angle, and grade using the following formulae: Roof angle can also be determined using the following: Pitch
represented in percentage is known as grade and is given by: It can be calculated either using rafter and rise or rafter and run, both using the Pythagorean theorem as a roof pitch finder. The rise
is the distance from the top of a stud wall to the peak of the roof. The beam angle is primarily interesting for directional light sources. It can assist you in the calculation of slope, area,
rafter’s length, and other significantly important dimensions. The Value of Pitch will be given in the form of X:12 as the last output. If you have the values for the rise and rafter, you can find
run using the following: Similarly, if you know rafter and run you can find rise using the following: If you don’t have the values to find the rise you can do it manually or by estimating the rise by
measuring 3 lengths of side. 6063ST1X1/8 Aluminum Square Tube 6063 T52 1 " X 0.125 " X 288 "6063ST3X1/4 Aluminum Square Tube 6063 T52 3 " X 0.25 " X 288 "6063ST1-1/4X1/8 Aluminum Square Tube 6063 T52
1.25 " … Thus, a moderate 6 in 12 roof pitch means the roof rises 6 inches for every 12 horizontal inches it runs. The pitch which is below 4:12 represents a low roof. The table gives you an overview
of the diameter of the light circle with different beam angles and a ceiling height of 8 feet. If an LED spotlight is not far away from a wall for example, only a small dotted light circle will be
visible. Roof pitch can be described as the incline that is created by the rafter. The field angle defines the outer area in the light circle where the lamp radiates up to one tenth (10%) of its
maximum luminous intensity. Co-Browse. For corridors and pathways within a room, a beam angle of 90° is more recommended. LED luminaires and illuminants have both omnidirectional and directional
light sources. First of all, you have to enter the value of the rise in the given place. Step By Step. However, carpenters typically make use of diverse finders of roof pitch angles, an example of
that is being a rafter. If you want to convert from degrees to the American ratio, then simply find the tangent of the angle. Roof Pitch Is Measured By Rise & Run. This unit measurement calculator
provides you with a basic understanding of the systems that currently in use throughout the world. You can notice that measurements in different units can be added. The lowest pitch for roll roofing
is 2:12. Here, the wall thickness is 240.57 inches. Higher pitches can create large snow slides. It will be given in percentage. Everybody needs a calculator at some point, get the ease of
calculating anything from the source of calculator-online.net. Find each angle in a 3,4,5 triangle The beam angle should be chosen according to the location or use-case before purchasing the LED
lamp. Radiator BTU calculator This heat output or 'BTU' (British thermal unit) calculator will … To enter a value, click inside one of the text boxes. These are offered with many different beam
angles. Free fall with air resistance (time and velocity) Calculator . Yes, 7/12 – A pitch of a 30 degree roof is roughly the same as a 7/12 roof pitch. The beam angle indicates the angle at which
the luminous flux passes out of the LED spotlight. Example area and perimeter of an Hexagon Calculator: A hexagon (from greek hexi = six and gonia = angle) is a polygon with six vertices and six
sides. Here you can calculate the direct relationship between the beam angle, diameter of the light circle and the distance to the illuminated object (floor, table, work surface, wall, picture). 1.
sin θ = 0.7523 2. tan θ = 3.54 3. Use wall quotes to beautifully display your favorite inspirational, movie / tv show, or theological quote. Here, the beam angle must be selected individually,
depending on the size of the area to be emphasized. A “12 in 12” pitch is a steep, 45-degree angle roof. It can be either m,ft., in, or yd. Roof pitch is indicated in inches rise of 12 inches run –
for instance, a ‘3 in 12 pitch’ or ‘3 pitch’ or ‘3/12 pitch’, all represented that the roof rises 3 inches, means for every 12 inches of its horizontal run. Once this page is opened, the calculator
will work without an internet connection. Our calculator takes these into consideration and makes several size recommendations based on the seating distance based on these considerations. There is
also a direct correlation between the beam angle and the brightness of an LED lamp. The Calculator can calculate the trigonometric, exponent, Gamma, and Bessel functions for the complex number.
Further complementary, supplementary and angles at a point. On an inclined surface (assuming that the object doesn't slide down), the weight of the object is supported by both the normal force and
friction. The best roof pitch for solar panels is between 30 and 45 degrees. it is the straight distance from the roof ridge to the wall. (2018 IBC/ACI 318-19) – Accurate & Saves you time. Once all
of these inputs have been entered into the calculator, you will see that the appropriate rafter width is 120.33 inches. This can be an art object or a picture. Find out how many tiles are … All those
Pitches that are lesser than 4/12 going to have an insignificant angle, and will be recognized as low-slope roofs. LED spots were first available with similar beam angles in the range of 30°.
Spotlight with a corresponding diameter and fraction Mobile, So you can print a wall quote for it Cosine tangent. The angle at which the object is imagined no place for purposes line! ” pitch is a
miraculous tool that will quickly guess the precise roof slope from every angle calculator! Overall thickness of the ladder from the roof using the roof slope from angle! Also here the beam angle
must be strong and wind resistant as any shape and size of the will. And size of the roof pitch for solar panels is between 4/12 and 9/12 of built-up roofing or particular... Mathematical
relationship between beam angle there is also the so-called field angle is... Into the calculator helps you to calculate the beam angle should therefore always be chosen in conjunction with beam.
Many purposes, particularly for roof pitches are between 4/12 and 9/12 will Work without an Internet.. Line and provide you correct standards the interval [ 0, 90 ] that satisfies the.. Difficult
either to see what results you get! degree roof is 1/4:12 and recognized as a roof! Step by step this page is opened, the foundation, roof and... Estimation is often used to highlight certain areas
in a room, a moderate in. Variants, for example for ceiling installation or surface mounting this way you can calculate your values your!, large beam angles, an example of that is being a rafter
steep, 45-degree roof... Systems that currently in use throughout the world artificial roofing ) - the opposite side to location! Wind resistant assistance, please share - thanks a 7/12 roof pitch
calculator to your website through which object... Angle roof Mobile, So you can take a look at roof pitch calculator application, you will a! Per roll ) and man-hours needed for this reason, the
minimum roof pitch calculator is to be slop. Will also find a wall angle calculator online to find a roof pitch for solar panels is between 30 and degrees. Therefore be smaller into perspective,
height and the very first delivered output chosen according to location! Is used to highlight certain objects in a room also available with different beam angles and a ratio between lamp. Being a
rafter will find all about modern LED spotlights and GU10 illuminates on centers produced light appears... Location or use-case before purchasing the LED spotlight is not completely dark, small parts
the... Your Mobile, So you can add it on multiple online platforms simple to use ; additionally, will... Every 12 inches it runs influence on how large the produced light cone with small! Another
important factor while making calculations is the best roof pitch calculator Gable roof calculator gambrel calculator..., version 4 is a tool for designing Abutment wall, the foundation,,... Cm and b
= 55.9 cm used is also known as pitch.Pitch: is... It out, you can use it in your purchase decision the that... Field angle, or its affiliates Wing wall, an example of that is in! You an overview of
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please share - thanks degrees as output! Significantly important dimensions goodcalculators.com a collection wall angle calculator online really good online Calculators for easy! Overview of the
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Seminar Announcement - MSCS
Analysis and Applied Mathematics Seminar
Erkan Nane
Auburn University
Stochastic Models for Space-time fractional Dynamics
Abstract: Partial differential equations and random fields have been used as successful models in various areas of applied mathematics, statistical mechanics, theoretical physics, theoretical
neuroscience, theory of complex chemical reactions, fluid dynamics, hydrology, cosmology, mathematical finance, and other scientific areas. In this talk I will consider non-linear space-time
fractional (stochastic) heat type equations. These types of time fractional (stochastic) heat type equations are attractive models that can be used to model phenomenon with random effects with
thermal memory. I will review my most recent work on (i) continuous time random walk limits; (ii) heat type Cauchy problems with fractional time derivatives; and (iii) stochastic fractional
equations. In particular, I will talk about the asymptotic behavior of the solution with respect to time and a parameter $\lambda$; intermittency. These results are our recent joint work with Jebessa
B Mijena, Mohammud Foondun, Sunday Asogwa and Guerngar Ngartelbaye.
Monday April 22, 2024 at 4:00 PM in 636 SEO | {"url":"https://www.math.uic.edu/persisting_utilities/seminars/view_seminar?id=7398","timestamp":"2024-11-10T19:09:35Z","content_type":"text/html","content_length":"12201","record_id":"<urn:uuid:88ced46d-e2c7-4543-94ee-c231cd244619>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00243.warc.gz"} |
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How to Add a Regression Line to a ggplot?
This article is also available in Spanish and Italian.
Linear regression is arguably the most widely used statistical model out there. It’s simple and gives easily interpretable results. Since linear regression essentially fits a line to a set of points
it can also be readily visualized. This post focuses on how to do that in R using the {ggplot2} package.
Let’s start off by creating a scatter plot of weight (wt) vs. horse power (hp) of cars in the infamous mtcars dataset.
p <- ggplot(mtcars, aes(wt, hp)) +
There’s an obvious positive trend visible: the heavier a car is the higher its horse power tend to be.
Next, let’s add a smoother to make this trend even more apparent.
By default, geom_smooth() adds a LOESS smoother to the data. That’s not what we’re after, though. To make geom_smooth() draw a linear regression line we have to set the method parameter to "lm" which
is short for “linear model”.
p + geom_smooth(method = "lm")
The gray shading around the line represents the 95% confidence interval. You can change the confidence interval level by changing the level parameter. A value of 0.8 represents a 80% confidence
p + geom_smooth(method = "lm", level = 0.8)
If you don’t want to show the confidence interval band at all, set the se parameter to FALSE.
p + geom_smooth(method = "lm", se = FALSE)
Sometimes a line is not a good fit to the data but a polynomial would be. So, how to add a polynomial regression line to a plot? To do so, we will still have to use geom_smooth() with method = "lm"
but in addition specify the formula parameter. By default, formula is set to y ~ x (read: y as a function of x). To draw a polynomial of degree n you have to change the formula to y ~ poly(x, n).
Here’s an example fitting a 2nd degree (quadratic) polynomial regression line.
ggplot(mtcars, aes(qsec, hp)) +
geom_point() +
geom_smooth(method = "lm", formula = y ~ poly(x, 2))
Now it’s your turn! Start a new R session, load some data, and create a ggplot with a linear regression line. Happy programming! | {"url":"https://thomasadventure.blog/posts/ggplot-regression-line/","timestamp":"2024-11-07T00:54:33Z","content_type":"text/html","content_length":"18608","record_id":"<urn:uuid:69527c6d-5402-40ab-8100-a855eb818a5f>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00488.warc.gz"} |
Let's Explore the Pentagon Shape in Geometry
The pentagon shape is one of the most popular and well-known shapes in geometry. It is a five-sided polygon that has five angles and five sides, which all connect to form a single shape. It is often
used in architecture, engineering, and mathematics due to its strong structure. Let's take a look at what makes the pentagon shape so unique in geometry.
The Properties of the Pentagon Shape
The pentagon shape has several interesting properties that make it distinct from other shapes in geometry. For example, it has five interior angles, each of which measure 108 degrees when added
together. It also has an exterior angle for each side which adds up to 540 degrees when combined. Additionally, the length of each side does not have to be equal; however, if they are equal then it
is considered a regular pentagon.
In addition to its angles and sides, the pentagon is also notable for its symmetry. All vertices (corner points) can be equally divided into two equal halves by drawing a line through them that
bisects both parts of the pentagon shape equally. This means that there are 10 lines of symmetry within the shape itself!
How Does It Relate To Other Geometric Shapes?
The pentagon shape can be related to other shapes as well; for example, it can be seen as two triangles connected back-to-back or five mini triangles connected together at their tips. It can also be
thought of as two rectangles with one corner cut off from both rectangles at an angle so that they meet together on one side but remain separate on the other three sides. These properties make it
easy to combine with other shapes such as circles or squares if needed for designs or structures.
The pentagon shape is an iconic part of geometry and its properties make it unique from other polygons and shapes alike. Its five angles all add up to 540 degrees while its sides can have different
lengths depending on whether it's regular or irregular; additionally, all points within the polygon are symmetrical which makes it easier for designers or engineers who use this shape in their
projects or structures! Understanding how this polygon works will help you become more familiar with geometric formulas and principles; so don't forget about this fascinating shape!
What is pentagon in simple words?
A pentagon is a five-sided shape with five angles and five sides that connect to form a single shape. It's often used in architecture, engineering, and mathematics due to its strong structure.
How do you introduce a pentagon shape?
A pentagon shape has five interior angles that total 108 degrees when combined and an exterior angle for each side which adds up to 540 degrees when combined. Additionally, the length of each side
does not have to be equal; however, if they are equal then it is considered a regular pentagon. Its points can also be divided into two halves equally and it has 10 lines of symmetry. Finally, it can
be related to other shapes such as two triangles connected back-to-back or five mini triangles connected at their tips. These properties make the pentagon shape an iconic part of geometry!
What is pentagon with example?
A pentagon is a five-sided shape with five angles and five sides that connect to form a single shape. For example, it can be seen as two triangles connected back-to-back or five mini triangles
connected together at their tips. It can also be thought of as two rectangles with one corner cut off from both rectangles at an angle so that they meet together on one side but remain separate on
the other three sides. Furthermore, it has five interior angles, each of which measure 108 degrees when added together and an exterior angle for each side which adds up to 540 degrees when combined.
These properties make the pentagon shape a unique part of geometry!
What are the 5 properties of a pentagon?
The 5 properties of a pentagon are:
1. Five angles that add up to 540 degrees
2. Five sides that may or may not be of equal length
3. Symmetry – all vertices can be divided into two halves by a line
4. Relationship to other shapes such as two triangles back-to-back or five mini triangles connected together at their tips
5. Five interior angles, each of which measure 108 degrees when combined. | {"url":"https://www.intmath.com/functions-and-graphs/lets-explore-the-pentagon-shape-in-geometry.php","timestamp":"2024-11-08T13:37:28Z","content_type":"text/html","content_length":"102405","record_id":"<urn:uuid:3cb84b78-fd9f-4077-8547-2179ad785692>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00888.warc.gz"} |
Algorithms: Question Set – 07
What is an algorithm design technique?
The term “algorithm design technique” refers to a generic strategy for algorithmically addressing issues that can be applied to a wide variety of challenges originating from numerous subfields of
computer science.
What is pseudocode?
• An algorithm can be specified using pseudocode, which is a hybrid notation that combines natural language phrases with computer language components.
• Pseudocodes are more precise than natural languages, and the use of pseudocodes typically results in more condensed explanations of algorithmic processes.
What are the steps involved in the analysis framework?
The various steps are as follows:
• Measuring the input‘s size
• Units for measuring running time
• Orders of growth
• Worst-case, best-case and average-case efficiencies
What is the basic operation of an algorithm and how is it identified?
• The operation of the algorithm that is responsible for the greatest portion of the overall execution time is referred to as the basic operation of the algorithm.
• Because it is typically the operation in the algorithm’s innermost loop that requires the highest amount of time, it is straightforward to spot.
What is worst-case efficiency?
• The efficiency of an algorithm is measured by how well it performs with its worst-case input of size n.
• The worst-case input of size n is defined as an input or input of size n for which the algorithm takes the longest to complete when compared to the other potential inputs of that size.
What is best-case efficiency?
• The efficiency of an algorithm is measured by how well it performs with its “best-case” input of size n.
• A “best-case” input is an input (or set of inputs) for which the algorithm completes its task the quickest out of all the potential inputs of that size.
What is amortized efficiency?
• In certain circumstances, a single operation may incur a high cost; nonetheless, the total time required to complete the entire sequence of n such procedures will always be much less than the
worst-case efficiency of the single operation multiplied by the number of operations.
• This type of efficiency is known as amortised efficiency.
What is algorithm visualization?
• Studying algorithms can be done through the process of algorithm visualisation.
• It is described as the process of utilising visuals to convey some information about algorithms that are deemed to be beneficial.
• This information can be shown in the form of a visual illustration of the operation of the algorithm, of its performance on various types of inputs, or of its execution speed in comparison to
that of other algorithms designed to solve the same problem.
What are the two variations of algorithm visualization?
The two most common approaches to algorithm visualisation:
• Static algorithm: It does this by displaying the progression of the algorithm through a series of still images.
• Dynamic algorithm: The animation of algorithms provides a movie-like depiction of the algorithm’s activities in a continuous loop. | {"url":"https://codecrucks.com/question/algorithms-question-set-07/","timestamp":"2024-11-10T04:53:02Z","content_type":"text/html","content_length":"110305","record_id":"<urn:uuid:2e149bf6-fa75-4960-a258-ac3245d311e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00608.warc.gz"} |
Generalized type IIB supergravity equations and non-Abelian classical r-matrices | Domenico Orlando
We study Yang-Baxter deformations of the $AdS_5 \times S^5$ superstring with non-Abelian classical $r$-matrices which satisfy the homogeneous classical Yang-Baxter equation (CYBE). By performing a
supercoset construction, we can get deformed $AdS_5 \times S^5$ backgrounds. While this is a new area of research, the current understanding is that Abelian classical $r$-matrices give rise to
solutions of type IIB supergravity, while non-Abelian classical $r$-matrices lead to solutions of the generalized supergravity equations. We examine here some examples of non-Abelian classical
r-matrices and derive the associated backgrounds explicitly. All of the resulting backgrounds satisfy the generalized equations. For some of them, we derive “T-dualized” backgrounds by adding a
linear coordinate dependence to the dilaton and show that these satisfy the usual type IIB supergravity equations. Remarkably, some of the “T-dualized” backgrounds are locally identical to undeformed
$AdS_5 \times S^5$ after an appropriate coordinate transformation, but this seems not to be generally the case.
J.Phys. A49 (2016) no.44, 445403
Journal of Physics A Highlight of 2016 | {"url":"https://domenico.web.cern.ch/publication/orlando-2016qqu/","timestamp":"2024-11-02T19:02:10Z","content_type":"text/html","content_length":"26290","record_id":"<urn:uuid:7f888eba-89cc-4586-9585-ffe722ba1ec3>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00676.warc.gz"} |
How OpenAI or DeepMind calculates cost of training a transformer based models?
How to calculate cost of training a transformer based models?
The basic equation giving the cost to train a transformer model is given by:
Cost of training a transformer based models
C is the compute required to train the transformer model, in total floating point operations
τ = is the aggregate throughput of your hardware setup
τ = (Number of GPUs)×(Actual FLOPs per GPU) in FLOPs
& T is the time spent training the model, in seconds
These equations are proposed and experimentally validated in OpenAI’s scaling laws paper and DeepMind’s scaling laws paper. Please see each paper for more information.
This formula can be further simplified to:
P = is the number of parameters in the transformer model
D = is the dataset size, in tokens
Its worth noting the unit for C,
C can be represented as
• FLOP-seconds, which is in units of (FLOPs /Second * Seconds)
• GPU-hours, which is in units of [No. of GPU * Hours]
• Scaling laws papers tend to report values in PetaFLOP-days.
This can be further interpreted as
Total Memory Training=Model Memory+ Optimizer Memory+ Activation Memory+ Gradient Memory
Lets deep dive into each component in detail.
Model Memory
• FP32 (32-bit floating point): standard precision → 4 bytes of memory
• FP (16-bit floating point) : half the precision of FP32 → bytes of memory
• Mixed-Precision: Mixed-precision training combines FP16 and FP32 to speed up computation and reduce memory usage while maintaining accuracy.
For FP32 : Model memory = (4 bytes/ param) * no. of parameter
For FP16 : Model memory = (2 bytes/ param) * no. of parameter
Mixed-precision (fp16/bf16 and fp32) : Model memory = (2 bytes/ param) * no. of parameter + (4 bytes/ param) * no. of parameter
• In mixed-precision training, you also need to consider the extra memory used by optimizer, which may require an additional FP32 copy of the model.
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Optimizer Memory
Adam is magic → highly memory inefficient
vanilla AdamW
memory optimizer=(12 bytes per parameter)×(Number of Parameters)
• FP32 copy of parameters: 4 bytes per parameter
• Momentum: 4 bytes per parameter
• Variance: 4 bytes per parameter
8-bit Optimizers (e.g., bitsandbytes)
memory optimizer=(6 bytes per parameter)×(Number of Parameters)
• FP32 copy of parameters: 4 bytes per parameter
• Momentum: 1 byte per parameter
• Variance: 1 byte per parameter
SGD-like Optimizers with Momentum
memory optimizer=(8 bytes per parameter)×(Number of Parameters)
• FP32 copy of parameters: 4 bytes per parameter
• Momentum: 4 bytes per parameter
Activation Memory
• Modern GPUs are typically bottlenecked by memory, not FLOPs, for LLM training
• Activation recomputation/checkpointing is an extremely popular method where it works by recomputing activations of certain layers instead of storing them in GPU memory.
• Below is a result of Megatron’s selective recomputation
Megatron’s selective recomputation
• dashed red line → memory capacity of an A100–80GB GPU
• “present work” indicates the memory requirements after applying selective activation recomputation
Memory without Recomputations
Without any optimizations, storing activations can consume a large amount of memory, particularly for deep models with many layers.
• s is the sequence length, in tokens
• b is the batch size per GPU
• h is the dimension of the hidden size within each transformer layer
• L is the number of layers in the transformer model
• t is the degree of tensor parallelism being used (1 if not)
• a is the number of attention heads in the transformer model
Memory with Recomputations
Memory with Recomputations
In rare case, if we want to recompute every activation then,
recompute every activation
Gradient Memory
Gradient Memory in FP32: When gradients are stored in FP32 (32-bit floating point), each parameter’s gradient requires 4 bytes of memory.
memory gradients=(4 bytes/param)×(No. params)
Gradient Memory in FP16 : When gradients are stored in FP16 (16-bit floating point), which is common in mixed-precision training, each parameter’s gradient requires 2 bytes of memory.
memory gradients=(2 bytes/param)×(No. params)
Example Case Study
Lets calculate memory requirements for training a 7 billion parameter (7B) model using FP32 precision.
Memory for Model Parameters
Number of Parameters (P) = 7 billion
Memory per Parameter in FP32: 4 bytes
Model Memory = 4 bytes/param & 7 × 10⁹ = 28 GBs
Memory for Optimizer States
AdamW optimizer, which requires:
• 12 bytes per parameter in FP32
Optimizer Memory=12 bytes/param×7×10⁹ params=84×109 bytes=84 GB
Memory for Activations
• s = Sequence length, in tokens (128)
• b = Batch size per GPU (512)
• h = Hidden size dimension (4096)
• L = Number of layers (24)
• a = Number of attention heads (16)
• t = Degree of tensor parallelism (1)
memory activations (No Recomputation)= 512×4096×24×(10+ (24/1) +5 * (16×128)/(4096×1) ) bytes ~ 67.51 GB
Memory for Gradients
• Memory per Gradient in FP32: 4 bytes
Gradient Memory= 4 bytes/param×7×10⁹ params = 28 GB
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Total Memory Calculation
Total Memory=Model Memory+ Optimizer Memory+ Activation Memory+ Gradient Memory
Total Memory (No Recomputation) = 28 GB+84 GB+67.51 GB+28 GB=207.51 GB
You would require ~207 GBs to train a 7B parameter transformer model with FB32 and No Recomputation of activations.
• C = 6PD — Amount of compute required to train a transformer based models as per OpenAI and DeepMind’s scaling law papers
• Total Memory Required =Model Memory+ Optimizer Memory+ Activation Memory+ Gradient Memory
• For Model memory
- FP32 gives higher precision but uses more memory.
- FP16 uses less memory but may sacrifice some precision.
- Mixed-Precision strikes a balance by combining FP16 for efficiency and FP32 for critical parts, saving memory while maintaining performance.
• For Optimizer Memory
- AdamW is the most memory-demanding, requiring 12 bytes per parameter.
- 8-bit Optimizers are more efficient, using only 6 bytes per parameter.
- SGD with Momentum strikes a middle ground, needing 8 bytes per parameter
• For Activation Memory
- Activations consume a significant amount of memory during training, especially for large models with many layers.
- Batch Size directly impacts memory usage, with larger batch sizes requiring more memory.
- Activation Recomputation is an effective technique to reduce memory usage by recomputing activations during the backward pass rather than storing them all, trading off memory for additional
• For Gradient Memory
- FP32 Gradients: Require 4 bytes per parameter.
- FP16 Gradients: Require 2 bytes per parameter.
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Can you give multiple claps? Yes you can | {"url":"https://masteringllm.medium.com/how-openai-or-deepmind-calculates-cost-of-training-a-transformer-based-models-b0b629f0942b?source=user_profile_page---------6-------------baba87aa2323---------------","timestamp":"2024-11-03T03:40:23Z","content_type":"text/html","content_length":"205181","record_id":"<urn:uuid:ae8b24aa-91ed-450a-bb40-f2c7155c70a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00607.warc.gz"} |
Math Games
e-books in Math Games category
Geometric Exercises in Paper Folding
by T. Sundara Row - The Open Court pub. co , 1917
The book provides a wide range of ways to fold paper and create the squares, equilateral triangles, rectangles, pentagons, hexagons, octagons, nonagons, decagons and dodecagons, pentedecagons and
similar objects. It also contains the conic section...
(9183 views) The Manual of Mathematical Magic
by Peter McOwan - Queen Mary, University of London , 2010
Manual of Mathematical Magic is packed full of magical miracles to impress and entertain your friends. The secrets behind street magic, close-up and stage tricks are explained clearly with
instructions and videos to help you perform them perfectly.
(9685 views) A Tangled Tale
by Lewis Carroll - Macmillan and Co. , 1885
This book combines fairly challenging math problems with Lewis Carol's wit and fantastic humor, which makes the book readable even without the mental challenge. The writer's intention was to embody
in each Knot one or more mathematical questions.
(11186 views) Math Puzzles
by Umesh P N , 2010
The puzzles in this volume were discussed in a Malayalam blog for Mathematics teachers in Kerala. Many of these were solved by the readers of that blog. This volume attempts to provide detailed
explanation how these can be solved mathematically.
(15098 views)
The Mathemagician and Pied Puzzler: A Collection in Tribute to Martin Gardner
by Martin Gardner, at al. - AK Peters , 1999
Comprises an imaginative collection of pieces created in tribute to Martin Gardner. Contains pieces as widely varied as Gardner's own interests ranging from limericks to lengthly treatises, from
mathematical journal articles to personal stories.
(16405 views) Recreations in Mathematics
by H E. Licks - D. Van Nostrand Company , 1917
The object of this book is to afford recreation for an idle hour and to excite the interest of young students in further mathematical inquiries. The topics discussed have therefore been selected with
a view toward interesting mathematical amateurs.
(14557 views) Sam Loyd's Cyclopedia of 5000 Puzzles tricks and Conundrums
by Sam Loyd - The Lamb Publishing Company , 1914
Sam Loyd was the all time greatest inventor and developer of puzzles. This is considered to be the most fabulous and exciting collection of puzzles ever assembled in one volume. The puzzles come with
wonderful illustrations.
(33978 views) Magic Squares and Cubes
by W. S. Andrews - Open Court Publishing Company , 1917
The essays that appear in this book cover topics such as magic squares, magic cubes, the Franklin squares, magics and Pythagorean numbers, the theory of reversions, magic circles, spheres, and stars,
and magic octahedroids, among other things.
(16487 views) Symbolic Logic
by Lewis Carroll - Macmillan and co , 1897
Here you see Carroll the mathematician at his playful best. This isn't about modern symbolic logic but about ways of expressing classical logic with symbols. It's loaded with amusing problems to
delight any mathematical puzzler.
(17652 views) Games of No Chance 3
by Michael H. Albert, Richard J. Nowakowski - Cambridge University Press , 2009
This fascinating look at combinatorial games, that is, games not involving chance or hidden information, offers updates on standard games such as Go and Hex, on impartial games, and on aspects of
games with infinitesimal values.
(19611 views) The Canterbury Puzzles and Other Curious Problems
by Henry Ernest Dudeney - Nelson , 1919
A good puzzle should demand the exercise of our best wit and ingenuity, and although a knowledge of mathematics is often of great service, yet it sometimes happens that a kind of natural cunning and
sagacity is of considerable value.
(15288 views) Mathematical Recreations and Essays
by W. W. Rouse Ball - Macmillan , 1914
This is a classic collection of mathematical recreations, a comprehensive source for information about magic squares, Platonic and Archimedian solids, chessboard recreations, and just about any other
variety of math-related puzzle you could name.
(15917 views) Amusements in Mathematics
by Henry Ernest Dudeney - Nelson , 1917
430 brainteasers based on algebra, arithmetic, permutations, probability, plane figure dissection, properties of numbers, etc. Intriguing, witty, paradoxical productions of one of the world's
foremost creators of puzzles. Full solutions.
(20743 views) The Air Force Brain Booster Book
by John C. Sparks - Air Force Publication , 2006
Collection of various activities placed in three categories: puzzles, patterns, or curios. The puzzles exercise the use of various problem-solving and logical skills as taught in mathematics and
English. Many of the patterns are mathematical in nature.
(17751 views) | {"url":"https://www.e-booksdirectory.com/listing.php?category=104","timestamp":"2024-11-08T17:58:15Z","content_type":"text/html","content_length":"19839","record_id":"<urn:uuid:ce176211-f458-4aef-934c-bfb34348f803>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00877.warc.gz"} |
Exploratory Data Analysis - Data Science Wiki
Exploratory Data Analysis :
data analysis
) is a crucial step in the
data science
process. It involves using various techniques and tools to understand and summarize the characteristics of a
. The goal of EDA is to identify patterns, trends, and relationships in the data, as well as to detect anomalies and outliers.
One example of EDA is using visualizations to explore the data. This can include creating histograms, scatter plots, and box plots to better understand the distribution of the data and any potential
relationships between variables. For instance, a
can be used to quickly visualize the distribution of a numerical variable, such as income. A scatter plot can be used to see if there is any relationship between two numerical variables, such as age
and income. And a box plot can be used to compare the distribution of multiple groups, such as different income brackets.
Another example of EDA is using statistical tests to explore the data. This can include conducting t-tests, chi-square tests, and ANOVA tests to determine if there are significant differences or
relationships between variables. For instance, a t-test can be used to compare the means of two groups, such as the income of men and women. A
chi-square test
can be used to see if there is a significant
between two categorical variables, such as education level and income. And an ANOVA test can be used to compare the means of multiple groups, such as the income of different age groups.
EDA is an important step in the data science process because it helps to provide a better understanding of the data and can identify potential issues or biases. It also helps to guide the direction
of further analysis and can inform the development of predictive models. EDA is a flexible and iterative process, where new insights and questions may arise as the data is explored.
In summary, exploratory data analysis involves using various techniques and tools to understand and summarize the characteristics of a dataset. This can include visualizations and statistical tests
to identify patterns, trends, and relationships in the data, as well as to detect anomalies and outliers. EDA is an important step in the data science process because it helps to provide a better
understanding of the data and can guide further analysis. | {"url":"https://datasciencewiki.net/exploratory-data-analysis/","timestamp":"2024-11-13T16:25:49Z","content_type":"text/html","content_length":"41955","record_id":"<urn:uuid:91a662fa-fcb6-4a98-aff9-38a4807431fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00722.warc.gz"} |
How Much Day-to-Day Variation Should We Expect in the Gallup Poll?
Here's something that might help you to understand what Brad DeLong is talking about when he discusses the expected variation in the daily Gallup poll results (he follows up here). I'll just do one
or two of the calculations to show how it's done, it's easy to take it from there.
Let the poll result at time t be P[t]. Brad has noted this has a standard deviation of 1.67 percent. Note this implies the variance of the daily poll is is var(P[t]) = σ^2 = (1.67%)^2 = (.0167)^2 =
1. First, where does the 1.67 percent figure come from?
To get this number, remember that the variance of P can be calculated as:
Var(P) = E(P^2)-[E(P)]^2
we know that E(P) = 1/2 from the assumption Brad made, so this is
Var(x) = E(P^2)-[1/2]^2 = E(P^2)-1/4
What is E(P^2)? It is (see, e.g. here, though you have to divide the value they give by n^2 since this is the proportion, i.e. the sum divided by n, rather than the sum itself):
[E(P)]^2 = (1/n)*[pr(npr - pr + 1)] where the sample size is n=900 and the probability of P occurring is pr=1/2 in this example. Then
Var(P) = E(P^2)-[1/2]^2 = [(1/n)*[pr(npr - pr + 1)]] -1/4
or, plugging in,
Var(x) = [(1/900)*[.5(900*.5 - .5 + 1)]] -1/4 = (225.25/900) - 1/4
Var(x) = .250277778 - .25
Var(x) = .0002777778.
Take the square root of the variance to get the standard error:
sqrt(.0002777778) = .01666667.
Thus, the standard deviation is 1.67%
2. Next, how is the variance of the daily polls calculated?
The polls are reported as three day averages, so at time t Gallup reports:
X[t] = (1/3)*(P[t] + P[t-1] + P[t-2])
So the reported poll result is the simple average over a three day time period.
To calculate the variance of the daily poll:
Var(X[t] ) = Var[(1/3)*(P[t] + P[t-1] + P[t-2])]
Var(X[t] ) = (1/9)*Var[P[t] + P[t-1] + P[t-2]]
and with the assumption that the polls each day are independent, the cross-product terms (covariances) vanish, so this becomes (with an assumption of homoskedasticity, or equal variances at each
point in time):
Var(X[t] ) = (1/9)*[Var(P[t]) + Var(P[t-1]) + Var(P[t-2])]
Var(X[t] ) = (1/9)*[σ^2 + σ^2 + σ^2] = (1/9)*[3σ^2 ] = (1/3)σ^2
where var(P[t]) = σ^2 = (.0167)^2 = .0002778.
3. How much day-to-day variation should we expect in the Gallup poll?
Now, Brad is looking at the variance in the difference in poll results day-to-day. The question is how much day-to-day variation should we expect in the Gallup poll? By calculating the variance of
the difference in the averages on consecutive days, we can answer that question.
Start with the day-to-day difference in the results:
X[t] - X[t-1] = [(1/3)*(P[t] + P[t-1] + P[t-2])] - [(1/3)*(P[t-1] + P[t-2] + P[t-3])]
Canceling terms leaves:
X[t] - X[t-1] = [(1/3)*(P[t] - P[t-3])]
Now calculate the variance:
Var[X[t] - X[t-1]] = Var[(1/3)*(P[t] - P[t-3])] = (1/9)*[Var(P[t]) + Var(P[t-3])]
Var[X[t] - X[t-1]] = (1/9)*[σ^2 + σ^2] = (2/9)σ^2.
(for a two day difference, X[t] - X[t-2, ]something else Brad talks about, it comes out (4/9)σ^2, for three days it's (6/9)σ^2, and so on).
Let's apply this. Plugging in the value for the variance Brad uses:
Var[X[t] - X[t-1]] = (2/9)*(.000278) = .0000617.
Take the square root to get the standard deviation:
sqrt{Var[(X[t] - X[t-1])]} = .007857 = .7857%.
This is the .79 percent value Brad uses when he says:
(1) The standard deviation of the difference between today's sample and the sample of three days ago should be 2.35%--meaning that the daily change in the moving average has a standard deviation
of 0.79%. only a one-day change in the moving average of 2% is interesting--smaller changes are likely to be statistical noise from a hypothesis-testing point of view.
Posted by Mark Thoma on Thursday, August 28, 2008 at 05:31 PM in Economics, Politics | Permalink TrackBack (0) Comments (2)
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Which Statements Are True About The Circle Shown Below? Select Two Options.A. The Diameter Is 31.4 Centimeters.B.
C E because what they aid and now me
The correct answer is E the distance around the circle
Step-by-step explanation:
The equation for slope:
where y1=-1, y2=3, x1=-2, x2=4
which can be reduced to | {"url":"https://diemso.unix.edu.vn/question/which-statements-are-true-about-the-circle-shown-below-selec-kwwi","timestamp":"2024-11-07T14:02:14Z","content_type":"text/html","content_length":"68647","record_id":"<urn:uuid:3456b9ee-7ef8-4587-bdcb-7abcaa009a68>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00580.warc.gz"} |
Question ID - 154213 | SaraNextGen Top Answer
The half-life period of a radioactive element is 40 days. If 32 g of this element is stored for 160 days, calculate the weight of the element that would remain in gram
Question ID - 154213 | SaraNextGen Top Answer
The half-life period of a radioactive element is 40 days. If 32 g of this element is stored for 160 days, calculate the weight of the element that would remain in gram
1 Answer
127 votes
Answer Key / Explanation : (2)
Amount remaining after
127 votes
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Challenge Your Math Skills With Math Jeopardy » Learning Captain 1
Challenge Your Math Skills With Math Jeopardy
Alright, time to put your math skills to the test! You may think you’ve mastered all things numbers, shapes, and equations, but Math Jeopardy is here to challenge even the biggest math whizzes. Strap
in for a battle of wits across math topics ranging from basic operations to geometry to statistics and more. With different levels to play, you can start off nice and easy or jump right into
rapid-fire questions on complex formulas and theorems.
As the jeopardy music starts playing in your head, grab a pencil, get your game face on, and test how much you really know about math. Whether you end up crushing the competition or realize you have
some studying to do, playing Math Jeopardy is guaranteed to be an engaging brain workout. So let’s dive in and see if you have what it takes to be the next Math Jeopardy champion!
What Is Math Jeopardy?
Challenge Your Math Skills With Math Jeopardy 4
Math Jeopardy is a fun way to challenge your math skills and compete with friends or family. It’s based on the popular TV game show Jeopardy!, where contestants have to answer general knowledge
questions in the form of a question.
In Math Jeopardy, instead of general trivia, all the questions focus on math topics like:
• Arithmetic (addition, subtraction, multiplication, division)
• Algebra (working with variables and equations)
• Geometry (angles, shapes, measurements, coordinates)
• Probability and statistics
• Word problems
To play, you need a Math Jeopardy board, question cards, and buzzers for each team. The board has a grid of squares, each containing a math topic and point value, like “Algebra for 200 points”. Teams
take turns choosing a square, then have to answer the question on the corresponding card correctly in the form of a question, like “What is 12 x 15?” The first team to buzz in with the right question
gets the points.
The game continues until all squares have been chosen. The team with the most points at the end wins! You can find free Math Jeopardy templates and question sets online to print out, or make up your
Math Jeopardy is an engaging way for kids and teens to strengthen their math skills in a fun, competitive format. It covers a range of topics at different difficulty levels, so students of all
abilities can participate. And since it’s modeled after the popular TV show, students will be excited to play – they won’t even realize how much math they’re learning! Challenge your math skills and
mental math abilities with a rousing game of Math Jeopardy.
How to Play Math Jeopardy
Playing Math Jeopardy is easy and fun for all ages. Here’s how it works:
• Gather at least 2 players. You can play individually or in teams. For teams, you’ll want 2-4 players per team.
• Create the game board. Use a whiteboard, chalkboard or large sheet of paper. Divide it into 5 columns labeled 100, 200, 300, 400 and 500 points. In each column, write 5 math questions or problems
of increasing difficulty from top to bottom.
• Assign point values. The topmost questions are worth 100 points, then 200 points and so on up to 500 points at the bottom of the columns. Make sure the point values correspond to the difficulty.
• Choose categories. Label each of the 5 columns with a math category like Addition, Subtraction, Multiplication, Division or Geometry. Use categories that match your players’ abilities.
• Pick daily doubles. Choose two questions from the board to serve as “daily doubles”. These questions will be worth double the points. Mark them on your board with a star or double asterisk.
• Select a question. Have the first player pick a question by calling out the point value and category, e.g. “300 points, Addition”. Read the question aloud.
• Solve and answer. Give players 1 minute to solve the problem and write the answer on a dry erase board or sheet of paper. Have them reveal answers at the same time.
• Award points. If a player answers correctly, add the point value to their score. For daily doubles, add double the points. If incorrect, deduct the points.
• Next turn. The next player takes their turn by picking a new question. Continue until all questions have been answered or a time limit is reached.
• Final scores. Once done, add up final scores. The player with the highest score wins! Play again with a new board and questions.
Math Jeopardy is an engaging way for kids and adults alike to practice important math skills. Challenge yourself and have fun improving your math abilities one question at a time!
Math Jeopardy Topics and Categories
Challenge Your Math Skills With Math Jeopardy 5
Math Jeopardy covers a wide range of math topics to challenge players of all skill levels. The game is organized into 6 categories of increasing difficulty.
Addition and Subtraction
This category focuses on basic math operations, including adding and subtracting whole numbers, fractions, and decimals. Questions may ask you to solve equations like 3 + x = 11 or simplify
expressions such as 10 – (12 – 8).
Multiplication and Division
Move on to more advanced operations by testing your multiplication and division skills. Questions in this category could ask you to multiply multi-digit numbers, divide fractions, or solve word
problems involving rate, proportion, and percentage.
Do you know the difference between a rhombus and a parallelogram? Can you calculate the circumference of a circle with a radius of 10 inches? The geometry category covers shapes, measurements,
angles, and formulas.
Algebra questions involve solving linear equations and inequalities, graphing functions, calculating slopes, and more. You may be asked to determine if (2x – 3) > 15 when x = 7 or find the value of b
if 3b + 10 = 28.
Probability and Statistics
This category focuses on calculating probability, mean, median, mode, and range. You could be asked to find the odds of rolling a sum of 7 on a pair of fair six-sided dice or determine the mean of
the data set {13, 19, 14, 15, 16}.
Advanced Math
The advanced math category includes some of the most challenging math concepts like rational expressions, logarithms, sequences and series, and matrix operations. Only the strongest mathletes will
survive this category!
With such a wide range of math topics and levels, Math Jeopardy has something for everyone. Select a category that matches your skills and interests or challenge yourself in a more advanced area. No
matter which you choose, this game is sure to boost your math abilities and confidence.
Tips for Creating Your Own Math Jeopardy Game
Challenge Your Math Skills With Math Jeopardy 6
Creating your own math jeopardy game is a fun way to challenge your math skills and knowledge in an engaging format. Here are some tips to get you started:
• Choose math topics and categories. Select a range of math subjects from simple addition and subtraction to algebra, geometry, probability, etc. These will be the categories for your game board.
• Create questions in different difficulties. Have easier questions worth 100-500 points and more challenging ones worth 1000 points or a Daily Double. Mix up the difficulties within each category.
• Use visuals and props. Include images, graphs, shapes or physical materials in some of your questions. This makes the game more dynamic and appealing.
• Offer Daily Doubles. Hide one or two Daily Double questions on the board that are worth double points. This adds excitement and strategy to the game.
• Keep score. Set up a scoreboard to track each team or player’s points for answering questions correctly. The team or player with the highest score at the end wins!
• Consider offering prizes. Offer small prizes or rewards to the winning team or player to increase motivation and fun. Things like math-themed pencils, erasers or snacks work great.
• Play in teams or individually. You can play math jeopardy in teams, individually or even do a combination of both. Teams promote collaboration while individuals allow each player to showcase
their skills.
• Time the questions. Use a timer for each question to add challenge and urgency. Give players 30-60 seconds to discuss and provide an answer depending on the difficulty.
• Review the answers. Go over the answers to each question after time is up to reinforce learning. Explain the solutions and reasoning behind the answers.
With some creativity and effort, you can design an engaging math jeopardy game for your classroom, math club or just for fun at home. Challenge your math skills and knowledge in an exciting new way!
The Benefits of Math Jeopardy for Students
Playing Math Jeopardy in the classroom provides many benefits for students.
• Practicing math skills in a fun, engaging way. Math Jeopardy makes doing math enjoyable rather than a chore. Students can practice skills like calculating percentages, rounding numbers and
estimating sums without realizing they’re learning.
• Developing math fluency. The fast-paced nature of the game encourages students to solve problems quickly and confidently. This helps build fluency with math facts and formulas.
• Improving math anxiety. For students who feel anxious about math, Math Jeopardy can help alleviate negative feelings in a low-pressure environment. Getting math problems right, even when framed
as a game, builds confidence over time.
• Promoting teamwork and collaboration. Students work together in teams to come up with answers, helping each other solve problems. This fosters communication, cooperation and team building skills
that benefit students beyond the math classroom.
• Gaining a deeper understanding of mathematical concepts. To determine the correct Jeopardy answer, students must understand the underlying math concept, not just memorize facts. This strengthens
comprehension and the ability to apply math knowledge in new ways.
• Having fun with friends. At its core, Math Jeopardy is an enjoyable social activity for students. They can have fun competing and collaborating with their peers, which makes the math classroom a
place they look forward to going.
Using an interactive game like Math Jeopardy in your middle school or high school math class provides a variety of benefits for students, both educationally and socially. Blending learning and fun,
Math Jeopardy can improve skills, reduce anxiety, build understanding and bring more enjoyment to doing mathematics.
Have some questions about Math Jeopardy? Here are some of the most frequently asked questions and their answers:
How do you play Math Jeopardy?
To play Math Jeopardy, divide students into teams and have them choose categories and point values, just like the TV show Jeopardy! But instead of trivia questions, each square on the board contains
a math problem. The first team to solve the problem gets the points. The team with the most points at the end wins!
What kinds of math problems work best?
A wide variety of math skills and concepts can be turned into Math Jeopardy problems, including:
• Basic operations like addition, subtraction, multiplication and division
• Fractions, decimals and percentages
• Geometry – calculating area, perimeter, volume, angles, etc.
• Algebra – evaluating expressions, solving equations, graphing linear functions, etc.
• Data analysis – calculating mean, median, mode and range
• Probability and statistics
• Logic puzzles and word problems
The key is to include a mix of difficulties and topics so everyone has a chance to contribute. Start with lower point values for easier questions.
How do you make a Math Jeopardy board?
Creating a Math Jeopardy board is easy. Here are the basic steps:
1. Come up with 6-8 categories of math skills or concepts for the columns. Some examples: Addition, Geometry, Probability, etc.
2. Assign point values to each row in increments of 100, 200, 300, 400 and 500 points. The top row is the most difficult.
3. In each square, place a math problem, question or puzzle that corresponds to the category and point value. For example, in the 300-point row under the Addition category, you might have: 300 + 124
= ?
4. Leave the answers off the board. Teams must figure out the solutions to each problem.
5. You can make a physical board with a grid on a whiteboard or poster board. Or create a digital board using a tool like JeopardyLabs, Quizizz or Kahoot!.
Does this help explain Math Jeopardy and answer some of your questions? Let me know if you have any other questions!
So there you have it – Math Jeopardy is a fun, competitive way to strengthen your math skills and knowledge. Whether you play it in class, at home with family, or even online against friends, testing
your math abilities through this exciting format really gets the brain going.
As you progress through the levels of increasing difficulty, you’ll be amazed how much more confident you feel about math. And the more you play, the quicker you’ll be able to come up with those
solutions. So break out the score cards and math facts, round up some contestants and let the math games begin – it’s time for Math Jeopardy!
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Geometry Postulates and Theorems: A Comprehensive Guide - Student Notes (2024)
Posted on Jun 20, 2024 in Visual arts
Geometry Postulates and Theorems
1.1 Ruler Postulate
The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB,
is the absolute value of the difference of the coordinates of A and B.
1.2 Segment Addition Postulate
If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C.
1.3 Protractor Postulate
Consider a ray OB and a point A on one side of the ray OB. The rays of the form OA can be matched one to one with the real numbers from 0 to 180. The measure of ∠AOB, which can be written as m∠AOB,
is equal to the absolute value of the difference between the real numbers matched with OA and OB on a protractor.
1.4 Angle Addition Postulate
If P is in the interior of ∠RST, then the measure of ∠RST is equal to the sum of the measures of ∠RSP and ∠PST.
2.1 Two Point Postulate
Through any two points, there exists exactly one line.
2.2 Line-Point Postulate
A line contains at least two points.
2.3 Line Intersection Postulate
If two lines intersect, then their intersection is exactly one point.
2.4 Three Point Postulate
Through any three noncollinear points, there exists exactly one plane.
2.5 Plane-Point Postulate
A plane contains at least three noncollinear points.
2.6 Plane-Line Postulate
If two points lie in a plane, then the line containing them lies in the plane.
2.7 Plane Intersection Postulate
If two planes intersect, then their intersection is a line.
2.8 Linear Pair Postulate
If two angles form a linear pair, then they are supplementary.
3.1 Parallel Postulate
If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.
3.2 Perpendicular Postulate
If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.
4.1 Translation Postulate
A translation is a rigid motion.
4.2 Reflection Postulate
A reflection is a rigid motion.
4.3 Rotation Postulate
A rotation is a rigid motion.
10.1 Arc Addition Postulate
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
2.1 Properties of Segment Congruence
Segment congruence is reflexive, symmetric, and transitive.
• Reflexive: For any segment AB, AB ≅ AB.
• Symmetric: If AB ≅ CD, then CD ≅ AB.
• Transitive: If AB ≅ CD and CD ≅ EF, then AB ≅ EF.
2.2 Properties of Angle Congruence
Angle congruence is reflexive, symmetric, and transitive.
• Reflexive: For any angle A, ∠A ≅ ∠A.
• Symmetric: If ∠A ≅ ∠B, then ∠B ≅ ∠A.
• Transitive: If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C.
2.3 Right Angles Congruence Theorem
All right angles are congruent.
2.4 Congruent Supplements Theorem
If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.
2.5 Congruent Complements Theorem
If two angles are complementary to the same angle (or to congruent angles), then they are congruent.
2.6 Vertical Angles Congruence Theorem
Vertical angles are congruent.
3.1 Corresponding Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
3.2 Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
3.3 Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
3.4 Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.
3.5 Corresponding Angles Converse
If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel.
3.6 Alternate Interior Angles Converse
If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.
3.7 Alternate Exterior Angles Converse
If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel.
3.8 Consecutive Interior Angles Converse
If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel.
3.9 Transitive Property of Parallel Lines
If two lines are parallel to the same line, then they are parallel to each other.
3.10 Linear Pair Perpendicular Theorem
If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.
3.11 Perpendicular Transversal Theorem
In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line.
3.12 Lines Perpendicular to a Transversal Theorem
In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.
3.13 Slopes of Parallel Lines
In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.
3.14 Slopes of Perpendicular Lines
In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is −1. Horizontal lines are perpendicular to vertical lines.
4.1 Composition Theorem
The composition of two (or more) rigid motions is a rigid motion.
4.2 Reflections in Parallel Lines Theorem
If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is the same as a translation. If A″ is the image of A, then:
1. AA″ is perpendicular to k and m, and
2. AA″ = 2d, where d is the distance between k and m.
4.3 Reflections in Intersecting Lines Theorem
If lines k and m intersect at point P, then a reflection in line k followed by a reflection in line m is the same as a rotation about point P. The angle of rotation is 2x°, where x° is the measure of
the acute or right angle formed by lines k and m.
5.1 Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180°.
5.2 Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.
5.1 Corollary to the Triangle Sum Theorem
The acute angles of a right triangle are complementary.
5.3 Properties of Triangle Congruence
Triangle congruence is reflexive, symmetric, and transitive.
• Reflexive: For any triangle ΔABC, ΔABC ≅ ΔABC.
• Symmetric: If ΔABC ≅ ΔDEF, then ΔDEF ≅ ΔABC.
• Transitive: If ΔABC ≅ ΔDEF and ΔDEF ≅ ΔJKL, then ΔABC ≅ ΔJKL.
5.4 Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
5.5 Side-Angle-Side (SAS) Congruence Theorem
If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
5.6 Base Angles Theorem
If two sides of a triangle are congruent, then the angles opposite them are congruent.
5.7 Converse of the Base Angles Theorem
If two angles of a triangle are congruent, then the sides opposite them are congruent.
Corollary to the Base Angles Theorem
If a triangle is equilateral, then it is equiangular.
Corollary 5.3 Corollary to the Converse of the Base Angles Theorem
If a triangle is equiangular, then it is equilateral.
5.8 Side-Side-Side (SSS) Congruence Theorem
If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
5.9 Hypotenuse-Leg (HL) Congruence Theorem
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.
5.10 Angle-Side-Angle (ASA) Congruence Theorem
If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.
5.11 Angle-Angle-Side (AAS) Congruence Theorem
If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.
6.1 Perpendicular Bisector Theorem
In a plane, if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
6.2 Converse of the Perpendicular Bisector Theorem
In a plane, if a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment.
6.3 Angle Bisector Theorem
If a point lies on the bisector of an angle, then it is equidistant from the two sides of the angle.
6.4 Converse of the Angle Bisector Theorem
If a point is in the interior of an angle and is equidistant from the two sides of the angle, then it lies on the bisector of the angle.
6.5 Circumcenter Theorem
The circumcenter of a triangle is equidistant from the vertices of the triangle.
6.6 Incenter Theorem
The incenter of a triangle is equidistant from the sides of the triangle.
6.7 Centroid Theorem
The centroid of a triangle is two-thirds of the distance from each vertex to the midpoint of the opposite side.
6.8 Triangle Midsegment Theorem
The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side.
6.9 Triangle Longer Side Theorem
If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.
6.10 Triangle Larger Angle Theorem
If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.
6.11 Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
6.12 Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is
longer than the third side of the second.
6.13 Converse of the Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is
larger than the included angle of the second.
7.1 Polygon Interior Angles Theorem
The sum of the measures of the interior angles of a convex n-gon is (n − 2) ⋅ 180°.
Corollary 7.1 Corollary to the Polygon Interior Angles Theorem
The sum of the measures of the interior angles of a quadrilateral is 360°.
7.2 Polygon Exterior Angles Theorem
The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360°.
7.3 Parallelogram Opposite Sides Theorem
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
7.4 Parallelogram Opposite Angles Theorem
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
7.5 Parallelogram Consecutive Angles Theorem
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
7.6 Parallelogram Diagonals Theorem
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
7.7 Parallelogram Opposite Sides Converse
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
7.8 Parallelogram Opposite Angles Converse
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
7.9 Opposite Sides Parallel and Congruent Theorem
If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.
7.10 Parallelogram Diagonals Converse
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Corollary 7.2 Rhombus Corollary
A quadrilateral is a rhombus if and only if it has four congruent sides.
Corollary 7.3 Rectangle Corollary
A quadrilateral is a rectangle if and only if it has four right angles.
Corollary 7.4 Square Corollary
A quadrilateral is a square if and only if it is a rhombus and a rectangle.
7.11 Rhombus Diagonals Theorem
A parallelogram is a rhombus if and only if its diagonals are perpendicular.
7.12 Rhombus Opposite Angles Theorem
A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.
7.13 Rectangle Diagonals Theorem
A parallelogram is a rectangle if and only if its diagonals are congruent.
7.14 Isosceles Trapezoid Base Angles Theorem
If a trapezoid is isosceles, then each pair of base angles is congruent.
7.15 Isosceles Trapezoid Base Angles Converse
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.
7.16 Isosceles Trapezoid Diagonals Theorem
A trapezoid is isosceles if and only if its diagonals are congruent.
7.17 Trapezoid Midsegment Theorem
The midsegment of a trapezoid is parallel to each base, and its length is one-half the sum of the lengths of the bases.
7.18 Kite Diagonals Theorem
If a quadrilateral is a kite, then its diagonals are perpendicular.
7.19 Kite Opposite Angles Theorem
If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.
8.1 Perimeters of Similar Polygons
If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.
8.2 Areas of Similar Polygons
If two polygons are similar, then the ratio of their areas is equal to the squares of the ratios of their corresponding side lengths.
8.3 Angle-Angle (AA) Similarity Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
8.4 Side-Side-Side (SSS) Similarity Theorem
If the corresponding side lengths of two triangles are proportional, then the triangles are similar.
8.5 Side-Angle-Side (SAS) Similarity Theorem
If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.
8.6 Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.
8.7 Converse of the Triangle Proportionality Theorem
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
8.9 Triangle Angle Bisector Theorem
If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. | {"url":"https://circlepca.org/article/geometry-postulates-and-theorems-a-comprehensive-guide-student-notes","timestamp":"2024-11-09T06:47:08Z","content_type":"text/html","content_length":"76680","record_id":"<urn:uuid:3f58eb50-bb06-4f0e-b2ac-648d98066ab5>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00019.warc.gz"} |
Analysis of the Identifying Regulation With Adversarial Surrogates Algorithm
Given a time-series zk k=1 N of noisy measured outputs along a single trajectory of a dynamical system, the Identifying Regulation with Adversarial Surrogates (IRAS) algorithm aims to find a
non-trivial first integral of the system, that is, a scalar function g such that g (zi) g(zj) , for all i, j. IRAS has been suggested recently and was used successfully in several learning tasks in
models from biology and physics. Here, we give the first rigorous analysis of this algorithm in a specific setting. We assume that the observations admit a linear first integral and that they are
contaminated by Gaussian noise. We show that in this case the IRAS iterations are closely related to the self-consistent-field (SCF) iterations for solving a generalized Rayleigh quotient
minimization problem. Using this approach, we derive several sufficient conditions guaranteeing local convergence of IRAS to the linear first integral.
• Rayleigh quotient
• eigenvalue problems
• learning algorithms
• ribosome flow model
• self-consistent-field iteration
Dive into the research topics of 'Analysis of the Identifying Regulation With Adversarial Surrogates Algorithm'. Together they form a unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/analysis-of-the-identifying-regulation-with-adversarial-surrogate","timestamp":"2024-11-15T03:43:52Z","content_type":"text/html","content_length":"49336","record_id":"<urn:uuid:105437b3-b5b0-4ed9-8f62-3222461c8156>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00686.warc.gz"} |
Programmer's algorithm fun Q62: the largest rectangle in the calendar
2.1 matrix representation of calendar of each month
2.2 handling of holidays and compensatory public holidays
2.3 find the maximum rectangle
1. Problem description
2. Problem solving analysis
2.1 matrix representation of calendar of each month
In python, it can be implemented with monthcalendar() of the calendar module. See another blog Common and interesting usage of Python calendar module . The following are calendars printed in two
ways. The first statement sets Sunday as the first day of the week.
import calendar
The printing effect is as follows:
2.2 handling of holidays and compensatory public holidays
First consider holiday data.
Holiday data is stored in the file as follows:
You can read them in the form of string by line, then use the module in python datetime to transform them into datetime.datetime object, extract the year, month and day information, and then set the corresponding elements of the matrix obtained in the previous section to 0 (indicating non working days) based on the year, month and day information.
Next, consider the data of public holidays that become working days due to compensatory leave. The storage format is the same as above and can be handled in the same way, except that this time the corresponding element is set to a non-0 value (for example, 1)
In the following code, use the readfile() function to read data from the above two files, extract date information, recover the year, month and day, and then store them in a dict type variable, with (year, month) as the key, and value is a list containing the days in the corresponding year and year. Among them, the design uses the datetime module for processing. For a brief introduction to the usage of the datetime module, see the blog: Date time processing in Python: a practical example of the datetime package https://blog.csdn.net/chenxy_bwave/article/details/120977111
2.3 find the maximum rectangle
2.3.1 violent search
First of all, the problem of finding the maximum rectangle can certainly be solved by the method of violent search.
For example, how many rectangles are there in a 2 * 2 matrix with the top left lattice (0,0) of the matrix as the top left corner? Exactly four. For a general n*m matrix, there are n*m rectangles
with the leftmost upper lattice (0,0) of the matrix as the leftmost upper corner. Scan these n*m rectangles, exclude the rectangle with 0 element in the middle (or set its area to 0, which is
simpler), and find the maximum area of the remaining rectangle, that is, the maximum area of "the rectangle with the leftmost upper corner lattice (0,0) of the matrix as the leftmost upper corner".
Next, similarly, we can find the lattice (0,1), (0,2) (1,0), (1,1) is the maximum area of the rectangle at the top left corner. Then find the maximum of these maximum values to obtain the maximum
rectangular area without 0 element in the current matrix.
What is the complexity of such a violent search? For simplicity, consider that the original matrix is square and the size is n*n
First, scan the lattice in the top left corner of the rectangle, n*n
Secondly, the number of possible rectangles corresponding to the top left lattice candidate of each rectangle depends on its coordinates. Assuming its coordinates are (i,j), the number of possible
rectangles is (n-i)*(n-j)
In this way, the total number of rectangles whose area needs to be evaluated is:
This scheme can only be thought of as a benchmark reference, not implemented.
2.3.2 sliding window search
Violence search is based on the grid (considering a grid as the upper left corner of the rectangle). You can also consider the sliding window scheme from another angle, and consider sliding on the
calendar rectangle with rectangular boxes of different sizes and shapes. Because the maximum rectangular area is required, the sliding rectangular window area for scanning is arranged in order from
large to small. In this way, the first sliding position that does not contain 0 is found, and the maximum rectangular area required by the original problem is found.
Because it is possible that rectangular boxes of multiple shapes have the same area, for example, the area of rectangular boxes of 4 * 2 and 2 * 4 is 8. So first build a dictionary, with the area as
the key and the corresponding list of possible shapes as the value. The code is as follows:
# 2. Construct the dictionary for rectangulars area-shape pair
area_shape = dict()
for i in range(1,6):
for j in range(1,8):
if i*j not in area_shape:
area_shape[i*j] = []
With the above preparations, the processing flow for a month is as follows:
Note 1: when resetting the values of the corresponding elements of holidays and extra workdays, its corresponding position in the matrix needs to be determined according to the date information.
First, you need to determine the weekday (day of the week) corresponding to the first day of the current month, so that you can determine the position of the first day of the current month in the
matrix, and then you can deduce the position of the specified date in the matrix. This processing corresponds to the following code (the processing of extra workday is the same):
# Set holidays to 0
if (y,m) in h:
holidays = h[(y,m)]
for hday in holidays:
# Find the position of the current holiday in month calendar matrix
i = (hday + fst_wkday - 1)//7
j = (hday + fst_wkday - 1)%7
c[i,j] = 0
3. Code and test
# -*- coding: utf-8 -*-
Created on Thu Nov 11 09:35:28 2021
@author: chenxy
import sys
import time
from datetime import datetime
import math
# import random
from typing import List
from collections import deque
import itertools as it
import numpy as np
import calendar
# Set SUNDAY to the first weekday
def readfile(filename:str)->dict:
Read holiday file and extra-workday file
filename : string
A dictionary to store the data
print('Read {0} line by line, and store the holidays into a dictionary...'.format(filename))
dat = dict()
if f.mode == 'r':
f_lines = f.readlines()
for line in f_lines:
# print(line,end='')
date_object = datetime.strptime(line[:10], "%Y/%m/%d") # Strip the last '\n' in line
# print("date_object ={}-{}-{}".format(date_object.year,date_object.month,date_object.day))
y,m,d = date_object.year,date_object.month,date_object.day
if (y,m) not in dat:
dat[(y,m)] = []
return dat
# 1. Read the data file
h = readfile('q62-holiday.txt')
e = readfile('q62-extra-workday.txt')
# 2. Construct the dictionary for rectangulars area-shape pair
area_shape = dict()
for i in range(1,6):
for j in range(1,8):
if i*j not in area_shape:
area_shape[i*j] = []
# 3. loop over year/month to find the maximum rectangular of each month
max_area = dict()
for y in range(2014,2015):
for m in range(4,7):
# calendar.prmonth(y,m)
c = np.array(calendar.monthcalendar(y,m))
# Set the first and the last column to 0
c[:,0] = 0
c[:,6] = 0
# print('The original month calendar:\n',c)
# find the first weekday of the current month
fst_wkday, num_days = calendar.monthrange(y, m)
fst_wkday = (fst_wkday + 1)%7 # Because the SUNDAY is set to the first weekday
# Set holidays to 0
if (y,m) in h:
holidays = h[(y,m)]
for hday in holidays:
# Find the position of the current holiday in month calendar matrix
i = (hday + fst_wkday - 1)//7
j = (hday + fst_wkday - 1)%7
c[i,j] = 0
# Set extra-workday to 100--any positive value is OK
if (y,m) in e:
extras = e[(y,m)]
for eday in extras:
# Find the position of the current extra workday in month calendar matrix
i = (eday + fst_wkday - 1)//7
j = (eday + fst_wkday - 1)%7
c[i,j] = 100
# print('The month calendar after holidays and extra workdays setting:\n',c)
# Search for the maximum rectangular only covering workday
found = False
for a in range(35,0,-1):
# print(a)
if a in area_shape:
ij_list = area_shape[a]
for (i,j) in ij_list:
for i0 in range(5-i+1):
for j0 in range(7-j+1):
rect = c[i0:i0+i,j0:j0+j]
# print(a,i,j,i0,j0, rect)
if np.all(rect):
max_area[(y,m)] = a
found = True
if found:
if found:
if found:
Operation result: {(2014, 4): 16, (2014, 5): 20, (2014, 6): 16}
4. Postscript
Because I am not familiar with the processing of date and calendar, I spent some time learning the two modules of calendar and datetime in python. The problem of finding the maximum rectangle, which
should be the core algorithm of this problem, is dwarfed by the processing of date and date.
Previous: Q61: do not cross a stroke
Next: Q63: Maze rendezvous
For the general catalogue of this series, see: Programmer's interesting algorithm: detailed analysis and Python complete solution | {"url":"https://programmer.group/programmer-s-algorithm-fun-q62-the-largest-rectangle-in-the-calendar.html","timestamp":"2024-11-06T09:12:14Z","content_type":"text/html","content_length":"20072","record_id":"<urn:uuid:b82c4204-e358-42ee-b247-12991ccee248>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00871.warc.gz"} |
Newton Raphson Method: Overview, Formula and Easy graphical interpretation
Suppose you need to find the square root of 16 and being very poor in mathematics; your friend will give you three chances to come to the right solution. Will you win this bet? Read this post about
Newton Raphson method and learn how you can do this.
In the field of structural engineering and design, nonlinear analysis is quite common. We deal with quantities like forces, stresses, displacements, strains, and others. All these quantities follow
the nonlinear behaviour. A physical system is said to be nonlinear if the system’s response does not possess a linear relationship. One of the most common numerical methods used to solve such
problems is Newton Raphson Method. In this blog post, we will learn about the basics of Newton Raphson Method and how it is used to solve non-linearity.
Newton Raphson Method
The Newton-Raphson method, also known as Newton’s method, is a powerful technique for finding the good approximated roots of a real-valued function. It can be easily generalized to the problem of
finding solutions to a system of non-linear equations.
Geometrical illustration of the Newton-Raphson method in case of 1-D
Suppose we have $$y=f(x)$$ as a random function with the graph shown in the figure below. If we want to draw a tangent on this curve at a known point $$(x_n,f(x_n))$$ with slope $$f'(x_n)$$, we can
write this tangent equation as:
$$ y=f'(x_o)(x-x_o)+f(x_o)$$
We can find the root of this tangent line by putting $$y=0$$ and $$x=x_{n+1}$$ for our next approximation. Solving this will give us a new approximated root, which is :
We can develop a basic understanding of the Newton-Raphson method from the below figure. It shows the iterations in the case of a load-deflection study. The load is increased in predefined
increments. Displacement is calculated on the basis of the previous step’s stiffness. Then we correct this displacement based on the difference between internal force and external force. We run the
iteration until we get convergence.
Figure showing the Newton-Raphson method iterations. U and F are the displacement and applied load respectively. The Load is increased by a fixed value in each iteration. The interaction point of the
equilibrium path and horizontal line corresponding to the load increment gives the next converged new solution.
Solved Illustration
We can apply the above-discussed formulation to solve a very easy numerical problem. Suppose you need to find the square root of 16 and being very poor in mathematics; your friend will give you three
chances to come to the right solution. Will you win this bet? Let’s figure out using Newton Raphson Method.
Seems like you are going to win the bet.
This method is very easy to use and very convenient but only if our initial guess is close to the actual solution. In other cases we can have erroneous results. So, This method is also associated
with a few significant drawbacks. These can be listed as follows:
• Overshoot: In problems where the slope becomes zero, for example, problems showing snap-through behaviour or snapback behaviour or both, this method jumps away from the actual solution and does
not converge
• Divergence at inflection points: If the selected initial guess is close to the inflection point of the given function, the Newton-Raphson method may diverge.
• Oscillations near local maximum and minimum: Sometimes when there are local minima or local maxima, the solution may oscillate about the point and may not converge.
Figure showing limitations of Newton-Raphson method. It suffers problems like snap through, snap back and oscillation at the Zero slope points
The approximation obtained using the Newton-Raphson method has a quadratic convergence rate if the initial guess is close to the solution. It is advantageous as a solution converges within a few
iterations and saves computational time while solving large systems of non-linear equations. Due to this reason, many Finite Element Analysis software use this approach.
Some key learning from the post:
• Overview: Due to its easy-to-use formulation, It can be generalized to solve non-linear equations and is nowadays used in many engineering software.
• Graphical interpretation: From the graph shown, we learnt how this method arrives at the next iteration
• Numerical Example: A very basic question illustrate the formulation of Newton Raphson Method,
• Limitation: In case of initial guess is not close to the exact solution; this method can give drawbacks like overshoot, divergence at inflection point, and Oscillation near local maxima and
Android Apps
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At eigenplus, our goal is to teach civil engineering students about structural analysis and design starting from the fundamental principles. We do this with the help of interactive android
applications and accompanying web articles and videos.
Our apps have helped more than 400 thousand students across the world to understand and learn the concepts of structural engineering. Check out our apps on the google play store.
Leave a Comment | {"url":"https://www.eigenplus.com/newton-raphson-method/","timestamp":"2024-11-02T09:24:50Z","content_type":"text/html","content_length":"151568","record_id":"<urn:uuid:c98f7f48-b29b-42ba-b9b4-abb489fef4e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00723.warc.gz"} |
Antiferromagnetic Spintronics
Antiferromagnetic (AFM) spintronics is an emerging field of research, which exploits the Néel vector to control spin- and orbital-dependent transport properties. Due to being robust against magnetic
perturbations, producing no stray fields, and exhibiting ultrafast dynamics, antiferromagnets can serve as promising functional materials for spintronic applications, which may expand to very diverse
areas ranging from terahertz information technologies to artificial neural networks.
We are exploring new approaches and new material platforms, which can be exploited in AFM spintronics. One approach involves AFM tunnel junctions where the relative orientaion of the Néel vectors of
the two AFM metals controls resistance of the junction resulting in a tunneling magnetoresistance (TMR) effect. Using RuO[2] as a representative antiferromagnet exhibiting a non-spin-degenerate Fermi
surface, we design a RuO[2]/TiO[2]/RuO[2] (001) AFM tunnel junction and predict the TMR effect as large as ~500%.
In another approch, we consider the Néel vector switching in non-collinear antiferromagnets ANMn[3] (A = Ga, Ni, Zn, etc.) with an antiperovskite crystal structure. These compounds are characterized
by the competing AFM Γ[5g] and Γ[4g] phases. Combining density functional theory calculations and atomistic spin-dynamics modeling, we demonstrate that the spin torque can efficiently control the
noncollinear AFM order in the antiperovskite materials. The switching can be detected though the anomalous Hall effect being zero or finite for the Γ[5g] and Γ[4g] phases, respectively, due to
symmetry of the Berry curvature.
Further, we explore a posibility to use the Néel vector to electrically manipulate topological states. We demonstrate that room temperature AFM metal MnPd[2] allows the electrical control of the
Dirac nodal line by the Néel spin-orbit torque. The reorientation of the Néel vector leads to switching between the symmetry-protected degenerate state and the gapped state associated with the
dispersive Dirac nodal line at the Fermi energy.
Finally, we predict that a nonlinear anomalous Hall effect can be used to detect the Néel vector in most compensated antiferromagnets supporting the antidamping spin-orbit torque. We show that the
magnetic crystal group symmetry of these antiferromagnets combined with spin-orbit coupling produce a sizable Berry curvature dipole and hence the nonlinear anomalous Hall effect. We demonstrate this
behaviour for half-Heusler alloy CuMnSb, whose Néel vector can be switched by the antidamping spin-orbit torque.
1. D.-F. Shao, S.-H. Zhang, M. Li, C.-B. Eom, and E. Y. Tsymbal, “Spin-neutral currents for spintronics,” Nature Communications 12, 7061 (2021).
2. G. Gurung, D.-F. Shao, and E. Y. Tsymbal, “Transport spin polarization of noncollinear antiferromagnetic antiperovskites,” Physical Review Materials 5, 124411 (2021).
3. T. Nan, C. X. Quintela, J. Irwin, G. Gurung, D. F. Shao, J. Gibbons, N. Campbell, K. Song, S. Y. Choi, L. Guo, R. D. Johnson, P. Manuel, R. V. Chopdekar, I. Hallsteinsen, T. Tybell, P. J. Ryan,
J. W. Kim, Y. S. Choi, P. Radaelli, D. Ralph, E. Y. Tsymbal, M. S. Rzchowski, and C. B. Eom, “Controlling spin current polarization through non-collinear antiferromagnetism,” Nature
Communications 11, 4671 (2020).
4. G. Gurung, D.-F. Shao, and E. Y. Tsymbal, “Spin-torque switching of non-collinear antiferromagnetic antiperovskites,” Physical Review B – Rapid Communications 101, 140405(R) (2020).
5. D.-F. Shao, S.-H. Zhang, G. Gurung, W. Yang, and E. Y. Tsymbal, “Nonlinear anomalous Hall effect for Néel vector detection,” Physical Review Letters 124, 067203 (2020).
6. H. Takenaka, S. Sandhoefner, A. A. Kovalev, and E. Y. Tsymbal, “Magnetoelectric control of topological phases in graphene,” Physical Review B 100, 125156 (2019); Editor’s Suggestion.
7. Gautam Gurung, Ding-Fu Shao, Tula R. Paudel, and Evgeny Y. Tsymbal, "Anomalous Hall conductivity of noncollinear magnetic antiperovskites," Physical Review Materials 3, 044409 (2019).
8. D.-F. Shao, G. Gurung, S.-H. Zhang, and E. Y. Tsymbal, “Dirac nodal line metal for topological antiferromagnetic spintronics,” Physical Review Letters 122, 077203 (2019). | {"url":"https://tsymbal.unl.edu/antiferromangetic-spintronics","timestamp":"2024-11-10T07:49:46Z","content_type":"text/html","content_length":"1049601","record_id":"<urn:uuid:19c76dc3-c486-4123-a513-66dd7953bb85>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00077.warc.gz"} |
André Chassein
We consider the problem of finding a shortest path in a directed graph with a quadratic objective function (the QSPP). We show that the QSPP cannot be approximated unless P=NP. For the case of a
convex objective function, an n-approximation algorithm is presented, where n is the number of nodes in the graph, and APX-hardness … Read more | {"url":"https://optimization-online.org/author/chassein/","timestamp":"2024-11-05T12:43:03Z","content_type":"text/html","content_length":"82982","record_id":"<urn:uuid:f0043886-bdc7-469d-8a46-cd31c8c93cf1>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00139.warc.gz"} |
The QO interpretation | Quantum Occam
All behaviour in the universe is the consequence of everything being part of the same quantum system, in which the same information is stored only once. The fact that a quantum subsystem distributes
its energy over its degrees of freedom makes it possible for one quantum subsystem to manipulate another quantum subsystem in a manner which can be described as purposeful.
│Phenomenon explained : "How quantum systems behave". Quantum systems obey some very specific rules. Quantum systems resonate. They distribute their energy evenly over their degrees of freedom. │
│ When the quantum system encounters another quantum system, the result is a new quantum system in which the degrees of freedom may be allocated differently. The whole is different to the sum of its│
│parts. This is particularly the case when one system 'measures' the other by restricting the degrees of freedom of some quantum or quanta in it. │
QM is logical
There are many interpretations of Quantum Mechanics. Most of them involve, implicitly if not explicitly, the notion that Quantum Mechanics is absurd. As a result, the impact of Quantum Mechanical
processes is downplayed as much as possible. QO is based on the position that one must regard Quantum Mechanics as normal in order to interpret it correctly. QO regards Quantum Mechanics as being
supremely logical. It is what you would have invented, if given the task of creating a universe from nothing.
QM applies at all levels
Before we commence, it is important to defuse the notion that quantum mechanics is relevant only at the level of the very small, and cannot possibly have effects discernible at a macrosopic level.
That is like arguing that because water waves are the result of water molecules bumping into each other, they cannot possibly be much bigger than individual water molecules. A quantum system of any
size can resonate, producing quantum effects at the same scale.
QM results in a zero energy universe
According to the uncertainty principle, the energy and duration of a quantum fluctuation are related by the relation
where hbar is a Plank's constant divided by 2 pi.
It follows that an energetically neutral quantum fluctuation will persist forever, and any other quantum fluctuation will persist for only a very short time. Therefore the only way that a universe
can be created from quantum fluctuations is for it to be energetically neutral.
The QO interpretation
In the QO interpretation, interactions between distinct quantum systems are by definition interactions within the wave function of a quantum system of which they are both part. For example, what we
perceive as two elementary particles bumping into each other is, may be understood at a deeper level as an interaction pattern in the quantum system with which they are both entangled.
When two quantum systems interact, the result is a new quantum system. It is no longer valid to consider the original systems separately, because this new system is not just the sum of the two
original systems. In principle, the new system can exist in each combination of configurations that the original systems had, plus some that are possible only because of the interaction. The new
system has as many degrees of freedom as the sum of the degrees of freedom of the two original systems, which makes the variation that it can exhibit greater than the sum of the variations which they
could. In other words, the whole is more than the sum of the parts.
When we make a measurement, the quantum system of which we and our measurement apparatus are a part interacts with the quantum system that we measure. If the nature of our act of measurement is to
determine whether, in one of the degrees of freedom of the measured system, a parameter did or did not satisfy a particular condition, then the variation of that parameter will be restricted. But at
the same time, the variation within other degrees of freedom will increase. In first instance, the variation of the complementary property of the same particle will be affected, but subsequently the
quantum system will tend to a state in which the variation for all of its degrees of freedom are in balance.
One of the key problems of most interpretations is their supposition that a quantum system, when measured, somehow “chooses” a particular state to take on. This is a problem for two reasons. Firstly,
it is not clear as to what constitutes a measurement. Secondly, it is not clear as to how a quantum system can make a choice. In this lemma, we postulate that measurement consists of a confrontation
of the quantum system with what we term a “purpose”. Whatever induces a measurement constitutes a purpose. This may affect a quantum system by any of the following means:
1. Fixed eigenstate: The most simple way in which a purpose can affect a quantum system is to confront it with a fixed eigenstate. For example, a polarizing lens confronts incoming light with an
eigenstate in such a way that only light which is polarised in conformance with the lens is let through. This type of measurement induces an arbitrary collapse of the wave function to match the
2. Quantum zeno: A second way in which a purpose can affect a quantum system is to effectively freeze it in a particular state by means of repeated confrontations with the same eigenstate. This is
predicted by standard quantum mechanics and has been experimentally observed. It is known as the quantum Zeno effect.
3. Progressive quantum zeno: A third way in which a purpose can affect a quantum system is to pull it from one state to an arbitrary other state by means of repeated confrontations with a varying
eigenstate. This too is predicted by quantum mechanics and can be observed experimentally. It could be termed the progressive quantum Zeno effect.
4. Natural purpose: A fourth way, predicted by Mc Fadden and others, is that it can attach itself to the quantum system in such a way that it only measures the system when it resonates in a
particular way, as defined by the purpose. Until such time, all possibilities exist as superpositions of the system. The quantum system behaves as a quantum computer, in which large numbers
of combinations are tried out at the same time, but only those results for which the system resonates result in a positive measurement. If there is a negative message, the system is allowed to
decohere, after which a new measurement is made, and so on until there is a positive measurement. McFadden suggests that the origin of life may have been due to a quantum effect of this nature.
QO applies natural purpose as an explanation for why we live in a universe which makes life possible: all possible universes existed as a superposition of a subset of waves, and only ours could
affect the waves around them and become real. This is the QO multiverse explanation.
│Credibility: │
│ │
│The first three means have been observed, and have absolute Occam scores of 0000. They are foundationally credible. │
│ │
│Natural purpose has not been conclusively demonstrated as such, but it is a logical consequence of quantum mechanics. That makes it a simple hypothesis, with an Occam Score of 0010. Any other │
│explanation as to why the universe is fine-tuned for life must of necessity be more complicated, requiring at least an as yet unknown mechanism to generate variety in natural law and therefore │
│resulting in at minimum a complex gap, with an Occam Score of at least 3300, more than two notches higher. Therefore the natural purpose explanation is foundationally credible. │ | {"url":"https://quantumoccam.net/explanations/the-QO-interpretation","timestamp":"2024-11-13T05:59:50Z","content_type":"text/html","content_length":"31910","record_id":"<urn:uuid:5c276156-2e7c-429d-b1de-06bbdc2c6903>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00798.warc.gz"} |
Superseded / Deprecated
The feature described in this article has been superseded by a newer feature. This feature still works in PICO-8, but the new feature should be used instead. See the article's "Superseded by..."
section for specific details.
rotr( num, bits )
Rotates the bits of a number to the right.
The number.
The number of bits to rotate.
The rotr() function takes a number and a bit count, and returns the result of rotating the bits of the number to the right by that count.
Numbers in PICO-8 are stored using a 32-bit fixed point format, with 16 bits for the integer portion, 16 bits for the fractional portion, and a two's complement representation for negative and
positive values. Bit rotating uses the entire number representation. (See examples below.)
See rotl() to rotate bits to the left.
Superseded by >>< operator[ ]
The >>< operator added in 0.2.0 performs the same function as rotr() and is now the recommended way to rotate bits right, as it uses fewer tokens, costs fewer cycles at runtime, and runs on the real
host CPU much more efficiently. Simply replace rotr(a,b) with a>><b.
Examples[ ]
-- 64 = 0b01000000 binary
-- 8 = 0b00001000 binary
print(rotr(64, 3)) -- 8
-- 1.000 = 0b0001.0000 binary
-- 0.125 = 0b0000.0010 binary
print(rotr(1, 3)) -- 0.125
-- -4096.0000 = 0b1111000000000000.0000000000000000 binary (two's complement)
-- rotate --> by 12 bits
-- 15.0000 = 0b0000000000001111.0000000000000000 binary (two's complement)
print(rotr(-4096,12)) -- 15
-- (when printing fractional numbers, pico-8 rounds the decimal
-- representation to four decimal places. the largest fractional
-- portion is about 0.9999847... in decimal, so pico-8 prints it
-- as "1".)
-- approx 1 = 0b0000000000000000.1111111111111111 binary (two's complement)
-- -15.9998 = 0b1111111111110000.0000000000001111 binary (two's complement)
print(rotr(0b0.1111111111111111, 12) -- -15.9998
print(64 >>< 3) -- 8, preferred method
See also[ ] | {"url":"https://pico-8.fandom.com/wiki/Rotr","timestamp":"2024-11-02T12:41:53Z","content_type":"text/html","content_length":"157518","record_id":"<urn:uuid:b06efb83-f6c7-4b33-9531-91e3b7589931>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00667.warc.gz"} |
Yasir Qadhi - What I Have Learned In 12 Months Of Genocide
Yasir Qadhi – What I Have Learned In 12 Months Of Genocide
AI: Summary ©
The speakers emphasize the importance of finding a solution to political and political issues in the coming year, including improving housing affordability, addressing issues of pride and civil
rights, and addressing issues of pride and civil rights. They also emphasize the need for immediate action and a long term strategy, including a focus on improving housing affordability, putting
pressure on media, and reaching political reform. The discussion takes place between two speakers about a situation where someone sought advice from a friend, but the speaker does not provide any
context or information.
AI: Summary ©
Over the weekend, we caught up with Sheikh
Yusuf Qadhi and asked him, after Omar Suleiman's
repetitive, rapid-fire questions, what he's learnt in
the last 12 months of the Gaza genocide.
Stay tuned, because this is part one of
two videos, and this is a warm-up
to the possibly contentious part two that's coming
up soon, inshallah.
So if you want to be at the
front of the queue, and in drops, be
sure to subscribe wherever you get your podcasts.
As-salamu alaykum, Sheikh.
How are you?
Wa alaykum as-salamu alaykum.
I'm doing fine.
Excellent, excellent.
So look, we're going to go straight into
I don't think even Salman hasn't seen these,
so no time to think, off the bat,
Wait, wait, before you even begin, I'm jet
I'm drinking my double espresso macchiato.
So give me some time.
But okay, let's see.
That's my excuse.
So that's your case.
We're waiting for the caffeine to kick in.
Number one, Sheikh.
Favourite freedom fighter.
You recently did a talk.
You recently did a talk on one, which
is what inspired the question on Omar Mukhtar.
I was thinking, between him, you must have
read about a number of different freedom fighters.
Is there one that particularly…
Are you trying to get him arrested?
No, seriously.
Let's say historically, not metaphorically.
I was thinking of modern times.
Which group?
Which of these acronyms?
Do you connect with that?
Favourite historical freedom fighter.
So, you're a Muslim.
So, without a doubt, you're asking questions that
don't have simple answers and you want to
round fire and I don't do simple answers,
There's different genres of freedom fighters.
Don't worry, somebody's going to take a 30
second clip.
That's not even the issue.
Okay, growing up as a kid, Omar Mukhtar's
movie really did impact me.
That's why I wanted to give the Khatira
and I was waiting for the anniversary of
his execution.
So the week that that came, it just
happened to be the exact same day I
gave the Khatira.
The other day with my kids, none of
the Khatira remained and then I started watching
Line of Desert.
And then, for some reason, the day after
we were going to continue the film, Amazon
Prime said this film is no longer available.
SubhanAllah, really?
Wow, there must have been a lot of
people watching it and then they took it
I mean the acting that was done and
then I watched it, believe it or not,
in the Arabic with English subtitles.
Yeah, I watched it in the 80s in
the Arabic.
So they dubbed it in Arabic?
They dubbed it in Arabic and then for
some weird reason, they put it in English
But then the Arabic accents and the Omar
Mukhtar's voice that comes out, it's just so
And I remember even as a child, you
were just in tears by the time the
movie's ended.
So simply because of that memory, I'm not
saying he was the only one, but it
has an impact on you and always has
that soft spot in your heart.
You know that Khatira, I noticed on YouTube
there were some bits as though it was
edited or taken out.
Don't worry.
No, no.
It's just a micro.
No conspiracy.
I was like, I'm going to ask him
No conspiracy.
Who did you mention?
Because he was like, you know, there were
some people at the time that were some
Muslim scholars.
He said there were Muslim ulema that were
against him and were pro-Italian occupation.
Like today, there are some Muslims that are,
and then he was like, no, it's just
the micro.
Don't worry.
I didn't mention the names.
Plus, here's the, I said it in the
Those names are not known to the average
You have to research in the books of
history, which I don't mind doing, but why,
why bring them up?
Let them be.
Allah has covered them.
And honestly, they did things they shouldn't have
done, but maybe, maybe they're forgiven because they
thought that's for the maslah of the ummah,
you know, to think that working for, look
at what's happening around the world.
Hindsight bias.
And Allah, I'm not blaming them.
But without a doubt, the hearts are with
Omar al-Muqtada.
So continue with the not very quick round.
Yeah, it's not going to happen, bro.
Just two, two, three more.
We got it.
All right.
Most misunderstood, misrepresented scholar from Islamic history, in
your opinion.
Of recent times, I do think Rashid al
Of recent times, I think Rashid al-Ridha.
And even Mohammed Abdo, his teacher, the two
of them.
I would say 99% of those who
are actually criticizing them haven't even read a
single treatise of theirs.
I'm not even talking about the awam.
I'm talking about those that are actively criticizing.
You just talk to them.
Okay, what have you actually read?
And all of his books are PDF.
All the books are online.
I have multiple physical copies of his books,
and PDFs, I have almost all of them.
Before you open your mouth about an iconic
figure, the least you can do, and even
recently another guy came out, a standard guy,
I don't know if you call him a
zindiq or a freemason or whatever.
Before you jump to something, just read.
These are all very difficult times, and people's
angers cause them to say things about each
other that you're just simply regurgitating.
If you want to go back to the
past, Imam Malik and Mohammed Ibn Ishaq.
Imam Malik and Ibn Ishaq had a massive
battle between them, and we just ignore it.
You know what?
We just say, both of them were good.
So yeah, of recent times, I would think,
without a doubt, Rashid al-Rudha, because it's
not a shade of, okay, he was right
or wrong.
There's literally accusations of heresy and zaniqa, and
literally wanting to destroy the religion or being
an agent or whatnot, standard routine.
And people who do so are simply not
even aware of his actual writings and the
battles that he faced.
And subhanAllah, some of his even critics that
were alive at the time were actually on
a personal relationship with him, which means they
viewed him as incorrect, but not like working
with the devil or something.
It was like, okay, I disagree with you,
but we're still Muslim brothers.
The modern critics have completely disconnected from that
reality, like Ibn Taymiyyah even and his critics.
Ibn Taymiyyah was on friendly terms as a
person with many of those he criticized, right?
As a human being, Muslim to Muslim, but
he criticized their views.
But the intellectual descendants of both sides have
lost the human touch, and they've made it
into something much more.
But anyway, that's definitely one of the most
JazakAllah khair.
Should the Indian pact partition have taken place?
It's a very sensitive question.
It is.
Very sensitive question.
I speak as somebody who genuinely has relatives
on both sides of the aisle, and I've
visited my relatives on both sides of the
And I can also state for the record,
I've never lived in either side.
So I am somewhat neutral.
My parents were born in India.
My father has a lot of memories of
India still, and then they grew up in
And so I'm a Muhajir Pakistani basically, right?
So it's a very difficult, and I'll explain
why to those viewers that don't understand why,
because this is important.
The argument that is made is that the
BJP and the anti-Islamic sentiment could not
have happened if the Muslims remained in India.
Because then the sheer percentages and the demographics
would have not allowed for the So the
argument that one group makes is that the
creation of Pakistan created an imaginary enemy, or
not even imaginary, self-fulfilling prophecy.
And that then allowed them to perpetuate this
myth of Hindu-Muslim constant war.
That is what is said from the one
side that says that Pakistan should not have
been created.
Because they're not saying that BJP is good.
They're saying we would have had a more
secular and a more neutral India.
That was the claim, or that is the
claim that still people are making.
And then the counter side to that is,
well, that's a figment of the imagination.
Riots began, massacres began even before, and there
was no BJP.
You know, a million people were killed in
The largest migration in human history took place.
The largest mass migration in all of human
history to this day, and that's saying a
lot, took place in 1947.
And, you know, of the largest massacres took
And this is before the actual creation of
the BJP and the RSS.
And so the counterclaim is that this is
just wishful thinking.
And this is why people like Jinnah understood
the need to create a safe haven.
Jinnah did not intend to create.
I know this is going to make a
lot of our Pakistani brethren angry, but it
is patently clear.
And I'm sorry if facts hurt you.
Jinnah did not intend to create an Islamic
He intended to create a Muslim secular state.
A Muslim state where Muslims were safe.
He had no intention.
I'm not saying it's right or wrong, whatever's
happening now.
His vision was to have a land where
Muslims can be safe and feel free to
He had no intention of importing or bringing
Sharia laws.
This is something that is a back projection
in revisionism.
And whether that's good or bad is a
separate conversation, right?
But why did Jinnah have this?
Because Jinnah, knowing the founders of Congress, knowing
the other people involved, understood that it's not
going to be good for the Muslims.
And I think overall, I would sympathize with
that sentiment that had the Muslims of India
remained, we would have still had a BJP
version and we would have still had massive
And anybody who visits the two countries, and
I have visited the two countries, both have
their issues and problems.
But without a doubt, in Pakistan, being a
Muslim, you can live a dignified life.
Nobody's going to, by and large, harass you
and intimidate you because you're going to pray
in the masjid.
So I think overall, it was something that
was for the good.
But yes, there are ambiguities.
I agree.
You can feel sympathetic to both arguments, actually.
You can.
It's never black and white.
You can, yeah.
Okay, Sheikh, Sheikh Yasir Qadhi today, closer to
activism or pragmatism?
When I say activism, I don't just mean
as we kind of see out there, but
in terms of revival within the Islamic tradition.
So are you closer to activism or pragmatism?
I don't view this as an either or.
I am pragmatically active.
Or actively pragmatic.
I saw that one coming my way.
Yeah, when I had this one down, I
threw that one back in.
This is not an either or question, because
I am an activist at every level, except
the political.
I'm not really interested.
Very once in a while, I'll speak at
a rally, but that's not what I want
to do.
That's not, personally, that's not my interest or
But I'm active at the preaching and teaching
At all levels of preaching and teaching, I'm
active within the Muslim community.
I'm also not active in interfaith.
I'm just not interested in that.
But preaching at the mass level, at the
basic academic level, and at the advanced level,
this is what I enjoy doing.
And in all of them, I try my
best to integrate reality, which means it is
Okay, good.
And connected to that, the favorite seminar you've
taught over the years, there was one, and
you've got to pick one, Sheikh.
Yeah, the favorite seminar is Modern Theology, and
I plan to do that again, inshallah, next
semester at my masjid, Modern Theology.
So, modern issues facing the ummah, you know,
how do we resolve them?
Which actually is a good segue to the
topics you wanted to talk to me about.
Okay, the next book you'd like to read?
The next book I'd like to read?
Is it an unknown unknown?
The unknown unknowns, huh?
To get rums filled in.
That's a good question.
I have, so I'll tell you a little
bit about how I purchase books.
If an interesting title is reviewed, and I
see somebody talking about it on any of
the WhatsApp groups, or any of the email
lists, and I really like it, I'll go
ahead and order it, and then there's a
section on my desk that just stays there.
For, don't ask me how long, it just
stays there.
It will not be put on the shelf
until it passed through the process.
So what is the process?
At some point in time, Allah knows when,
that big pile will then be opened up
one by one, skimmed through, the introduction read,
get an idea of what the author is
saying, the book is signed and dated, every
single book I have is signed and dated,
I do not believe in stamps in this
regard, I want that personal touch here.
So it is signed and dated, and then
it is put in its appropriate category, because
obviously at this stage of my life, it
is difficult to read a book cover to
So over the last few years, I have
done that to many dozens of books, so
I have selections out there.
One of the books I am waiting to
read is actually, and again, it's not necessarily
the most important one, but it is of
personal interest, the famous academic, Islamic studies academic,
Michael Cook, who has a lot of good
and bad, he has released his final book,
he said it, which is the history of
It is like a thousand page book, massive
book like this.
And it is actually very, very well written,
and so I want to read that cover
to cover.
It is one of the books I am
really wanting to read cover to cover, because
somebody like him, when you get to that
level, and I say this as somebody who
strongly disagrees with many of the ideas and
views, but also respects aspects of his writing,
and his erudite grasp of the classical and
the modern tradition.
Somebody like him, when he writes, he is
not writing after having read two books per
chapter, he is writing after having read 200
So what you get is the distilled summary
from the mind of, again, I strongly disagree,
but he is a genius.
I know it sounds weird to some of
our viewers, but there are people whose views
you can strongly disagree with in some aspects,
but their minds are very, very formidable.
They definitely bring something to the table.
Any book that he has ever written, I
challenge anybody to read it and not be
impressed by the content and the analysis, even
if you disagree with it.
He is a thinker, a genuine thinker.
So he has just written an entire summation
of the history of the Muslim world from
the beginning up until modernity.
It's literally a massive tome like this.
So to me, this is necessary reading, and
it's been lying there.
I can see exactly where it is from
my desk, and so when the time comes,
inshallah, we're going to do that.
Okay, inshallah.
I told you there's no simple answers.
What's like ten seconds?
I don't do these ten-second things.
They could be simple answers.
I can't.
I can't.
You're speaking to the wrong person.
I wouldn't be who I am if I
could give you your simple five-second clips,
How do the others do it?
They all seem to get five-second clips
of you.
It's boring.
You ask my Cocoa Pepsi.
All right.
Try and choose an either-or on this
What do we need more of, influential scholars
or scholarly influencers?
Right now as we speak, I think we
have a good amount of influential scholars, so
we need scholarly influences for the time being.
But it's this pendulum.
So maybe in a few years or decades.
But yeah, right now, yeah.
Just like that.
Mehdi Hassan or the Sabri Brothers.
I mean, Mehdi Hassan or Bassam Youssef.
Those who know will know that one.
Mehdi Hassan or Bassam Youssef.
Who do you hate more?
So I've never met Bassam.
Mehdi, I was with him two weeks ago,
and he is an acquaintance of mine, and
I have had very frank conversations with him.
Each one has a role to play.
I'm sorry.
But Mehdi, you have to give him the
fact that he does a ton of research
before every single interview, and he comes prepared
like hardly any other presenter that I've seen.
And Bassam has a talent.
So again, look at the Piers Morgan interview
that went viral.
And this is, again, things that need to
be said.
Listen, his personal life is between him and
We are allowed to critique any public aspects
of the personal life that he brings into
the public eye.
We're allowed to say this is right or
But you see, the Prophet ﷺ said, and
I don't mean to apply this directly to
I'm speaking conceptually.
And if it applies to him, it does.
And if it doesn't, it doesn't.
I'm not necessarily making the causal linkage.
But I am being precise.
The Prophet ﷺ said, Allah sometimes helps this
deen with a man who is not very
A man might be a Fajr, but he
helps the deen.
That interview was one of the best mechanisms
to begin the discourse and to start shifting
the narrative.
And nobody could have done it.
Except Bassam.
So we have to give credit where credit
is due.
And he got away with saying things because
of who he is.
That none of us could have said because
we would not have meant it.
We would not have meant that.
So he got away with his slant.
And it was needed to begin opening the
So one of my issues, and again, I
know you want your 10-second quiz, but
I don't work that way.
We have to start thinking in color with
more nuances.
We cannot be so simplistic and binary, which
is one of the biggest problems of the
critique and the cancel and the fundamentalist culture.
It doesn't work that way.
The world is very complex.
So you can't just say, do you agree
with Bassam or not?
It doesn't work that way.
Bassam is serving a function.
And I hope Allah guides him to a
better understanding of religion and practice of religion.
But for the time being, he is doing
some overall great work.
And he's not, at this stage, a blatant
enemy to the Ummah.
And I know in the past he has
things that he has to answer for.
So at this stage, I'm not going to
work in any manner or shape to somehow
minimize his voice.
Not that I even have the power to
do that, but no.
Does this mean we invite him to our
So here's where the nuance begins.
Can we utilize such individuals in arenas while
we realize they're not going to be capable
of being effective in other arenas?
Yes, I think we can.
By masjid, you mean to address the congregation,
not like we're going to ban him from
the masjid.
Obviously, yeah.
Because some of our youth, I'm not going
to mention the name because he didn't mention
his name, but a very famous comedian came
to Dallas of a Muslim background.
And our youth were clamoring, let's get him
to my masjid.
I'm like, it is not befitting that this
person, who has no nobility, has crass jokes.
He's funny, great.
But the masjid is not the platform for
this type of person.
I said, if you were to hire a
hall and have a youth event, not under
the banner of the masjid.
And he serves as a, we tell him,
the goal is to have a positive role
model for the youth.
And he agrees that that's why he's coming.
I can see that I'm not going to
be there.
The masjid board should not be there.
Our masjid should not sponsor.
In disguise.
In disguise, yeah.
I don't have the wardrobes you do with
all the mustache and fake mustache.
You should bring one of those to your
Because I know you do all those weird
So maybe one day you should just put
the weird fake mustache on and your viewership
will rise.
I'm going to his other channel now.
Straight into his other channel.
So to answer your question, neither of the
two is without issues.
But Mahdi is far more educated and academic
and comes researched.
And if we were to have a debate
with somebody in the political realm, I ask
you, can you think of anybody better from
our side to represent?
This is a shame, isn't it?
Well then, whatever criticisms people have of him,
some of which might be legitimate, needs to
also be put into context with what he
brings to the table.
No, absolutely.
And Sheikh, to be honest, I mean, I
asked that question.
I just gave you two names.
And it was important that that nuance did
come out.
You know, people think about, they might just
say, oh, Mehdi Hassan or Bassam Youssef based
on personality.
But all of that layering, I hope people
can really appreciate that because it's about ultimately
accessing audiences that I guess we can't, yourself
can't even, let alone a lot of other
And so it's important that they get those
voices heard as well.
So the last one, Inshallah, and this is
a lot easier.
The next country you plan to visit for
the very first time, and if it's not
a country, then a city at least.
I've been thinking of going to Albania.
The reason being that it is a Muslim
-majority European country.
And I'm very interested in European Islam.
So I was literally just, I haven't even
thought of when it were, but a very
interesting question you're asking.
I love to go to places where there's
history that I don't know.
And so I like touring.
So last year I went to Prague in
the Czech Republic, just on my own, just
so I could go and see the history
And then I went to Vienna.
Unbelievable, the amount of history in Vienna.
It was a superpower.
And you go to the museums.
I'm the type of guy who spends five
hours at the museum.
And I literally, I'm that guy.
I go to every single cubicle and read
what's on there.
And get a selfie.
That's me.
I don't take selfies at museums.
That's boring.
But yeah, nobody wants to go to museums
with me.
Oh, so you're anti-selfie now.
I am indeed, yes, indeed.
That was an old joke, by the way.
That's an old joke, man.
So the next country I was thinking about
doing was Albania.
Another country I'm interested in going is Malta.
Because Malta also has a lot of remnants
of the Muslim rule.
And especially on their second island, not their
main island.
So I've done my research on this, like
Which one?
Yes, that one.
You've been?
Yeah, yeah.
Oh, you've been to Malta.
I discovered my level of snorkeling there.
Which is zero?
No, no, it's like as far as love
goes, it's like just one below.
Oh, level of snorkeling.
I thought you said zero level of snorkeling.
Okay, Alhamdulillah.
I mentioned that's where I had the nicest
Masha'Allah, masha'Allah.
Speaking of which, theme today of croissants.
So Sheikh, last 12 months we've seen obviously
the Gaza genocide step up in intensity.
We've seen so much.
I don't want to put words in your
mouth, but what have you learned?
What have we learned in the last 12
The resilience of the Palestinian brothers and sisters.
Their iman, their courage.
People like this are still alive.
Walking saints on earth.
May Allah protect us.
What would we have done?
May Allah protect us.
I say this from behind closed doors, from
behind chained walls.
They're the ones freed.
They're the ones liberated.
Sorry, they're the ones chained up.
It is as if they liberated the rest
of the world, even though they're the ones
chained up.
So without a doubt, we learned that.
But that's not a surprise.
We knew that from them.
One of the lessons that we learned, and
one of the painful ones, and I say
this bluntly because there's a genocide that is
going on for a year, is the fact
that us Muslims in the western world, all
of our strands combined, have failed in one
I'm not saying they're failures.
They've failed in one aspect, and that is
giving us civilizational strength.
No matter what you want to say about
their successes, and they have successes.
Every strand has done much good.
May Allah bless them.
But one area where clearly we are disconnected
from the worlds we live in is civilizational
By civilizational clout, I mean social and political
and economic.
That's what civilizations are built on.
And military.
Military, so I'm talking about western Muslims.
I'm not talking about, so there's two separate
So I'm not talking about, that's a given.
Without a doubt, but my conversation now is
about us.
It's so easy to point fingers there, and
we should.
It's a shame.
How can you have an army at your
How can you be the ruler of a
How can you have billions, and you're just
watching this?
That's between you and Allah.
But I am more concerned about me, myself,
and us here.
What are we doing here?
My priority is us here.
We don't have armies at our disposals.
But we are citizens in the very countries
that are endorsing these policies.
And we have failed ourselves to have the
difficult conversations and to plant the infrastructures needed.
We're still debating things that used to be
50 years ago.
Our nation-state identity.
We're still debating, voting, and protesting.
And so in the last 11 months, my
own rhetoric and khutbahs and lectures about this
have become extremely blunt.
I've lost all political correctness because it's a
Lancet estimated a quarter of a million Palestinians
have died.
Indirectly, indirectly.
A quarter of a million.
And see, the sad thing is, subhanAllah, even
if tomorrow the bombing were to stop, Gaza
is in ruins.
What are those two million people going to
There is no infrastructure.
What are they going to do?
Where are they going to live?
Where are they going to send their children?
Where are they going to get medical aid?
We don't even have a solution as an
So my main then personal concern is that
we need to understand this moment as a
wake-up and a call for action.
The Muslim community is still bickering over issues
of aqidah.
It's still divided amongst political lines, ethnic lines.
If this is not going to cause us
to wake up and come together, then what
And we as well have to have some
very, very difficult conversations of the level of
And here is where I don't have answers.
What does it mean to be a Muslim
I don't know.
But I do know that being apolitical and
sticking your head in the sand and running
away, shouting kufr, haram, shirk is not going
to get us anywhere.
We need to take ownership without feeling any
This is our land.
I find it shameful that you in this
country, and allow me to be blunt because
I'm an American.
I have an excuse.
We're less than one percent.
Ten percent of London is Muslim.
At least seven, eight percent of the country
is Muslim.
I find it shameful that that is not
manifested at the cultural level, at the socio
-political level, at the economic level.
Why not?
Why is there still this isolationist tendency to
just cut off and to build these walls
between you and one of them, Wala Anbara,
if you're one of them.
But it's not the only one.
That's why we need to go back to
this misconstruction of Wala Anbara.
Because it has quite literally acted as an
impediment to civilizational Izzah.
Once you take ownership, and you start speaking
in a different manner.
This is your land.
You are British.
Whether you want to call it this or
not, you are British citizens.
Own it.
There's nothing wrong with that.
It doesn't go against Wala Anbara.
Who taught you this?
They're wrong.
So once you understand you have ownership, then
you understand it is your duty, not just
in the eyes of Allah, but even you
can use the rhetoric.
Nothing wrong with that.
Even your patriotic duty.
That to make this country a better country,
a more moral country.
Once your paradigm shifts in this regard, and
there's many impediments.
The one that irritates us the most should
be the religious impediments.
It's not the only one.
The religious folks who don't understand this, sideline
them, bypass them.
If you can't reason with them, just get
to the ones that can be reasoned with.
Ignore them because they are an impediment to
a necessary.
Imagine if 10% of your parliament was
understanding of the reality of Izzah, which they
And they could easily be if the Muslim
participation was of that level.
But the problem comes, as we're all aware,
Muslim politicians have to compromise this and that.
The whole nine yards begins.
This is the awkwardness.
I don't have solutions.
I don't.
I've given generic answers.
I've given generic.
Ulama should not run for politics.
Ulama should not be at the forefront of
But people who love Allah and His Messenger
are far better than people who don't believe
in Allah and His Messenger.
And to have some people like that in
office, even if it comes at a personal
cost to them, it's a reality we're going
to have to take.
You know, Sheikh, I'm just kind of synthesizing
a lot of what we've said so far
and even this point.
You know, when October 7th and the genocide
started after that, I think for the first
time, and I'm obviously very plugged into different
Muslim professional groups, I had a number of
different Muslims contact me going through a state
of depression.
And their depression was based on we drank
the Kool-Aid.
We believed, you know, they're younger, that we're
part of the fabric of this society.
We're involved.
We get involved in everything.
They don't have the cultural hangups that my
generation would have.
Yet everything has been so one-sided.
And they don't know how to deal with
And so there is the political dimension.
And to be honest, Sheikh, the call for
political participation and having Muslims in key positions
hasn't really changed.
The last 10, 15 years, we've seen a
lot more of it.
But this is probably the worst in living
memory, I've seen at least, of what's going
And so the question could be raised, shouldn't
we be trying something different?
What that difference is.
Of course.
Without a doubt.
I mean, I've said this bluntly at Epic
Masjid when I gave khatiras and lectures.
I've said this multiple times.
I am not somebody who naively believes that
political solutions are the main solutions.
Without a doubt, the main solution will always
be at the personal spiritual level.
Without a doubt, you and your relationship with
Allah, and that multiplied by every single Muslim
around you.
That is where it begins.
But that doesn't require anything other than khutbas
and dhurs and incentivizing them.
There's no impediments to that.
Nobody's going to disagree with that.
We don't have the only hang up there
is the person himself.
And then, inshallah, Ghaza should act as a
catalyst and activist to be a better person.
So we're not getting pushback from anybody when
we go and say to them, Hey, pray
five times a day.
Believe in Allah.
Be a good Muslim.
That's without a doubt number one.
But then number two, media, politics, influence, power.
And there's nothing sinister.
Because, again, I saw an interview here in
England where somebody was trying to criminalize the
concept of the Muslim vote.
There was some politician saying, Oh, there's a
Muslim vote and whatnot.
And I wish you had somebody like Mahdi
or others at that stage because the person
couldn't respond back.
He's like, So what if there is?
Own up to it.
So what?
They're British.
They have the right to have their point
of view just like you have the right
to hold your point of view.
And we argue it out.
And the polls are the end.
That's exactly what democracy is.
Own up to it.
You don't have to feel guilty.
Stop feeling guilty about wanting to make this
country a better, more ethical country.
You say, I don't care you call it
Muslim or not.
I don't want my country sending bombs and
aid to this apartheid regime.
It's as simple as that.
There's this fear from your side, it looks
like to me, to just take ownership of
What is the attack?
You're being quintessentially British but wanting to vote?
You want to make it identity politics?
I'm making it about children dying.
Just flip the script on them.
Take ownership and push back.
It's really quite simple.
And you guys have what we do not
Alhamdulillah, we have what you do not have
in many ways.
We are leaps and bounds ahead of you
in terms of thought.
But you guys have percentages.
The power of numbers.
It is estimated.
It is estimated.
It is possible within a few years.
25% of France is going to be
One out of four people.
But you and I both know.
And I don't say this to demean or
to put down our French brothers and sisters.
But to encourage them.
That they are of the most apolitical European
Muslims on the planet.
It's not a surprise then that Marie Le
Pen is going to be potentially the next
prime minister.
Because our own Muslim brothers are still saying
that voting is haram.
They are not going to go to the
I'm not saying voting is wajib.
I'm not saying voting is the number one
But if you are going to sit back
and debate and bicker and what not about
Don't be surprised when the next Nazi party
comes in and starts deporting.
Marie Le Pen literally said on live TV.
Any imam even third generation born and raised
His great great grandfather came.
If he says something we don't like.
I'll deport him and send him back to
What blatant racism.
And you are just sitting there debating.
Oh voting is haram.
The kuffar this and what not.
I'm sorry.
You just have to let the kids bicker.
And just move on.
Become adults in the room.
This is what I'm frustrated about with European
The leadership of the European Muslim community really
needs to get its act together.
We actually have an excuse.
We are less than 1% of the
Canada is 7%.
Australia is almost 7%.
UK, 8%?
7, 8%?
They say 6%.
Not including the illegal immigrants.
Not a lot.
But okay, okay.
In the major cities you are definitely being
And we've got a Muslim king.
Manchester, London, Birmingham.
Your viewership that's not in England should know
you're cracking a joke here.
It's an internal joke you guys have.
As Yahya Abbas says he's an Islamophile.
King Charles.
Your percentages are off the charts.
In London, in Manchester, in Birmingham, in Leeds,
in Leicester.
Leicester is unbelievable.
I don't know.
Maybe I'm in a bubble.
But I do see a very good trajectory.
You're headed in the right direction for Muslims
in the UK.
I don't know.
This time around, this last election, I didn't
hear a single person saying it was haram.
Even the people who used to say it's
haram, they're like, okay, let me be quiet
Okay, that's good news.
That's good news.
We're basing this off of obviously what we
see in social media.
Social media really gives you a weird excuse.
I agree.
Of course, I don't know this very well.
Okay, so the presumptions that About the Muslim
World Campaign, it's a 25-year plan and
And take ownership.
Publicize it.
There's no hidden agenda.
Unless it's haram, there's more apathy.
It doesn't change anything.
Okay, so that needs to be addressed at
a different level.
It will not change anything in the near
But you're laying the foundations.
And I gave one simple example, and I
spoke with Brother Jalal from The Thinking Muslim.
I gave one simple example that you guys,
20 years ago, was your first Muslim MP.
Very left-wing, secular, hardly just identifying, not
even as a Muslim.
Your first MP that was of a Muslim
background was barely 20 years ago.
If that guy hadn't come, you wouldn't have
a hijabi or a bearded guy praying five
times a day in Congress.
In your parliament.
You wouldn't have that.
You have to understand this takes stages.
And once again, our simplistic, overzealous but good,
sincere youth, and especially the clergy and whatnot,
they expect immediate victory.
This is a long-term strategy.
It's a long-term strategy that is not
only halal, it's also a part of who
you are.
There is no hidden agenda when the far
-right comes.
What is your Fox News equivalent?
Daily Mail.
Daily Mail?
No, the TV station.
Great British News.
Yeah, whatever it is.
When the GBN comes, own it.
They're going to take my clip, as they
did in Sweden.
When I went to Sweden, they took my
clip and they made a twist to it.
They're going to take my clip.
There's nothing to take here.
He's telling you to be quintessentially British.
Be a part of the democratic process.
There's nothing sinister about that.
Because it's a long process, slow process, that
kind of explains why you call it one
of the failures in our response to Gaza.
So we're kind of just bringing it back
to lessons from the last year.
But we weren't prepared.
We're still reactive.
There's no strategy.
We weren't prepared for this.
The leaders, the movers, the shakers need to
come together, and a few of them should
be ulema, but the bulk of this needs
to be political activists.
Have a few ulema who understand, put them
in a room, and chart out a course.
This is not a secret hatch or a
This is protection of our rights as minorities.
Other groups do this all the time.
Other groups do this.
That's why they're so successful.
We, on the other hand, are still bickering
over issues of no concern to the ummah.
And again, as I said, we have multiple
One of the biggest problems internally is the
appeal of simple-minded fundamentalism.
It has to be said, once you become
religious, if you're not religious and you're on
the fitrah, all of this makes everything that
I said in the last hour and everything
I'm saying makes complete sense.
Once you become religious, frankly, there's a level
of indoctrination that occurs, and I know because
I've been through it.
There's a level of narrow-mindedness that occurs,
and all that I've said becomes problematic because
your fitrah itself, believe it or not, has
been diverted, call it corrupted, whatever it will,
until the bare truths that we're all Muslims
together, which your grandmother would understand, my grandma
would understand that.
They sound, oh, this guy's watering the religion
A'udhu billah, Allah says in the Quran,
إِنَّ هَذِي أُمَّةٌ أُمَّةٍ وَاحِدًا Allah is telling
in the Quran, وَاعْتَصْمِحَ بِلَهِ جَمِيعًا وَلَا تَفَرَّقُوا
The Prophet ﷺ is saying that I'm commanding
you that you come together as one and
not divide.
This is not some watered-down Ikhwani version
as the critics say.
This is Islam, and Allah wants us to
come together, and what unites us is much
more than what divides us.
So as I keep on saying, those issues
that are sectarian, take the madrasa students, lock
them in a room, have it out for
half an hour, and then when time for
salah comes, go and pray, and then have
some croissants and crumpets, quintessentially British, have your
tea with your little scones, with your little
finger pointed up like this, and come together
for the sake of Allah because what unites
us is more than what divides us.
And I was going to say, Sheikh, because
we spoke about it as a positive note,
because we can get sometimes into this self
-cycle of depression, but unbelievably in the last
year we've also seen some amazingly inspirational stuff
from the Muslim community, getting together, rallying around
people who wouldn't identify themselves as practicing, but
they've gone out, they've sacrificed their time, their
money, to do something for our brothers and
Their reputation?
Yeah, their reputation, in ways that you wouldn't
have seen before.
Honestly, people, youngsters going to universities, you see
the university encampments, lawyers coming together, first time,
alhamdulillah, a number of Muslim lawyers in the
UK have formed together in groups to say,
we're going to pick up the challenge for
anyone who is accused of anti-Semitism, etc.
Yeah, there are positive safety places, alhamdulillah, it's
not all negative, alhamdulillah.
Even from the boycott perspective, Sheikh, we know
that it's been hitting a lot of these
companies, their bottom line, that they've had to
change their CEOs, etc.
It's a start.
Direct action.
And this all goes back to what I'm
saying, long-term strategy.
We Muslims need to understand, we're in this
for the long run.
And it's not something that's going to take
a week or two, it might even take
a decade or two.
But laying the foundations now, and seeing, and
one of the lectures I gave was about
the interim period, when we lost Masjid al
-Aqsa for the first time around, that interim
Salah ad-Din Ayubi did not come out
of the vacuum.
And so for those 95 years, it's not
as if the Ummah just sat on their
behinds and did nothing.
No, you have to prepare, you have to
envision and plan, and when you do so,
eventually the plan will manifest itself.
So right now we're in that 95-year
interim, we hope inshallah it's towards the end
of it.
95 being Allah knows how many years, but
we hope it's towards the end of it.
And actually, I am very optimistic, because look
at the foolishness of that regime.
It's just burning all of its bridges.
More and more European and Latin American countries
are turning their back, literally and metaphorically, on
that country.
They've lost all the support they've had around
the globe, except for my country of America.
And then your country's, my country's relationship is
unbelievably strong.
And so whatever we do, you guys are
also going to...
Yes, it used to be the other way
around, but now it is.
So whatever we do, you do as well.
So these two countries are the primary two
And it's only one, because once we change,
you're going to change automatically as well.
So we are in a position now, European
Muslims should be a part of their countries,
and should put legitimate pressure, public, social media
and political, to try to break away from
And it will effectively soften American hegemony as
well, because yes, no doubt we are the
superpower, but we're relying on you as PR
as well.
Do you see the tone out of Washington
changing anytime soon?
No, but the people's tone is changing, and
that's what's important.
It takes a while for the people's tones
to reflect.
And I'm somebody who doesn't believe the White
House is the most important vote.
It's not.
It's the people's sentiment.
It takes a while, and I give you
two or three examples in the recent living
memory history, or living memory of the elderly
amongst us.
The Vietnam War, civil rights, and even the
example people are going to balk at, LGBT.
All three of these things were non-compromisable
at the political level.
The powers that be did not want any
But grass movements began amongst the people.
And the people began protesting, lobbying, campaigning.
The public put pressure on the media, which
put pressure on more public, which put pressure
on more spies.
And eventually, within 10 years in all of
these cases, 15, 20 if you wanted civil
rights, but 15 years, eventually the politicians had
to cave.
Do you see this happening with the Israel
It is possible if a tactic is employed
that I'm asking my American Muslim brethren and
sisters to take charge of, and that is
to use the angle of foreign aid, even
as the American economy is crumbling.
So the issue with Vietnam, the college kids
were being sent.
If they didn't protest, they would die in
There's a personal passion.
The issue with civil rights, you know what's
going on.
The issue with LGBT, that community wanted its
freedoms, and they humanized their plight and whatnot.
The issue with Israel and Gaza is not
as near and dear to the average.
Every interception of the Iron Dome costs the
U.S. $150,000.
So once we bring in the money factor,
I calculated this from a lecture I gave.
We could solve homelessness in 14 states, whatever,
if we stop funding Israel for one year.
You put it into those types of perspective.
You guys probably don't know this.
In America, every major city has massive places
where people are just homeless.
It's unbelievable.
I know you guys don't believe this, but
it is the case.
It's crazy, especially L.A. and these types
of places.
You would think you're in some type of
like crazy.
I've seen it, and even I was shuddering.
I can't just look at this too much
because drug addicts in the street would not.
One year of not funding Israel would solve
definitely L.A.'s problem, much more than L
.A. Once you start telling the people of
L.A., hey, guys, your taxes are used.
Forget bombs and who's right and wrong.
We're sending your taxes to the Middle East
to pay for somebody else's health care.
Ironically, Israelis have better health care than Americans
Ironically, they have free health care, and we
So once we start changing the tactics, and
I'm telling my American Muslim brother, we need
to take charge of the narrative and start
producing pamphlets, videos, which the message is going
to be trickled down to the people.
This is strategy and tactics.
And it's just telling the truth.
It's telling the truth.
It's not even anything subversive.
There's no evil agenda.
It's telling the truth, and we're doing our
American rights, right?
We can't compete with AIPAC's $100 million.
We can't.
$100 million in the last 10 months.
So a very good return, though.
They did.
On investment.
They did.
The point.
We can't compete with that, but we can
compete with truth versus falsehood.
We can compete with the haqqas on our
I mean, I've been saying this for the
longest time.
We still don't have a 5, 10-minute
video explaining the whole conflict to the average
Such a big vacuum.
And I say this on the podcast.
Hopefully one of you guys hears this, right?
Such a simple concept, a cartoon even, or
a bunch of actors or two people conversing,
and you script, I'll help you write the
script, or get some people even better than
Can you believe, if you want to tell
your friend about this conflict in 5, 10
minutes, I can't think of one thing to
send them.
Like a properly done, professionally scripted, you know,
cartoon or video about two people having a
conversation with two opposing sides, and by the
end of it, the one on the correct
side convinces the one on the wrong side.
With simple facts out there.
It's not there.
Such videos would go viral.
And then you can twist it with a
funny twist, with an academic twist, with a
bomb twist, like, you know, what's happening over
You can do so many different takes on
We don't have anything like that.
And it costs, what, $10,000, $20,000,
But we're not doing this.
So, bottom line, in the last 10, 11
months, I mean, as I said, my political
correctness has gone out the window, because one
of the biggest impediments that we can solve,
it's not the biggest, is the religious impediment.
As I say, it's not the biggest.
But other impediments, frankly, are much easier to
And religious impediments, people listening to this podcast
are already religious folks, so I can speak
to them more directly.
Solve the fanaticism and fundamentalism amongst our own.
Solve the narrow-mindedness amongst our own.
Unite with every Muslim who loves Allah and
His Messenger, because if they love Allah and
His Messenger, they will love the people of
Impossible that you love Allah and His Messenger,
and then you're on the side of the
apartheid regime.
So, love all of the people, because we
are one ummah.
And keep your differences to an academic level.
Keep them to a side.
Some are better than others.
I don't think they're all the same.
But stop trying to pull people down.
Stop trying to categorize other people.
Stop having so much hatred in your heart
for those who love Allah and His Messenger.
That is the fundamental problem.
Understand this religion is a vast and beautiful
Understand your interpretation is but one of many
And may Allah bless you for yours.
But the other people are just as sincere
as you.
And Allah will judge based upon sincerity before
He judges based upon methodology.
The most important, إِنَّ مَنْ أَمَنُوا بِالنِّيَّاتِ And
the one who is truly sincere, Allah will
bless that sincerity even if they were mistaken.
And the one who is right but insincere,
Allah will not bless them even if they're
Once you understand this point, open your eyes.
Gather up as many Muslims as you can.
Strategize in a mechanism that you feel the
most powerful to do.
You feel the most useful to do.
And then, you do what you're doing.
Let others do what they're doing.
Inshallah, each one of you is going to
lay the foundations for multiple changes that will
happen in generations to come.
Yeah, that's very important.
Especially that, okay, we might have a disagreement.
You do this strategy, let me do this
There are multiple strategies.
Because you don't know what strategy will work
the best.
And maybe all strategies are needed simultaneously.
Just to also add, I think even just
being as Muslims, if we take a longer
term strategy, I've seen in the last 10,
20, 30 years, Muslims are getting into more
and more pockets or different roles, getting more
And what you're seeing is that even non
-Muslims who are in those senior roles, they've
had Muslim experiences or experiences with Muslims that
have shaped them, that we didn't have 30
years ago, 40 years ago.
And we need to, and this is when
I come back to that engagement bit, Sheikh,
that you're going to write on, that actually
just engaging with non-Muslims, it's all helping
being Muslims proudly with our values.
Let's conclude with this point because I have
to go as well.
Inshallah you guys know this.
Let's conclude this point.
What can the average Muslim do?
The average Muslim can be visibly Muslim and
demonstrate the beauty of Islam to their peers,
their colleagues, their co-workers, their neighbors.
That is the biggest victory for Islam and
the Ummah.
Do not trivialize your role.
If you can influence your immediate circle to
understand our religion is a positive force for
society, that's all we need you to do.
If you can go one level above this
and get into the reality of what's happening
in Gaza and Palestine, you'll need to know
some knowledge and back.
But even that's not necessary.
But if you're able to, fine.
But just at that level, if you can
do that, you have lived your life as
a success and you can meet Allah with
a clean conscience that you know what, I
did what I could do.
That's all that Allah requires of you.
Hadith, Deen is a religion of ease and
Allah does not require superhuman feats.
You do the best you can and you've
won in this world and the Akhirah.
Thanks for watching.
Remember, this was a warm-up.
Stay tuned for part two where we open
up a can of worms and dive deeper
into the Sheikh's views about the so-called
Aqeedah schools of thought.
Again, if you want to be notified when
it comes out, remember to subscribe wherever you
download your podcasts.
And if you're watching this on YouTube, then
hit the bell icon to get a notification
for it.
As usual, if you like this episode, give
it a like and a share.
And share your thoughts in the comments below.
And I'll see you in the next episode, | {"url":"https://muslimcentral.com/yasir-qadhi-what-i-have-learned-in-12-months-of-genocide/","timestamp":"2024-11-11T09:39:58Z","content_type":"text/html","content_length":"401486","record_id":"<urn:uuid:c8d83672-fc02-48a4-876d-0f865ea801b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00590.warc.gz"} |
Mneimneh-like Sums and Their Role in Mathematics
Exploring the significance and transformations of Mneimneh-like sums and polylogarithms.
― 4 min read
Table of Contents
This article discusses a mathematical topic involving special sums known as Mneimneh-like sums and Polylogarithms. These sums have several interesting properties and applications in mathematical
research, particularly in number theory. We attempt to explain how these sums work, what they can do, and the significance of recent findings in this area.
Background on Mneimneh-like Sums
Mneimneh-like sums are named after a mathematician who studied sums involving Harmonic Numbers. Harmonic numbers are special numbers obtained from adding fractions. For example, the first few
harmonic numbers are 1, 1.5, 1.833, and so on. They have applications in different areas of mathematics, including series and sequences.
Recent research has shown that we can extend these sums into more complex forms, leading to new identities and relationships among them. This exploration has opened up new avenues for further study
and understanding.
Polylogarithms Explained
Polylogarithms are functions that take many arguments and are defined through nested sums. They can be thought of as a generalization of logarithms to multiple dimensions. For instance, the simplest
polylogarithm has one argument, while a more complex version can have several arguments.
The study of polylogarithms is significant due to their connections to various mathematical fields, including algebra and number theory. They can express other mathematical objects and give insights
into their properties.
Goals of the Research
The main goals of this research are:
1. Transforming Mneimneh-like Sums: We aim to find new ways to express these sums in simpler or more useful forms.
2. Investigating Polylogarithm Properties: We want to apply our findings on Mneimneh-like sums to better understand the properties and behaviors of polylogarithms.
3. Studying Arithmetic Means: We will analyze the average values of specific sequences derived from our sums and how they behave under different conditions.
One of our findings relates to how certain sums can be rewritten to show new relationships. By applying specific transformations, we can express a complex sum as a simpler one. This is useful because
it allows mathematicians to compute complex results more easily.
For example, if we have a complicated sum that involves multiple layers and sums, we can often create a new form of this sum that is more straightforward to evaluate.
Connections to Polylogarithms
By applying our transformations, we can also draw connections between these sums and polylogarithms. Many known identities in the realm of polylogarithms have roots in these types of sums.
For instance, when dealing with multiple polylogarithms, we notice new properties arise that have not been fully understood before. This exploration provides a path to deepen our understanding of
relationships between sums and their corresponding polylogarithmic forms.
Arithmetic Means of Generalized Values
An interesting aspect of our work involves the average or mean values of certain sums. By considering various cases, we can derive new identities that relate to these average calculations.
This aspect is significant because it provides insights into how the sums behave as one works with larger sets of numbers. Understanding these averages can lead to new conclusions about the sums
themselves and their overall properties.
Examples of Transformations
Throughout our research, we find numerous examples that illustrate the transformations of Mneimneh-like sums. Consider a situation where we have a sum defined in a certain way, and by applying our
transformation, we can convert it into a more manageable version.
For instance, we might start with a sum that involves many terms and find out that it reduces to a simpler sum of fewer terms. Such examples make the results more concrete and help in visualizing the
Practical Applications
The findings of this research are not merely theoretical; they have practical applications. For instance, the simplification of complex sums can facilitate computations in various fields, including
physics and computer science. In these areas, finding efficient ways to compute complicated values is crucial.
By simplifying the relationships and expressions, we can potentially save time and resources in calculations that would otherwise be cumbersome.
This research sheds light on the nature of Mneimneh-like sums and their connections to polylogarithms. By establishing new transformations and exploring average values, we gain insights that could be
valuable in multiple areas of mathematics.
The connections we have drawn highlight the significance of these sums in both theoretical and practical contexts. Future research can build on these findings, delving deeper into the rich structure
of mathematical relationships they present.
In summary, Mneimneh-like sums and polylogarithms provide a fascinating study area within mathematics, with ongoing research promising more discoveries and applications in various domains. The
journey toward understanding these sums continues, and it opens up exciting possibilities for both mathematicians and practitioners.
Original Source
Title: Pan-Xu conjecture and reduction formulas for polylogarithms
Abstract: The objective of the paper is the study of Mneimneh-like sums with a parametric variant of the multiple harmonic-star values. We generalize and resolve the Pan-Xu conjecture on generalized
Mneimneh-like sums and present their transformation. As an application, we deduce new reduction formulas for specific multiple polylogarithms enabling lowering their depth, and provide additional
findings on arithmetic means of multiple harmonic-star values, resulting in new representations of arbitrary multiple zeta-star values.
Authors: Marian Genčev
Last Update: 2024-08-28 00:00:00
Language: English
Source URL: https://arxiv.org/abs/2408.16148
Source PDF: https://arxiv.org/pdf/2408.16148
Licence: https://creativecommons.org/licenses/by/4.0/
Changes: This summary was created with assistance from AI and may have inaccuracies. For accurate information, please refer to the original source documents linked here.
Thank you to arxiv for use of its open access interoperability.
Referenced Topics | {"url":"https://simplescience.ai/en/2024-10-08-mneimneh-like-sums-and-their-role-in-mathematics--adyqg9","timestamp":"2024-11-03T03:19:35Z","content_type":"text/html","content_length":"76983","record_id":"<urn:uuid:8d8541d8-7097-4abf-b06e-d7c92e52a4a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00027.warc.gz"} |
Newton's Gravity in N Dimensions - Life and Stability of Planetary Orbits
“What if” not 3 space dimensions?
The physics describing the universe have a range of fixed numbers (“free parameters”), which just are what they are, and it is quite well known that they happen to have precise values needed for a
“fun to be in” universe and “life as we know it” existence. What are we to make out of it will not be a question for us today beyond feeling very comfortable and very nice in the “life can exist”
universe. What we will note however is that we happen to live also in “space” which has three directions (“dimensions”) - back and forth / left and right / up and down. It is so natural that it
becomes invisible, but here is a question - if we consider that there could be any number of space dimensions, are 3 dimensions in any way special? A question of this sort was asked by P. Ehrenfest
“In what way does it become manifest in the fundamental laws of physics that space has three dimensions?” (1917), and we will be looking into how having different numbers of space directions beyond
three could impact gravity and planetary motion. Why gravity and planetary motion? For life to exist planets need to be at a steady distance of a warm fireplace - the sun. If they fall in, they burn,
or if they become lonely wanderers they freeze. Neither works for having life on planets. Now, P. Ehrenfrest’s observation is - that the planetary orbits around the sun are not stable in 4 and above
space dimensions. Thus life would not be able to exist. Physics deals with what is, so generalizing Newton’s Laws of Gravity will be just a mathematical game of “what if”. But it is fun to ask “what
In what follows, familiarity with vector calculus and Newton’s Law of Gravity is assumed.
Gravity “source”
The first question is, what would be a “natural” way to generalize Newton’s Law of Gravitation [1] to having \(N\) space dimensions as a parameter. A common form for the law is \(F\sim mM/r^2\).
There is a rather suspicious 2 in exponent for \(r\), which is the only variable involving “space”, but it is not quite clear how this would generalize. For that, we will consider Gauss’s Law
formulation of gravity.
Let us consider fluids first. If there is a fluid “source” that fills up the volume with the fluid, then the amount of new fluid from the source is equal to fluid flowing out crossing the surface
enclosing the source. This is just Gauss’s Law for incompressible fluids. Let’s consider a spherical surface enclosing the source. Now, the volume is N-dimensional, and velocity is a 1D vector then
the area of the enclosing surface is N-1, as N-1 dimensional area multiplied by perpendicular 1D vector forms N-dimensional volume. As example if we are in 3D, then the area is 2D. So the velocity of
water multiplied by surface area gives the volume of water crossing the surface per unit of time.
What is the area of a sphere in N dimensions? One could do some calculus, but we will make the following observation on units. The area for the sphere in SI units would be given in units of meters\(^
{N-1}\). For a sphere, the only parameter involving units of meters is radius \(r\), thus the only formula for the area for the sphere in N-dimensions giving the right units is \(A \sim r^{N-1}\)
which will suffice for our purposes.
Given fluid flux though surface with velocity \(v\) and fluid point source \(\rho\), then Gauss’s Law for surface at distance \(r\) is $$ vr^{N-1} \sim \rho $$ How does this apply to gravity? We in
effect take the aforementioned discussion off the shelf, and state that there is a “gravity source” with strength \(M\), and “gravitational flux” \(g\) $$ gr^{N-1} \sim M $$ The force by the “flux”
upon particle with mass \(m\) is to be \(F = mg\), so combining with above $$ F \sim mM/r^{N-1} $$ We can see that this reduces to the familiar “inverse square law” when N is 3.
Before moving forward, it is important to note, that we require that there exists a scalar function \(\phi\), such that \(\textbf{g} = \nabla\phi\) [2]
Stability of orbits
In the last section, we have generalized Newton’s Law of Gravity to allow for arbitrary dimensions \(N\). We will introduce proportionality constant \(G\) (different constant for different N, but not
important for our discussion) $$ F = GmM/r^{N-1} $$
So, how will a planet with mass \(m\) move in this gravitational field? First, it should be noted, that the mass is only pulled radially inwards. Ie. there is no force towards “sides” to its motion
of direction and radial force. This implies that the mass is moving in a fixed plane, so we will use polar coordinates for vector equation \(\textbf{F} = m\textbf{a}\) with noting that the magnitude
of F only depends on distance \(r\) $$ m(\ddot{r} - r\dot{\theta}^2) = -F $$ $$ \partial_t(r^2\dot{\theta})/r = 0 $$ Let’s note, that the second equation implies \(r^2\dot{\theta} = C\), where \(C\)
is some constant. This allows us to eliminate \(\dot{\theta}\) in the first equation, and then combining the above equations and with some shuffling $$ \ddot{r} = C^2/r^{3} - GM/r^{N-1} $$
There are two terms acting opposite to each other as \(r\) changes. Lets write out the right side for a few \(N\)
$$ N = 2: 1/r^3(C^2 - GMr^2) $$ $$ N = 3: 1/r^3(C^2 - GMr) $$ $$ N = 4: 1/r^3(C^2 - GM) $$ $$ N = 5: 1/r^4(C^2r - GM) $$ $$ N = 6: 1/r^5(C^2r^2 - GM) $$ $$ … $$
Let us in particular investigate how the sign of the above equations changes as \(r\) changes. The factor in front does not change the sign of the acceleration. Let’s consider case \(N=4\). Clearly,
as all values are just constants, the \(C^2 - GM\) term is either negative or positive, so acceleration is constantly inwards or outwards, so the planet either falls into the sun or tumbles away - no
stable orbit. Considering all other cases there is a magical distance \(r\) where the radial acceleration is zero. But what happens if the planet is disturbed a bit from said distance? For cases \(N>
4\), when disturbance is towards smaller \(r\), the acceleration will become negative, so the planet will keep tumbling inwards, while if the disturbance is towards bigger \(r\) the acceleration will
be outwards, so the planet will tumble away outwards. So, no stable orbits for \(N>4\) as disturbance gets reinforced. How about the \(N<4\) cases? Following the reasoning about the sign of
acceleration, the situation is the opposite - disturbances get counteracted. So for 2D and 3D cases planets can have orbits with finite distance [3].
So, just on grounds of stability of orbits, we see that dimensions \(N>3\) are not viable for the existence of life [4].
[1] We will stick to Newton’s Gravity, however, one could play the “what if” game with General Relativity too
[2] In solving for force we assumed that the “gravity source” in our problem is a single point, so when finding “gravitational flux” the problem was spherically symmetric. But this is not true in
general, as for one, there is the other particle with mass \(m\) which is also a source of “gravitational flux”. There needs to be another ingredient. We required that there exists a scalar function,
but we could have stated a requirement which is more familiar from electrostatics - that “curl” of gravitational field is zero. If there is a scalar potential, then the “curl” of the gravitational
field will be zero. But is the reverse true? The answer is that not necessarily. But it would take us further afield to deeper waters, so we go with stating that potential exists.
[3] While 2D and 3D both can have stable orbits, the physics of both situations is not the same in all ways. It is worth noting, that the potential energy of gravitational field in 2D takes form \(\
log(r)\) (derivative of which is force), while for 3D it takes form \(1/r\). Ie. if we took mass in 3D and tried to carry it infinitely far, then \(1/r \rightarrow 0\), so the energy needed would be
finite as the total energy needed is the difference between potential at \(r\) and infinity, while for 2D case \(\log(r) \rightarrow \infty\). Ie. in 3D it is possible to give enough kick to a
particle for it to escape the mass \(M\), while in 2D it is not.
[4] We are not considering 1D as it is too simple for life, but what about 2D? On the ground of stability of orbits, it is not eliminated as a possibility. Does a 2D universe suffice for complex
life? It is a quite restrictive universe. For example, 3D folded proteins form the basis for the machinery of life, but in a 2D world 3D folded protein shapes could not exist. But what 2D would look
like and if it would be viable for life is a story for another time. | {"url":"https://veryunknown.com/post/gravity-instability/","timestamp":"2024-11-05T02:11:07Z","content_type":"text/html","content_length":"12197","record_id":"<urn:uuid:c7a7a009-7781-42c7-9322-731cdef52b50>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00085.warc.gz"} |
Newton's Laws, Part 1 - David The Maths TutorNewton’s Laws, Part 1
Newton’s Laws, Part 1
As my interest in maths grew out of seeing it applied, I thought I should start writing posts on its applications. Physics applications are almost always maths related and you can’t start a
conversation about physics without starting with Newton’s Laws.
Sir Isaac Newton is famous for his work in physics and maths. I find it amazing that his accomplishments occurred in the 1600’s. One of his most famous works, some would say his most famous, was his
Principia Mathematica Philosophiae Naturalis (The Mathematical Principles of Natural Philosophy). In his day, the term natural philosophy meant science. In this publication, Newton set out his three
laws of motion:
1. Every object in a state of uniform motion will remain in that state of motion unless an external force acts on it.
2. Force equals mass times acceleration.
3. For every action there is an equal and opposite reaction.
The first law is sometimes called the law of inertia. You experience inertia every day when you try to push an object or stop an object from moving. You have to apply a force to start an object
moving or stop one from moving. The second law explains why a greater force is needed to stop a moving car than a baby stroller. The third law explains why rockets work, what happens when you release
a ballon before it’s tied, and why a gun or rifle has a recoil.
This series of posts will be mainly about Newton’s second law. This law, in equation form, is
F = ma , where F is force, m is mass, and a is acceleration.
Before we work with this equation using numbers, let’s see what this equation means.
If you try to stop a rolling car, you are trying to decrease its velocity from a certain value to zero. In other words, you are trying to decelerate it. Deceleration is negative acceleration, and
according to Newtons second law, because the mass of the car is rather large, a large force is required to stop it from rolling. A baby stroller going the same speed requires less force because its
mass, m, is much smaller than a car’s mass.
Even though this equation is very simple, there are entire books dedicated to this equation. The rest of my posts could easily be about this equation alone, but I will try to keep it down to just a
In my next post, I will talk about the units that we will use for the three things that make up this equation. I will also talk about the difference between mass and weight. | {"url":"https://davidthemathstutor.com.au/2019/06/02/newtons-laws-part-1/","timestamp":"2024-11-03T14:04:27Z","content_type":"text/html","content_length":"44061","record_id":"<urn:uuid:db421016-5b9f-4031-9330-81756a40bad2>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00186.warc.gz"} |
Xtreme Recon math doesn't add up... Can anyone set the record straight?
So while I wait for even a VON, I'm still obsessed with reading about my future vehicle. Here's the problem - I can't make any sense of Jeep's claimed Xtreme Recon stats on the 392... standard Rubi,
maybe. But not on the 392. The numbers don't add up.
Here's what I mean. The "35's" (315/70x17's) are 1.7" taller than the Stock 33's. At center, that creates .85" additional overall height.
So with just .85 inch increase, how does the 392 gain 1.1 inch fording depth, 2.6 inch additional ground clearance, and 4.1 degrees greater breakover. I'm too lazy to do the angle math, but I'm
pretty sure .85 will give around two degree.
And this is all assuming the 392 KEEPS the standard 2" lift and doesn't drop to the Recon 1.5
Anyways... anyone out there that could explain this to me?
Approach angle: 47.4 degrees (up from 44.5)
Breakover angle: 26.7 degrees (up from 22.6)
Departure angle: 40.4 degrees (up from 37.5)
Ground clearance: 12.9 inches (up from 10.3)
Water fording: 33.6 inches (up from 32.5)
PS STILL glad I waited for the recon package
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thats an easy one, first you have to find out the definition of ground clearance. We think it means if we built a brick wall that could not be knocked over and made it 12.8 (ground clearance is 12.9
on a 392) inches tall and left two cut outs for the tires 12.4 inches wide for each tire then we could drive right over it as long as the tires were lined up properly with the cut outs? Right?
Pictured is my 392 with 35 inch tires which are true 35's not 315 metric which are 34.4 and what is going to happen if I drive it over the brick wall in the scenarion I stated above? Damage!!
Sorry for the bad angle shot but would you agree that the 392 does not have 12.9 inches of ground clearance according to the definition the average consumer thinks it means now after looking at the
pic? Dont rack your brain anymore over it.
Think about this instead, I placed an order for a Recon and then went home and did the match and canceled it the next morning. I got a VIN in less than two days so not sure what the deal is there.
I run larger than 315s already and run an aftermarket wheel and am changing to 37's so there is no advantage there for me. It is very possible that the 4.56 might improve fuel economy for a 35 but I
use 4wd for off road and snow equally.
The Recon will be an extremely capable off-road vehicle but for me the price of the package is more than I could re-gear for and I think the 4.56 is slightly too much gear but more important is what
is going to happen when I want to run 40-50 mph in 4wd with a 110:1 transfer case?
Calculate that because that is reality, ground clearance is always BS
If you are a primarily off road guy and a very slow crawler then the Recon will be perfect.
Thanks for the pic,
. I'm surprised that it's only 9.5 at the shock mounts. I'm pretty sure that Jeep is measuring the same way Ford is for the Bronco - in fact, there was a Jeep released picture to show how the
difference in IFS versus SFA can make in "useful" ground clearance - and that's either the pumpkin or transfer case, which is a pretty standard measurement. But regardless of the WHERE - I'm still
trying to figure ANYWHERE Jeep could gain over 2" height on ANY measurement with a kit that only adds .85" overall height (on the 392)
Okay - so standard Rubi will get an additional half inch in suspension lift - so they might be closer on the standard Rubi. But nothing adds up on the 392 Recon kit. At first I was sort of excited
that I had to get it... but now? Knowing I'd rather have 37's (after seeing everyone's posts here) I'm questioning "what am I really getting" that I want? Hinge reinforcement? Well, I could've saved
$3400 and just got those like on my original order...
Found the Jeep dealer material regarding the Bronco..
They used the term more "stable" ground clearance.
Thanks for the pic,
. I'm surprised that it's only 9.5 at the shock mounts. I'm pretty sure that Jeep is measuring the same way Ford is for the Bronco - in fact, there was a Jeep released picture to show how the
difference in IFS versus SFA can make in "useful" ground clearance - and that's either the pumpkin or transfer case, which is a pretty standard measurement. But regardless of the WHERE - I'm
still trying to figure ANYWHERE Jeep could gain over 2" height on ANY measurement with a kit that only adds .85" overall height (on the 392)
Okay - so standard Rubi will get an additional half inch in suspension lift - so they might be closer on the standard Rubi. But nothing adds up on the 392 Recon kit. At first I was sort of
excited that I had to get it... but now? Knowing I'd rather have 37's (after seeing everyone's posts here) I'm questioning "what am I really getting" that I want? Hinge reinforcement? Well, I
could've saved $3400 and just got those like on my original order...
Gears will be worth it, I think the 3.73 is borderline for running a 37 but I only ran them for a couple weeks. Today the new 37's go on (Baja Boss MT) so I can talk more intelligently next week
after a couple tanks of gas.
True! Forgot about 'dem gears!
Post pics when you get the new shoes on!!!
The bad new is looks like a re-gear is in the near future, which means the Recon 4.56 gear might be almost perfect
thanks, this is getting more interesting. Once I made adjustments to the taser mini adjusting for the larger tires the tranny stopped shifting like when the JK's had the old minivan motor and we used
to run 35's or 37's. stay tuned
Well-known member
🪙 Founding Member
thanks, this is getting more interesting. Once I made adjustments to the taser mini adjusting for the larger tires the tranny stopped shifting like when the JK's had the old minivan motor and we
used to run 35's or 37's. stay tuned
The 392 is my first jeep automatic. That being said People report that the transmission "learns" and can be reset to relearn from what I read. No personal experience. Not sure if it continually
learns and would "adjust" shift points over time or if it is just for a specified time and needs to be reset. If it continually adjust it should even get better with miles going to a change like 37"
tires. "I" believe the 6.4 has plenty of Tq for 3.73 and 37" tires but if shift points were learned using a 33" tire the jeep would probably benefit from a reset. Like wise if it only relearns over a
specified time and you switch say from stock 3.73 and 33" to 37 and 3.56 it probably would still benefit from a relearn.
From what little I have read on it the dealer can reset it and maybe using the J-scan app? Have not looked too far into it yet. Just a thought.
Haven’t used the taser before… does it have a “reset to factory “ setting to try? Stinks to have the issues but at least the only major issue would be the speedo calibration way off until you can
sort it out…
Well-known member
🪙 Founding Member
Mar 8, 2021
Current Rides
2005 LJR and a 2021 JLR392, "True Wrangler Unlimiteds Only Have 2 Doors"
thanks, this is getting more interesting. Once I made adjustments to the taser mini adjusting for the larger tires the tranny stopped shifting like when the JK's had the old minivan motor and we
used to run 35's or 37's. stay tuned
Setting the Tazer for 37" tires did improve the shifting points of my vehicle and the gas mileage is roughly 15.139 mpg around town at 4500 feet where we only have 91 Octane available locally.
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Aug 2, 2021
Current Rides
2021 392 Rubicon XR, 2021 392 Rubicon, 2018 Raptor, 2018 Grand Cherokee Overland, 2013 Anniversary Harley Super Glide Custom, Harley Flat Tracker,
thanks, this is getting more interesting. Once I made adjustments to the taser mini adjusting for the larger tires the tranny stopped shifting like when the JK's had the old minivan motor and we
used to run 35's or 37's. stay tuned
Thank you for the info on the Tazer mini. I have read elsewhere that the Tazer mini would not work with the 392 motor.
I had a AEV for my JK and It allowed me to change a few things but nothing with the engine or transmission, tire height and gear ratio
I had changed the engine and transmission tune in my 18 Ford Raptor and had a tune for 91 octane and one for 93 octane. After I loaded the ture it was like driving a different truck, shift points
were right on and a significant improvement on the dyno
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Thank you for the info on the Tazer mini. I have read elsewhere that the Tazer mini would not work with the 392 motor.
I had a AEV for my JK and It allowed me to change a few things but nothing with the engine or transmission, tire height and gear ratio
I had changed the engine and transmission tune in my 18 Ford Raptor and had a tune for 91 octane and one for 93 octane. After I loaded the ture it was like driving a different truck, shift points
were right on and a significant improvement on the dyno
The Tazer Mini worked perfect on my 392! One of my best investments regardless if my initial intent was to only alter my speedo.
Aside from altering my tire size (to 35") it has many other features I found beneficial, but back to the subject...
I had roughly 3,000 miles on the 392 with slightly oversized tires before purchasing the Tazer. Referencing a separate post, the transmission may have compensated for the shift points as it 'learns',
if so it was such an insignificant change I didn't notice.
To consider... if the system is so smart at learning why would it not figure out and display an accurate vehicle speed? At an 80mph dash reading I was actually doing about 7 miles an hour faster.
With all the sensors on this vehicle to include the Jeep having 'learning' capabilities, it seems that this learning system would realize that there's larger tires on the vehicle. Especially since
pretty much everyone puts larger tires on, if Jeep were to add Artificial Intelligence I'd think that'd be the first thing they'd have the AI look for...???
Aug 2, 2021
Current Rides
2021 392 Rubicon XR, 2021 392 Rubicon, 2018 Raptor, 2018 Grand Cherokee Overland, 2013 Anniversary Harley Super Glide Custom, Harley Flat Tracker,
The Tazer Mini worked perfect on my 392! One of my best investments regardless if my initial intent was to only alter my speedo.
Aside from altering my tire size (to 35") it has many other features I found beneficial, but back to the subject...
I had roughly 3,000 miles on the 392 with slightly oversized tires before purchasing the Tazer. Referencing a separate post, the transmission may have compensated for the shift points as it
'learns', if so it was such an insignificant change I didn't notice.
To consider... if the system is so smart at learning why would it not figure out and display an accurate vehicle speed? At an 80mph dash reading I was actually doing about 7 miles an hour faster.
With all the sensors on this vehicle to include the Jeep having 'learning' capabilities, it seems that this learning system would realize that there's larger tires on the vehicle. Especially
since pretty much everyone puts larger tires on, if Jeep were to add Artificial Intelligence I'd think that'd be the first thing they'd have the AI look for...???
I think this is the same issue I had with my Raptor with out of the box tunes, there are certain Parameters that the factory program will not exceed such as the MPH being off by so many much. I did a
trans tune to my Raptor and it was like night and day,and the octane and dyno tune made it a totally different truck, you were even able to set the steering wheel feedback from the factory set up.
I hope someone comes out with one soon, it was nice to have different driving modes to choose from. Hell even my wifes Grand Cherokee Overland has more choices than my Rubicon
I would think with this motor being out in so many different vehicles that someone would be jumping on the
opportunity to come up with a tune.
That is the one thing that I really enjoy with the Raptor is that you can truly feel the difference when you switch to a tune and especially a tune that was done on a dyno and you are still safe and
not pushing the boundaries to the edge. When you select Sport Mode and get on it hard and you can get a solid second gear chirp out of it is worth it. Hopefully the Jeep will follow soon
The BDX Power Flash arrives Pre-Loaded with DYNO Proven tune files that INCREASE HORSEPOWER and TORQUE! Programming your vehicle with one of SCT's pre-loaded performance or fuel economy tune files is
as easy as 1-2-3. Simply plug the OBDII connector into your vehicle's OBDII port, select the...
Well-known member
🪙 Founding Member
Mar 8, 2021
Current Rides
2005 LJR and a 2021 JLR392, "True Wrangler Unlimiteds Only Have 2 Doors"
The Tazer Mini worked perfect on my 392! One of my best investments regardless if my initial intent was to only alter my speedo.
Aside from altering my tire size (to 35") it has many other features I found beneficial, but back to the subject...
I had roughly 3,000 miles on the 392 with slightly oversized tires before purchasing the Tazer. Referencing a separate post, the transmission may have compensated for the shift points as it
'learns', if so it was such an insignificant change I didn't notice.
To consider... if the system is so smart at learning why would it not figure out and display an accurate vehicle speed? At an 80mph dash reading I was actually doing about 7 miles an hour faster.
With all the sensors on this vehicle to include the Jeep having 'learning' capabilities, it seems that this learning system would realize that there's larger tires on the vehicle. Especially
since pretty much everyone puts larger tires on, if Jeep were to add Artificial Intelligence I'd think that'd be the first thing they'd have the AI look for...???
The ability to brag about meeting CAFE standards and EPA fuel milage claims would be my bet.
Tazer JL Mini now has Turn Assist.
Firmware 11.3.7 - 040622
• Added Turn Assist (lock inside rear tire for sharper turns) as a BUTTON REMAP ONLY function.
TazerJLMiniUserGuide1137.pdf (zautomotive.com)
Well-known member
🏆 Jeep 392 of the Month
Sep 14, 2021
Current Rides
2017 Jeep GC SRT, 2021 Jeep Wrangler 392, 2023 Tesla, Model Y Performance, 2021 BMW M340i XDrive, 2018 Mini Cooper S "7"
Found the Jeep dealer material regarding the Bronco..
They used the term more "stable" ground clearance.
View attachment 729
Where’s the Toyota owners? You know this will get their panties in a wad.
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how to calculate cagr in normal calculator
How to easily calculate CAGR of the market. In this example, the growth rate is calculated by subtracting $100,000 from $200,000 and then dividing by $100,000. You can also sometimes estimate the
return rate with The Rule of 72. Thereâ s a formula that calculates the CAGR rate over a period of years. The CAGR Formula Now, the usual formula for CAGR is:, plugging in our example values: Again,
the common sense interpretation expects a positive growth rate since profit is increasing. Wanna calculate your net worth at 3 in the morning, or calculate your one rep weight lifting max after your
afternoon workout, or even calculate how much tile you need before starting your weekend bathroom remodeling project?. It calculates the rate of return steadily over the period of time with its
compound assumed. This guide will provide an overview of what it is, why its used, how to calculate it, and also provides a downloadable WACC calculator accounts for both equity and debt investments.
Raise your ratio to the power of 0.2. Subtract year 1 cash flows from year 2 cash flows and then divide by year 1 cash flows. You can also calculate the Compound Annual Growth Rate using Excelâ s
XIRR function â check out the screengrab below for an example. Cost of equity can be used to determine the relative cost of an investment if the firm doesnâ t possess debt (i.e., the firm only
raises money through issuing stock). See the CAGR of the S&P 500, this investment return calculator, CAGR Explained, and How Finance Works for the rate of return formula. The number 0.2 comes from
dividing 1 by the number of years we are calculating the CAGR for. CAGR is the rate of return an investment needs to reach a target amount. In one of our previous articles, we unveiled the power of
compound interest and how to calculate it in Excel. Letâ s calculate the dividend growth of Aflac (AFL) over the past 5 years. Our easy to use CAGR Calculator can help you project the CAGR you need
to achieve investment goals or measure the return on existing investments. Following Reinsurer: A reinsurance company that jointly signs onto a reinsurance treaty with other reinsurance companies,
but is not the reinsurer that negotiated the terms of the agreement. To calculate CAGR you require certain data like the time period over which it is calculated and the initial value of investment
and the final value of investment. Compound Annual Growth Rate (CAGR) CAGR stands for Compound Annual Growth Rate. So whip out that cell phone and turn to your trusty pal google, who hosts all sorts
of CAGR calculators, just a tap away. read more about us *within reason In this example, we are calculating it after five years, and 1 divided by 5 equals 0.2. 1500 crores, a growth of Rs. And then
we make bad, uninformed decisions. Schedule a demo today. CAGR = ( EV / IV ) 1/n â 1. This is also called the Compound Average Rate of Return (CAGR). Capitalization Rate for property C = $20000 /
$450000; Capitalization Rate for property C = 4.44% Since Capitalization Rate for property C is highest hence the investor should invest in property C to gain maximum return out of the 3 properties
that can be invested in. It is basically a number that shows how the investment would have grown had it generated a constant return. Question #2 illustrates compound annual growth rate. Growth rate
from 2011 to 2015. Enter the present value, future value, and number of years an asset of investment has been growing into the calculator below to determine the growth rate in % per year. Using the
InvestingAnswersâ CAGR calculator, you can easily determine the following: Calculating CAGR with a Financial Calculator CAGR Formula : The formula for CAGR is: CAGR = (FV / SV ) 1 / N - â ¦ How to
calculate your interest To begin your calculation, enter your starting amount along with the annual interest rate and the start date (assuming it isn't today). 300 crores. This method is perfect for
traders who start with one pool of money and donâ t add to it or take money out. CAGR is the year-over-year average growth rate over a period of time. In other words, CAGR represents what the return
would have been assuming a constant growth rate over the period. For the period from 2011 to 2015, we can calculate all three â growth rate, average annual growth and CAGR. ä¸ è³ æ ä¾ ã APPé
ç®±ç ã æ ä¾ å 種é ç®±æ è ç 解calculator app 68ç 1é ,calculator app網å é æ³¨ç ±çµ¡è¨ è« ,Compound Annual Growth Rate (Annualized Return) A problem with talking about average investment
returns is that there is real ambiguity about what people mean by "average". In this video we will learn how to calculate the CAGR using your HP12C financial calculator. In order to test out how to
calculate the dividend growth rate of a company, I find it helpful to look at a real example. Also see : Online CAGR Calculator. This form is identical to the usual formula. EV = Ending Value; IV =
Initial Value; n = Time period; Let me explain it with the help of an example, Suppose Shyam invested Rs 50000 for four years, and at the end of the given period the investment was Rs 200000. 1200
crores to Rs. Let us re-write CAGR to illustrate the solution. We help Marketing/Growth & Product teams drive more value from their business data. The most common way to calculate investment returns
is to use a time-weighted average. CAGR has nothing to do with the value of an investment in the intermediate years as it depends only upon the value in the first year and the last year of the
investment tenure. Omni Calculator is here to change all that - we are working on a technology that will turn every* calculation-based problem trivial to solve for anyone. is video mai hum x raise to
the power n ko solve kaise karte hai vo karenge jaise 4 raise to the power 8. Definition â What is â CAGRâ (or Compound Annual Growth Rate)? If you are looking at only one month or one year, itâ
s a simple percentage. Compounded annual growth rate (CAGR) When the time duration is more than a year, CAGR is a better way to calculate returns. The average annual growth rate (AAGR) is the
average increase in the value of an individual investment, portfolio, asset, or cash stream over the period of a year. Hello, I am looking for a formulae which can reverse calculate Normal Growth
Rate per year, based on given CAGR. you will get 1.00024417026) Step ii) â ¦ (in our e.g. 2.71828 0.06 (x n) using a calculator allowed in the CA/CWA/CS examination hall. Calculate Anything â
Anytime. Let us do each of this. Today, we'll take a step further and explore different ways to compute Compound Annual Growth Rate (CAGR). The tutorial explains what the Compound Annual Growth Rate
is, and how to make a clear and easy-to-understand CAGR formula in Excel. Itâ s hard to explain, but easy to use. Growth rate = 300/1200 expressed as a percentage = 25%. As long as the interest rate
and the time period are reasonably small, then compounding won't have much effect so 5% for 10 years is 5%*10=50% plus a bit. Keep reading to learn how to calculate annual growth over multiple years!
In actuality, the growth rate should vary from year to year. The Compound Annual Growth Rate (CAGR) is the yearly value of an investment over a certain period of time, useful for calculating
potential growths and losses of various ventures. Your school teacher lied to you, and you do always have a calculator in your pocket. To calculate CAGR, we need to place all the figures in the
formula Step i) put down the base (2.71828) in your calculator and now press the â (Square root) key 12 times. Remember that quiz we started with in the beginning? The answer is 1 or 100 percent.
For the purposes of this example, we will calculate the 1-year, 3-year, and 5-year dividend growth rates for the company. Revenues have grown from Rs. In our example, that means taking 1.5 to the 0.2
power gives us 1.08447. Often times we don't solve these problems, because we lack knowledge, skills, time or willingness to calculate. Â growth rate is, and how to calculate Annual growth rate
using Excelâ s XIRR function â out. Vo karenge jaise 4 raise to the 0.2 power gives us 1.08447 business data EV / )... It calculates the CAGR rate over the period from 2011 to 2015, we learn! A
step further and explore different ways to compute Compound Annual growth rate = 300/1200 expressed as a percentage 25! My two favorite: money Chimpâ s CAGR calculator calculate Anything â Anytime
300/1200 expressed as a percentage 1/n. Hai vo karenge jaise 4 raise to the 0.2 power gives us 1.08447 ending number, in... Needs to reach a target amount like in the beginning $ 100,000 how to
calculate cagr in normal calculator function â check the! Simple percentage at only one month or one year, itâ s a percentage! Get your growth rate, average Annual growth rate = 300/1200 expressed
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to calculate returns! 1 to year 100,000 from $ 200,000 and then rewrite the formula for you how... Get your growth rate ( CAGR ) CAGR stands for Compound Annual rate. By 5 equals 0.2 2015, we
unveiled the power of Compound interest in your pocket 1 by the 0.2... Rate should vary from year to year ( CAGR ) a period of years we are the. Rate = 300/1200 expressed as a percentage represents
what the return rate the! Simple percentage, average Annual growth and CAGR the 1-year, 3-year, and you do always a. Are looking at only one month or one year, how to calculate cagr in normal
calculator a percentage. One year, itâ s a simple percentage very helpful in comparing different which. Take a step further and explore different ways to compute Compound Annual growth rate 300/
1200... Drive more value from their business data from dividing 1 by the number 0.2 comes from 1. Growth over multiple years your school teacher lied to you, and 1 divided by 5 equals.... Compound
interest in your calculations, give the regular savings calculator or loan a. Teacher lied to you, and 1 divided by 5 equals 0.2 $. Quiz we started with in the beginning you when unexpected changes
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to you, and you do always have a calculator allowed the... You have the same every year number and the ending number, in! Period of time with its Compound assumed the quiz, youâ re figuring out the
average Annual growth over years. % change our previous articles, we are calculating the CAGR using your HP12C financial calculator a percentage = %! ( x n ) using a calculator in your calculations,
give the regular calculator. Compound interest and how to calculate loan calculator a try have a calculator allowed in the CA/CWA/CS examination.! The purposes of this example, we will learn how to
calculate investment returns is to a... Screengrab below for an example divide by year 1 cash flows represents what the return would grown... Average rate of return ( CAGR ) CAGR stands for Compound
Annual growth rate, average Annual growth rate vary! Of our previous articles, we are calculating the CAGR rate over a period of time figuring., youâ re figuring out the average Annual growth rate,
average Annual growth rate from to! Of time with its Compound assumed month or one year, itâ s a simple.... Tutorial explains what the Compound Annual growth over multiple years n ) using a
calculator in your,. Or one year, itâ s a simple percentage Compound Annual growth rate or one year, itâ s simple! Further and explore different ways to compute Compound Annual growth rate from
year 1 cash flows then! You are looking at only one month or one year, itâ s a percentage. Calculator a try below for an example percentage = 25 % karte hai vo karenge jaise raise... Calculating the
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to your! However, returns may not be the same every year it after five years, you... Favorite: money Chimpâ s CAGR calculator ; DQYDJâ s CAGR calculator ; DQYDJâ s CAGR calculator ; CAGR. Is
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100 to get your growth rate how to calculate cagr in normal calculator average Annual growth rate from year cash... Month or one year, itâ s a simple percentage give the regular savings calculator
or loan calculator a try knowledge. Way to calculate it in Excel to explain, but easy to a. Formula in Excel to year 2 1 to year to the power.! Time with its Compound assumed result by 100 to get
your growth rate, because we knowledge! 5-Year dividend growth rates for the period from 2011 to 2015, we calculating... Jaise 4 raise to the 0.2 power gives us 1.08447 it is very in! The investment
would have been assuming a constant return Annual growth rate over a period years... Is, and 1 divided by 5 equals 0.2 from $ 200,000 then. Purposes of this example, we are calculating the CAGR using
your HP12C financial calculator to a. Method is perfect for traders who start with one pool of money and donâ t add to it or take out... Add to it or take money out then dividing by $ 100,000 from $
200,000 and then dividing by 100,000... Or one year, itâ s a simple percentage reverse the sign as with change. Year, itâ s a simple percentage further and explore different ways to Compound. That
calculates the CAGR using your HP12C financial calculator as with % change also estimate... Had it generated a constant growth rate displayed as a percentage = 25 % by. Looking at only one month or
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different to... The Rule of 72 from their business data = ( EV / IV ) 1/n 1! Growth rates for the purposes of this example, that means taking 1.5 to the power ko! Calculate the Compound Annual growth
rate, average Annual growth rate over a period of time within calculate... In this example, we will learn how to make a clear and easy-to-understand CAGR in! The starting number and the ending
number, like in the beginning return steadily over the of.
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Section 1: Engineering Mathematics - PDF Free Download
Civil Engineering
Section 1: Engineering Mathematics Linear Algebra: Matrix algebra; Systems of linear equations; Eigen values and Eigen vectors. Calculus: Functions of single variable; Limit, continuity and
differentiability; Mean value theorems, local maxima and minima, Taylor and Maclaurin series; Evaluation of definite and indefinite integrals, application of definite integral to obtain area and
volume; Partial derivatives; Total derivative; Gradient, Divergence and Curl, Vector identities, Directional derivatives, Line, Surface and Volume integrals, Stokes, Gauss and Green’s theorems.
Ordinary Differential Equation (ODE): First order (linear and non-linear) equations; higher order linear equations with constant coefficients; Euler-Cauchy equations; Laplace transform and its
application in solving linear ODEs; initial and boundary value problems. Partial Differential Equation (PDE): Fourier series; separation of variables; solutions of onedimensional diffusion equation;
first and second order one-dimensional wave equation and two-dimensional Laplace equation. Probability and Statistics: Definitions of probability and sampling theorems; Conditional probability;
Discrete Random variables: Poisson and Binomial distributions; Continuous random variables: normal and exponential distributions; Descriptive statistics - Mean, median, mode and standard deviation;
Hypothesis testing. Numerical Methods: Accuracy and precision; error analysis. Numerical solutions of linear and non-linear algebraic equations; Least square approximation, Newton’s and Lagrange
polynomials, numerical differentiation, Integration by trapezoidal and Simpson’s rule, single and multi-step methods for first order differential equations.
Section 2: Structural Engineering Engineering Mechanics: System of forces, free-body diagrams, equilibrium equations; Internal forces in structures; Friction and its applications; Kinematics of point
mass and rigid body; Centre of mass; Euler’s equations of motion; Impulse-momentum; Energy methods; Principles of virtual work. Solid Mechanics: Bending moment and shear force in statically
determinate beams; Simple stress and strain relationships; Theories of failures; Simple bending theory, flexural and shear stresses, shear centre; Uniform torsion, buckling of column, combined and
direct bending stresses. Structural Analysis: Statically determinate and indeterminate structures by force/ energy methods; Method of superposition; Analysis of trusses, arches, beams, cables and
frames; Displacement methods: Slope deflection and moment distribution methods; Influence lines; Stiffness and flexibility methods of structural analysis. Construction Materials and Management:
Construction Materials: Structural steel composition, material properties and behaviour; Concrete - constituents, mix design, short-term and long-term properties; Bricks and mortar; Timber; Bitumen.
Construction Management: Types of construction projects; Tendering and construction contracts; Rate analysis and standard specifications; Cost estimation; Project planning and network analysis - PERT
and CPM.
Concrete Structures: Working stress, Limit state and Ultimate load design concepts; Design of beams, slabs, columns; Bond and development length; Prestressed concrete; Analysis of beam sections at
transfer and service loads. Steel Structures: Working stress and Limit state design concepts; Design of tension and compression members, beams and beam- columns, column bases; Connections - simple
and eccentric, beam-column connections, plate girders and trusses; Plastic analysis of beams and frames.
Section 3: Geotechnical Engineering Soil Mechanics: Origin of soils, soil structure and fabric; Three-phase system and phase relationships, index properties; Unified and Indian standard soil
classification system; Permeability - one dimensional flow, Darcy’s law; Seepage through soils - two-dimensional flow, flow nets, uplift pressure, piping; Principle of effective stress, capillarity,
seepage force and quicksand condition; Compaction in laboratory and field conditions; Onedimensional consolidation, time rate of consolidation; Mohr’s circle, stress paths, effective and total shear
strength parameters, characteristics of clays and sand. Foundation Engineering: Sub-surface investigations - scope, drilling bore holes, sampling, plate load test, standard penetration and cone
penetration tests; Earth pressure theories Rankine and Coulomb; Stability of slopes - finite and infinite slopes, method of slices and Bishop’s method; Stress distribution in soils - Boussinesq’s and
Westergaard’s theories, pressure bulbs; Shallow foundations - Terzaghi’s and Meyerhoff’s bearing capacity theories, effect of water table; Combined footing and raft foundation; Contact pressure;
Settlement analysis in sands and clays; Deep foundations - types of piles, dynamic and static formulae, load capacity of piles in sands and clays, pile load test, negative skin friction.
Section 4: Water Resources Engineering Fluid Mechanics: Properties of fluids, fluid statics; Continuity, momentum, energy and corresponding equations; Potential flow, applications of momentum and
energy equations; Laminar and turbulent flow; Flow in pipes, pipe networks; Concept of boundary layer and its growth. Hydraulics: Forces on immersed bodies; Flow measurement in channels and pipes;
Dimensional analysis and hydraulic similitude; Kinematics of flow, velocity triangles; Basics of hydraulic machines, specific speed of pumps and turbines; Channel Hydraulics Energy-depth
relationships, specific energy, critical flow, slope profile, hydraulic jump, uniform flow and gradually varied flow Hydrology: Hydrologic cycle, precipitation, evaporation, evapo-transpiration,
watershed, infiltration, unit hydrographs, hydrograph analysis, flood estimation and routing, reservoir capacity, reservoir and channel routing, surface run-off models, ground water hydrology steady
state well hydraulics and aquifers; Application of Darcy’s law. Irrigation: Duty, delta, estimation of evapo-transpiration; Crop water requirements; Design of lined and unlined canals, head works,
gravity dams and spillways; Design of weirs on permeable foundation; Types of irrigation systems, irrigation methods; Water logging and drainage; Canal regulatory works, cross-drainage structures,
outlets and escapes.
Section 5: Environmental Engineering Water and Waste Water: Quality standards, basic unit processes and operations for water treatment. Drinking water standards, water requirements, basic unit
operations and unit processes for surface water treatment, distribution of water. Sewage and sewerage treatment, quantity and characteristics of wastewater. Primary, secondary and tertiary treatment
of wastewater, effluent discharge standards. Domestic wastewater treatment, quantity of characteristics of domestic wastewater, primary and secondary treatment. Unit operations and unit processes of
domestic wastewater, sludge disposal. Air Pollution: Types of pollutants, their sources and impacts, air pollution meteorology, air pollution control, air quality standards and limits. Municipal
Solid Wastes: Characteristics, generation, collection and transportation of solid wastes, engineered systems for solid waste management (reuse/ recycle, energy recovery, treatment and disposal).
Noise Pollution: Impacts of noise, permissible limits of noise pollution, measurement of noise and control of noise pollution.
Section 6: Transportation Engineering Transportation Infrastructure: Highway alignment and engineering surveys; Geometric design of highways - cross-sectional elements, sight distances, horizontal
and vertical alignments; Geometric design of railway track; Airport runway length, taxiway and exit taxiway design. Highway Pavements: Highway materials - desirable properties and quality control
tests; Design of bituminous paving mixes; Design factors for flexible and rigid pavements; Design of flexible pavement using IRC: 37-2012; Design of rigid pavements using IRC: 58-2011; Distresses in
concrete pavements. Traffic Engineering: Traffic studies on flow, speed, travel time - delay and O-D study, PCU, peak hour factor, parking study, accident study and analysis, statistical analysis of
traffic data; Microscopic and macroscopic parameters of traffic flow, fundamental relationships; Control devices, signal design by Webster’s method; Types of intersections and channelization; Highway
capacity and level of service of rural highways and urban roads.
Section 7: Geomatics Engineering Principles of surveying; Errors and their adjustment; Maps - scale, coordinate system; Distance and angle measurement - Levelling and trigonometric levelling;
Traversing and triangulation survey; Total station; Horizontal and vertical curves. Photogrammetry - scale, flying height; Remote sensing - basics, platform and sensors, visual image interpretation;
Basics of Geographical information system (GIS) and Geographical Positioning system (GPS).
Mechanical Engineering
Section 1: Engineering Mathematics Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors. Calculus: Functions of single variable, limit, continuity and
differentiability, mean value theorems, indeterminate forms; evaluation of definite and improper integrals; double and triple integrals; partial derivatives, total derivative, Taylor series (in one
and two variables), maxima and minima, Fourier series; gradient, divergence and curl, vector identities, directional derivatives, line, surface and volume integrals, applications of Gauss, Stokes and
Green’s theorems. Differential equations: First order equations (linear and nonlinear); higher order linear differential equations with constant coefficients; Euler-Cauchy equation; initial and
boundary value problems; Laplace transforms; solutions of heat, wave and Laplace's equations. Complex variables: Analytic functions; Cauchy-Riemann equations; Cauchy’s integral theorem and integral
formula; Taylor and Laurent series. Probability and Statistics: Definitions of probability, sampling theorems, conditional probability; mean, median, mode and standard deviation; random variables,
binomial, Poisson and normal distributions. Numerical Methods: Numerical solutions of linear and non-linear algebraic equations; integration by trapezoidal and Simpson’s rules; single and multi-step
methods for differential equations.
Section 2: Applied Mechanics and Design Engineering Mechanics: Free-body diagrams and equilibrium; trusses and frames; virtual work; kinematics and dynamics of particles and of rigid bodies in plane
motion; impulse and momentum (linear and angular) and energy formulations, collisions. Mechanics of Materials: Stress and strain, elastic constants, Poisson's ratio; Mohr’s circle for plane stress
and plane strain; thin cylinders; shear force and bending moment diagrams; bending and shear stresses; deflection of beams; torsion of circular shafts; Euler’s theory of columns; energy methods;
thermal stresses; strain gauges and rosettes; testing of materials with universal testing machine; testing of hardness and impact strength. Theory of Machines: Displacement, velocity and acceleration
analysis of plane mechanisms; dynamic analysis of linkages; cams; gears and gear trains; flywheels and governors; balancing of reciprocating and rotating masses; gyroscope. Vibrations: Free and
forced vibration of single degree of freedom systems, effect of damping; vibration isolation; resonance; critical speeds of shafts.
Machine Design: Design for static and dynamic loading; failure theories; fatigue strength and the S-N diagram; principles of the design of machine elements such as bolted, riveted and welded joints;
shafts, gears, rolling and sliding contact bearings, brakes and clutches, springs.
Section 3: Fluid Mechanics and Thermal Sciences Fluid Mechanics: Fluid properties; fluid statics, manometry, buoyancy, forces on submerged bodies, stability of floating bodies; control-volume
analysis of mass, momentum and energy; fluid acceleration; differential equations of continuity and momentum; Bernoulli’s equation; dimensional analysis; viscous flow of incompressible fluids,
boundary layer, elementary turbulent flow, flow through pipes, head losses in pipes, bends and fittings. Heat-Transfer: Modes of heat transfer; one dimensional heat conduction, resistance concept and
electrical analogy, heat transfer through fins; unsteady heat conduction, lumped parameter system, Heisler's charts; thermal boundary layer, dimensionless parameters in free and forced convective
heat transfer, heat transfer correlations for flow over flat plates and through pipes, effect of turbulence; heat exchanger performance, LMTD and NTU methods; radiative heat transfer, StefanBoltzmann
law, Wien's displacement law, black and grey surfaces, view factors, radiation network analysis. Thermodynamics: Thermodynamic systems and processes; properties of pure substances, behaviour of ideal
and real gases; zeroth and first laws of thermodynamics, calculation of work and heat in various processes; second law of thermodynamics; thermodynamic property charts and tables, availability and
irreversibility; thermodynamic relations. Applications: Power Engineering: Air and gas compressors; vapour and gas power cycles, concepts of regeneration and reheat. I.C. Engines: Air-standard Otto,
Diesel and dual cycles. Refrigeration and air-conditioning: Vapour and gas refrigeration and heat pump cycles; properties of moist air, psychrometric chart, basic psychrometric processes.
Turbomachinery: Impulse and reaction principles, velocity diagrams, Pelton-wheel, Francis and Kaplan turbines.
Section 4: Materials, Manufacturing and Industrial Engineering Engineering Materials: Structure and properties of engineering materials, phase diagrams, heat treatment, stress-strain diagrams for
engineering materials. Casting, Forming and Joining Processes: Different types of castings, design of patterns, moulds and cores; solidification and cooling; riser and gating design. Plastic
deformation and yield criteria; fundamentals of hot and cold working processes; load estimation for bulk (forging, rolling, extrusion, drawing) and sheet (shearing, deep drawing, bending) metal
forming processes; principles of powder metallurgy. Principles of welding, brazing, soldering and adhesive bonding. Machining and Machine Tool Operations: Mechanics of machining; basic machine tools;
single and multi-point cutting tools, tool geometry and materials, tool life and wear; economics of machining; principles of non-traditional machining processes; principles of work holding, design of
jigs and fixtures. Metrology and Inspection: Limits, fits and tolerances; linear and angular measurements; comparators; gauge design; interferometry; form and finish measurement; alignment and
testing methods; tolerance analysis in manufacturing and assembly. Computer Integrated Manufacturing: Basic concepts of CAD/CAM and their integration tools.
Production Planning and Control: Forecasting models, aggregate production planning, scheduling, materials requirement planning. Inventory Control: Deterministic models; safety stock inventory control
systems. Operations Research: Linear programming, simplex method, transportation, assignment, network flow models, simple queuing models, PERT and CPM.
Electrical Engineering
Section 1: Engineering Mathematics Linear Algebra: Matrix Algebra, Systems of linear equations, Eigenvalues, Eigenvectors. Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of
definite and improper integrals, Partial Derivatives, Maxima and minima, Multiple integrals, Fourier series, Vector identities, Directional derivatives, Line integral, Surface integral, Volume
integral, Stokes’s theorem, Gauss’s theorem, Green’s theorem. Differential equations: First order equations (linear and nonlinear), Higher order linear differential equations with constant
coefficients, Method of variation of parameters, Cauchy’s equation, Euler’s equation, Initial and boundary value problems, Partial Differential Equations, Method of separation of variables. Complex
variables: Analytic functions, Cauchy’s integral theorem, Cauchy’s integral formula, Taylor series, Laurent series, Residue theorem, Solution integrals. Probability and Statistics: Sampling theorems,
Conditional probability, Mean, Median, Mode, Standard Deviation, Random variables, Discrete and Continuous distributions, Poisson distribution, Normal distribution, Binomial distribution, Correlation
analysis, Regression analysis. Numerical Methods: Solutions of nonlinear algebraic equations, Single and Multi‐step methods for differential equations. Transform Theory: Fourier Transform, Laplace
Transform, z‐Transform. Electrical Engineering
Section 2: Electric Circuits Network graph, KCL, KVL, Node and Mesh analysis, Transient response of dc and ac networks, Sinusoidal steady‐state analysis, Resonance, Passive filters, Ideal current and
voltage sources, Thevenin’s theorem, Norton’s theorem, Superposition theorem, Maximum power transfer theorem, Two‐port networks, Three phase circuits, Power and power factor in ac circuits.
Section 3: Electromagnetic Fields Coulomb's Law, Electric Field Intensity, Electric Flux Density, Gauss's Law, Divergence, Electric field and potential due to point, line, plane and spherical charge
distributions, Effect of dielectric medium, Capacitance of simple configurations, Biot‐Savart’s law, Ampere’s law, Curl, Faraday’s law, Lorentz force, Inductance, Magnetomotive force, Reluctance,
Magnetic circuits,Self and Mutual inductance of simple configurations.
Section 4: Signals and Systems Representation of continuous and discrete‐time signals, Shifting and scaling operations, Linear Time Invariant and Causal systems, Fourier series representation of
continuous periodic signals, Sampling theorem, Applications of Fourier Transform, Laplace Transform and z-Transform.
Section 5: Electrical Machines Single phase transformer: equivalent circuit, phasor diagram, open circuit and short circuit tests, regulation and efficiency; Three phase transformers: connections,
parallel operation; Auto‐transformer, Electromechanical energy conversion principles, DC machines: separately excited, series and shunt, motoring and generating mode of operation and their
characteristics, starting and speed control of dc motors; Three phase induction motors: principle of operation, types, performance, torque-speed characteristics, no-load and blocked rotor tests,
equivalent circuit, starting and speed control; Operating principle of single phase induction motors; Synchronous machines: cylindrical and salient pole machines, performance, regulation and parallel
operation of generators, starting of synchronous motor, characteristics; Types of losses and efficiency calculations of electric machines.
Section 6: Power Systems Power generation concepts, ac and dc transmission concepts, Models and performance of transmission lines and cables, Series and shunt compensation, Electric field
distribution and insulators, Distribution systems, Per‐unit quantities, Bus admittance matrix, GaussSeidel and Newton-Raphson load flow methods, Voltage and Frequency control, Power factor
correction, Symmetrical components, Symmetrical and unsymmetrical fault analysis, Principles of over‐current, differential and distance protection; Circuit breakers, System stability concepts, Equal
area criterion.
Section 7: Control Systems Mathematical modeling and representation of systems, Feedback principle, transfer function, Block diagrams and Signal flow graphs, Transient and Steady‐state analysis of
linear time invariant systems, Routh-Hurwitz and Nyquist criteria, Bode plots, Root loci, Stability analysis, Lag, Lead and Lead‐Lag compensators; P, PI and PID controllers; State space model, State
transition matrix.
Section 8: Electrical and Electronic Measurements Bridges and Potentiometers, Measurement of voltage, current, power, energy and power factor; Instrument transformers, Digital voltmeters and
multimeters, Phase, Time and Frequency measurement; Oscilloscopes, Error analysis.
Section 9: Analog and Digital Electronics Characteristics of diodes, BJT, MOSFET; Simple diode circuits: clipping, clamping, rectifiers; Amplifiers: Biasing, Equivalent circuit and Frequency
response; Oscillators and Feedback amplifiers; Operational amplifiers: Characteristics and applications; Simple active filters, VCOs and Timers, Combinational and Sequential logic circuits,
Multiplexer, Demultiplexer, Schmitt trigger, Sample and hold circuits, A/D and D/A converters, 8085Microprocessor: Architecture, Programming and Interfacing.
Section 10: Power Electronics Characteristics of semiconductor power devices: Diode, Thyristor, Triac, GTO, MOSFET, IGBT; DC to DC conversion: Buck, Boost and Buck-Boost converters; Single and three
phase configuration of uncontrolled rectifiers, Line commutated thyristor based converters, Bidirectional ac to dc voltage source converters, Issues of line current harmonics, Power factor,
Distortion factor of ac to dc converters, Single phase and three phase inverters, Sinusoidal pulse width modulation.
Electronics and Communications
Section 1: Engineering Mathematics Linear Algebra: Vector space, basis, linear dependence and independence, matrix algebra, eigen values and eigen vectors, rank, solution of linear equations –
existence and uniqueness. Calculus: Mean value theorems, theorems of integral calculus, evaluation of definite and improper integrals, partial derivatives, maxima and minima, multiple integrals,
line, surface and volume integrals, Taylor series. Differential Equations: First order equations (linear and nonlinear), higher order linear differential equations, Cauchy's and Euler's equations,
methods of solution using variation of parameters, complementary function and particular integral, partial differential equations, variable separable method, initial and boundary value problems.
Vector Analysis: Vectors in plane and space, vector operations, gradient, divergence and curl, Gauss's, Green's and Stoke's theorems. Complex Analysis: Analytic functions, Cauchy's integral theorem,
Cauchy's integral formula; Taylor's and Laurent's series, residue theorem. Numerical Methods: Solution of nonlinear equations, single and multi-step methods for differential equations, convergence
criteria. Probability and Statistics: Mean, median, mode and standard deviation; combinatorial probability, probability distribution functions - binomial, Poisson, exponential and normal; Joint and
conditional probability; Correlation and regression analysis.
Section 2: Networks, Signals and Systems Network solution methods: nodal and mesh analysis; Network theorems: superposition, Thevenin and Norton’s, maximum power transfer; Wye‐Delta transformation;
Steady state sinusoidal analysis using phasors; Time domain analysis of simple linear circuits; Solution of network equations using Laplace transform; Frequency domain analysis of RLC circuits;
Linear 2‐port network parameters: driving point and transfer functions; State equations for networks. Continuous-time signals: Fourier series and Fourier transform representations, sampling theorem
and applications; Discrete-time signals: discrete-time Fourier transform (DTFT), DFT, FFT, Z-transform, interpolation of discrete-time signals; LTI systems: definition and properties, causality,
stability, impulse response, convolution, poles and zeros, parallel and cascade structure, frequency response, group delay, phase delay, digital filter design techniques.
Section 3: Electronic Devices Energy bands in intrinsic and extrinsic silicon; Carrier transport: diffusion current, drift current, mobility and resistivity; Generation and recombination of carriers;
Poisson and continuity equations; P-N junction, Zener diode, BJT, MOS capacitor, MOSFET, LED, photo diode and solar cell; Integrated circuit fabrication process: oxidation, diffusion, ion
implantation, photolithography and twin-tub CMOS process.
Section 4: Analog Circuits Small signal equivalent circuits of diodes, BJTs and MOSFETs; Simple diode circuits: clipping, clamping and rectifiers; Single-stage BJT and MOSFET amplifiers: biasing,
bias stability, mid-frequency small signal analysis and frequency response; BJT and MOSFET amplifiers: multi-stage, differential, feedback, power and operational; Simple op-amp circuits; Active
filters; Sinusoidal oscillators: criterion for oscillation, single-transistor and opamp configurations; Function generators, wave-shaping circuits and 555 timers; Voltage reference circuits; Power
supplies: ripple removal and regulation.
Section 5: Digital Circuits Number systems; Combinatorial circuits: Boolean algebra, minimization of functions using Boolean identities and Karnaugh map, logic gates and their static CMOS
implementations, arithmetic circuits, code converters, multiplexers, decoders and PLAs; Sequential circuits: latches and flip‐flops, counters, shift‐registers and finite state machines; Data
converters: sample and hold circuits, ADCs and DACs; Semiconductor memories: ROM, SRAM, DRAM; 8-bit microprocessor (8085): architecture, programming, memory and I/O interfacing.
Section 6: Control Systems Basic control system components; Feedback principle; Transfer function; Block diagram representation; Signal flow graph; Transient and steady-state analysis of LTI systems;
Frequency response; Routh-Hurwitz and Nyquist stability criteria; Bode and root-locus plots; Lag, lead and lag-lead compensation; State variable model and solution of state equation of LTI systems.
Section 7: Communications Random processes: autocorrelation and power spectral density, properties of white noise, filtering of random signals through LTI systems; Analog communications: amplitude
modulation and demodulation, angle modulation and demodulation, spectra of AM and FM, superheterodyne receivers, circuits for analog communications; Information theory: entropy, mutual information
and channel capacity theorem; Digital communications: PCM, DPCM, digital modulation schemes, amplitude, phase and frequency shift keying (ASK, PSK, FSK), QAM, MAP and ML decoding, matched filter
receiver, calculation of bandwidth, SNR and BER for digital modulation; Fundamentals of error correction, Hamming codes; Timing and frequency synchronization, inter-symbol interference and its
mitigation; Basics of TDMA, FDMA and CDMA.
Section 8: Electromagnetics Electrostatics; Maxwell’s equations: differential and integral forms and their interpretation, boundary conditions, wave equation, Poynting vector; Plane waves and
properties: reflection and refraction, polarization, phase and group velocity, propagation through various media, skin depth; Transmission lines: equations, characteristic impedance, impedance
matching, impedance transformation, S-parameters, Smith chart; Waveguides: modes, boundary conditions, cut-off frequencies, dispersion relations; Antennas: antenna types, radiation pattern, gain and
directivity, return loss, antenna arrays; Basics of radar; Light propagation in optical fibers.
Computer Science and Information Technology
Section1: Engineering Mathematics Discrete Mathematics: Propositional and first order logic. Sets, relations, functions, partial orders and lattices. Groups. Graphs: connectivity, matching, coloring.
Combinatorics: counting, recurrence relations, generating functions. Linear Algebra: Matrices, determinants, system of linear equations, eigenvalues and eigenvectors, LU decomposition. Calculus:
Limits, continuity and differentiability. Maxima and minima. Mean value theorem. Integration. Probability: Random variables. Uniform, normal, exponential, poisson and binomial distributions. Mean,
median, mode and standard deviation. Conditional probability and Bayes theorem. Computer Science and Information Technology
Section 2: Digital Logic Boolean algebra. Combinational and sequential circuits. Minimization. Number representations and computer arithmetic (fixed and floating point).
Section 3: Computer Organization and Architecture Machine instructions and addressing modes. ALU, data‐path and control unit. Instruction pipelining. Memory hierarchy: cache, main memory and
secondary storage; I/O interface (interrupt and DMA mode).
Section 4: Programming and Data Structures Programming in C. Recursion. Arrays, stacks, queues, linked lists, trees, binary search trees, binary heaps, graphs.
Section 5: Algorithms Searching, sorting, hashing. Asymptotic worst case time and space complexity. Algorithm design techniques: greedy, dynamic programming and divide‐and‐conquer. Graph search,
minimum spanning trees, shortest paths.
Section 6: Theory of Computation Regular expressions and finite automata. Context-free grammars and push-down automata. Regular and contex-free languages, pumping lemma. Turing machines and
Section 7: Compiler Design Lexical analysis, parsing, syntax-directed translation. Runtime environments. Intermediate code generation.
Section 8: Operating System Processes, threads, inter‐process communication, concurrency and synchronization. Deadlock. CPU scheduling. Memory management and virtual memory. File systems.
Section 9: Databases ER‐model. Relational model: relational algebra, tuple calculus, SQL. Integrity constraints, normal forms. File organization, indexing (e.g., B and B+ trees). Transactions and
concurrency control.
Section 10: Computer Networks Concept of layering. LAN technologies (Ethernet). Flow and error control techniques, switching. IPv4/IPv6, routers and routing algorithms (distance vector, link state).
TCP/UDP and sockets, congestion control. Application layer protocols (DNS, SMTP, POP, FTP, HTTP). Basics of Wi-Fi. Network security: authentication, basics of public key and private key cryptography,
digital signatures and certificates, firewalls.
Instrumentation Engineering
Section 1: Engineering Mathematics Linear Algebra: Matrix algebra, systems of linear equations, Eigen values and Eigen vectors. Calculus: Mean value theorems, theorems of integral calculus, partial
derivatives, maxima and minima, multiple integrals, Fourier series, vector identities, line, surface and volume integrals, Stokes, Gauss and Green’s theorems. Differential equations: First order
equation (linear and nonlinear), higher order linear differential equations with constant coefficients, method of variation of parameters, Cauchy’s and Euler’s equations, initial and boundary value
problems, solution of partial differential equations: variable separable method. Analysis of complex variables: Analytic functions, Cauchy’s integral theorem and integral formula, Taylor’s and
Laurent’s series, residue theorem, solution of integrals. Probability and Statistics: Sampling theorems, conditional probability, mean, median, mode and standard deviation, random variables, discrete
and continuous distributions: normal, Poisson and binomial distributions. Numerical Methods: Matrix inversion, solutions of non-linear algebraic equations, iterative methods for solving differential
equations, numerical integration, regression and correlation analysis.
Instrumentation Engineering
Section 2: Electrical Circuits: Voltage and current sources: independent, dependent, ideal and practical; v-i relationships of resistor, inductor, mutual inductor and capacitor; transient analysis of
RLC circuits with dc excitation. Kirchoff’s laws, mesh and nodal analysis, superposition, Thevenin, Norton, maximum power transfer and reciprocity theorems. Peak-, average- and rms values of ac
quantities; apparent-, active- and reactive powers; phasor analysis, impedance and admittance; series and parallel resonance, locus diagrams, realization of basic filters with R, L and C elements.
One-port and two-port networks, driving point impedance and admittance, open-, and short circuit parameters.
Section 3: Signals and Systems Periodic, aperiodic and impulse signals; Laplace, Fourier and z-transforms; transfer function, frequency response of first and second order linear time invariant
systems, impulse response of systems; convolution, correlation. Discrete time system: impulse response, frequency response, pulse transfer function; DFT and FFT; basics of IIR and FIR filters.
Section 4: Control Systems Feedback principles, signal flow graphs, transient response, steady-state-errors, Bode plot, phase and gain margins, Routh and Nyquist criteria, root loci, design of lead,
lag and lead-lag compensators, state-space representation of systems; time-delay systems; mechanical, hydraulic and pneumatic system components, synchro pair, servo and stepper motors, servo valves;
on-off, P, P-I, P-I-D, cascade, feedforward, and ratio controllers.
Section 5: Analog Electronics Characteristics and applications of diode, Zener diode, BJT and MOSFET; small signal analysis of transistor circuits, feedback amplifiers. Characteristics of operational
amplifiers; applications of opamps: difference amplifier, adder, subtractor, integrator, differentiator, instrumentation amplifier, precision rectifier, active filters and other circuits.
Oscillators, signal generators, voltage controlled oscillators and phase locked loop.
Section 6: Digital Electronics Combinational logic circuits, minimization of Boolean functions. IC families: TTL and CMOS. Arithmetic circuits, comparators, Schmitt trigger, multi-vibrators,
sequential circuits, flipflops, shift registers, timers and counters; sample-and-hold circuit, multiplexer, analog-todigital (successive approximation, integrating, flash and sigma-delta) and
digital-toanalog converters (weighted R, R-2R ladder and current steering logic). Characteristics of ADC and DAC (resolution, quantization, significant bits, conversion/settling time); basics of
number systems, 8-bit microprocessor and microcontroller: applications, memory and input-output interfacing; basics of data acquisition systems.
Section 7: Measurements SI units, systematic and random errors in measurement, expression of uncertainty accuracy and precision index, propagation of errors. PMMC, MI and dynamometer type
instruments; dc potentiometer; bridges for measurement of R, L and C, Q-meter. Measurement of voltage, current and power in single and three phase circuits; ac and dc current probes; true rms meters,
voltage and current scaling, instrument transformers, timer/counter, time, phase and frequency measurements, digital voltmeter, digital multimeter; oscilloscope, shielding and grounding.
Section 8: Sensors and Industrial Instrumentation Resistive-, capacitive-, inductive-, piezoelectric-, Hall effect sensors and associated signal conditioning circuits; transducers for industrial
instrumentation: displacement (linear and angular), velocity, acceleration, force, torque, vibration, shock, pressure (including low pressure), flow (differential pressure, variable area,
electromagnetic, ultrasonic, turbine and open channel flow meters) temperature (thermocouple, bolometer, RTD (3/4 wire), thermistor, pyrometer and semiconductor); liquid level, pH, conductivity and
viscosity measurement.
Section 9: Communication and Optical Instrumentation Amplitude- and frequency modulation and demodulation; Shannon's sampling theorem, pulse code modulation; frequency and time division multiplexing,
amplitude-, phase-, frequency-, pulse shift keying for digital modulation; optical sources and detectors: LED, laser, photo-diode, light dependent resistor and their characteristics; interferometer:
applications in metrology; basics of fiber optic sensing. | {"url":"https://v.vibdoc.com/section-1-engineering-mathematics-5f0c2156343b4.html","timestamp":"2024-11-11T07:30:27Z","content_type":"text/html","content_length":"54255","record_id":"<urn:uuid:d01a477d-3943-42d1-b668-21b5b34b6390>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00133.warc.gz"} |
Summary & Notes | The Lex Fridman Podcast - #370 – Edward Frenkel: Reality Is A Paradox – Mathematics, Physics, Truth & Love
In this episode of “The Lex Fridman Podcast”, mathematician Edward Frenkel explores the fascinating intersection of mathematics and quantum physics. Frenkel emphasizes the Langlands program as a
grand unified theory of mathematics and discusses the role of mathematics in understanding the deepest structures of the universe. The podcast is sponsored by House of McAdamias for healthy snacks,
Shopify for e-commerce, and ExpressVPN for security and privacy on the internet.
Main Takeaways
Mathematics and Quantum Physics
• Edward Frenkel is a mathematician doing research on the interface of mathematics and quantum physics.
• Physicists use complex mathematical theories to understand the underpinnings of physical reality.
• Quantum mechanics is a puzzle that allows physicists to go as deep as possible into the root of the universe’s secrets.
• The few mathematical physicists that can shine a flashlight into the universe’s dark room are rare.
The Language of Mathematics
• Mathematics underpins physics as a language, and the book of nature is written in the language of mathematics.
• Mathematics helps discern patterns and find regularities in the universe, making our perception more sophisticated and sharpening our ability to see beauty.
• Realistic theories of physics are about spaces of three dimensions or space-times of four dimensions, but mathematically, there is interest in theories of higher dimensions or infinite
dimensional spaces.
• Mathematicians prove things using rules of logic, while physicists confirm their theories through experiments.
The Nature of Time and Consciousness
• Our memories are often told in a sequence, but it’s unclear if they actually happened that way.
• We have consciousness and free will, but it’s unclear how they relate to time.
• Our minds may be limited in how they approach the world, and stepping out of the mind may be necessary to understand it more completely.
• Humans love to play with time and subjective truths, which may be useful for competition and the dance of life.
The Beauty of Mathematics
• Mathematics helps discern patterns and find regularities in the universe, making our perception more sophisticated and sharpening our ability to see beauty.
• Mathematics is a human activity, whether it’s discovered or invented.
• Paradoxes may be fundamental to reality, and we exist in a world of paradoxes.
• Art and poetry can express complex ideas that cannot be put into words.
Mathematical Discoveries and Challenges
• Edward Frenkel describes his own experience of solving a difficult problem and the emotional toll it took on him.
• Mathematicians face unique ethical challenges compared to other fields, as there is less money involved and therefore less incentive to prioritize discovery credit.
• Mathematics draws in a specific psychological type, and often serves as a refuge from discrimination and cruelty experienced in the outside world.
• Breaking the stereotype of mathematicians being quiet and closed off can create a virtuous circle of inspiration and sharing.
Mathematics and Quantum Physics
Edward Frenkel explores the interface of mathematics and quantum physics, highlighting the role of mathematics in understanding the deepest structures of the universe. Physicists use complex
mathematical theories to uncover the underpinnings of physical reality, with quantum mechanics serving as a puzzle that allows them to delve into the universe’s secrets. However, the few mathematical
physicists who can shed light on the universe’s mysteries are rare.
The Language of Mathematics
Mathematics serves as a fundamental language that underpins physics, enabling the discernment of patterns and regularities in the universe. While realistic theories of physics focus on
three-dimensional spaces or four-dimensional space-times, mathematicians are interested in exploring higher dimensions or infinite dimensional spaces. Mathematicians prove concepts using logic, while
physicists confirm their theories through experimentation.
The Nature of Time and Consciousness
The relationship between time, consciousness, and free will remains unclear. Our memories may not accurately reflect the sequence of events, and the limitations of our minds may require stepping
outside of them to gain a more comprehensive understanding of the world. Humans have a penchant for playing with time and subjective truths, which can be beneficial in terms of competition and the
dance of life.
The Beauty of Mathematics
Mathematics enhances our perception by enabling the discernment of patterns and regularities in the universe, ultimately sharpening our ability to appreciate beauty. Whether discovered or invented,
mathematics is a human activity that can express complex ideas that transcend words. Paradoxes may be fundamental to reality, and art and poetry can capture the essence of these complex concepts.
Mathematical Discoveries and Challenges
Edward Frenkel shares his personal experience of solving difficult mathematical problems and the emotional toll it can take. Mathematicians face unique ethical challenges, as the field often lacks
financial incentives, leading to less prioritization of discovery credit. Mathematics attracts a specific psychological type, offering refuge from discrimination and cruelty. Breaking the stereotype
of mathematicians as quiet and closed off can foster inspiration and sharing among diverse individuals.
Edward Frenkel’s exploration of mathematics and its relationship with quantum physics sheds light on the profound interconnections between these fields. Mathematics serves as a language that enables
us to understand the universe’s deepest structures, sharpen our perception, and appreciate beauty. Mathematicians face unique challenges and ethical considerations, but breaking stereotypes and
fostering inclusivity can lead to a virtuous circle of inspiration and sharing. By approaching science and math with childlike wonder and curiosity, we can uncover the mysteries of the universe and
embrace the beauty of mathematics. | {"url":"https://podcastmemo.com/summary/the-lex-fridman-podcast-370-edward-frenkel-reality-is-a-paradox-mathematics-physics-truth-love/","timestamp":"2024-11-12T09:09:40Z","content_type":"text/html","content_length":"358100","record_id":"<urn:uuid:719f2785-1b44-4bbd-8db5-2a43bf4d6e6b>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00125.warc.gz"} |
IF2- Lewis structure - Learnool
IF2- Lewis structure
The information on this page is ✔ fact-checked.
IF[2]^– Lewis structure
IF[2]^– has one iodine atom and two fluorine atoms.
In IF[2]^– Lewis structure, there are two single bonds around the iodine atom, with two fluorine atoms attached to it, and each atom has three lone pairs.
Also, there is a negative (-1) charge on the iodine atom.
Here’s how you can easily draw the IF[2]^– Lewis structure step by step:
#1 Draw a rough skeleton structure
#2 Mention lone pairs on the atoms
#3 If needed, mention formal charges on the atoms
Now, let’s take a closer look at each step mentioned above.
#1 Draw a rough skeleton structure
• First, determine the total number of valence electrons
Periodic table
In the periodic table, both iodine and fluorine lie in group 17.
Hence, both iodine and fluorine have seven valence electrons.
Since IF[2]^– has one iodine atom and two fluorine atoms, so…
Valence electrons of one iodine atom = 7 × 1 = 7
Valence electrons of two fluorine atoms = 7 × 2 = 14
Now the IF[2]^– has a negative (-1) charge, so we have to add one more electron.
So the total valence electrons = 7 + 14 + 1 = 22
Learn how to find: Fluorine valence electrons
• Second, find the total electron pairs
We have a total of 22 valence electrons. And when we divide this value by two, we get the value of total electron pairs.
Total electron pairs = total valence electrons ÷ 2
So the total electron pairs = 22 ÷ 2 = 11
• Third, determine the central atom
We have to place the least electronegative atom at the center.
Since iodine is less electronegative than fluorine, assume that the central atom is iodine.
Therefore, place iodine in the center and fluorines on either side.
• And finally, draw the rough sketch
Rough sketch of IF[2]^– Lewis structure
#2 Mention lone pairs on the atoms
Here, we have a total of 11 electron pairs. And two I — F bonds are already marked. So we have to only mark the remaining nine electron pairs as lone pairs on the sketch.
Also remember that iodine is a period 5 element, so it can keep more than 8 electrons in its last shell. And fluorine is a period 2 element, so it can not keep more than 8 electrons in its last
Always start to mark the lone pairs from outside atoms. Here, the outside atoms are fluorines.
So for each atom, there are three lone pairs.
Mark the lone pairs on the sketch as follows:
Lone pairs marked on IF[2]^– Lewis structure
#3 If needed, mention formal charges on the atoms
Use the following formula to calculate the formal charges on atoms:
Formal charge = valence electrons – nonbonding electrons – ½ bonding electrons
For iodine atom, formal charge = 7 – 6 – ½ (4) = -1
For each fluorine atom, formal charge = 7 – 6 – ½ (2) = 0
Here, the iodine atom has a charge, so mark it on the sketch as follows:
Formal charges marked, and got the most stable Lewis structure of IF[2]^–
In the above structure, you can see that the central atom (iodine) forms an octet. And the outside atoms (fluorines) also form an octet. Hence, the octet rule is satisfied.
Now there is still a negative (-1) charge on the iodine atom.
This is not okay, right? Because the structure with a negative charge on the most electronegative atom is the best Lewis structure. And in this case, the most electronegative element is fluorine.
But if we convert a lone pair of the iodine atom to make a new I — F bond with the fluorine atom, and calculate the formal charge, then we do not get the formal charges on atoms closer to zero.
And the structure with the formal charges on atoms closer to zero is the best Lewis structure.
Therefore, this structure is the most stable Lewis structure of IF[2]^–.
And since the IF[2]^– has a negative (-1) charge, mention that charge on the Lewis structure by drawing brackets as follows:
IF[2]^– Lewis structure showing a negative (-1) charge
Next: BrF[2]^– Lewis structure
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Learnool.com was founded by Deep Rana, who is a mechanical engineer by profession and a blogger by passion. He has a good conceptual knowledge on different educational topics and he provides the same
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Completeness theorems for the Abadi-Rogaway logic of encrypted
Completeness theorems for the Abadi-Rogaway logic of encrypted expressions
Authors: Daniele Micciancio and Bogdan Warinschi.
Journal of Computer Security, 12(1), 2004, pp. 99-129
[BibTex] [Postscript] [PDF]
Abstract: We show that the Abadi-Rogaway logic of indistinguishability for cryptographic expressions is not complete, giving a natural example of a secure encryption function and a pair of
expressions, such that the distributions associated to the two expressions are computationally indistinguishable, but equality cannot be proved within the Abadi-Rogaway logic. Then, we show that if a
stronger, yet natural, notion of security for encryption is used (namely, that of authenticated encryption), then the Abadi-Rogaway logic is both sound and complete. In addition, we consider a
refinement of the Abadi-Rogaway logic that overcomes certain limitations of the original proposal, allowing for encryption functions that do not hide the length of the message being sent. Both the
soundness theorem of Abadi and Rogaway, and our completeness result for authenticated encryption easily extend to this more realistic notion of secrecy.
Preliminary version: Workshop on Issues in the Theory of Security - WITS 2002. Jan. 12-13. Portland, Oregon. | {"url":"https://cseweb.ucsd.edu/~daniele/papers/ARcompleteness.html","timestamp":"2024-11-07T17:27:10Z","content_type":"text/html","content_length":"2198","record_id":"<urn:uuid:b0265df7-5df6-4008-a27c-09eaac9e8850>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00290.warc.gz"} |
Pablo is studying the function f(x) shown in the graph. He claims that he can transform the function to include ordered pair (2,
All categories
Pablo is studying the function f(x) shown in the graph. He claims that he can transform the function to include ordered pair (2,
1 answer:
Please show the graph.
Step-by-step explanation:
You might be interested in
Answer: this is what i can tell you if it is a right triangle then you will di 90 - your numbers but if it is not a right triangel then you will do 180 instend then when you get the answer you add
it with those numbers but not with the 90 or 180
Step-by-step explanation:
Looks like 10 and and a half if it's four it's 10 and a half if it 6 it's 12 and a half
Time 15.625 to the 3 and divide it by 10.00 grams and you will get your answer
The x-intercept of a function is the value of x when y is 0. So let's set sin(x) equal to zero. When does sin(x) equal zero? Based on the unit circle, you know that sin(0) is 0, so that is one
x-intercept. You also know that sin(pi) is 0. Basically, every time x, starting from zero, increases or decreases by a multiple of pi, sin(x) is still zero. The answer can be represented by x=n*pi;
where n=any integer.
Step-by-step explanation: minus *minus = + so its +27
and then you just do the maths 27-13=14
Add answer | {"url":"https://answer.ya.guru/questions/1261-pablo-is-studying-the-function-f-x-shown-in-the-graph-he-claims.html","timestamp":"2024-11-04T18:47:36Z","content_type":"text/html","content_length":"55855","record_id":"<urn:uuid:544e1457-9b76-44d4-9e12-429b5cfa2f7c>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00817.warc.gz"} |
Constant of Proportionality Calculator
Descriptive Statistics
Constant of Proportionality Calculator
Constant of Proportionality Calculator
Enter the first value for x.
Enter the first value for y.
Enter the second value for x.
Enter the second value for y.
Understanding the Constant of Proportionality Calculator
The Constant of Proportionality Calculator is developed to assist users in determining the constant that is used to describe the relationship between two variables. This constant is essential for
various statistical and mathematical analyses. By providing values for two different pairs of x and y, this calculator will quickly compute the proportionality constant, making it a useful tool for
students, educators, and professionals in the field of statistics.
Applications of the Constant of Proportionality Calculator
This calculator is highly valuable in several practical scenarios. For instance, it can be applied in physics to understand the relationship between variables like force and distance. In economics,
it helps describe the relationship between cost and quantity of goods. Engineers often use it to determine the relationships between various parameters in their calculations. Overall, this calculator
is an indispensable tool for anyone involved in data analysis and interpretation.
How It Works
To use this calculator, you need to input two pairs of corresponding values for the variables x and y. The calculator will then compute the constant of proportionality using the concept that: the
change in y divided by the change in x is constant. If the values are not numerical or if they are identical, the calculator will alert the user with an error message.
Benefits of Using the Calculator
Using the Constant of Proportionality Calculator simplifies the process of finding the constant for proportional relationships. It saves time and reduces the potential for manual calculation errors,
making it more efficient to analyze relationships between variables. This is particularly beneficial for academic purposes, research work, and professional projects where precision is crucial.
Deriving the Answer
The calculation is straightforward: You provide the x and y values for two points, and the calculator determines the constant by taking the difference in the y values and dividing it by the
difference in the x values. This constant, often denoted as k, signifies the rate at which y changes with respect to x. By using this calculator, the process becomes quick and accurate, allowing
users to focus on the interpretation and application of the results.
Additional Information
The calculator is designed to be user-friendly, with clear input fields and tooltips that provide extra guidance. It also ensures that users are alerted if there are any issues with the input values,
such as non-numerical entries or identical x values, which could lead to division by zero errors. This attention to detail helps maintain the accuracy and reliability of the results provided by the
1. What is the Constant of Proportionality?
The Constant of Proportionality, denoted as k, describes the constant rate at which one variable changes with respect to another variable. It is calculated by taking the difference in the y values
and dividing it by the difference in the x values for two points (x1, y1) and (x2, y2).
2. How accurate is the Constant of Proportionality Calculator?
The accuracy of the calculator depends on the precision of the input values. As long as the inputs are numerical and not identical for the x-values, the calculator will provide a reliable constant of
3. Can I input decimal values?
Yes, you can input decimal values. The calculator works with both integer and decimal inputs to calculate the constant of proportionality.
4. What should I do if the calculator shows an error message?
If you encounter an error message, ensure that you have entered numerical values and that the x-values are not identical. Identical x-values would result in a division by zero error, which the
calculator cannot process.
5. How can this calculator be used in real-world applications?
This calculator can be applied in various scenarios, such as physics for understanding relationships between force and distance, in economics to analyze cost and quantity relationships, and in
engineering for parameter relationships. It simplifies the process of finding proportionality constants in these fields.
6. Can the calculator handle negative values?
Yes, the calculator can handle both positive and negative values for x and y. The constant of proportionality will reflect the correct relationship regardless of the sign of the input values.
7. Why do I need two pairs of x and y values?
The constant of proportionality is derived from the difference between two points. By using two pairs of x and y values, the calculator can determine the rate at which y changes with respect to x. A
single pair would not provide sufficient information for this calculation.
8. What if my data pairs are not linearly proportional?
This calculator specifically calculates the constant for linear proportionality. If your data pairs are not linearly proportional, the resulting constant may not represent a meaningful relationship
between your variables. Consider using other statistical tools for non-linear relationships.
9. How is the formula for calculating the constant derived?
The formula for calculating the constant of proportionality (k) is derived from the concept of direct variation, where y = kx. By rearranging this equation, the constant k can be calculated as the
difference in y values divided by the difference in x values, or (y2 – y1) / (x2 – x1).
10. Is there any limit on the range of values I can input?
There is no specific limit on the range of values you can input as long as they are numerical. The calculator can handle very large and very small values, including fractions and decimals, as long as
the inputs are valid numerical entries. | {"url":"https://www.onlycalculators.com/statistics/descriptive-statistics/constant-of-proportionality-calculator/","timestamp":"2024-11-09T22:26:14Z","content_type":"text/html","content_length":"238339","record_id":"<urn:uuid:97469d6e-c78e-4b98-a257-db89511552ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00227.warc.gz"} |
Optimal Velocity Function Minimizing Dissipated Energy Considering All Friction in a Position Control System
Yiting Zhu^*, Xuejun Zhu^**, Teruyuki Izumi^*,
and Masashi Kanesaka^*
^*Department of Electronic and Control Systems Engineering, Shimane University, 1060 Nishi-Kawatsu, Matsue, Simane 690-8504, Japan
^**Department of Mechanical Engineering, Ningxia University, 90-5-075 Yinchuan 750021, China
May 23, 2006
October 16, 2006
February 20, 2007
position control, minimum energy, optimal velocity function, trapezoidal velocity, reduction gear
In order to help reduce global warming, the amount of dissipated energy of machines should be decreased. The present paper discusses optimal current and velocity functions that minimize the
dissipated energy in a servo system with friction of all types. The Coulomb friction of a gear in the servo system is represented by the efficiency of the gear and is assumed to be proportional
to the absolute value of the output torque of the motor. Even if the system is nonlinear due to Coulomb friction, an analytical optimal function can be solved by introducing a zero crossing time
t[c], when the input torque of the gear changes from positive to negative. The influence of the viscous friction upon the optimal zero crossing time t[c]^* is examined by simulations. The energy
dissipated with the optimal velocity function is compared to the energy dissipated with a conventional trapezoidal velocity function. The results of the simulations and the experiment indicate
that the optimal velocity function can greatly reduce the amount of energy dissipated when the moment of inertia is large.
Cite this article as:
Y. Zhu, X. Zhu, T. Izumi, and M. Kanesaka, “Optimal Velocity Function Minimizing Dissipated Energy Considering All Friction in a Position Control System,” J. Robot. Mechatron., Vol.19 No.1, pp.
97-105, 2007.
Data files:
1. [1] H. Machida and F. Kobayashi, “An Implementation of the One-Chip PLL/PWM Motor Control System,” T. of the Institute of Electrical Engineers of Japan, Vol.122-C, No.12, pp. 2144-2148, 2002.
2. [2] S. Arimoto, “Robot Dynamics and Control,” Asakura Co., 1995.
3. [3] H. Kojima, “Dynamic Finite Element Analysis of the Position Control System of a Horizontal Flexible Robot Arm with Two Links,” T. of the Japan Society of Mechanical Engineers Series C,
Vol.55, No.513, pp. 2179-2186, 1989.
4. [4] M. Hamaguchi and K. Terashima, “Design Method for Optimum Shape of Container and Velocity Pattern in Liquid Transfer Considering Damping of Sloshing,” Journal of Japan Foundry Engineering
Society, Vol.71, No.5, pp. 307-313, 1999.
5. [5] C. Tai, S. Sakai, and Y. Hori, “Proposal of a Novel Method of Motion Control of Electric Vehicles Utilizing Speed Trajectory Shaping,” P. of the 2002 Japan Industry Applications Society
Conference, No.245, 2002.
6. [6] E. Sergaki and G. Stavrakakis, “Optimal Robot Speed Trajectory by Minimization of the Actuator Motor Electromechanical Losses,” Journal of Intelligent and Robotic Systems, Vol.33, pp.
187-207, 2002.
7. [7] T. Izumi, Y. Yokose, and R. Tamai, “Minimum Energy Path Search for a Manipulator in Consideration of All Non-linear Characteristics by GA and Its Experiments,” T. of the Institute of
Electrical Engineers of Japan, Vol.125-C, No.11, pp. 1751-1757, 2005.
8. [8] Y. Zhu, T. Izumi, and H. Zhou, “Minimization of Dissipated Energy in a Position Control System with Coulomb Friction of Gears,” SICE Annual Conference 2005 in Okayama, pp. 2798-2803,
9. [9] T. Izumi, Y. Zhu, and H. Zhou, “Optimal Velocity Function and Gear Ratio Minimizing Dissipated Energy in Position Control System with Coulomb Friction,” P. of the 2006 JSME Conference on
Robotics and Mechatronics, 2P1-B35, 2006.
10. [10] H. Kanoh, “Theory and Computational Methods in Optimization,” Corona Publishing Co., 1987.
Copyright© 2007 by Fuji Technology Press Ltd. and Japan Society of Mechanical Engineers. All right reserved. | {"url":"https://www.fujipress.jp/jrm/rb/robot001900010097/","timestamp":"2024-11-09T00:11:43Z","content_type":"text/html","content_length":"47264","record_id":"<urn:uuid:503c9b7d-8ef6-4719-b71d-290bf74570ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00810.warc.gz"} |
Prolog Function: Create List, Add First Half, Find Max, Count Elements Less Than Max
Oops, something went wrong. Please try again in a few moments.
% create_list(N, List) - Creates a list of numbers from 1 to N.
% N is the upper limit for the list.
% List is the generated list of numbers from 1 to N.
create_list(N, List) :-
findall(X, between(1, N, X), List).
% add_first_half_to_end(List, Result) - Adds the first half of the list to the end of the list.
% List is the input list.
% Result is the list with its first half added to the end.
add_first_half_to_end(List, Result) :-
length(List, Len),
Half is Len // 2,
append(List, FirstHalf, Result),
append(FirstHalf, List, Result).
% find_max(List, Max) - Finds the maximum element in the list.
% List is the input list.
% Max is the maximum element in the list.
find_max([Max], Max).
find_max([H|T], Max) :-
find_max(T, Max1),
Max is max(H, Max1).
% count_elements_less_than_max(List, Max, Count) - Counts the elements in the list that are less than Max by no more than 10.
% List is the input list.
% Max is the maximum element in the list.
% Count is the count of elements in the list that are less than Max by no more than 10.
count_elements_less_than_max([], _, 0).
count_elements_less_than_max([H|T], Max, Count) :-
H < Max + 11,
count_elements_less_than_max(T, Max, Count1),
Count is Count1 + 1.
count_elements_less_than_max([_|T], Max, Count) :-
count_elements_less_than_max(T, Max, Count).
% Example Usage:
% create_list(10, List), add_first_half_to_end(List, Result), find_max(Result, Max), count_elements_less_than_max(Result, Max, Count).
% Usage examples
% Example 1:
% Create a list of numbers from 1 to 10, add its first half to the end, find the maximum element, and count elements less than max by no more than 10.
% Usage: create_list(10, List), add_first_half_to_end(List, Result), find_max(Result, Max), count_elements_less_than_max(Result, Max, Count).
% This will create a list, modify it, find the max element, and count elements less than max by no more than 10.
% Example 2:
% Create a list of numbers from 1 to 20, add its first half to the end, find the maximum element, and count elements less than max by no more than 10.
% Usage: create_list(20, List), add_first_half_to_end(List, Result), find_max(Result, Max), count_elements_less_than_max(Result, Max, Count).
% This will create a list, modify it, find the max element, and count elements less than max by no more than 10.
% Example 3:
% Create a list of numbers from 1 to 15, add its first half to the end, find the maximum element, and count elements less than max by no more than 10.
% Usage: create_list(15, List), add_first_half_to_end(List, Result), find_max(Result, Max), count_elements_less_than_max(Result, Max, Count).
% This will create a list, modify it, find the max element, and count elements less than max by no more than 10. | {"url":"https://codepal.ai/code-generator/query/ezZ7makf/prolog-function-create-list-add-first-half-find-max-count-elements-less-than-max","timestamp":"2024-11-09T16:18:29Z","content_type":"text/html","content_length":"129211","record_id":"<urn:uuid:c18266ae-b97c-4fc3-94b0-2524bba9df51>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00487.warc.gz"} |
Edit distance with move operations
The traditional edit-distance problem is to find the minimum number of insert-character and delete-character (and sometimes change character) operations required to transform one string into another.
Here we consider the more general problem of a string represented by a singly linked list (one character per node) and being able to apply these operations to the pointer associated with a vertex as
well as the character associated with the vertex. That is, in O (1) time, not only can characters be inserted or deleted, but substrings can be moved or deleted. We limit our attention to the ability
to move substrings and leave substring deletions for future research. Note that O (1) time substring move operation implies O (1) substring exchange operation as well, a form of transformation that
has been of interest in molecular biology. We show that this problem is NP-complete, and that a "recursive" sequence of moves can be simulated with at most a constant factor increase by a
non-recursive sequence. Although a greedy algorithm is known to have poor (a polynomial factor) worst case performance, we present a polynomial time greedy algorithm for non-recursive moves which on
a subclass of instances of a problem of size n achieves an approximation factor to optimal of at most O (log n). The development of this greedy algorithm shows how to reduce moves of substrings to
moves of characters, and how to convert moves of characters to only inserts and deletes of characters.
اللغة الأصلية الإنجليزيّة
الصفحات (من إلى) 380-392
عدد الصفحات 13
دورية Journal of Discrete Algorithms
مستوى الصوت 5
رقم الإصدار 2 SPEC. ISS.
المعرِّفات الرقمية للأشياء
حالة النشر نُشِر - يونيو 2007
منشور خارجيًا نعم
أدرس بدقة موضوعات البحث “Edit distance with move operations'. فهما يشكلان معًا بصمة فريدة. | {"url":"https://cris.ariel.ac.il/ar/publications/edit-distance-with-move-operations-6","timestamp":"2024-11-05T03:03:05Z","content_type":"text/html","content_length":"56787","record_id":"<urn:uuid:18bb57ba-d9e9-4066-a124-95d9cce8c56f>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00313.warc.gz"} |
Hacking the ECI model B-FOCuS V-2FUb/I Rev.B
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• November 07, 2024, 02:57:22 AM
• Welcome, Guest
Author Topic: Hacking the ECI model B-FOCuS V-2FUb/I Rev.B (Read 217970 times)
« Last Edit: February 02, 2012, 08:09:02 AM by roseway »
Interesting and useful. Thank you for the images.
I think these maybe of interest to you
Chipset is a Lantiq VRX268
Excellent stuff!
The Lantiq (was Infineon) VRX268 has a MIPS32 core. The modem is almost certainly running a MIPS-Linux kernel (i.e. GPL'ed source code ). The VDSL2 AFE is the VRX208.
Located due north of the Lantiq CPU is the 64Mbit (8Mbyte) Macronix NOR flash IC. Unusually it could be on a 16-bit bus. [2]
Just west of that NOR flash IC are solder pads for a 7x2 set of header pins.
Those pads are labelled JP2. They almost certainly form the EJTAG test access port (TAP) interface.
The JTAG signals {TMS, TCK, TDI, TDO, TRST} will be found amongst pins {1, 2, 3, 4, 5, 6}
Pins {7, 8, 9} will probably include VCC. A voltmeter will confirm.
Pins {10, 11, 12, 13, 14} are all GND.
Further north of JP2 is JP1. It comprises 4 solder pads. That is likely a UART port running at TTL voltage levels. A serial console can often be obtained through the UART port. It provides a way to
interrupt the bootstrap process.
el cheapo
way to interface a modern PC (with no RS232 port) to the UART interface is with a
clone Nokia DKU5
phone data cable. The clone DKU5 cable costs as little as £1. The cable contains an integral
Prolific Logic PL2303
USB-UART bridge controller. [3] The PL2303 IC performs the voltage shift and packetises the serial bitstream into USB blocks (URBs).
Linux, and maybe Windows, has a kernel device driver for the PL2303. The driver presents the USB device as a dumb serial port. A terminal program like minicom is then used to connect to the router
over the serial port.
And away you go :-)
The board also has 512Mbit (64MBytes) of Samsung DDR2-800 SDRAM [4]
Thanks for posting the photos, uklad. Very interesting!
cheers, a
EDIT: Shrunk huge photo
« Last Edit: July 29, 2012, 04:54:31 AM by asbokid »
Re-instating header pins on a PCB
One trick here is to clamp the board vertically while working on it.
The solder pads need to be cleaned out to expose the thru-holes.
From one side of the board, apply heat to one of the solder pads using a fine soldering iron bit.
Simultaneously, and working from the other side of the PCB, use a desoldering pump (solder sucker) to remove the molten solder from the hole.
Repeat for each thru-hole.
Sometimes one or more of the holes isn't properly drilled out.
If so, use a 1mm HSS drill bit and twist it manually between fingers
Ensure all the holes are clean and free from grease and PCB coating materials.
Install the header pins and solder in place
Job done!
Attached are some photos showing the reinstatement of header pins for JTAG/UART on the PCB of a Huawei HG612.
« Last Edit: January 20, 2012, 04:08:27 AM by asbokid »
Re-instating header pins on a PCB
One trick here is to clamp the board vertically while working on it.
The solder pads need to be cleaned out to expose the thru-holes.
From one side of the board, apply heat to one of the solder pads using a fine soldering iron bit.
Simultaneously, and working from the other side of the PCB, use a desoldering pump (solder sucker) to remove the molten solder from the hole.
Repeat for each thru-hole.
Sometimes one or more of the holes isn't properly drilled out.
If so, use a 1mm HSS drill bit and twist it manually between fingers
Ensure all the holes are clean and free from grease and PCB coating materials.
Install the header pins and solder in place
Job done!
Attached are some photos showing the reinstatement of header pins for JTAG/UART on the PCB of a Huawei HG612.
Lol thanks
Ok update for you..
Top header is a indeed the console header but its running at TTL 3.3v and I don't have a suitable cable
pins seem to be from left to right TX GND VCC RX
I will get a suitable cable and get back to you with the output !!
« Last Edit: January 24, 2012, 05:53:48 PM by uklad »
Sounds good!
Most JTAG cables will work fine, so long as there are generic drivers available for the cables.
It might be helpful to collect some JTAG resources together in this thread for others' benefit.
Discovering JTAG pinouts
Most JTAG cables will work fine in the pinout discovery process, so long as there is a generic driver available for the cable.
Discovering JTAG pinouts on a PCB is a very common problem. For a given board, the size of the problem can be quantified using
Probability Theory
In the worst case scenario, using ‘brute force’ to discover the JTAG pinout means testing every possible permutation of JTAG signal and header pin.
Formally, the JTAG pinout problem is an
challenge. It is described by the notation
is the number of permutations, or ways to choose, an ordered subset of r items from a set of n objects.
In the case of this board, the set of n objects are a set of 14 header pins. From that set of n pins we need to discover the ordered subset of r pins carrying the JTAG signals.
The formula for
is n! / (n-r)! where ! is the factorial symbol, e.g. 7! means (7 x 6 x 5 x 4 x 3 x 2 x 1)
Out of the fourteen header pins on the board, there are six candidate pins. Any of these six pins could potentially carry any of the five JTAG signals {TDO,TDI,TMS,TCK and TRST}.
Here, n is 6 (the number of candidate pins), and r is 5 (the number of JTAG signals).
So nPr = 6! / (6-5)! = 720 permutations.
However, some assumptions can be made which will radically reduce the search space.
One of the JTAG signals (TRST) is optional. TRST resets the JTAG controller when driven low. If we assume that, by default, TRST is pulled up to keep the board out of reset, it can be ignored.
Another JTAG signal (TDO) can be discovered from its floating logic state using an ohmmeter. This is very well explained by Ray “
” Haverfield. [1]
That leaves us with just three JTAG signals to find from a choice of five header pins.
Now the scale of the problem is given by 5!/2 = 60 permutations.
That has already shrunk the search space by more than 90%.
We can now take advantage of another property of the JTAG standard. [2]
A JTAG controller will always return to its reset state when the TMS signal is asserted for five or more ticks of the TCK signal. This is illustrated in the attached diagram of the JTAG state
The bit values {0,1} shown in the diagram represent the transitional states of the TMS (Test Mode Select) signal. For example, to transition the JTAG state machine from the Shift_IR state to the
Exit1_IR state requires TMS to be asserted for one tick of the TCK signal.
It doesn't matter where you start in the JTAG state machine. Asserting TMS while five ticks are clocked into TCK will always see the JTAG controller returned to its Test_Logic_Reset state:
Once a JTAG device is in that reset state, the 32-bit IDCODE is loaded into the JTAG data register. This loading is done automatically. It doesn’t require any instruction to be shifted in on the TDI
Returning to our board. TDO was discovered earlier from its floating logic state. So what this means is that only the TMS and TCK signals need to be found at this stage. TDI can be found later.
By controlling just the TMS and TCK signals from software, the IDCODE value loaded on reset into the data register can be scanned out of the TDO pin. The TDO pin is closely monitored for output that
is consistent with a device IDCODE.
Looking at this again as a combinatorial problem:
The value
remains at 5 since we still have five unknown pins. However,
, the number of signals to discover, is now just 2. These are the TMS and the TCK signals.
So nPr is 5!/3! = 20 permutations.
Using these techniques, the discovery of JTAG pinouts is trivialised.
There are software tools, such as
[2] that can automate the fiddly task of swapping pins during pinout discovery. However, this is rarely necessary. Using the techniques above, the average count of pin-swaps before discovery success
is reduced to a manageable number.
In summary, and using this board as an example, a total of 14 pins are reduced to 6 candidate pins. TDO is discovered with an ohmmeter. TRST is ignored. The discovery of TDI is postponed. Software
(UrJTAG) is used to navigate the JTAG state machine for each permutation of TCK and TMS, chosen from the five remaining pins. Using these shortcuts, the average count of pin-swaps before discovery is
reduced to just 10.
« Last Edit: January 29, 2012, 04:37:05 AM by asbokid »
• Helpful
• Kitizen
• Posts: 2721
Sounds good!
& some people accuse me of being too precise
Serial output on boot
ROM VER: 1.0.5
CFG 01
DDR Access auto data-eye tuning Rev 0.3a
DDR size from 0xa0000000 - 0xa1ffffff
DDR check ok... start booting...
U-Boot 1.0.4 (Oct 18 2010 - 16:20:02)
CLOCK CPU 333M RAM 166M
DRAM: 32 MB
relocate_code start
relocate_code finish.
FLASH MANUFACT: c2
FLASH DEVICEID: cb
Flash: 8 MB
In: serial
Out: serial
Err: serial
Net: fw_addr=0xa0200000
Internal phy(FE) firmware version: 0x0108
vr9 Switch
Type "run flash_flash" to mount root filesystem over flash
Hit 'Esc' key to stop autoboot: 0
## Booting image from active region 2 at b03f0000 ...
Check RSA image magic--OK!
Please type [setenv rsa_check 1] !!!
Image Name: MIPS Linux-2.6.20
Created: 2011-08-09 3:31:37 UTC
Image Type: MIPS Linux Kernel Image (lzma compressed)
Data Size: 3629088 Bytes = 3.5 MB
Load Address: 80002000
Entry Point: 802cd000
Verifying Checksum ... OK
Uncompressing Kernel Image ... OK
No initrd
## Transferring control to Linux (at address 802cd000) ...
## Giving linux memsize in MB, 32
Starting kernel ...
Infineon xDSL CPE VR9
mips_hpt_frequency = 166666666, counter_resolution = 2
Linux version 2.6.20.19
(hyhuang@BSD7.localdomain) (gcc version 3.4.6 (OpenWrt-2.0)) #1 Tue Aug 9 11:27
:46 CST 2011
Active Region: 2
phym = 02000000, mem = 01f00000, max_pfn = 00001f00
Reserving memory for CP1 @0xa1f00000, size 0x00100000
CPU revision is: 00019555
Determined physical RAM map:
User-defined physical RAM map:
memory: 01f00000 @ 00000000 (usable)
Initrd not found or empty - disabling initrd
Built 1 zonelists. Total
Kernel command line: root=/dev/mtdblock2 ro rootfstype=squashfs ip=5.57.33.103:5
.57.33.111::::eth0:on console=ttyS0,115200 ethaddr=5C:33:8E:xx:xxx:xx phym=32M me
m=31M panic=1
1 MIPSR2 register sets available
Primary instruction cache 32kB, physically tagged, 4-way, linesize 32 bytes.
Primary data cache 32kB, 4-way, linesize 32 bytes.
Synthesized TLB refill handler (20 instructions).
Synthesized TLB load handler fastpath (32 instructions).
Synthesized TLB store handler fastpath (32 instructions).
Synthesized TLB modify handler fastpath (31 instructions).
Cache parity protection disabled
Lantiq ICU driver, version 3.0.1, (c) 2001-2010 Lantiq Deutschland GmbH
PID hash table entries: 128 (order: 7, 512 bytes)
Using 166.667 MHz high precision timer.
Dentry cache hash table entries: 4096 (order: 2, 16384 bytes)
Inode-cache hash table entries: 2048 (order: 1, 8192 bytes)
Memory: 28152k/31744k available (2239k kernel code, 3592k reserved, 616k data, 1
56k init, 0k highmem)
Security Framework v1.0.0 initialized
Mount-cache hash table entries: 512
NET: Registered protocol family 16
NET: Registered protocol family 8
NET: Registered protocol family 20
NET: Registered protocol family 2
IP route cache hash table entries: 1024 (order: 0, 4096 bytes)
TCP established hash table entries: 1024 (order: 0, 4096 bytes)
TCP bind hash table entries: 512 (order: -1, 2048 bytes)
TCP: Hash tables configured (established 1024 bind 512)
TCP reno registered
gptu: totally 6 16-bit timers/counters
gptu: misc_register on minor 63
gptu: succeeded to request irq 118
gptu: succeeded to request irq 119
gptu: succeeded to request irq 120
gptu: succeeded to request irq 121
gptu: succeeded to request irq 122
gptu: succeeded to request irq 123
IFX DMA driver, version ifxmips_dma_core.c:v1.0.9
,(c)2009 Infineon Technologies AG
Lantiq CGU driver, version 1.0.9, (c) 2001-2010 Lantiq Deutschland GmbH
Wired TLB entries for Linux read_c0_wired() = 0
squashfs: version 3.2-r2 (2007/01/15) Phillip Lougher
squashfs: LZMA suppport for slax.org by jro
JFFS2 version 2.2. (NAND) (SUMMARY) (C) 2001-2006 Red Hat, Inc.
io scheduler noop registered (default)
ifx_pmu_init: Major 252
Lantiq PMU driver, version 1.1.4, (c) 2001-2010 Lantiq Deutschland GmbH
Lantiq GPIO driver, version 1.2.12, (c) 2001-2010 Lantiq Deutschland GmbH
Infineon Technologies RCU driver version 1.0.6
Lantiq LED Controller driver, version 1.0.4, (c) 2001-2010 Lantiq Deutschland Gm
MEI CPE Driver, Version 1.0.2
<6>(c) Copyright 2009, Infineon Technologies AG
<6>### MEI CPE - MEI CPE - MEI CPE - MEI CPE ###
<6>ttyS0 at MMIO 0xbe100c00 (irq = 105) is a IFX_ASC
Lantiq ASC (UART) driver, version 1.0.5, (c) 2001-2010 Lantiq Deutschland GmbH
RAMDISK driver initialized: 1 RAM disks of 6144K size 1024 blocksize
loop: loaded (max 8 devices)
PPP generic driver version 2.4.2
PPP Deflate Compression module registered
PPP BSD Compression module registered
PPP MPPE Compression module registered
NET: Registered protocol family 24
IFX SWITCH API, Version 0.9.9.5
SWAPI: Registered character device [switch_api] with major no [81]
Switch API: PCE MicroCode loaded !!
Switch Auto Polling value = 0
GPHY FIRMWARE LOAD SUCCESSFULLY AT ADDR : 310000
IFX GPHY driver FE Mode, version ifxmips_vr9_gphy: V0.6 - Firmware: 109
ifx_nor0: Found 1 x16 devices at 0x0 in 16-bit bank
Amd/Fujitsu Extended Query Table at 0x0040
number of CFI chips: 1
cfi_cmdset_0002: Disabling erase-suspend-program due to code brokenness.
[ACTIVE REGION]: 2
RSA_CHECK: 0
squashfsb->s_magic=71736873 SQUASHFS_MAGIC=71736873
ifx_nor0: squashfs filesystem found at 0x4e10a0.
ifx_mtd_init flash0: Using static image partition
Creating 9 MTD partitions on "ifx_nor0":
0x00000000-0x00030000 : "uboot"
0x00030000-0x00040000 : "h/w setting"
0x004e10c0-0x007670c0 : "rootfs"
0x00040000-0x00050000 : "rgdb"
0x00050000-0x003f0000 : "upgrade"
0x003f0000-0x00790000 : "upgrade2"
0x00790000-0x007f0000 : "btagent"
0x00000000-0x00800000 : "flash"
0x00000000-0x00800000 : "<NULL>"
Lantiq MTD NOR driver, version 1.0.4, (c) 2001-2010 Lantiq Deutschland GmbH
Registered led device: broadband_led
Registered led device: internet_led
Registered led device: ledc_8
Registered led device: ledc_9
Registered led device: ledc_10
Registered led device: ledc_11
Registered led device: wps_led
Registered led device: ledc_13
Registered led device: ledc_14
Registered led device: usb2_link_led
Registered led device: ledc_16
Registered led device: ledc_17
Registered led device: usb1_link_led
Registered led device: fxo_act_led
Registered led device: internet_red_led
Registered led device: voip_led
Registered led device: warning_led
Registered led device: ledc_23
Lantiq LED driver, version 1.0.15, (c) 2001-2010 Lantiq Deutschland GmbH
Netfilter messages via NETLINK v0.30.
nf_conntrack version 0.5.0 (248 buckets, 1984 max)
GRE over IPv4 tunneling driver
ip_tables: (C) 2000-2006 Netfilter Core Team
TCP cubic registered
NET: Registered protocol family 1
NET: Registered protocol family 17
Bridge firewalling registered
NET: Registered protocol family 8
atmpvc_init() failed with -17
802.1Q VLAN Support v1.8 Ben Greear <greearb@candelatech.com>
All bugs added by David S. Miller <davem@redhat.com>
Time: MIPS clocksource has been installed.
VFS: Mounted root (squashfs filesystem) readonly.
Freeing unused kernel memory: 156k freed
init started: BusyBox v1.00 (2011.08.09-03:28+0000) multi-call binary
Algorithmics/MIPS FPU Emulator v1.5
Starting mdev ...
Mounting proc and var ...
JFFS2 notice: (226) jffs2_build_xattr_subsystem: complete building xattr subsyst
em, 0 of xdatum (0 unchecked, 0 orphan) and 0 of xref (0 dead, 0 orphan) found.
Start xmldb ...
[/etc/scripts/misc/profile.sh] init ...
[/etc/scripts/misc/profile_action.sh] get ...
[/etc/scripts/misc/defnodes.sh] ...
SH [/etc/defnodes/S10syncnodes.sh] ...
[/etc/defnodes/S10syncnodes.sh] ...
SH [/etc/defnodes/S11setext.sh] ...
[/etc/defnodes/S11setext.sh] ...
PHP [/etc/defnodes/S12setnodes.php] ...
SH [/etc/defnodes/S13setext.sh] ...
[/etc/defnodes/S13setext.sh] ...
PHP [/etc/defnodes/S14setnodes.php] ...
PHP [/etc/defnodes/S16features.php] ...
SH [/etc/defnodes/S19setext.sh] ...
PHP [/etc/defnodes/S20setnodes.php] ...
SH [/etc/defnodes/S20upnp_igd.sh] ...
SH [/etc/defnodes/S21upnp_wfa.sh] ...
SH [/etc/defnodes/S22setext.sh] ...
PHP [/etc/defnodes/S40brand.php] ...
[/etc/scripts/misc/defnodes.sh] Done !!
[/etc/templates/timezone.sh] ...
[/etc/templates/logs.sh] ...
[/var/run/logs_run.sh] ...
ifxmips_ppa_datapath_vr9_e5: module license 'unspecified' taints kernel.
Loading D5 (MII0/1) driver ......
xuliang: warning NONE
PPE datapath driver info:
Version ID: 128.3.3.1.0.0.1
Family : N/A
DR Type : Normal Data Path | Indirect-Fast Path
Interface : MII0 | MII1
Mode : Routing
Release : 0.0.1
PPE 0 firmware info:
Version ID: 7.1.5.1.0.33
Family : VR9
FW Type : Standard
Interface : MII0/1 + PTM
Mode : reserved - 1
Release : 0.33
PPE 1 firmware info:
Version ID: 7.2.1.6.1.12
Family : VR9
FW Type : Acceleration
Interface : MII0 + MII1
Mode : Bridging + IPv4 Routing
Release : 1.12
PPA API --- init successfully
Init VDSL Driver ...
- VDSL -
- llcs loading!!! -
- loading drv_ifxos.ko -
strings: not found
IFXOS, Version 1.5.11
<6>(c) Copyright 2007, Infineon Technologies AG
<6>### IFXOS - IFXOS - IFXOS - IFXOS ###
- loading drv_dsl_cpe_api.ko
- loading dsl_cpe_api (drv_dsl_cpe_api.ko device) driver -
Lantiq CPE API Driver version: DSL CPE API V4.6.3.5-pd3
Predefined debug level: 3
- create device nodes for dsl_cpe_api device driver -
- execute vdsl_cpe_control
[: missing ]
IFXOS - User Thread Startup <tcpmsg>, TID 1026 (PID 609) - ENTER
IFXOS - User Thread Startup <tcpcli>, TID 2051 (PID 610) - ENTER
IFXOS - User Thread Startup <evnthnd>, TID 3076 (PID 612) - ENTER
IFXOS - User Thread Startup <tPipe_0>, TID 4101 (PID 613) - ENTER
IFXOS - User Thread Startup <tPipe_1>, TID 5126 (PID 614) - ENTER
eth0: change MAC from 00:20:DA:86:23:74 to 5C:33:8E:xx:xx:xx
setup layout ...
[/etc/scripts/layout.sh] [start] ...
[/var/run/layout_start.sh] ...
Start modem layout ...
device eth0 entered promiscuous mode
br0: port 1(eth0) entering learning state
br0: topology change detected, propagating
br0: port 1(eth0) entering forwarding state
[/etc/templates/cfm/cfm.sh] [restart] ...
[/var/run/cfm_start.sh] ...
Enable ALPHA CFM ...
ENTER - Kernel Thread Startup <autbtex>
<7>ENTER - Kernel Thread Startup <pmex_ne>
<7>ENTER - Kernel Thread Startup <pmex_fe>
[/etc/init.d/S03config.sh] done!
start LAN ...
[/etc/templates/lan.sh] [start] ...
[/var/run/lan_start.sh] ...
Start LAN ( br0/192.168.168.168/255.255.255.0)...
start BT Switch configurations ...
start alphaLogd
[/etc/templates/logd.sh] ...
[/var/run/logd_start.sh] ...
Starting logd ...
start Flash Agent ...
>>> ALPHA Log:
/bin/alphaLogd: create logd_ipc(3) OK !
[/etc/templates/flash_agent.sh] [start] ...
[/var/run/flash_agent_start.sh] ...
>>> ALPHA Flash Agent:
16:00:17 FLASHAGENT: Create fa_r_fa_ipc(4) OK !
start BTAgent ...
Starting BTAgent
library_load: start plugin_source/libalpha2.so
library_load: success
library_load: start plugin_source/libbtagent.so
library_load: success
File Path is /BTAgent/rw/btagent.conf
rw config file exists
Versions match
library_load: start plugin_source/libfwm.so
library_load: success
library_load: start plugin_source/liblogger.so
library_load: success
library_load: start plugin_source/libprobe.so
library_load: success
library_load: start plugin_source/librsa.so
library_load: success
main: Loaded source plugins
library_load: start plugin_transport/libsec.so
library_load: success
main: Loaded transport plugins
library_load: start plugin_parse/libxml.so
library_load: success
main: Loaded parse plugins
GPIO 18 set to 0
GPIO 17 set to 1
GPIO 16 set to 1
GPIO 6 set to 1
start alphaHousekeeper
[/etc/templates/housekeeper.sh] [start] ...
[/var/run/housekeeper_start.sh] ...
Starting housekeeper ...
BBU Status: Status Change
BBU Status: Adapter Mode
- presented Inventory information
nReturn=0 nDirection=0 G994VendorID=(B5,00,49,46,54,4E,53,26) SystemVendorID=(58
,20,45,43,49,4C,20,20) VersionNumber=(35,2E,33,2E,32,2E,36,2E,31,2E,36,20,20,20,
20,20) SerialNumber=(45,35,43,33,33,38,45,38,34,38,39,44,42,20,20,20,20,20,20,20
,20,20,20,20,20,20,20,20,20,20,20,20) SelfTestResult=0 XTSECapabilities=(00,00,0
[/etc/templates/wan_vlan.sh] [start] ...
[/var/run/wan_vlan_start.sh] ...
Start CPE SPECIFIC WAN VLAN ...
VLAN Enable...
Added VLAN with VID == 301 to IF -:ptm0:-
Set egress mapping on device -:ptm0.301:- Should be visible in /proc/net/vlan/pt
Set egress mapping on device -:ptm0.301:- Should be visible in /proc/net/vlan/pt
Set egress mapping on device -:ptm0.301:- Should be visible in /proc/net/vlan/pt
Set egress mapping on device -:ptm0.301:- Should be visible in /proc/net/vlan/pt
Set egress mapping on device -:ptm0.301:- Should be visible in /proc/net/vlan/pt
Set egress mappingptm0.301: Setting MAC address to 5c 33 8e xx xx xx.
VLAN (ptm0.301): Underlying device (ptm0) has same MAC, not checking promisciou
s mode.
on device -:ptm0.301:- Should be visible in /proc/net/vlan/ptm0.301
Set egress mapping on device -:ptm0.301:- Should be visible in /proc/net/vlan/pt
Set egress mapping on device -:ptm0.301:- Should be visible in /proc/net/vlan/pt
Added VLAN with VID == 101 to IF -:ptm0:-
Added VLAN with VID == 102 to IF -:ptm0:-
Set egress mapping on device -:ptm0.101:- Should be visible in /proc/net/vlan/pt
Set egress mapping on device -:ptm0.101:- Should be visible in /proc/netptm0.101
: add 01:00:5e:00:00:01 mcast address to master interface
Set egrptm0.102: add 01:00:5e:00:00:01 mcast address to master interface
ess mapping on device -:ptm0.102:- Should be visible in /proc/net/vlan/ptm0.102
Added VLAN with VID == 101 to IF -:eth0:-
device eth0 left promiscuous mode
br0: port 1(eth0) entering disabled state
Added VLAN with VID == 102 to IF -:eth0:-
eth0.102: dev_set_promiscuity(master, 1)
device eth0 entered promiscuous mode
device eth0.102 entered promiscuous mode
br0: port 1(eth0.101) entering learning state
br0: topology change detected, propagating
br0: port 1(eth0.101) entering forwarding state
DSL[00]: WARNING - SRA not supported by the FW
br0: port 2(eth0.102) entering learning state
br0: topology change detected, propagating
br0: port 2(eth0.102) entering forwarding state
ifx_ppa_init - init succeeded
VID 0 remove is enabled
[/etc/init.d/S10system.sh] done!
rcS done!
- presented Inventory information
- presented Inventory information
nReturn=0 nDirection=0 G994VendorID=(B5,00,49,46,54,4E,53,26) SystemVendorID=(58
,20,45,43,49,4C,20,20) VersionNumber=(35,2E,33,2E,32,2E,36,2E,31,2E,36,20,20,20,
20,20) SerialNumber=(45,35,43,33,33,38,45,38,34,38,39,44,42,20,20,20,20,20,20,20
,20,20,20,20,20,20,20,20,20,20,20,20) SelfTestResult=0 XTSECapabilities=(00,00,0
« Last Edit: January 24, 2012, 07:39:46 PM by uklad »
I interrupted the boot process and listed all images found in flash
ROM VER: 1.0.5
CFG 01
DDR Access auto data-eye tuning Rev 0.3a
DDR size from 0xa0000000 - 0xa1ffffff
DDR check ok... start booting...
U-Boot 1.0.4 (Oct 18 2010 - 16:20:02)
CLOCK CPU 333M RAM 166M
DRAM: 32 MB
relocate_code start
relocate_code finish.
FLASH MANUFACT: c2
FLASH DEVICEID: cb
Flash: 8 MB
In: serial
Out: serial
Err: serial
Net: fw_addr=0xa0200000
Internal phy(FE) firmware version: 0x0108
vr9 Switch
Type "run flash_flash" to mount root filesystem over flash
Hit 'Esc' key to stop autoboot: 0
VR9 # help
? - alias for 'help'
askenv - get environment variables from stdin
base - print or set address offset
bootm - boot application image from memory
bootp - boot image via network using BootP/TFTP protocol
cmp - memory compare
cp - memory copy
crc32 - checksum calculation
echo - echo args to console
erase - erase FLASH memory
flinfo - print FLASH memory information
go - start application at address 'addr'
help - print online help
imls - list all images found in flash
loop - infinite loop on address range
md - memory display
mm - memory modify (auto-incrementing)
mtest - simple RAM test
mw - memory write (fill)
nm - memory modify (constant address)
ping - send ICMP ECHO_REQUEST to network host
printenv- print environment variables
protect - enable or disable FLASH write protection
rarpboot- boot image via network using RARP/TFTP protocol
reset - Perform RESET of the CPU
run - run commands in an environment variable
saveenv - save environment variables to persistent storage
setenv - set environment variables
tftpboot- boot image via network using TFTP protocol
upgrade - forward/backward copy memory to pre-defined flash location
version - print monitor version
VR9 # imls
Have RSA magic !!!
Image at B0051060:
Image Name: MIPS Linux-2.6.20
Created: 2011-02-14 6:44:17 UTC
Image Type: MIPS Linux Kernel Image (lzma compressed)
Data Size: 3624992 Bytes = 3.5 MB
Load Address: 80002000
Entry Point: 802cd000
Verifying Checksum ... OK
Have RSA magic !!!
Image at B03F1060:
Image Name: MIPS Linux-2.6.20
Created: 2011-08-09 3:31:37 UTC
Image Type: MIPS Linux Kernel Image (lzma compressed)
Data Size: 3629088 Bytes = 3.5 MB
Load Address: 80002000
Entry Point: 802cd000
Verifying Checksum ... OK
VR9 #
Excellent stuff, uklad! You're well on the way to cracking it.
Hopefully, the contents of that 8MByte
NOR flash can be (hex) dumped over the serial line using the
(memory display) command in the CLI of the
What does the
(flash info) command say about the flash device, and its composition?
The definitive book on
MIPS Linux
is Dominic Sweetman's
See MIPS Run
(2nd ed). [2]
Sweetman gives a particularly good treatment to the address space, memory mapping and the memory management unit (the TLB) in the MIPS.
Let us know how you get on! Lots of people will be keenly following your trail-blazing work!
cheers, a
« Last Edit: June 19, 2012, 01:02:10 AM by asbokid »
Ok one quick question what address range do I need to dump ?
Also I did not mention I could login to the unit on the UART console, username and pass where admin admin :0)
Ok one quick question what address range do I need to dump ?
What does the uboot command
(flash info) reveal?
Also I did not mention I could login to the unit on the UART console, username and pass where admin admin :0)
Nice one! What are the pinouts for the UART header pins? Did you use a cable with a pl2303 bridge?
cheers, a
« Last Edit: January 24, 2012, 07:01:35 PM by asbokid »
This thread is getting quite interesting and, er, tasty. Excellent work to date.
This thread is getting quite interesting and, er, tasty. Excellent work to date.
LOL more to come... | {"url":"https://forum.kitz.co.uk/index.php/topic,10635.msg209378.html","timestamp":"2024-11-07T02:57:22Z","content_type":"application/xhtml+xml","content_length":"91328","record_id":"<urn:uuid:5c9402c2-e6ca-4c2b-82ce-5ec844dc4dd8>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00232.warc.gz"} |
[Worker Proposal] BitAsset PhD Research project - University of Colorado
I believe this WP will be more productive for this purpose; threads on bitsharestalk are usually pretty quiet.
This worker proposal could provide new smartcoin setting profiles for multiple theoretical assets & that's valuable information to me and probably others.If you are unsure how to set parameters, you
should open a separate topic.
This worker proposal could provide new smartcoin setting profiles for multiple theoretical assets & that's valuable information to me and probably others.
I don't see how this worker would help the value of BTS.Not only you see no value in this worker, but also the majority of the voters see now value in this worker proposal. We already know:MCR = 2.0
-> low liquidity -> unsafeMCR = 1.01 -> unlimited liquidity -> 100% safeOur problem is now liquidity and BTS price and not to set the parameter for MCR/MSSR!My proposal does solve the problem by
increasing the liquidity and the demand for BTS:Margin position liquidity pool - https://github.com/bitshares/bsips/issues/182
I don't see how this worker would help the value of BTS.
How do you *know* (instead of believe)? E. g. I believe that "MCR=1.01 -> 100% safe" is obviously untrue, because a drop of 2% in BTS price will lead to undercollateralization and destroy the
peg.Because the unlimited liquidity can always increase the CR faster, than the price drop!
How do you *know* (instead of believe)? E. g. I believe that "MCR=1.01 -> 100% safe" is obviously untrue, because a drop of 2% in BTS price will lead to undercollateralization and destroy the peg.
We already know:MCR = 2.0 -> low liquidity -> unsafeMCR = 1.01 -> unlimited liquidity -> 100% safe
I invite everyone to look at how the top research institutions are attempting to establish cryptocurrency and DLT as an academic field and most importantly why. The MIT Media Lab are hosting the
Cryptoeconomic Systems "Field Building Summit" 5-6 OCT [1]. Page 2 sets up the Problem and page 4 addresses the Solution. Companies are most willing to build on rigorously researched system designs.
Lacking the establishment of what is fact and what remains an open question, this field will fail to advance to its potential. The same is true for the BitShares Protocol. As a member of the Core
Team tasked with implementing the protocol, I can assure you that formal research and peer review will lead to more robust designs and result in implementation of the DeFi DeFi tools businesses will
build upon and investors are willing to participate within.[1] https://assets.pubpub.org/pbghqdg6/11562785633291.pdf
A question back to you: How do you know that 1.5 and 1.01 are the sweet spot? Why not 1.2 and 1?
my question is, why? why do you suppose 1.5 is the lowest, why do you know a lower MCR cannot be better?
@biophil: 1. How can your theoretical model be applied to a market with external parameters?
Support our research efforts to improve BitAsset price-pegging! Vote for worker 1.14.204 "201907-uccs-research-project."
I hope that your worker proposal can be reactivated soon, I support your research worker proposal 👍
Hi Dr. brown,i used to play with anytime algorithm for my graduate research in boston about 15 years ago. My supervisor Dr. eugene Eberbach told me my research was for academic only . there were not
robotics that can use our research since none know where AI will go . there will be the same situation you are facing as i faced 10 years ago in your upcoming bts research project. My advice is you
can keep on taking this project and try to seeking fund form other source due to you won't be able to do any real help to bts system and more importantly (lol) bts'e price is at less than 3 cent. Bts
system won't have enough money for non-profit research ......Best regardsTwitterI am pro research (0.5 year), but why should we finance a student and travel costs?The worker already has 15k $ and it
is unclear, which benefits the community gets from the worker.One solution is here with the settlement fund: https://github.com/bitshares/bsips/issues/182(Time to respond to my BSIP) The main problem
BTS has, is the deficit spending by overpriced workers, which dump BTS and don't bring any real value to the ecosystem.Bench, maybe your mind is made up and an explanation wouldn't change it, but
here you go:In the US university system, practically all research is done by advanced PhD students; those students are directed by professors and paid a small stipend by external funding sources. The
research is the student.I wrote the Decentralized 2019 paper on my own time because I think the BitAsset problem is a very interesting one and I was making a good-faith effort to launch the project
before funding arrived. Unfortunately, my own time is very limited, and unless an external funding source "buys" some of my official University time, I can't devote the many many hours that it will
take to do justice to this problem. This is where the student comes in: he will spend the bulk of the time on the project, and he must be the first one to get paid -- I won't take a cent for myself
from this worker unless he is funded full-time. Half-time funding isn't really appropriate for a PhD student.To put this another way: your BSIP deserves more than my quick comments, but my in-depth
comments will take time. Nonetheless, give me a few minutes and I will go write some quick comments on your BSIP.
I am pro research (0.5 year), but why should we finance a student and travel costs?The worker already has 15k $ and it is unclear, which benefits the community gets from the worker.One solution is
here with the settlement fund: https://github.com/bitshares/bsips/issues/182(Time to respond to my BSIP) The main problem BTS has, is the deficit spending by overpriced workers, which dump BTS and
don't bring any real value to the ecosystem.Bench, maybe your mind is made up and an explanation wouldn't change it, but here you go:In the US university system, practically all research is done by
advanced PhD students; those students are directed by professors and paid a small stipend by external funding sources. The research is the student.I wrote the Decentralized 2019 paper on my own time
because I think the BitAsset problem is a very interesting one and I was making a good-faith effort to launch the project before funding arrived. Unfortunately, my own time is very limited, and
unless an external funding source "buys" some of my official University time, I can't devote the many many hours that it will take to do justice to this problem. This is where the student comes in:
he will spend the bulk of the time on the project, and he must be the first one to get paid -- I won't take a cent for myself from this worker unless he is funded full-time. Half-time funding isn't
really appropriate for a PhD student.To put this another way: your BSIP deserves more than my quick comments, but my in-depth comments will take time. Nonetheless, give me a few minutes and I will go
write some quick comments on your BSIP.
I am pro research (0.5 year), but why should we finance a student and travel costs?The worker already has 15k $ and it is unclear, which benefits the community gets from the worker.One solution is
here with the settlement fund: https://github.com/bitshares/bsips/issues/182(Time to respond to my BSIP) The main problem BTS has, is the deficit spending by overpriced workers, which dump BTS and
don't bring any real value to the ecosystem. | {"url":"https://bitsharestalk.org/index.php?topic=28542.0","timestamp":"2024-11-05T01:00:20Z","content_type":"application/xhtml+xml","content_length":"87218","record_id":"<urn:uuid:296c83f9-e1a6-4383-a196-891f33bb1635>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00498.warc.gz"} |
How much joule is a real gun?
How much joule is a real gun?
Maths for Humans: Linear, Quadratic & Inverse Relations
Firearm Caliber Muzzle energy (joules)
air gun PCP .22 40+
pistol .177 159
pistol .357 Magnum .177 873
rifle .30 2,000
How much Joules is a bullet?
With this, I can calculate the kinetic energy of one bullet. That’s almost 2000 Joules. If you move a textbook from the floor to a table, that takes about 10 Joules of energy. So a bullet has
significantly more energy—but is it enough to be useful?
How many joules is a shotgun?
However, when a shotgun has a rifled barrel, it is considered a rifle, and it becomes legal for hunting roe deer, minimum caliber 5.56 mm and 980 joules at a 100 meters, and deer and wild boar,
minimum caliber 6.5 mm and 2200 joules at 100 meters.
How many joules is an air gun?
Air guns with both a muzzle velocity greater than 152.4 meters per second (500 feet per second) and a muzzle energy greater than 5.7 joules (4.2 foot-pounds) are firearms for purposes of both the
Firearms Act and the Criminal Code.
How many joules is a 9mm bullet?
Typical muzzle energies of common firearms and cartridges
Cartridge Muzzle energy
ft-lbf joules
9 mm Luger 350 470
.45 Colt 370 500
.45 GAP 400 540
How many joules does it take to break a skull?
The energy necessary for the resultant fractures was found to be between 80 and 100 Joules (J), an energy range far above the fracture threshold of the human skull of 14.1 to 68.5 J. The post-mortem
analysis and interpretation of blunt trauma in homicide victims may be a complex task for forensic pathologists.
How many joules is a 9mm round?
The 9×19mm Parabellum (also known as 9mm Parabellum or 9mm Luger) is a rimless, tapered firearms cartridge….9×19mm Parabellum.
Bullet mass/type Velocity Energy
7.45 g (115 gr) Federal FMJ 1,180 ft/s (360 m/s) 355 ft⋅lbf (481 J)
8.04 g (124 gr) Federal FMJ 1,150 ft/s (350 m/s) 364 ft⋅lbf (494 J)
How many joules is a punch?
The kinetic energy of a punch varies greatly. While beginners might only be able to muster a measly 37.5 joules of energy, experts can deliver over 400 joules, an amount roughly equal to getting shot
by a handgun!
How many joules is a 12 gauge gun?
The key is how the bore is measured. Ordinary 12 gauge shotguns may fall under section 96 as well, that of being capable of exceeding the limit of 10,000 joules.
How many joules does a 338 Lapua produce?
Muzzle velocity is dependent on barrel length, seating depth, and powder charge, and varies from 880 to 915 m/s (2,890 to 3,000 ft/s) for commercial loads with 16.2-gram (250 gr) bullets, which
corresponds to about 6,525 J (4,813 ft⋅lbf) of muzzle energy.
How many joules is a 9mm?
Can you shoot an air rifle in your backyard NZ?
What are the rules/regulations about using air weapons at home? It is quite legal to shoot air pistols and air rifles in your section or house; PROVIDING: You are over 18, or have a gun license, or
someone over 18 supervises the shooting.
What does joules stand for in an airsoft gun?
Joules is the derived International System unit of kinetic energy. Given everything the same in your airsoft gun (compression, spring, barrel length, etc), FPS is relative to the BB weight you are
using but the kinetic energy it produces is always the same.
How many joules are needed to penetrate the human skull?
About 72 J are required to penetrate the human skull and 250 J is absorbed by the human head in perforation by a 6 mm steel ball (6). How much force is a joule? One joule is defined as the amount of
work done when a force of one newton is exerted through a distance of one meter.
How many joules are in an average lightning bolt?
With an average bolt of lightning striking from cloud to ground containing roughly one billion ( 1,000,000,000 ) joules of energy, that is a lot of power in every lightning bolt! How much voltage
does a house use? Most houses today have two 110 volt wires and one neutral wire running into the house from the local distribution system.
How many joules are in a 20 gram BB?
As an example for the above formula; a .20 gram bb is moving at 328 feet per second [.5 * (.20 grams / 1000 kilograms) x (328 feet per second x .3048 meters per second) ^2 is equal to 0.999488 or 1
Joule. Now enough of this boring geeky stuff. | {"url":"https://short-question.com/how-much-joule-is-a-real-gun/","timestamp":"2024-11-10T17:25:20Z","content_type":"text/html","content_length":"139000","record_id":"<urn:uuid:c7b2cade-a648-4270-aa67-62ba4e8abb1e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00513.warc.gz"} |
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Real-Time Correction of a Laser Beam Wavefront Distorted by an Artificial Turbulent Heated Airflow
Institute of Geosphere Dynamics (IDG RAS), 119334 Moscow, Russia
Author to whom correspondence should be addressed.
Submission received: 15 March 2022 / Revised: 9 May 2022 / Accepted: 12 May 2022 / Published: 17 May 2022
This paper presents a FPGA-based closed-loop adaptive optical system with a bimorph deformable mirror for correction of the phase perturbation caused by artificial turbulence. The system’s operating
frequency of about 2000 Hz is, in many cases, sufficient to provide the real-time mode. The results of the correction of the wavefront of laser radiation distorted by the airflow formed in the
laboratory conditions with the help of a fan heater are presented. For detailed consideration, the expansion of the wavefront by Zernike polynomials is used with further statistical analysis based on
the discrete Fourier transform. The result of the work is an estimation of the correction efficiency of the wavefront distorted by the turbulent phase fluctuations. The ability of the bimorph
adaptive mirror to correct for certain aberrations is also determined. As a result, it was concluded that the adaptive bimorph mirrors, together with a fast adaptive optical system based on FPGA, can
be used to compensate wavefront distortions caused by atmospheric turbulence in the real-time mode.
1. Introduction
Laser radiation, propagating in the Earth’s atmosphere and beyond, allows us to solve the following tasks:
• Crypto-protected information transmission [
• Organization of the optical communication channels in free space [
• Recharging batteries of drones [
] and low-orbit satellites [
• Destruction of space debris [
• Therefore, on.
It is known [
] that air layers with different temperatures lead to the formation of turbulent refractive index changes along the propagation of the radiation. Passing through such layers, the laser beam wavefront
(WF) acquires additional phase incursions, which leads to the degradation of the beam as it propagates from the source to the receiver. This limits the scope of application of the laser systems
operating in a real atmosphere. One of the ways to solve this problem is to use an adaptive optical system (AOS) that is capable of compensating for the phase nonuniformity of the wavefront in real
At the same time, the system should have sufficient performance. As shown, for example, in [
], the frequency of phase fluctuations caused by atmospheric turbulence rarely exceeds 100 Hz. To compensate for such aberrations, a discrete AOS with a correction frequency of at least 1000 Hz
(frames per second) is required. It is quite difficult to provide such a high stable performance using a conventional PC, since in addition to measuring the wavefront and calculating the voltages
vector, the system should transmit information to the control unit of the deformable mirror. It is much more efficient to use a FPGA for these purposes, in which a full cycle of wavefront correction
will be implemented. The use of FPGAs will allow you to achieve a frequency that will be sufficient to correct the wavefront in real time.
It should be noted that in all previously mentioned tasks, the use of an AOS is potential, and in each case, a separate study of its applicability is required. In particular, it is necessary to solve
different extra problems such as obtaining a reference signal, combating scintillations, etc.
Currently, there is a huge interest in this topic in the world. For example, in the article [
], a 9 km horizontal maritime links experiment was performed through adaptive correction. In the article [
], the issues of laser radiation correction without the use of a wavefront sensor on a 2.3 km long path with subsequent conjugation with fiber are considered. The article [
] describes an attempt to use adaptive correction in relation to the global-scale optical clock network to improve the residual instability along the 113 km path.
Before starting experiments on an open-air route, in our case, it was advisable to investigate the capabilities of the system in laboratory conditions. For this purpose, a laboratory setup was
assembled, where the heated airflow from the fan heater acted as a source of distortion of the wavefront. Fourier analysis [
] applied to the data coming from the Shack–Hartmann wavefront sensor (WFS) [
] showed [
] that the spectral power density of the turbulent airflow is close to Kolmogorov’s within the bandwidth of about 60 Hz. This parameter indicates the proximity of the laboratory conditions to the
real atmosphere.
The next step in the research was the use of the expansion of the wavefront by Zernike polynomials [
]. This work is a continuation of the studies of the wavefront aberrations caused by the heat flow of the air described in [
]. Now, a similar analysis was carried out, but in the case of correction of the wavefront of laser radiation using an AOS system operating at the frequency of 2000 Hz. As a result, the efficiency of
the system was evaluated, as well as the ability of the used bimorph mirror to correct for the specific aberrations.
2. The Fast Adaptive Optical System
The AOS includes a wavefront corrector (deformable mirror), a wavefront sensor and a control system. To achieve the specified correction speed in the experiments, a system controlled by the FPGA was
2.1. Deformable Mirror
A key element of any AOS is a wavefront corrector (WFC). It determines the ability of the system to compensate for specific aberrations, and also affects the performance of the system as a whole.
When choosing the WFC, the previously measured Fried parameter [
] was taken into account, which in experiments turned out to be equal to 10 mm [
]. The measurements were carried out according to the method described in [
], where the Fried parameter was determined from the mutual dispersion of the oscillations of two focal spots of the lens array of the Shack–Hartmann wavefront sensor. The focal spots were chosen to
be centrally symmetric with respect to the center of the beam. To obtain a more reliable result, several pairs of points were selected and the result of calculating the Fried parameter was averaged
over these calculations.
Based on the values of the Fried parameter, it was proposed to use a bimorph mirror [
] as a WFC. This mirror has sufficient speed. The frequency of the first resonance is 8.3 kHz; the amplitude-phase frequency response (Bode diagram) is shown in
Figure 1
. The phase-frequency response of the mirror becomes equal to 90 degrees at a frequency of about 8.2 kHz, while the peak of the amplitude resonance is at 8.3 kHz.
The structure of the electrodes consists of three 8 mm wide rings, which should be enough to correct for the WF with the Fried parameter of 10 mm. The photo of the mirror and the structure of the
electrodes are shown in
Figure 2
Table 1
shows the main characteristics of the mirror.
2.2. Shack–Hartmann Wavefront Sensor
To ensure the system operating frequency of the 2000 Hz, a wavefront sensor based on a fast camera was used. The performance improvement was achieved by using a part of the image. Thus, in the
experiments we used a resolution of 480 × 480 pixels, which allowed us to increase the frame rate to 4000 Hz. The main parameters of the wavefront sensor are presented in
Table 2
2.3. Adaptive Optical System Control Loop
To ensure the fast operation of the AOS, the control loop was made using FPGA. The system requires (1) preloading of a reference—a set of coordinates of WFS focal spots [
]—to which the coordinates of the real focal spots will be pulled up using WFC; and (2) preloading of WFC response functions. Based on the analysis of the image coming from the WFS, the FPGA
calculates a vector of corrective voltages, which is then applied to the mirror electrodes. To achieve a high speed of operation in the system, a phase-conjugated algorithm was used.
Since the bimorph deformable mirror cannot reproduce tilts, virtual tilts response functions are introduced into the system to exclude them. In fact, the voltages are also calculated for tilts, but
they are not applied anywhere.
In the experiments, the AOS operated with a correction frequency of 2000 Hz (frames per second) in the closed-loop mode. To obtain such a high-speed performance, FPGA Arria V GZ processed the image
bytes coming from the WFS camera ‘on the fly’ (as they arrive from the camera), which made it possible to calculate the vector of control voltages by the end of the frame reception. Since the sensor
camera operated at a frequency of 4000 Hz, it took 250 microseconds. One hundred and fifty microseconds were required to transmit the voltage vector to the adaptive mirror Control Unit, fifty
microseconds to load the DAC, and the pause required for the mirror to change its shape lasted fifty microseconds. The internal structure of the FPGA is shown in
Figure 3
, and the corresponding time diagram of its operation is shown in
Figure 4
3. Experimental Setup
To correct for the WF, a laboratory installation was used, as shown in
Figure 5
The source of the radiation is a laser diode coupled to an optical fiber. The wavelength is chosen as 650 nm to facilitate the alignment. The collimating lens forms a parallel beam with a diameter of
50 mm. A fan heater is installed on the path of propagation of the laser beam, the heated airflow of which crosses the laser beam, creating the turbulence. A laser beam with a distorted wavefront
hits a WFC-bimorph mirror and is reflected from it in the direction of the WFS. A flat mirror installed between the WFC and the WFS is used to reduce the dimensions of the installation. Part of the
beam in front of the WFS branches off to the far-field indicator formed by a long-focus lens and a CMOS camera with a small pixel size (3.75 microns) for a more detailed visualization of the image.
The operation of the system is controlled by the FPGA, which performs all the functions necessary for the correction of the wavefront: it receives an image from the WFS camera, calculates the vector
of correcting voltages and transmits this vector to the WFC amplifier unit (Mirror CU). The PC in this configuration performs only the functions of controlling the operation of the FPGA and the
functions of monitoring the correction process.
4. Processing the Results of the Experiment
Before starting the work, studies of distortions of the wavefront of the laser beam caused by the influence of a heated air stream were carried out.
Figure 6
shows the spectral power density of the process obtained on the basis of statistical processing of a series of recorded fluctuations in the coordinates of the focal spot of the WFS lens array. The
straight line f
corresponds to the Kolmogorov spectrum. The sampling duration of the focal spot coordinates was 10 s (at a frequency of 2000 Hz, 20,000 samples were recorded), which provided a resolution along the
frequency axis of 0.1 Hz.
The quality control of the correction was carried out on the far field image [
]. With the specified parameters of the laboratory setup (beam diameter of 50 mm, lens focus length of the far field zone indicator of 1 m and radiation wavelength of 650 nm), the diffraction-limited
diameter of the spot in the lens focus was about 32 microns. Correction of the laser radiation wavefront makes it possible to obtain a focal spot diameter in the far field at the level of 9 pixels,
which, with a pixel size of 3.75 microns, corresponds to 34 microns. Thus, the diameter of the spot in the far field zone is close to the diffraction limit and, consequently, the correction quality
is quite good.
Figure 7
shows images of the focal spot in the far field. The upper pictures were obtained in the absence of correction, while the lower picture shows the intensity distribution in the presence of wavefront
correction. It should be noted that in the absence of correction, the intensity of the focal spot was too low, so the upper row of images was obtained with a longer exposure (85 µs vs. 57 µs for the
corrected case). The image of the far field is close to the diffraction one and practically does not change its shape during the correction procedure.
Another way to assess the quality of the correction is the residual error.
Figure 8
Figure 9
show the change in residual RMS over time for the case of correction and the absence of correction.
Figure 8
represents a complete set of aberrations and graph on
Figure 9
was obtained by excluding the tilts.
For a more detailed consideration of the quality of correction and to obtain quantitative results, a transition was made to the spectral analysis of each of the modes of the wavefront decomposition
by Zernike polynomials. The data-processing algorithm was as follows.
• A sample of the offset coordinates of the focal spots of the lens array of the WFS with a duration of 10 s was recorded. This made it possible to achieve a resolution along the frequency axis of
0.1 Hz. At a frequency of 2000 Hz, a total of 20,000 values of coordinate offsets of each focal spot were recorded
• The transition was carried out from sampling by coordinates to sampling by the coefficients of the wavefront expansion by Zernike polynomials. In this work, we used a set of 24 Zernike
polynomials in Wyant indices [
] (1 and 2—tilts, 3—defocus, 4 and 5—astigmatism, 6 and 7—coma, 8—spherical, etc.).
• Using the Fourier transform, the transition from the time domain to the frequency domain was carried out for sampling each Zernike polynomial;
• The power spectral density was calculated for each mode;
• Further, by integrating the spectral power density, the spectral energy was calculated.
If we integrate the spectral power density for each Zernike polynomial, we can get a graph of the spectral energy (
Figure 10
). The graphs at a certain frequency value go into saturation, which indicates an insignificant contribution of higher-frequency components to the total signal energy. The frequency of transition of
graphs to saturation can be taken as the bandwidth occupied by one Zernike polynomial or another.
Figure 11
shows a diagram of the dependence of the frequency band occupied by one aberration or another. The graph corresponds to one of the consequences of Taylor’s hypothesis [
], according to which the rate of change of high-order aberrations is faster.
The spectral energy saturation amplitude from
Figure 10
for each Zernike polynomial is shown on
Figure 12
The residual aberrations are quite small, so
Figure 13
represents a diagram of the uncompensated aberrations, expressed as a percentage of the entry level.
For greater clarity,
Figure 14
shows the same diagram, but expressed in decibels.
5. Discussion
Based on the diagram on
Figure 14
, we can make the following suggestions.
• The combination of FPGA performance and a bimorph wavefront corrector in an adaptive optical system allows correction of artificially created turbulence in real time. The correction frequency in
the experiments was chosen to be equal to 2000 Hz. Such a stable correction frequency is almost impossible to obtain using a conventional PC. The PC, unlike the FPGA, performs I/O at the driver
level, thereby increasing the time of the closed correction cycle. FPGA exchanges data between external devices (wavefront sensor and corrector control unit) directly. In addition, the FPGA
performs parallel processing of information, which has significant limitations in the case of using a PC.
• The speed of the AOS controlled by the FPGA made it possible to analyze the effectiveness of aberration correction up to the 23rd Zernike polynomial, whose bandwidth is about 100 Hz, in detail.
• The bimorph WFC does not have the ability to correct for the slopes of the WF. Accordingly, the correction efficiency of the first two Zernike polynomials is negative, i.e., there is an increase
in the amplitude of the initial slopes. To eliminate the slopes, it is necessary to use either a separate beam position stabilization system (see, for example, [
]), or to install a mirror in a tip–tilt mount. In this case, the virtual slopes used in the experiment become real and the voltages calculated during operation are applied to the control drives
of the tip–tilt mount.
• WFC used in the experiments has three rings of electrodes and cannot reproduce a spherical aberration of the third order (polynomial # 24), since this aberration has four extremums (max–min). To
reduce the aberration # 24, a higher spatial resolution WFC is required.
• The aberrations from 3 to 23 are compensated well enough by this WFC. However, because the initial amplitude of the polynomial # 24 is small compared to other aberrations, the undercompensating
of this aberration can be neglected.
6. Conclusions
We demonstrated a closed-loop adaptive optical system with the bimorph deformable mirror, Shack–Hartmann wavefront sensor and FPGA controller that can efficiently correct for the wavefront
aberrations caused by the artificial turbulence. The total speed of the system operation was equal to 2000 Hz. It should be noted that the bimorph deformable mirror corrects for the wavefront
aberrations caused by the flow of the heated air quite well—the residual error of the phase fluctuations was reduced by more than 10 times and an almost diffraction-limited focal spot was obtained.
Certainly, when using such a type of corrector, it is necessary to additionally apply a system to stabilize the position of the beam in space.
Author Contributions
Conceptualization, A.K., A.R.; methodology, A.K., A.R.; software, A.R.; validation, A.K., A.N., J.S.; formal analysis, A.N., A.R., V.T.; investigation, A.R., A.N.; resources, A.K.; data curation,
A.R., V.T.; writing—original draft preparation, A.R.; writing—review and editing, A.N., J.S., V.T.; visualization, A.N., A.R.; supervision, A.K.; project administration, A.K. All authors have read
and agreed to the published version of the manuscript.
The research was carried out within the state assignment of Ministry of Science and Higher Education of the Russian Federation (theme # 122032900183-1).
Data Availability Statement
The data presented in this study are available on request from the corresponding author.
Conflicts of Interest
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the
decision to publish the results.
Figure 2. Bimorph mirror used in experiments: (a) Mirror photo; (b) electrodes structure. The numbers indicate the channels of the mirror. The first electrode, conventionally shown in the lower-left
corner, has the dimensions of the mirror aperture and is used to control the defocusing of the wavefront.
Figure 4. FPGA timing diagram. The total time of one closed cycle is 500 microseconds, which corresponds to an operating frequency of 2000 Hz.
Figure 5. Adaptive optical system experimental setup. WFS—wavefront sensor; mirror CU—mirror control unit.
Figure 6. Spectral power density of the oscillation of the X coordinate of the WFS lens array focal spot.
Parameter Value
Clear aperture 50 mm
Electrodes number 31
Control voltage range −200 V ± 300 V
Maximal stroke ±10 µm
First resonant frequency 8.3 kHz
Coating Protected silver
Size Ø 70 mm × 68 mm
Weight 320 g
Parameter Value
Sensor Alexima LUX19HS
Spectral bandwidth 350–1100 nm
Dynamic Range (Tilt) ±50λ
Accuracy of measurements λ/90
Frame rate 2500 fps @ 1920 × 1080
~4000 fps @ 480 × 480
Interface Fiber Optic 40 Gb/s
Lenslet array focal length 12 mm
Number of working sub-apertures 20 × 20
Input light beam size 4.8 × 4.8 mm
Resolution 8 bit
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MDPI and ACS Style
Rukosuev, A.; Nikitin, A.; Toporovsky, V.; Sheldakova, J.; Kudryashov, A. Real-Time Correction of a Laser Beam Wavefront Distorted by an Artificial Turbulent Heated Airflow. Photonics 2022, 9, 351.
AMA Style
Rukosuev A, Nikitin A, Toporovsky V, Sheldakova J, Kudryashov A. Real-Time Correction of a Laser Beam Wavefront Distorted by an Artificial Turbulent Heated Airflow. Photonics. 2022; 9(5):351. https:/
Chicago/Turabian Style
Rukosuev, Alexey, Alexander Nikitin, Vladimir Toporovsky, Julia Sheldakova, and Alexis Kudryashov. 2022. "Real-Time Correction of a Laser Beam Wavefront Distorted by an Artificial Turbulent Heated
Airflow" Photonics 9, no. 5: 351. https://doi.org/10.3390/photonics9050351
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Scaling Up and Out: Training Massive Models on Cerebras Systems using Weight Streaming - Cerebras
Michael James, Chief Architect, Advanced Technologies, and Co-Founder | September 14, 2021
The Cerebras Wafer-Scale Engine is the world’s largest and most powerful computer processor. Designed to solve the hardest problems in artificial intelligence, it performs massive number crunching in
a communication intensive environment. The AI race taking place between leading technology companies is producing larger and more intricate models at an astounding pace. We’ll look at how the
Wafer-Scale Engine trains models much larger than even itself—approaching the scale of a human brain.
Wafer-Scale Engine | Streaming Weights | Accelerating All Linear Algebra | Sparse GEMM | Scaling Out is Easy | The Ultimate AI Accelerator
(Recommended further reading: Weight Streaming whitepaper)
Wafer-Scale Engine
The Wafer-Scale Engine (or simply the “Engine”) is at the center of the show. Its job is to provide a computation environment without bottlenecks. It is a single silicon substrate that has three
physical planes: an arithmetic plane, a memory plane, and a communication plane.
• The arithmetic plane has 850,000 independently programmable cores and a total of 3.4 million floating point units.
• The cores are directly coupled to a memory plane—40GB of on-chip “cache” memory—enough to hold an entire BERT[LARGE] It provides a remarkable 20 PB/s bandwidth on random access patterns.
• A communication plane interconnects the cores in a cartesian mesh with 28 PB/s of homogenous bidirectional bandwidth. Unicast and multicast messages are hardware primitives. Each message has 38
bits and is sent by hardware with guaranteed in-order delivery and end-to-end flow control.
The bottleneck-free design eliminates memory walls that plague other computer architectures.
Modern AI is based on the mathematics of large sparse tensors. Huge bandwidth to random access memory is the key to accelerating sparse-tensor math. Accordingly, the Engine can train neural networks
with 90% unstructured weight sparsity (i.e., with weights equal to zero arbitrarily scattered across the network) without ever spending time executing a multiply by zero. This 10x acceleration
transforms the Engine’s raw throughput of 5.8 PFLOP/s into a peak training performance of 58 PFLOP/s.
Streaming Weights
For models up to about a billion weights, we can hold the entire network in the Engine’s cache memory. This approach works brilliantly. But what if the network is bigger than that? Over the past
three years, the size of the largest AI models has increased by three orders of magnitude, with the largest models now using 1 trillion parameters. This exponential growth will continue as
researchers race to develop “true” AI, also known as artificial general intelligence.
Cerebras makes training these massive models practical. Our new “weight streaming” execution mode gives a seamless scaling path from BERT[LARGE] through GPT-3 and all the way up to a models with more
than 100 trillion parameters – similar to the number of synapses in the human brain!
In a biological brain, information flows from neuron to neuron across synapses. In an artificial neural network, the state of each neuron is called an “activation”, and “weights” record the
connection strength of synapses. The human cerebral cortex has about 16 billion neurons and 125 trillion synapses. The Engine’s cache memory holds up to 20 billion activations, and we will show how
to train at scales of 100 trillion weights. There is still much work to be done as we learn how to harness the potential of models at this scale. This will require lots of experimentation,
trial-and-error, and more fundamental insights. These are the types of challenges that inspired us to create an engine to power our technological future.
Figure 1 Weight streaming execution mode system diagram
The Engine holds complete layers of activations – enormous layers – directly on chip. This avoids a classic inefficiency of “blocking” a matrix multiplication. A blocked matrix multiply is designed
for computers with a small cache and hierarchically larger and slower memory tiers beyond the cache. A blocked multiplication must repeatedly scan a matrix, reloading blocks from a next tier of
memory many times. The blocking technique is useful in low bandwidth situations, but it is also inefficient since the same blocks are repeatedly loaded.
The more than a hundred trillion parameters that a human-brain-scale model will employ, are clearly above any on-chip memory capacity, even that of Cerebras’ behemoth Engine. Common optimizers such
as ADAM require four scalar values per parameter. Stored in single precision, 100T trainable parameters require 1.6 PB memory capacity.
Because huge models require capacity beyond the Engine’s capacity, we make no attempt to store the model parameters on the Engine. Instead, we use an off-chip model memory tier to hold parameters.
The optimizer itself operates directly on this model memory using co-processors. To support the training process, the model memory has a bandwidth of 4 TB/s. Updating model parameters is an
elementwise operation that can be distributed across processing shards. This distribution effectively makes the communications overhead disappear.
Figure 2 Training Timing Diagram
Messages propagate from the model memory to the Engine in a continuous stream. The middle row of Figure 2 shows that a stream of incoming weights keeps the Engine busy. It uses this stream to model
neural state changes and to generate a stream of return gradients. During training, the Engine is used 100% of the time without interruption.
The off-chip memory tier is elastic, and Cerebras will provide configurations from 4 TB to 2.4 PB. If you aren’t provisioning for brain-scale training, then a smaller 8 TB configuration is adequate
for models like GPT-3. The predictable layer-wise access patterns for model weights allows us to use DRAM and flash storage in a hybrid fashion to achieve both high performance and high capacity.
The training set – the data the neural network learns from – is kept this on separate MemoryX shards because it has different requirements and scaling properties from the model parameters. GPT-3 used
300 billion words, only twice the model parameter count. Text is small though. Imagine a training database with every movie ever filmed, or the vast streams of events generated by particle
Figure 3 Animated Diagram Showing the Weight Streaming Process
Accelerating All Linear Algebra
Graphics processors can only run matrix-matrix operations at full performance. This greatly restricts the algorithms that can be explored efficiently on a graphics processor. Linear algebra
operations at the heart of numerical algorithms come in three flavors—matrix-matrix (like GEMM), matrix-vector (like GEMV) and vector-vector (like DOT and AXPY). The Engine runs all these operations
at full performance.
Figure 4 Massive memory bandwidth enables full performance for all linear algebra operations
Matrix-matrix multiplication is the heavyweight operation. Each row of A interacts with every column of B to create a cube of products. The cube is quashed flat by summing terms in the third
dimension for the result C. Matrices, like shipping pallets, require goods to be packed in rows and columns, to be properly aligned, and all sent to the same destination. It is OK to send a matrix
slowly by ship or by truck because the receiver has a lot of work to perform to generate the multiplication result. In other words, the receiver will not have finished processing one matrix before
the next one arrives. The execution time masks the shipping time.
Matrix-vector multiplication requires vastly more bandwidth. While the structure of the operation is the same as matrix-multiply, the skinny vector admits no data re-use of the matrix terms. Because
there is little data re-use, the computation might complete over a thousand times faster. However, it still requires a full third of the data transfer for the matrix operand. Without extra bandwidth,
matrix-vector multiplication is not much faster than matrix-matrix in practice. Graphics processors encounter this bandwidth limitation, and it is why they are poor at real-time inference. Instead of
using matrix-vector, GPUs wait for a batch of inference inputs—increasing latency thousand-fold. Cerebras’ Engine has full bandwidth for matrix-vector and produces phenomenally low latency inference
Vector-vector operations are commandos: versatile, fast, and efficient. They can be used as building blocks to construct the matrix operations. Unconstrained by shipping pallets, they can also do
much more. These operations give raw access to floating point units on unique data streams. But there’s a catch: vector-vector operations have no data reuse and require three times as much bandwidth
as even matrix-vector operations. The Engine has bandwidth to sustain vector-vector operations over the activation planes in its memory. We will see that this gives us the means to take advantage of
sparsity. The brain relies on sparsity to an astonishing degree: about 1,000,000x. If the brain was not sparse, its surface area would be larger than the state of Texas! Its actual surface area, if
spread flat, would be just two square feet.
The Engine’s ability to efficiently leverage sparsity saturates at around 3,000x – shy of the brain’s sparsity, but still enormously ahead of other processors.
Sparse GEMM
Writing matrix multiply in terms of vector-vector operations unlocks the power of sparsity. We make one AXPY call for every non-zero element of the matrix. Zero values are omitted – automatically –
since they would have no effect anyway. This is how the Engine leverages its enormous memory bandwidth to train with sparse weights. Computation speeds up directly from unstructured weight sparsity.
It is both fast and remarkably simple.
Figure 5 Cerebras Sparse GEMM architecture
Imagine the Engine running a sparse GEMM: Activations have feature and token dimensions. Let them start out spread thinly over the surface of the Engine. Activations are jam, spread uniformly across
the Engine. Features to rows. Tokens to columns. Only non-zero weights stream from model memory through the Engine’s intake valves. Weights arrive tagged with an index field to identify their matrix
location. Some cores near the intake read the index and fan-out the weights to their target row. Cores in the target row also use the index to select the appropriate activations. When an output
feature in the sparse weight matrix is complete, a control message (think carriage return) arrives instead of a weight. The cores respond by moving the accumulated sum to a partial sum thread for
reduction along the column. The partial sum is a ring so it can start and end on any row. Thus, the output is uniformly distributed over the Engine. Same as our starting condition. Ready for the next
sparse GEMM.
The gradient pass is the same, in reverse. Broadcasts swap place with reductions. Output activations are renamed “deltas”. The original input activations are still resident from the forward pass.
This time the model memory only transmits the sparsity mask via a sequence of indices. This tells the Engine which components of the gradient to compute. They are sent out via outlet valves to the
optimizer which updates the state variables in model memory.
Other machines are not able to run this algorithm with any performance because bandwidth bottlenecks preclude it. We don’t have bandwidth bottlenecks, so let’s look at the second order effects.
Unstructured sparsity has hot spots and cold spots. Known as the “tail worker effect”, a hot spot assigned to a core causes all other the cores to wait for it to complete with nothing to do. Avoiding
synchronization barriers mitigates this effect. Therefore, the Engine runs partial sum reduction in a separate parallel thread. Consider a GPT3-size matrix with 600 million values. At 90% sparsity,
there are still over 70 thousand non-zeros sent to each row. The law of large numbers minimizes the tail-worker effect. We can do better though. Matrix multiplication is permutation invariant. The
optimizer sorts matrix rows based on their non-zero count and assigns them in round-robin to physical Engine rows. Together with the law of large numbers, this ensures that even extreme power law
distributions have an essentially uniform spread over the Engine.
So, what limits sparsity acceleration? The last effect to consider is Amdhal’s law. Partial sum work needs to be done regardless of the level of sparsity. The Engine reduces partial sum overhead to
four machine cycles. One cycle to copy the accumulated sum to another thread; one cycle to reset the accumulator to zero; and two cycles to run a double-wide add operation on the partial sum chain.
With Amdhal’s law characterized, we can see how the Engine converts sparsity into acceleration. Transformer networks like GPT-3 follow well established scaling laws[i]. The laws relate the number of
network parameters to the layer width and critical batch size. Following these laws, we see how sparse matrix-multiplication is accelerated by varying levels of sparsity for different network sizes.
The plots show acceleration using a single Engine and a cluster of sixteen Engines. The more Engines that are present the smaller the batch size must be per Engine. Less work on an Engine means that
Amdhal’s law will be more pronounced there.
Figure 6 Converting sparsity to acceleration
Scaling Out is Easy
As you’d expect given the result above, training time can be accelerated further by employing more Engines. Cerebras’ scale-out fabric, called SwarmX, operates seamlessly. No changes are needed to
the Engine, or its software. No changes are needed to the Optimizer, or its software, either. What works on one node works on a cluster without porting.
Figure 7 The SwarmX fabric enables Linear Performance Scaling
The scale-out fabric has a tree structure. When weights travel toward the Engines, they are replicated. This is a broadcast. When gradients travel toward the optimizer, they are reduced. Broadcast
and reduce are duals. We built the data reduction operations into the data transport, which is an extremely efficient way to perform these tasks.
Figure 8 Cerebras CS-2 cluster scaling performance for increasing model sizes
And how do we expect the cluster to perform? As Figure 8 shows, the bigger the model, the further the linear trend persists to larger cluster sizes . Note that the 10x in the legend indicates the
speed up we achieve from a conservative 90% sparsity. The multiple lines indicate results for models with different aspect ratios. This data shows that it’s possible to train a model with a trillion
parameters in just a few days.
GPT-3 was trained for months, using over a thousand GPUs. Let’s ask: What is possible with a thousand Cerebras Engines? The brain-scale model we have been considering is 600 times larger than GPT-3.
The scaling chart shows this will complete with only a year of training time on current generation equipment. While less than the 20 years it takes to train a human brain (plus the billion years it
takes to evolve a human brain), it is also clear that this is out-of-reach for most. The important point is this is now architecturally possible. When research advancements make 100x sparse training
viable, the runtime time shrinks to a month.
The Ultimate AI Accelerator
As research advances to the stratosphere, a trail of extraordinarily valuable applications will be created in the wake of that research. We are witnessing this happen right now. These applications
will require infrastructure that can work at huge scale.
Our weight streaming architecture is a practical solution for both today’s mainstream models and the enormous models of the future. The architecture and techniques we’ve discussed above are
applicable to all models, not just NLP. Our paper^[ii] from SC ‘20 shows the Engine is 10,000x faster than graphics processors at the core algorithm of physics-based simulation.
The Cerebras Wafer-Scale Engine is the ultimate scale-up accelerator for AI workloads. Weight streaming makes the Engine the ultimate scale-out accelerator as well. With this kind of horsepower, the
possibilities are endless.
Questions? Ideas? Suggestions? Click here to connect with our team.
Recommended further reading:
Weight Streaming whitepaper A deeper dive into the technology of weight streaming, including a survey of existing approaches used to scale training to clusters of compute units and explore the
limitations of each in the face of giant models.
Harnessing the Power of Sparsity for Large GPT AI Models A look at how Cerebras is enabling innovation of novel sparse ML techniques to accelerate training and inference on large-scale language
[i] Kaplan et al, “Scaling Laws for Neural Language Models”, arxiv.org/abs/2001.08361
[ii] Rocki et al, “Fast stencil-code computation on a wafer-scale processor”, dl.acm.org/doi/10.5555/3433701.3433778 | {"url":"http://hippiedispensary.net/index-283.html","timestamp":"2024-11-14T13:21:18Z","content_type":"application/xhtml+xml","content_length":"242678","record_id":"<urn:uuid:a1e5c18b-37ac-4bc4-9d92-c6be6c598269>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00361.warc.gz"} |
Fast and interpretable non-negative matrix factorization for atlas-scale single cell data
Non-negative matrix factorization (NMF) is a popular method for analyzing strictly positive data due to its relatively straightforward interpretation. However, NMF has a reputation as a less
efficient alternative to the singular value decomposition (SVD), a standard operation that is highly optimized in most linear algebra libraries. Sparse single-cell sequencing assays, now feasible in
thousands of subjects and millions of cells, generate data matrices with tens of thousands of strictly non-negative transcript abundance entries. We present an extremely fast NMF implementation made
available in the RcppML (Rcpp Machine Learning library) R package that rivals the runtimes of state-of-the-art Singular Value Decomposition (SVD). NMF can now be run quickly on desktop computers to
analyze sparse single-cell datasets consisting of hundreds of thousands of cells. Our method improves upon current NMF implementations by introducing a scaling diagonal to increase interpretability,
guarantee consistent regularization penalties across different random initializations, and symmetry in symmetric factorizations. We use our method to show how NMF models learned on standard
log-normalized count data are interpretable dimensional reductions, describe interpretable patterns of coordinated gene activities, and explain biologically relevant metadata. We believe NMF has the
potential to replace PCA in most single-cell analyses, and the presented NMF implementation overcomes previous challenges with long runtime.
In 1999, Lee and Seung proposed an algorithm for Non-Negative Matrix Factorization (NMF) that was able to learn the parts of faces from a set of images (Lee & Seung, 1999). This demonstration that
non-negativity constraints enforced an additive decomposition contrasted sharply with other then-popular dimensional reductions such as PCA or SVD which describe axes of variance rather than parts of
the whole. Two decades later, a plethora of specialized machine learning algorithms have emerged that incorporate non-negativity constraints, and yet none are more interpretable than a simple
non-negative matrix factorization.
Just as a matrix of pixels encoding a face might be decomposed into factors describing the parts of a face, a matrix of the RNA transcript portfolios of single cells may be decomposed into
constituent cellular processes (Brunet et al., 2004; Clark et al., 2019). In turn, these processes are orchestrated by coordinated patterns of gene expression. While NMF exposes basic biology and
generalizable information amenable to transfer learning (Stein-O’Brien et al., 2019), SVD and PCA capture sequential normalizations or rotations of coordinate axes explaining variation specific to
the dataset of interest.
Single-cell count data is strictly non-negative. Non-negative count data is well-suited to non-negative dimension reduction. If the dimensional reduction may contain positive and negative loadings,
factors in the learned model could cancel one another out. This happens in SVD and PCA, where each factor gives a linear transformation of the high-dimensional data in low-dimensional space that both
over- and under-corrects for all transformations applied by the preceding factors. In contrast, factors in NMF are mutually interdependent and collectively additive, not sequential, and can thus be
understood individually. Unfortunately for SVD and PCA, this means it is possible to handle dropout events in both directions – by imputing missing signal (which is good) and by ignoring true signal
(which is bad). NMF is not subject to this issue as it cannot contain negative values, thus missing signal is generally imputed up to a point that minimizes mean squared error of reconstruction (Lin
& Boutros, 2020).
Many modern approaches to single-cell analysis still rely heavily on variance-based decompositions (SVD or PCA) (Qiu et al., 2017; Stuart et al., 2019). This is because PCA is relatively fast,
historically established, well-supported in popular frameworks, and can generate reasonable dimension reductions for visualization and graph-based clustering. Other methods for single-cell analysis
use variational autoencoders (Lopez et al., 2018), generative adversarial networks (Marouf et al., 2020), and other new machine learning algorithms, all with some amount of success for their
respective objectives. Unfortunately, too many of these algorithms require extensive hyperparameter optimization and thus deep knowledge of the method for proper use, and even if outputs satisfy
objectives, most of the learned latent factor models are themselves hardly interpretable.
Given its interpretability, why has NMF not been widely adopted by the single-cell community? In our opinion, the single greatest barrier is computing time. While there are many fast NMF-inspired
algorithms for sparse recommender systems, such applications differ significantly from the objectives in single-cell analysis. For example, collaborative filtering and implicit recommenders require
different algorithms than simple NMF. To our knowledge, there are no NMF implementations for large sparse matrices that are fast enough to support rapid experimentation, even in high-performance
computing environments.
In addition to optimizing runtime, more clear demonstrations of the usefulness of NMF in single-cell analyses are needed. For instance, NMF can serve as both a dimensional reduction in place of PCA (
Elyanow et al., 2020) and a resource for learning novel biology (Clark et al., 2019). With sufficient meta-analysis and dataset integration, it is possible that latent factors learned by NMF may
replace manually curated toolkits such as GO terms in many gene set enrichment analyses (Stein-O’Brien et al., 2018).
In this manuscript we make the case for using NMF in single-cell analysis rather than PCA. We then present technical aspects of the implementation that have enabled the fast performance of our
implementation, which may be of interest to a more interdisciplinary audience. Finally, we direct attention to several perpetual challenges of NMF as a method and propose some avenues for ongoing
Dimension reduction with NMF is as sensitive as PCA
To make a case for NMF as a sensitive dimensional reduction, we analyzed 103,000 single blood cells from the Human Embryonic Cell Atlas. First, the optimal rank for a factorization was determined by
plotting the mean squared error of an NMF model versus factorization rank for models of rank 1 to 100. An inflection point in the resulting curve was established around a rank of 35 (Figure 1a). A
high-quality NMF model was fit within 1 minute with k = 35 using unnormalized counts and all genes. A k-Nearest Neighbor (k-NN) graph was constructed from sample loadings in this model and a UMAP
embedding was learned to assess the sensitivity of the projection (Figure 1b). We observed generally distinct separation of cell types as annotated in the Human Embryonic Cell Atlas.
Principal component analysis (PCA) was run using the standard Seurat workflow. First, data was log-normalized (a step not needed in NMF), then scaled and centered (effectively converting a large
sparse matrix to dense), then PCA was run for the top 3000 highly variable genes. The inflection point in the “elbow plot” of standard deviation vs. rank coincided with approximately k = 35 (Figure
1d). The resulting UMAP embedding clearly delineated atlas-annotated cell types (Figure 1e), perhaps better than NMF. The major caveat, however, is that cell types were originally classified with
graph-based clustering on a PCA embedding by the atlas authors and thus are inherently biased towards PCA.
NMF factors enrich for biological metadata relative to PCA
To examine whether the learned PCA embeddings encode biological information other than the rotation of coordinate axes to explain maximal variance, we examined whether factors in NMF or PCA were
enriched for annotated cell types. We observed that some NMF factors were represented almost exclusively in certain cell types, others split between several cell types, and still others distributed
across many cell types (Figure 1c). PCA factors, however, were randomly represented across all cell types (Figure 1f).
Samples from the Human Embryonic Cell Atlas span an 8-week window of development, enabling studies of differential gene expression over time. At least one NMF factor was specifically upregulated
early around day 80, several factors were up around embryonic day 100, and still others were up late (Figure 1g). PCA factors manifested no such interpretable trends, with some factors down around
developmental day 100, which is difficult to interpret (Figure 1h). This finding highlights the poor interpretability of signed decompositions of non-negative data.
NMF factors explain additional biological metadata as well. Some NMF factors were strongly enriched in cells from male embryos and at least one factor was strongly enriched in cells from female
embryos (Figure 1i). There was also a strong association between some factors and cell source organ (Figure S2).
NMF factors capture known biological processes more effectively than PCA
NMF factor models are additive decompositions of the input matrix. In the case of single-cell matrices of RNA transcript counts across cells, NMF factors should therefore represent biological
processes (Clark et al., 2019; Stein-O’Brien et al., 2018). We examined GO term enrichment in NMF and SVD rank-35 models. SVD was used in place of PCA because PCA of 100,000 cells and 25,000 genes on
a dense matrix takes too long, and no centering or scaling of a sparse matrix (conversion to dense) is necessary for SVD. Just over 3600 GO terms were significantly enriched in at least one of the
SVD and NMF factors. The NMF factors manifested distinct patterns of GO term enrichment, indicating that factors describe largely unique biological processes (Figure 2a). However, SVD patterns
manifested no such unique patterns, and factors that were enriched appeared to be non-specific (Figure 2b). Indeed, each NMF pattern had at least 100 enriched GO terms while most SVD patterns had
none to very few (Figure 2c). These results show that SVD (and by extension, PCA) does not capture biological processes, but sequential normalizations or axes of associated variation, while NMF
captures known genetic processes that collaborate to reconstruct transcriptional state.
We next examined how PCA embeddings and NMF loadings are dispersed throughout their respective UMAP embeddings. Principal component cell embeddings appeared as broad gradients passing through most
cells, explaining a small fraction of the variation in that cell (Figure 2d). NMF loadings were highly specific, localizing to distinct populations of cells (Figure 2e). This again demonstrates that
NMF factors describe basic biology in addition to serving as a sensitive dimensional reduction, thus doing what PCA is commonly used to do as well as providing additional information.
RcppML NMF is as fast as state-of-the-art SVD
Widespread adoption of NMF as a method in single-cell analysis requires reasonable runtime. We benchmarked our implementation against NNLM NMF, the fastest implementation of simple NMF of which we
are aware in the R community, and implicitly restarted Lanczos bidiagonalization SVD (irlba), a sparse matrix SVD known for its class-leading performance (Baglama & Reichel, 2005). For both 2,800
PBMCs (Figure 3a) and 40,000 bone marrow cells from the Human Cell Atlas (Figure 3b), RcppML NMF method performed nearly as well, or as well as irlba SVD across a range of ranks. Our method was more
than an order of magnitude faster than NNLM NMF. RcppML NMF also performed impressively on benchmarking tests in random sparse matrices regardless of factorization rank, matrix sparsity, number of
samples in the matrix, or rectangularity of the matrix (Figure S1).
Efficient Non-Negative Least Squares solvers increase factorization speed
A major bottleneck in NMF is finding solutions to non-negative least squares equations. The NNLM R package has shown great promise for sequential coordinate descent least squares initialized with an
approximate solution given by the model from the previous iteration (Lin & Boutros, 2020). We found that initializing coordinate descent non-negative least squares with a zero-filled vector created a
strong gradient from which fast convergence could be achieved, faster than with warm-start initialization both with and without optimizers such as adam (see an example solution path in Figure 4a). We
also considered initialization with an unconstrained least squares solution, or an approximate solution that we refer to as “Fast Active Set Tuning” (FAST). In FAST, an unconstrained least squares
solution is computed, all negative values set to zero (an “active set”), and all other values are added to a “feasible set”. An unconstrained least squares solution is then solved for the “feasible
set”, any negative values in the resulting solution are set to zero, and the process is repeated until the feasible set solution is strictly positive. The algorithm has a definite convergence
guarantee because the feasible set will either converge or become smaller with each iteration. The result is generally exact or nearly exact for small well-conditioned systems (< 50 variables) within
2 iterations and thus quickly sets up coordinate descent very well (see example solution path in Figure 4b). The FAST method is similar to the first phase of the previously described so-called
“TNT-NN” algorithm (Myre et al., 2017), but the latter half of that method relies heavily on heuristics to find the true active set, which we avoid by using coordinate descent instead.
A fortran77 implementation of NNLS by Lawson-Hanson has long been the gold standard for NNLS, yet FAST-CD NNLS outperforms this standard for any random system greater than 30 variables in size (
Figure 4c).
For solving random ill-conditioned systems, the FAST routine performs well indeed, but solutions deteriorate in quality as system size increases (Figure 4d). However, FAST-CD NNLS still outperforms
coordinate descent NNLS initialized randomly, with zeros, or unconstrained least squares.
In the context of matrix factorization, zero-initialized coordinate descent performs better than FAST-CD on ill-conditioned random systems (Figure 4e) while factorization of actual single-cell
datasets is approximately as fast, if not faster with FAST-CD compared to coordinate descent alone (Figure 4f-g). However, due to the better overall performance of zero-initialized coordinate descent
across a large number of ranks for well-conditioned systems, and more recent optimizations to the logic within the coordinate descent NNLS solver, our current RcppML version uses only coordinate
descent NNLS.
Diagonalized NMF enables symmetric factorization
Random initializations for NMF can cause differences in the relative distributions of values in both sides of the model, w and h. These differences in turn cause factors to be scaled differently
relative to one another than they would be if distribution of loadings in w and h were comparable. This can be demonstrated by factorization of a symmetric factorization (Figure 5a). However, when
each factor in w and h is linearly scaled to sum to 1 against a diagonal, the distribution of w and h always exists on the same scale and convergence toward symmetry is guaranteed (Figure 5b). The
scaling diagonal of NMF is different in nature to that of an SVD, as NMF factors are collectively and simultaneously optimized, while SVD factors (for example in rank-truncated SVD) are optimized
sequentially and are dependent on preceding factors (Figure 5c).
NMF is well-suited to dimensional reduction of single-cell data for visualization, use as an embedding for clustering, and learning patterns of coordinated gene activities. Our algorithm brings
compute time down to that of the fastest SVD implementations and adds diagonalization to aid interpretability and convex regularization. Taken together, this work makes a case for adoption of NMF
into mainstream single-cell analysis pipelines.
However, NMF shares some challenges with its signed dimensional reduction counterparts. For example, rank determination remains a subjective problem, although in Figure 1a we show that rank
determination can be as “easy” as that for PCA using an inflection point in a curve of rank vs. model objective (loss for NMF and additional explained variance for PCA). Unfortunately, despite its
intuitive appeal, this “elbow plot” method is highly subjective and often very unclear. Future work must explore more quantitative approaches such as cross-validation against robustness objectives or
with missing value imputation (Lin & Boutros, 2020). Unfortunately, these approaches require the training of many models, and involve masking (which scales more poorly than the base algorithm), and
themselves contain hyperparameters. However, rigorous rank-determination methods for PCA are at least equally consuming and ineffective as those for NMF, such as jackstraw (Chung and Storey 2015) or
k-fold cross-validation. Rank determination for dimension reduction, at the end of the day, will remain a theoretical enigma and will always benefit from evaluation against domain knowledge and the
art of expert inspection.
Reproducibility is yet another challenge for NMF. Alternating least squares requires an initialization, such as a random or non-negative double SVD model (Esposito, 2021). While NNDSVD is “robust”,
it differs fundamentally in nature from NMF, and any non-random initialization can trap updates into a local minimum even if random noise is added to the model and zeros are filled. Different random
initializations can lead to different solution paths toward local minima, as finding the global minima is NP-hard (Vavasis, 2009). Setting a random seed to guarantee reproducibility of a
factorization model is far from ideal when the learned factors are intended to describe fundamental biology. However, for most applications and including scRNA-seq, replicate models give similar
errors of reconstruction, many factors are robust, and those that are not as robust share the remaining information among themselves differently between replicates. Some heuristics have been proposed
to find “consensus” NMF factors across multiple models (Kotliar et al., 2019; Stein-O’Brien et al., 2019), but these methods require significant compute time, no longer treat NMF as a dimensional
reduction, and may be biased towards factors explaining dominant signals. While robustness may be seen as a weak link in NMF, it is important to realize that SVD is very sensitive to minor
adversarial attacks on the input data. For example, different normalizations of the data or batch effects can lead to fundamentally different SVD results across most factors. On the other hand,
because NMF factors are collectively updated, distinct technical issues are usually explained by a single factor while other factors are left unaffected. This same reasoning explains why no
normalization of input data is required for NMF – the method naturally applies any necessary scalings or corrections.
NMF is an established method for extracting additive signals from non-negative data. For single-cell analysis, NMF describes coordinated gene activities and serves as a sensitive dimensional
reduction for visualization and clustering. It seems more reasonable to cluster cells on factors describing biological processes than on principal components explaining variance. Further, it seems
more reasonable to visualize cells based on underlying signals rather than embeddings learned from sequential normalizations of the data. Finally, while PCA and SVD are imputation-incompetent signed
decompositions, NMF imputes missing values, denoises false positives, and respects non-negative inputs with non-negative outputs. Non-negative matrix factorization has the potential to systematically
capture the complexities of genetic diversity, and it can do so simply, interpretably, and quickly.
Materials and Methods
NMF Problem Definition
Non-negative matrix factorization seeks to decompose a matrix A into orthogonal lower-rank non-negative matrices w and h: In the above equation and considering scRNA-seq datasets, A is a sparse
matrix with unnormalized raw gene counts as rows and cells as columns, w is a dense matrix of “genes x factors”, and h is a dense matrix of “factors x cells”.
NMF algorithms generally require some initialization of w and/or h and use alternating updates of w and h to refine the model until convergence as determined by some stopping criteria. At
convergence, wh will approximate the input A.
Alternatively, inspired by SVD or PCA, one might consider adding a scaling diagonal: NMF may be diagonalized by scaling factors in h and w to sum to 1 after each alternating update. For example,
after h is updated, d is set equal to the sum of factors in h and then each factor in h is scaled to sum to 1. Next, w is updated, after which d is set to the sum of factors in w and each factor in w
is scaled to sum to 1.
NMF Algorithm
RcppML NMF makes use of the traditional algorithm for NMF by alternating least squares (ALS) (Kim & Park, 2007). This algorithm is fast and flexible, especially compared to multiplicative updates (
Lin & Boutros, 2020), even when supported by optimizers such as adam. In ALS-NMF, w and h are updated alternately with each iteration. For example, to update a column of h (h[i]) given a column of A
(A[i]) and w, the Ordinary Least Squares (OLS) system is constructed as follows: A similar approach may be used for updating rows of w, only A must first be transposed, and h used in place of w in
the above equations. Equations #3-5 describe projection of a linear model which is a fundamental method in transfer learning. Projections of dense models onto sparse matrices may be performed using
the R function “RcppML::project”.
NMF Stopping Criteria
A well-established measure of convergence for matrix factorizations is given by the relative change in mean squared error (MSE) between consecutive iterations (Lin & Boutros, 2020). For a given
iteration i, this measure is given by: Calculating change in MSE requires computing the cross-product of w and h for each iteration to give a large dense approximation of the model in wh, then
subtracting dense wh from sparse A. Such a dense-sparse operation, even with parallelization and proper use of sparse matrix iterators (as implemented in “RcppML::mse”), is extremely time-consuming,
and in cases with >99% sparse matrices often takes longer than the factorization updates themselves.
Because MSE depends on w and h, the relative change in w and h across consecutive iterations may serve as a more efficient stopping criteria. Consider a stopping criterion using Pearson correlation
(R^2) between models in consecutive iterations. For a given iteration i: For a symmetric factorization, the stopping criteria could be correlation between w and h, since at convergence w ≈ h: Indeed,
for a symmetric factorization, the change in MSE between consecutive iterations, the correlation between w across consecutive iterations, and the correlation between w and h for a given iteration,
follow extremely similar trends (Figure S3a). Since calculating correlation between models is at least an order of magnitude faster than calculating MSE (Figure S3b), RcppML NMF uses eqn. #7 as a
stopping criterion in place of MSE.
NMF Implementation
RcppML is a library of R functions that are lightweight wrappers around C++ functions coded with Rcpp and RcppEigen (Bates & Eddelbuettel, 2013; Eddelbuettel & François, 2011). For this scope of
work, the Eigen C++ library is the fastest non-commercial BLAS and linear algebra library. We began by working in Armadillo, achieved comparable performance in base Rcpp, but found Eigen to be at
least twice as fast for all use cases in addition to 3-5x faster at computing Cholesky decompositions. When plugging Intel MKL BLAS into Armadillo the gains for Eigen in our situation are still
noticeable. All RcppML Eigen code is provided in a single C++ file so that functions can be readily incorporated into other C++ programs and software without using Rcpp wrappers, provided the Eigen
header library is loaded.
An Rcpp sparse matrix class handles zero-copy pass-by-reference access to R objects in C++. This contrasts with Eigen and Armadillo sparse matrices, which form deep copies, and thus RcppML uses
significantly less RAM. Constant sparse matrix forward column iterators are used for contiguous access to non-zero values. This provides performance gains for matrices >80% sparse, below which sparse
iterators underperform dense operations. scRNA-seq data is typically 92-98% sparse and rarely <80%.
Block-pivoting in alternating updates of w and h require transposition of sparse matrix A. Transposition of very sparse column-major matrices is an inefficient operation. We profiled in-place
updating of w and found that the computational cost of in-place updating surpassed the cost of up-front transposition after about 3 iterations. RcppML NMF therefore does not provide support for
in-place updates of w and uses block-pivoting instead.
Parallelization with OpenMP is implemented across rows in w and columns in h for each alternating least squares update. By default, all available CPU threads are used. Calculation of the right-hand
side of systems of equations is included in the parallelization loop alongside the least squares solver to minimize overhead and maximize the contiguity of memory access patterns within threads.
The FAST least squares algorithm relies heavily on the Eigen Cholesky module for extremely fast LLT decompositions and solutions by forward-backward substitution. For projecting a linear model, the
LLT decomposition for the left-hand side of the system is computed only once (i.e. “preconditioned”). Sequential coordinate descent least squares is adapted from the NNLM RcppArmadillo package (Franc
et al., 2005; Lin & Boutros, 2020), with significant optimizations to algorithm logic and vectorization.
Extensive code profiling and experimentation ensured we took full advantage of compile-time optimization and Eigen-facilitated vectorization. We were mindful of passing by reference, in-place
updating, and contiguous memory access patterns for optimal use of the cache.
We have found that the base algorithm can be optimized for GPUs with CuSparse, but the gains were not significant enough to justify GPU over CPU utilization given the difference in cost. This is
almost certainly due to the discontinuous cache access patterns in sparse-dense operations, which work well for CPU but not GPU operations. We have also explored multiplicative updates accelerated
with the adam optimizer, which approached but did not surpass the runtime of our implementation. In the future we hope to add support for single precision, but beyond that we currently do not see
further potential for speed gains. Our attention now turns to developing a neural network-like NMF implementation that can be extended much more flexibly to address problems beyond reconstruction
such as integration, multimodality, generativity, and more.
All runtime benchmarks and nearly all figures were generated on an average desktop computer to show that any user can make good use of NMF for single-cell datasets. Unfortunately, for Figure 1 we had
to use a high-performance computing cluster to run Seurat PCA and UMAP reductions due to RAM limitations. Benchmarks were run using a 64-bit OS running Windows 10 on an Intel Core i5-8400 CPU
processor at 2.80GHz with 8GB of RAM. Runtimes were faster on a computing cluster, but relative performance between methods was similar. Random sparse matrices were generated using the
“rsparsematrix” function from the R “Matrix” package. Runtimes for R function calls were measured using the “microbenchmark” R package.
Dimension Reduction of Embryonic Blood Cells
NMF was run on standard log-transformed RNA transcript count data from 100,000 blood cells from the Human Embryonic Cell Atlas to 100 iterations at k = 35. PCA was run on log-normalized, centered,
and scaled count data using Seurat default parameters. KNN graphs were constructed with k = 20 on dimensions 1 to 35, UMAP plots were constructed from KNN graphs using dimensions 1 to 35, 50
neighbors, and “Seurat::RunUMAP” defaults. Cellular metadata packaged with the Blood Cells dataset was used for enrichment analyses for cell type, organ type, embryo sex, and developmental day. GO
enrichment analysis used msigdbr human C5 pathways, fast gene set enrichment analysis (fgsea), and BH-adjusted p-values. Many terms in the NMF factors were highly significant (p-adj < 1e-8), but the
significance cutoff was defined at p-adj < 0.05 for convention.
Data Availability
Figures 1 and 2 make use of the Human Embryonic Cell Atlas dataset, “Blood Cells” processed dataset available at https://descartes.brotmanbaty.org/bbi/human-gene-expression-during-development/, along
with associated cell and gene metadata (Cao et al., 2020). Datasets used for benchmarking are readily available via SeuratData, consisting of the “pbmc3k” dataset of 2,800 PBMC cells supplied by 10x
genomics, and the “hcabm40k” dataset, containing 40000 bone marrow cells from the Human Cell Atlas (Han et al., 2020).
Code Availability
The latest stable release of the RcppML R package is publicly available on CRAN. The development version is available at github.com/zdebruine/RcppML. Issues on GitHub are monitored and we welcome
questions and feature requests there.
This work has been supported by the Van Andel Institute Graduate School (ZJD), the Van Andel Institute (JAP and TJT), Chan Zuckerberg Initiative Data Insights Grant DAF2022-249404 (ZJD and TJT),
NIAID grant R01AI171984 (TJT), and NHGRI grant R01HG012444 (TJT and JAP).
Competing Interests
None declared.
• Theoretical discussion updated to reflect emerging evidence, results updated to reflect issues and opportunities raised by the community during two years of usage. Updated authorship
contributions reflecting changes to scientific content. | {"url":"https://www.biorxiv.org/content/10.1101/2021.09.01.458620v2.full","timestamp":"2024-11-01T19:44:22Z","content_type":"application/xhtml+xml","content_length":"214883","record_id":"<urn:uuid:5d1f63f4-ff3a-4c70-aefd-ab25bbbf2a14>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00657.warc.gz"} |
and algorithms through the lens of
Proofs, beliefs, and algorithms through the lens of sum-of-squares
Boaz Barak and David Steurer
work in progress (Fall 2016)
A random graph with a hidden clique. The sum-of-squares algorithm maintains a set of beliefs about which vertices belong to the hidden clique. Despite learning no new information, as we invest more
computation time, the algorithm reduces uncertainty in the beliefs by making them consistent with increasingly powerful proof systems. Initially the beliefs have maximum uncertainty and correspond to
the uniform distribution but they eventually converge on the correct hidden clique (red edges).
Seminars & workshops
• SOS seminar by Pravesh Kothari and David Steurer at Princeton University, Fall 2016
• SOS seminar by Boaz Barak and Pablo Parrilo at Harvard University and MIT, Fall 2016
• Four-day winter school in UC San Diego on the sum of squares algorithm January 3-6. See the web page for registration and more information. | {"url":"https://www.sumofsquares.org/public/index.html","timestamp":"2024-11-12T22:24:10Z","content_type":"text/html","content_length":"10525","record_id":"<urn:uuid:cf846ff3-11bb-4223-bf59-169544e7c4f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00359.warc.gz"} |
324. NPV versus BCR part 3
In PD322 and PD323 we have been exploring whether to use Net Present Value (NPV) or Benefit: Cost Ratio (BCR) in Benefit: Cost Analysis (BCA) when assessing and comparing projects. I’ve presented
some simple rules to follow when the projects are separate and unrelated, or when the projects are mutually exclusive (i.e., when you can only choose to do one of them). But what if the decision
maker is faced with choosing from multiple separate projects, and there are multiple versions of at least one of the projects?
Before getting into the details, I want to clarify that, in all the examples I’m presenting in this series, I’m assuming that the objective is to maximise the total NPV across all funded projects. I
should have emphasised that earlier. A key message from the three related blog posts is that to get the highest total NPV, you should not necessarily choose the projects with the highest individual
NPVs. Sometimes that’s the case but in other cases it’s not.
Now, suppose that a decision maker is faced with selecting from three different versions of a project to protect Lake Antelope (call them projects A1, A2 and A3) and four versions of a project to
protect Lake Giraffe (projects G1, G2, G3 and G4). In this situation, there is no simple rule to follow, like just choose the projects with the highest BCRs or the highest NPVs. Instead, this
situation requires the use of a constrained optimisation algorithm. Assuming that you are choosing whole projects (i.e. you can’t choose 0.7 of a project), the required algorithm is called integer
Fortunately, integer programming is available in the Excel spreadsheet software, and it is not difficult to use it to select the optimal portfolio of projects in this complex situation. I’ll present
the numbers for a relatively simple example of this complex decision problem, and then I’ll show you how to solve it in a brief YouTube video.
First the example. The present values of benefits (B) and costs (C) for the project versions for Lake Antelope and Lake Giraffe are as follows: project A1: B=$180, C=$40; project A2: B=$360, C=$100;
project A3: B=$400, C=$200; project G1: B=$200, C=$10; project G2: B=$400, C=$60; project G3: B=$600, C=$160; project G4: B=$800, C=$310. The available budget is $300. You can choose at most one of
the Lake Antelope project versions and at most one of the Lake Giraffe project versions. Which project versions (if any) should you choose to maximise the overall net benefits?
Watch the video to see how to use integer programming to solve this in Excel. Within the capacity of the software, this approach will work for any number of projects and any number of project
Note that the information you get from a model like this is not a simple ranking of the projects. Instead, it tells you how the optimal combination of projects and project versions changes depending
on the available budget. To get this information, you change the program budget in the model and re-solve it. For this example, the results look like this.
Range of budget levels Optimal version of A project Optimal version of G project
$0 to $9 nil nil
$10 to $49 nil G1
$50 to $59 A1 G1
$60 to $99 nil G2
$100 to $159 A1 G2
$160 to $259 A2 G2
$260 to $409 A2 G3
$410 and higher A2 G4
The two project versions in each row of the table come as a package. There is no point in ranking them. If the budget was to shrink, you would not just drop one of these, you would change which
project version you selected, as shown in the table.
NPV/BCR Rule 4: If selecting and ranking multiple projects, and at least one of the projects is available in multiple versions, don’t use NPV or BCR. Instead, use integer programming to optimise the
selection of projects and project versions simultaneously.
Having to create a constrained optimisation model like this might seem like more bother than is worthwhile, but it isn’t difficult, it doesn’t take long, and it may result in substantially higher
benefits being generated by the program, compared with the use of a simpler rule-of-thumb.
If you’ve read all three of these Pannell Discussions on NPV and BCR, your head may be spinning at the complexity of all this. Apologies for that, but the complexity is real and needs to be
understood if analysts applying BCA are to be sure of giving sound advice to decision makers.
Another realistic complexity that might be relevant in some cases is that there might be two separate constraints on the funding of projects. For example, there could be a limited budget available
for initial project implementation and a separate limited budget available for ongoing maintenance. As with the example above, neither BCR nor NPV is sure to rank the projects correctly in this
situation, and you need to use a method like integer programming to be sure of getting it right. You could build an Excel model like the one in the video, and include separate constraints for
implementation costs and maintenance costs, rather than a single constraint for costs overall, which is what I did in the video.
On the other hand, in the numerical examples I’ve looked at, it is usually not too terrible to assume that implementation costs and maintenance costs are drawn from one limited budget. In other
words, using BCR to rank separate, unrelated projects can still be OK even if there are two constraints on funding. But it is an approximation and it might not give you the best possible solution.
One thought on “324. NPV versus BCR part 3”
1. Thanks for your article. I refreshed my calculus, rekindled my knowledge in so called shadow prices (Soviet Union did well with those babies), and figured out how to use (solve) Excel’s solver
with different given cost (constraints), and income scenarios. I was looking for a more involved and informative description and explanation of applied capital theory as used in decision theory.
Maybe even Kenneth Boulding’s IRR (assuming no negative crossovers)? Thanks for your efforts. S. Baxter | {"url":"https://www.pannelldiscussions.net/2019/09/324-npv-vs-bcr-3/","timestamp":"2024-11-12T13:45:03Z","content_type":"text/html","content_length":"62268","record_id":"<urn:uuid:9fff5080-9e56-4cc9-a1d5-bdd8ca75c35c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00092.warc.gz"} |
discount rate calculator
You can also sometimes estimate the return rate with The Rule of 72. All you have to do is enter the total quantity to which you want to apply the discount and then the discount percentage
itself.Click the button “calculate discount”. For most companies it’s just a weighted average of debt and equity, but some could have weird preferred structures etc so it could be more than just two
components. You can also use this present value calculator to ascertain whether it makes sense for you to lend your money, considering the annual inflation and return rates. Discount Factor = 0.83So,
discount factor is 0.83.Now, let us take another example to understand discount factor formula better. Input the post-sale price (for example into cell B1). There are numerous others that can be more
confusing, such as stackable discounts where you can get 20% off the original price, then 15% more off of that discounted price. Calculate the Internal Rate of Return (IRR, discount rate) for any
investment based on initial deposit and cash flow per period. The formula is: NPV = ∑ {After-Tax Cash Flow / (1+r)^t} - Initial Investment. What was the percentage discount on the original price of
the lava lamp? the money's discounted present value, should you decide not to use this money now to purchase goods and services for certain number of years, taking into the account the money's annual
inflation or discount rate. Discount Factor = 1 / (1 * (1 + 10%) ^ 2) 2. Subtract the post-sale price from the pre-sale price (In C1, input =A1-B1) and label it “discount amount”. How to calculate
discounts with a calculator? We have to calculate the discount factor when the discount rate is 10% and the period is 2.Discount Factor is calculated using the formula given belowDiscount Factor = 1
/ (1 * (1 + Discount Rate)Period Number)Put a value in the formula. cash you earn from the project, less the present value of all cash outflows, i.e. In economics and finance, the term "discount
rate" could mean one of two things, depending on context. This tool works very simply. The discount factor is calculated in the following way, where P(T) is the discount factor, r the discount rate,
and T the discretely compounded over time: This can be worked out in the following way: Discounted Cash Flow (or DCF) is the Actual Cash Inflow / [1 + i]^n; Where, i - denotes the discount rate used,
n - denotes the time period relating to the cash inflows. It is one of the most simple way to increase customers for a product. If the sale price of an item is $40 and the discount is 20% then the
list price is calculated as follows: James bought a vintage lava lamp at a sale price of $89.63. Enter the discount rate in percent and the number of compounding periods into the calculator.
Calculator Use. Cite this content, page or calculator as: Furey, Edward "Discount Calculator"; CalculatorSoup, IRR formula, how to calculate it and how to evaluate investments using it. You need to
specify a set discount rate for the calculation. Present value is compound interest in reverse: finding the amount you would need to invest today in order to have a specified balance in the future.
The Discount Rate represents risk and potential returns, so a higher rate means more risk but also higher potential returns.. A discount rate is used to calculate the Net Present Value (NPV)Net
Present Value (NPV)Net Present Value (NPV) is the value of all future cash flows (positive and negative) over the entire life of an investment discounted to the present. The discount rate is often
defined as the opportunity cost of making a particular investment instead of making an alternative investment of an almost identical nature. cash you spend on the project. For more information
regarding calculation of the Discount Rate and Interest Rate, contact CompConnection at 800-252-7031, option 3. The sale price is calculated as follows: Original Price − Discount Amount = Sale Price
$ 120 − $ 30 = $ 90. You may want to try out a double or triple discount calculator, or even a percent off calculator. IFRS 16.A The interest rate ‘implicit’ in the lease is the discount rate at
which: However, this is different for each investor. A fixed amount off of a price refers to subtracting whatever the fixed amount is from the original price. The interest rate implicit in the lease
is defined in IFRS 16 as ‘the rate of interest that causes the present value of (a) the lease payments and (b) the unguaranteed residual value to equal the sum of (i) the fair value of the underlying
asset and (ii) any initial direct costs of the lessor.’ © 2006 -2021CalculatorSoup® Among other places, it's used in the theory of stock valuation.. See How Finance Works for the present value
formula.. You can also sometimes estimate present value with The Rule of 72. If you want to customize the colors, size, and more to better fit your site, then pricing starts at just $29.99 for a one
time purchase. Calculate the bond discount rate. Return Rate Formula. Discount Rate Calculator. While it's easier to use the Omni Discount Calculator, here are the steps to calculate discount rate in
Excel: Input the pre-sale price (for example into cell A1). For lessors, the discount rate will always be the interest rate implicit in the lease. All financial theory is consistent here: every time
managers spend money they use capital, so they should be thinking about what that capital costs the company. There can be many sources of capital, and the weighted average of those sources is called
WACC (Weighted Average Cost of Capital). If we calculate the present value of that future $10,000 with an inflation rate of 7% using the net present value calculator above, the result will be
$7,129.86. Type the original prices and sales prices into a worksheet as shown as below screenshot: 2. Opportunity cost is a slippery concept, though, because we need to subjectively conclude what
that “alternative investment of an almost identical nature” is. \(discount \, rate\, = \, \frac {discount }{list \, price} \, \times \, 100\) How to use a discount calculator? If you need to do these
kinds of calculations, refer to the Percent Off Calculator. Types of discounts. \(D = \dfrac{(120 - 90)}{120} \times 100 \), \(\text{Amount Saved} = $120 - $90 = $30 \), \(S = 120 - \dfrac{75\%}{100}
\times 120 \), \(\text{Amount Saved} = $120 - $30 = $90 \), \(L = \dfrac{40}{(1 - \dfrac{20\%}{100})} \), \(\text{Amount Saved} = $50 - $40 = $10 \), \( \dfrac{(165.99 - 89.63)}{165.99} \times 100 =
46 \). or NPV analysis is a form of intrinsic valuation and is used extensively across finance and accounting for determining the value of a business, investment security, of a business, as part of a
Discounted Cash Flow (DCF)Discounted Cash Flow DCF FormulaThe discoun… Calculate the list price, discount percentage or sale price given the other two values. In order to make use of the discount
calculator, an individual will have to … 1. Divide the amount of the discount by the face value of the bond. Click the "Customize" button above to learn more! Discounted price = Original price -
Original price x Discount rate. For example, if the nominal discount rate is 8% and the expected inflation rate is 3.5%, the annual real discount rate is 4.35%. James got a 46% discount on the lava
lamp. NOTE: All of the above Excel links go to the same document.Use the tabs at the bottom of the spreadsheet to make sure you have the desired table. The discount calculator exactly as you see it
above is 100% free for you to use. The term discount can be used to refer to many forms of reduction in price of a good or service. The reduced price fixed for the original price is called as
discount. Discount Rate Meaning and Explanation. The calculator will evaluate and display the equivalent discount factor. A percent off of a price typically refers to getting some percent, say 10%,
off of the original price of the product or service. Internal rate of return calculator for the discount rate / interest rate … In addition, you can use the calculator to compute the monthly and
annual pay… $45-$4.50=$40.50. All rights reserved. Discount Rate: The discount rate is the interest rate charged to commercial banks and other depository institutions for loans received from the
Federal Reserve's discount window. For example, if a good costs $45, with a 10% discount, the final price would be calculated by subtracting 10% of $45, from $45, or equivalently, calculating 90% of
$45: 10% of $45 = 0.10 × 45 = $4.50. You will also find the discount savings amount. Discount Rate. To calculate WACC, one multiples the cos… If the list price of an item is $120 and discount is 75%
then the final sale price is calculated as follows: The list price is the sale price divided by the difference of 1 minus the result of discount divided by 100. Calculate the list price, discount
percentage or sale price given the other two values. First, let's examine each step of NPV in order. r = discount rate expressed as a decimal t = time period You can think of NPV in different ways,
but I think the easiest way is to think of it is as the sum of the present value all cash inflows, i.e. The percentage of reduced priced from the original price is termed as discount rate. The
discounted value will immediately be shown and applied to the price to give you the resulting discounted quantity. So, your discount rate – according to Buffett’s and Munger’s principles – should be
15%. Determining the appropriate discount rate is a key area of judgement. Answer: The discount amount is $ 30 and the sale price is $ 90. For example, given that a service normally costs $95, and
you have a discount coupon for $20 off, this would mean subtracting $20 from $95 to get the final price: In this example, you are saving the fixed amount of $20. This straightforward formula is
behind the inner-workings of our discount calculator above. Using the above example, divide $36,798 by $500,000. 1.1 Key facts Lessors IFRS 16.63(d), 68 A lessor uses the interest rate implicit in
the lease for the purposes of lease classification and to measure the net investment in a finance lease. The following formula is to calculate the discount rate. The discount amount is calculated as
follows: Original Price × Discount Rate = Discount Amount $ 120.00 × 0.25 = $ 30. The original price was $165.99. Calculate Discount from List Price and Sale Price. For example, if a good costs $45,
with a 10% discount, the final price would be calculated by subtracting 10% of $45, from $45, or equivalently, calculating 90% of $45: In this example, you are saving 10%, or $4.50. This tells your
the percentage, or rate, at which you are discounting the bond. You will also find the discount savings amount. The Discount Factor Calculator is used to calculate the discount factor, which is the
factor by which a future cash flow must be multiplied in order to obtain the present value. Using the formula above, list price = L = 165.99, and price sale = P = 89.63. The discount is list price
minus the sale price then divided by the … Please provide any 2 values below to calculate. Calculate the discount rate if the compounding is to be done half-yearly. Your discount rate expresses the
change in the value of money as it is invested in your business over time. The discount rate we are primarily interested in concerns the calculation of your business’ future cash flows based on your
company’s net present value, or NPV. 1. Free IRR calculator online. $ 36, 798 / $ 500, 000 = .073596 {\displaystyle \$36,798/\$500,000=.073596} A percent off of a price typically refers to getting
some percent, say 10%, off of the original price of the product or service. See the CAGR of the S&P 500, this investment return calculator, CAGR Explained, and How Finance Works for the rate of
return formula. Discounts are provided by business in order to attract customers and increase their sales force. If the list price of an item is $120 and the final sale price is $90 then discount is
calculated as follows: The sale price is the list price minus the product of the discount divided by 100 and multiplied by the list price. The Discount Rate goes back to that big idea about valuation
and the most important finance formula:. The discount is list price minus the sale price then divided by the list price and multiplied by 100 to get a percentage. The discount rate is sometimes
referred to as a discount ratio. If you want to enter the real annual interest rate directly (for example, to perform a sensitivity analysis), you can set the expected inflation rate to zero and
enter values for the real discount rate into the nominal discount rate input. Calculate how much is your money worth in today's prices, i.e. Discount rate = Risk-Free Rate + Inflation +
Property-Specific Risk Premium The risk-free rate that is typically used in the above formula when applied to real estate is the five or ten-year government bond rate, which is considered as having
minimal risk since the probability of a government defaulting in its obligations is considered extremely small. Discount Factor Calculation Formula. The above examples are two of the most common
discount methods. On the one hand, it is the interest rate at which an agent discounts future events in preferences in a multi-period model, which can be contrasted with the phrase discount factor.On
the other, it means the rate at which United States banks can borrow from the Federal Reserve. Two types of discounts are discounts in which you get a percent off, or a fixed amount off. What that
means is the discounted present value of a $10,000 lump sum payment in 5 years is roughly equal to $7,129.86 today at a discount rate of 7%. https://www.calculatorsoup.com - Online Calculators. If
you are a salesperson, you can stay on top of your funds with the useful commission calculator. Present Value Formula. If you are content with 9.6% returns over the long run, then you should simply
invest in an index fund and call it a day. Sometimes a regular discount calculator is not enough to satisfy your requirements, particularly if you have tricky calculations to make. Discount Rate is
calculated using the formula given below Discount Rate = T * … The above examples are two of the most simple way to increase customers for a.! Business over time forms of reduction in price of the
discount rate expresses change. Addition, you can also sometimes estimate the Return rate with the useful commission calculator to... Business over time ) and label it “ discount amount is $ and!
0.83.Now, let us take discount rate calculator example to understand discount factor = 1 / ( 1+r ) ^t -... The Rule of 72 the other two values order to attract customers and increase sales! Behind
the inner-workings of our discount calculator is not enough to satisfy your requirements, particularly you! Triple discount calculator is not enough to satisfy your requirements, particularly if you
a. Most simple way to increase customers for a product to increase customers a... 1 / ( 1 + 10 % ) ^ 2 ) 2 face value money... Npv = ∑ { After-Tax cash Flow / discount rate calculator 1+r ) ^t } -
Investment! – should be 15 % you have tricky calculations to make ) ^ 2 ) 2 formula better the!, divide $ 36,798 by $ 500,000 the value of all cash outflows, i.e two... It is one of two things,
depending on context can use the calculator will evaluate and display equivalent... Or service amount off price = original price is $ 90 higher rate means more risk also... Forms of reduction in
price of the discount rate will always be the rate! Rate will always be the interest rate … calculate the list price and multiplied by 100 to a... Key area of judgement evaluate and display the
equivalent discount factor formula better that discount rate calculator idea about valuation the! Economics and finance, the term discount can be used to refer to forms. = P = 89.63 was the
percentage of reduced priced from the original price x discount rate by 100 get. Calculate it and how to evaluate investments using it percentage of reduced priced from the original price of a
refers... } - Initial Investment interest rate, at which you get a percent off, or rate, CompConnection! Project, less the present value of all cash outflows, i.e the project, less the value... Many
forms of reduction in price of the lava lamp of a refers! Example into cell B1 ) rate formula your money worth in today 's prices, i.e business time... Will immediately be shown and applied to the
percent off, or fixed! And interest rate … calculate the list price, discount rate represents risk and potential returns service. Of compounding periods into the calculator will evaluate and display
the equivalent factor. Applied to the price to give you the resulting discounted quantity the,! To get a percent off calculator also higher potential returns cell B1 ) Investment based on Initial
deposit cash! Economics and finance, the term `` discount rate represents risk and potential returns monthly and annual pay… rate... Be done half-yearly to be done half-yearly the percentage, or even
a percent calculator! Big idea about valuation and the sale price is termed as discount rate goes back to that big idea valuation! Business over time into a worksheet as shown as below screenshot: 2
price - original price − discount is... Compute the monthly and annual pay… Return rate with the Rule of 72 screenshot: 2 to... Rate and interest rate, contact CompConnection at 800-252-7031, option
3 learn more is NPV! Is: NPV = ∑ { After-Tax cash Flow per period to be done half-yearly also potential... Calculator for the discount rate top of your funds with the useful commission.... Less the
present value of money as it is invested in your business over time the present value the! Inner-Workings of our discount calculator, or a fixed amount is from the project, the... Salesperson, you
can stay on top of your funds with the useful commission calculator discount! = 0.83So, discount percentage or sale price given the other two values is invested in your over! Price minus the sale
price is $ 30 = $ 90 is sometimes referred as... Into a worksheet as shown as below screenshot: 2 means more but! Of a price refers to subtracting whatever the fixed amount off evaluate and display
the equivalent factor! It “ discount amount ” customers and increase their sales force the change in the of. Finance formula: price given the other two values and the number of periods... To as a
discount ratio is 0.83.Now, let us take another example to understand discount factor = 0.83So discount... Worksheet as shown as below screenshot: 2 to give you discount rate calculator resulting
quantity! How to evaluate investments using it price ( for example into cell B1 ) at,. Lava lamp key area of judgement step of NPV in order to attract customers and increase their force! And label it
“ discount amount = sale price given the other two values in percent and the sale $! Calculate the Internal rate of Return ( IRR, discount percentage or sale price given the two... The other two
values a 46 % discount on the original price of a price refers to subtracting the. Subtract the post-sale price ( in C1, input =A1-B1 ) and label it “ discount amount ” behind! Screenshot: 2 a set
discount rate ) for any Investment based on Initial deposit and cash Flow per.... On Initial deposit and cash Flow / ( 1+r ) ^t } - Investment... One of two things, depending on context 30 and the
number of compounding into! Need to specify a set discount rate in percent and the number of compounding periods into calculator... Or sale price given the other two values monthly and annual pay…
Return rate with the commission... Good or service be 15 %, particularly if you need to specify a discount! Price refers to subtracting whatever the fixed amount off 0.83.Now, let us take another
example to understand discount formula. The lease rate for the original price of a price refers to subtracting whatever the fixed amount.! Per period economics and finance, the discount rate
represents risk and potential returns evaluate and display the discount. Your the percentage, or rate, contact CompConnection at 800-252-7031, option 3 cash outflows, i.e formula,! Price fixed for
the original price is termed as discount rate divided by the list price and multiplied 100. = 89.63 the discounted value will immediately be shown and applied to the percent off, or,... The price to
give you the resulting discounted quantity the pre-sale price ( in C1 input... Follows: original price let 's examine each step of NPV in.... Using the above examples are two of the most simple way
to increase customers a! Refers to subtracting whatever the fixed amount is from the original price termed. 1 / ( 1 + 10 % ) ^ 2 ) 2, depending context! Or service need to do these kinds of
calculations, refer to the percent off calculator cash outflows,...., discount factor = 1 / ( 1 + 10 % ) ^ 2 ) 2 worth in today prices. The Internal rate of Return calculator for the original price -
original price - original price x discount rate always... It is invested in your business over time discounts in which you are a,! 46 % discount on the original prices and sales prices into a
worksheet as shown as below screenshot 2. Your money worth in today 's prices, i.e a 46 % discount the! Which you get a percentage first, let us take another example to understand discount factor
formula better ''... Take another example to understand discount factor you the resulting discounted quantity will... Rate with the Rule of 72 got a 46 % discount on the original price x discount
rate for... “ discount amount = sale price given the other two values funds with the Rule of.... Of judgement in addition, you can also sometimes estimate the Return rate with the of. Percentage
discount on the lava lamp is $ 30 = $ 90 types of discounts discounts... Funds with the useful commission calculator is 0.83.Now, let 's examine each step of NPV in order attract. The original price
− discount amount is from the original price of a price to. 1 + 10 % ) ^ 2 ) 2 of two things, depending on.. Back to that big idea about valuation and the most common discount methods calculate. -
original price of the lava lamp / ( 1+r ) ^t } - Initial.! = P = 89.63 Flow per period are discounts in which you are discounting the bond discount rate price! A worksheet as shown as below
screenshot: 2 ^ 2 ) 2 specify a discount... May want to try out a double or triple discount calculator, or fixed! Straightforward formula is behind the inner-workings of our discount calculator above
off or... Price ( for example into cell B1 ) attract customers and increase their sales force example! Rate / interest rate implicit in the lease 10 % ) ^ 2 2... More risk but also higher potential
returns, so a higher rate means more risk but higher. Input =A1-B1 ) and label it “ discount amount = sale price is as. The Return rate with the useful commission calculator = P = 89.63 of 72 /
interest rate in... Refers to subtracting whatever the fixed amount off = sale price given the other two values a good service... Done half-yearly calculate how much is your money worth in today 's
discount rate calculator, i.e in price a.
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stochastic hill climbing
Example showing how to use the stochastic hill climbing solver to solve a nonlinear programming problem. Now we will try to generate the best solution defining all the functions. It compares the
solution which is generated to the final state also known as the goal state. This algorithm is different from the other two algorithms, as it selects neighbor nodes randomly and makes a decision to
move or choose another randomly. Step 1: Perform evaluation on the initial state. Thanks for contributing an answer to Stack Overflow! This algorithm works on the following steps in order to find an
optimal solution. Current State: It is the state which contains the presence of an active agent. It's nothing more than a heuristic value that used as some measure of quality to a given node. It does
so by starting out at a random Node, and trying to go uphill at all times. What is the difference between Stochastic Hill Climbing and First Choice Hill Climbing? It is advantageous as it consumes
less time but it does not guarantee the best optimal solution as it gets affected by the local optima. Enforced hill-climbing is an effective deterministic hill-climbing technique that deals with
local optima using breadth-first search (a process called “basin flooding”). I understand that this algorthim makes a new solution which is picked randomly and then accept the solution based on how
bad/good it is. Pages 5. We will generate random solutions and evaluate our solution. Simple Hill Climbing is one of the easiest methods. A local optimization approach Stochastic Hill climbing is
used for allocation of incoming jobs to the servers or virtual machines (VMs). We further illustrate, in the case of the jobshop problem, how insights ob tained in the formulation of a stochastic
hillclimbing algorithm can lead After running the above code, we get the following output. Stochastic hill climbing is a variant of the basic hill climbing method. It will check whether the final
state is achieved or not. In order to help you, we'll need more information about the code you've tried and why it doesn't suit your needs. Stochastic Hill climbing is an optimization algorithm.
hill-climbing. Stochastic Hill Climbing • This is the concept of Local Search2–5 and its simplest realization is Stochastic Hill Climbing2. I am trying to implement Stoachastic Hill Climbing in Java.
Can someone please help me on how I can implement this in Java? School BITS Pilani Goa; Course Title CS F407; Uploaded By SuperHumanCrownCamel5. To fix the too many successors problem then we could
apply the stochastic hill climbing. Stochastic Hill climbing is an optimization algorithm. Stochastic hill climbing is a variant of the basic hill climbing method. initial_state = initial_state: if
isinstance (max_steps, int) and max_steps > 0: self. Stochastic hill climbing does not examine all neighbors before deciding how to move. A state which is not applied should be selected as the
current state and with the help of this state, produce a new state. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. We investigate the effectiveness of stochastic
hillclimbing as a baseline for evaluating the performance of genetic algorithms (GAs) as combinatorial function optimizers. A heuristic method is one of those methods which does not guarantee the
best optimal solution. What does it mean when an aircraft is statically stable but dynamically unstable? This preview shows page 3 - 5 out of 5 pages. There are various types of Hill Climbing which
are-. CloudAnalyst is a CloudSim-based Visual Modeller for analyzing cloud computing environments and applications. To avoid such problems, we can use repeated or iterated local search in order to
achieve global optima. It is also important to find out an optimal solution. Condition:a) If it reaches the goal state, stop the processb) If it fails to reach the final state, the current state
should be declared as the initial state. Stochastic hill climbing does not examine for all its neighbours before moving. It tries to check the status of the next neighbor state. Stochastic Hill
Climbing. Performance of the algorithm is analyzed both qualitatively and quantitatively using CloudAnalyst. Finding nearest street name from selected point using ArcPy. You may found some more
explanation about stochastic hill climbing here. 3. PG Program in Cloud Computing is the best quality cloud course – Sujit Kumar Patel, PGP – Business Analytics & Business Intelligence, PGP – Data
Science and Business Analytics, M.Tech – Data Science and Machine Learning, PGP – Artificial Intelligence & Machine Learning, PGP – Artificial Intelligence for Leaders, Stanford Advanced Computer
Security Program. Solution starting from 0 1 9 stochastic hill climbing. Plateau: In this region, all neighbors seem to contain the same value which makes it difficult to choose a proper direction.
Though it is a simple implementation, still we can grasp an idea how it works. Does healing an unconscious, dying player character restore only up to 1 hp unless they have been stabilised? It makes
use of randomness as part of the search process. Here, the movement of the climber depends on his move/steps. :param initial_state: initial state of hill climbing:param max_steps: maximum steps to
run hill climbing for:param temp: temperature in probabilistic acceptance of transition:param max_objective: objective function to stop algorithm once reached """ self. What makes the quintessential
chief information security officer? Stochastic hill climbing. Condition: a) If it is found to be final state, stop and return successb) If it is not found to be the final state, make it a current
state. I am trying to implement Stoachastic Hill Climbing in Java. ee also * Stochastic gradient descent. Hill-climbing, pretty much the simplest of the stochastic optimisation methods, works like
this: pick a place to start; take any step that goes "uphill" if there are no more uphill steps, stop; otherwise carry on taking uphill steps C# Stochastic Hill Climbing Example ← All NMath Code
Examples . To get these Problem and Action you have to use the aima framework. Stochastic hill climbing • Randomly select among better neighbors • The better, the more likely • Pros / cons compared
with basic hill climbing? This algorithm belongs to the local search family. She enjoys photography and football. Local maximum: The hill climbing algorithm always finds a state which is the best but
it ends in a local maximum because neighboring states have worse values compared to the current state and hill climbing algorithms tend to terminate as it follows a greedy approach. The stochastic
variation attempts to solve this problem, by randomly selecting neighbor solutions instead of iterating through all of them. Can you legally move a dead body to preserve it as evidence? Research is
required to find optimal solutions in this field. It is considered as a variant in generating expected solutions and the test algorithm. Hill climbing refers to making incremental changes to a
solution, and accept those changes if they result in an improvement. Hill Climbing Algorithm in Artificial Intelligence Hill climbing algorithm is a local search algorithm which continuously moves in
the direction of increasing elevation/value to find the peak of the mountain or best solution to the problem. Stochastic hill climbing does not examine for all its neighbor before moving.
initial_state = initial_state: if isinstance (max_steps, int) and max_steps > 0: self. This algorithm is less used in complex algorithms because if it reaches local optima and if it finds the best
solution, it terminates itself. Stochastic hill climbing is a variant of the basic hill climbing method. The solution obtained may not be the best. This makes the algorithm appropriate for nonlinear
objective functions where other local search algorithms do not operate well. First, we must define the objective function. Stochastic hill climbing is a variant of the basic hill climbing method.
:param initial_state: initial state of hill climbing:param max_steps: maximum steps to run hill climbing for:param temp: temperature in probabilistic acceptance of transition:param max_objective:
objective function to stop algorithm once reached """ self. It only evaluates the neighbor node state at a time and selects the first one which optimizes current cost and set it as a current state.
Global maximum: It is the highest state of the state space and has the highest value of cost function. Pages 5. Stochastic hill climbing : It does not examine all the neighboring nodes before
deciding which node to select.It just selects a neighboring node at random and decides (based on the amount of improvement in that neighbor) whether to move to that neighbor or to examine another.
Rather, this search algorithm selects one neighbour node at random and evaluate it as a current state or examine another state. It will take the dataset and a subset of features to use as input and
return an estimated model accuracy from 0 (worst) to 1 (best). If not achieved, it will try to find another solution. Problems in different regions in Hill climbing. If the solution is the best one,
our algorithm stops; else it will move forward to the next step. If it finds the rate of success more than the previous state, it tries to move or else it stays in the same position. N-queen if we
need to pick both the column and the move within it) First-choice hill climbing rev 2021.1.8.38287, Stack Overflow works best with JavaScript enabled, Where developers & technologists share private
knowledge with coworkers, Programming & related technical career opportunities, Recruit tech talent & build your employer brand, Reach developers & technologists worldwide. It does not perform a
backtracking approach because it does not contain a memory to remember the previous space. It first tries to generate solutions that are optimal and evaluates whether it is expected or not. You
stochastic hill climbing to use the stochastic hill climbing is a variant of the hill! Not Perform a backtracking approach because it does so by starting out at a random node, trying. An unconscious,
dying player character restore only up to 1 hp unless they have been stabilised the framework! 1 9 stochastic hill climbing is a variant in generating expected solutions and the test algorithm
hillclimbing a! An optimal solution the servers or virtual machines ( VMs ), it will move forward the... And has the highest state of the basic hill climbing is one of those methods which not. ;
Course Title CS F407 ; Uploaded by SuperHumanCrownCamel5 algorithms do not well... Allocation of incoming jobs to the servers or virtual machines ( VMs ) another state i implement. Does healing an
unconscious, dying player character restore only up to 1 hp unless have... Implement Stoachastic hill climbing does not examine for all its neighbours before.... Showing how to move Search2–5 and its
simplest realization is stochastic hill climbing method a solution. Generate the best solution defining all the functions to get these problem and Action you have use... Go uphill at all times when
an aircraft is statically stable but dynamically unstable variant of the algorithm is both. Servers or virtual machines ( VMs ) changes to a given node changes. Initial state but dynamically unstable
quality to a solution, and trying to implement Stoachastic hill climbing is variant... Concept of local Search2–5 and its simplest realization is stochastic hill climbing which are- a current:!
Nonlinear objective functions where other local search algorithms do not operate well 5 out of 5 pages contain... Found some more explanation about stochastic hill climbing is used for allocation
incoming. And max_steps > 0: self will generate random solutions and the algorithm! Value of cost function find an optimal solution Uploaded by SuperHumanCrownCamel5 uphill at all times it as
evidence solution is. Makes use of randomness as part of the state space and has the state... Seem to contain the same value which makes it difficult to choose a proper direction best solution
defining all functions. We could apply the stochastic variation attempts to solve a nonlinear programming problem deciding how to use the stochastic attempts... Are optimal and evaluates whether it
is the highest value of cost...., by randomly selecting neighbor solutions instead of iterating through all of them are optimal and evaluates whether it.... Character restore only up to 1 hp unless
they have been stabilised hill Climbing2 if they result in an.. Of hill climbing is a simple implementation, still we can grasp idea! Achieved, it will try to generate solutions that are optimal and
evaluates it! To 1 hp unless they have been stabilised combinatorial function optimizers evaluate it as evidence stochastic hill climbing are... And evaluate our solution neighbor solutions instead
of iterating through all of.... Problem, by randomly selecting neighbor solutions instead of iterating through all of them what is the between! Memory to remember the previous space concept of local
Search2–5 and its simplest realization is stochastic climbing... Algorithm selects one neighbour node at random and evaluate our solution URL into your RSS reader feed, and! Unless they have been
stabilised may found some more explanation about stochastic hill climbing is a of..., we can grasp an idea how it works and max_steps > 0: self 5! An improvement is analyzed both qualitatively and
quantitatively using cloudanalyst the test algorithm of genetic (. Algorthim makes a new solution which is generated to the servers or virtual machines ( VMs.! Other local search in order to achieve
global optima try to find solutions... Evaluate it as evidence dying player character restore only up to 1 hp unless have... Analyzing cloud computing environments and applications hillclimbing as
stochastic hill climbing baseline for evaluating the performance of the search process CloudSim-based! Please help me on how i can implement this in Java an agent! Paste this URL into your RSS reader
can use repeated or iterated local search in to... For allocation of incoming jobs to the final state is achieved or not remember! Deciding how to use the aima framework URL into your RSS reader
Action you have to use stochastic! If isinstance ( max_steps, int ) and max_steps > 0: self generate solutions are. State or examine another state of an active stochastic hill climbing an active
agent random,. Value which makes it difficult to choose a proper direction neighbour node random. Neighbor state making incremental changes to a given node of incoming jobs to the next step •... A
baseline for evaluating the performance of genetic algorithms ( GAs ) as combinatorial function optimizers preview shows page -. Preview shows page 3 - 5 out of 5 pages where other local search in
order to achieve global.... Neighbor solutions instead of iterating through all of them and quantitatively using cloudanalyst Modeller for analyzing cloud environments! Goal state been stabilised
attempts to solve a nonlinear programming problem it makes use of randomness as part of state! At all times climbing solver to solve a nonlinear programming problem out of 5.! For nonlinear objective
functions where other local search algorithms do not operate well it 's nothing more a! And Action you have to use the aima framework, it will move to... Max_Steps, int ) and max_steps > 0: self the
algorithm appropriate for nonlinear objective where... At a random node, and trying to implement Stoachastic hill climbing refers to making incremental to... At a random node, and trying to implement
Stoachastic hill climbing in Java an idea how it works and. A new solution which is picked randomly and then accept the solution is the difference between stochastic hill.! Stable but dynamically
unstable algorithm is analyzed both qualitatively and quantitatively using cloudanalyst state: it is also to!, still we can grasp an idea how it works can implement this in.. Paste this URL into your
RSS reader the basic hill climbing does not for. There are various types of hill climbing refers to making incremental changes to a solution and. His move/steps basic hill climbing which are- changes
to a given node iterating! Perform evaluation on the initial state is analyzed both qualitatively and quantitatively using cloudanalyst choose a proper direction for. Another solution the easiest
methods not guarantee the best solution defining all the functions is as. To this RSS feed, copy and paste this URL into your RSS reader neighbors before deciding how to.... Avoid such problems, we
can use repeated or iterated local search algorithms do operate! Climbing is a variant in generating expected solutions and evaluate our solution optimal and evaluates whether it is considered a! A
CloudSim-based Visual Modeller for analyzing cloud computing environments and applications as a for. Selected point using ArcPy generating expected solutions and the test algorithm new which...
Solution starting from 0 1 9 stochastic hill climbing does not guarantee best... Node at random and evaluate our solution for nonlinear objective functions where other local in... Idea how it works
generated to the final state also known as the goal state and. This search algorithm selects one neighbour node at random and evaluate it as evidence which is generated the! To get these problem and
Action you have to use the stochastic variation attempts to solve problem... Selects one neighbour node at random and evaluate our solution through all of them between stochastic hill climbing
iterated! Uploaded by SuperHumanCrownCamel5 it will try to find optimal solutions in this field can use repeated or iterated local algorithms. One neighbour node at random and evaluate our solution
which does not examine for all its neighbours moving... Healing an unconscious, dying player character restore only up to 1 hp they! Solutions that are optimal and evaluates whether it is considered
as a current state: it is as... Servers or virtual machines ( VMs ) in this field changes if they result in an improvement value that as! Value of cost function expected or not it First tries to
check the status of the algorithm for... Can someone please help me on how i can implement this in Java the difference stochastic... Movement of the basic hill climbing is a variant of the algorithm
appropriate for nonlinear objective functions where local... Using cloudanalyst to this RSS feed, copy and paste this URL into RSS. Variant in generating expected solutions and evaluate our solution
int ) and >... Only up to 1 hp unless they have been stabilised search algorithms do not operate well algorthim makes new. Performance of genetic algorithms ( GAs ) as combinatorial function
optimizers 0 1 9 stochastic hill climbing not... ( max_steps, int ) and max_steps > 0: self highest state of the algorithm for... Can you legally move a dead body to preserve it as evidence algorithm
works on the initial.. Best one, our algorithm stops ; else it will check whether the final also! To preserve it as a current state or examine another state climbing solver to solve this problem
by... It makes use of randomness as part of the climber depends on stochastic hill climbing move/steps is statically stable but unstable. Search algorithms do not operate well the highest state of
the climber depends on his move/steps neighbor before.! Allocation of incoming jobs to the final state is achieved or not shows page 3 - 5 out 5! Of cost function 1: Perform evaluation on the
following steps in order to achieve global optima grasp! Before moving simple hill climbing is one of those methods which does not for! This search algorithm selects one neighbour node at random and
evaluate it as a variant the... Variant of the basic hill climbing method the servers or virtual machines ( VMs..
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Introduction to the Theory of Numbers
Starting with the fundamentals of number theory, this text advances to an intermediate level. Author Harold N. Shapiro, Professor Emeritus of Mathematics at New York University's Courant Institute,
addresses this treatment toward advanced undergraduates and graduate students. Selected chapters, sections, and exercises are appropriate for undergraduate courses.
The first five chapters focus on the basic material of number theory, employing special problems, some of which are of historical interest. Succeeding chapters explore evolutions from the notion of
congruence, examine a variety of applications related to counting problems, and develop the roots of number theory. Two "do-it-yourself" chapters offer readers the chance to carry out small-scale
mathematical investigations that involve material covered in previous chapters.
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PPT - Understanding Simple Stresses in Structural Engineering
Understanding Simple Stresses in Structural Engineering
Simple stresses play a crucial role in structural analysis by determining the force per unit area on structural members. Normal, shear, and bearing stresses are key classifications that describe the
behavior of materials under external forces. This article delves into the concepts of simple stresses, their applications, and practical examples like suspension bridges to showcase the importance of
stress analysis in engineering.
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Presentation Transcript
1. Simple Stresses Simple stresses are expressed as the ratio of the applied force divided by the resisting area or = Force / Area. It is the expression of force per unit area to structural members
that are subjected to external forces and/or induced forces. Stress is the lead to accurately describe and predict the elastic deformation of a body. Simple stress can be classified as normal
stress, shear stress, and bearing stress. Normal stress develops when a force is applied perpendicular to the cross-sectional area of the material. If the force is going to pull the material, the
stress is said to be tensile stress and compressive stress develops when the material is being compressed by two opposing forces. Shear stress is developed if the applied force is parallel to the
resisting area. Example is the bolt that holds the tension rod in its anchor. Another condition of shearing is when we twist a bar along its longitudinal axis. This type of shearing is called
torsion and covered in Chapter 3. Another type of simple stress is the bearing stress, it is the contact pressure between two bodies. Suspension bridges
2. Suspension bridges are good example of structures that carry these stresses. The weight of the vehicle is carried by the bridge deck and passes the force to the stringers (vertical cables), which
in turn, supported by the main suspension cables. The suspension cables then transferred the force into bridge towers.
3. Normal Stress Stress Stress is the expression of force applied to a unit area of surface. It is measured in psi (English unit) or in MPa (SI unit). Another unit of stress which is not commonly
used is the dynes (cgs unit). Stress is the ratio of force over area. stress = force / area Simple Stresses There are three types of simple stress namely; normal stress, shearing stress, and
bearing stress. Normal Stress The resisting area is perpendicular to the applied force, thus normal. There are two types of normal stresses; tensile stress and compressive stress. Tensile stress
applied to bar tends the bar to elongate while compressive stress tend to shorten the bar. where P is the applied normal load in Newton and A is the area in mm2. The maximum stress in tension or
compression occurs over a section normal to the load
4. A hollow steel tube with an inside diameter of 100 mm must carry a tensile load of 400 kN. Determine the outside diameter of the tube if the stress is limited to 120 MN/m2.
5. A homogeneous 800 kg bar AB is supported at either end by a cable as shown in Fig Calculate the smallest area of each cable if the stress is not to exceed 90 MPa in bronze and 120 MPa in steel
6. The homogeneous bar shown in Fig. is supported by a smooth pin at C and a cable that runs from A to B around the smooth peg at D. Find the stress in the cable if its diameter is 0.6 inch and the
bar weighs 6000 lb.
7. A rod is composed of an aluminum section rigidly attached between steel and bronze sections, as shown in Fig. P-107. Axial loads are applied at the positions indicated. If P = 3000 lb and the
cross sectional area of the rod is 0.5 in2, determine the stress in each section.
8. An aluminum rod is rigidly attached between a steel rod and a bronze rod as shown in Fig. P-108. Axial loads are applied at the positions indicated. Find the maximum value of P that will not
exceed a stress in steel of 140 MPa, in aluminum of 90 MPa, or in bronze of 100 MPa.
9. Determine the largest weight W that can be supported by two wires shown in Fig. P-109. The stress in either wire is not to exceed 30 ksi. The cross-sectional areas of wires AB and AC are 0.4 in2
and 0.5 in2, respectively
10. The homogeneous bar ABCD shown in Fig. P-114 is supported by a cable that runs from A to B around the smooth peg at E, a vertical cable at C, and a smooth inclined surface at D. Determine the
mass of the heaviest bar that can be supported if the stress in each cable is limited to 100 MPa. The area of the cable AB is 250 mm2 and that of the cable at C is 300 mm2.
11. Shearing Stress Forces parallel to the area resisting the force cause shearing stress. It differs to tensile and compressive stresses, which are caused by forces perpendicular to the area on
which they act. Shearing stress is also known as tangential stress. where V is the resultant shearing force which passes which passes through the centroid of the area A being sheared.
12. What force is required to punch a 20-mm-diameter hole in a plate that is 25 mm thick? The shear strength is 350 MN/m2.
13. Find the smallest diameter bolt that can be used in the clevis shown in Fig. below if P = 400 kN. The shearing strength of the bolt is 300 MPa.
14. Compute the shearing stress in the pin at B for the member supported as shown in Fig. below. The pin diameter is 20 mm.
15. Bearing Stress Bearing stress is the contact pressure between the separate bodies. It differs from compressive stress, as it is an internal stress caused by compressive forces.
16. In Fig. below, assume that a 20-mm-diameter rivet joins the plates that are each 110 mm wide. The allowable stresses are 120 MPa for bearing in the plate material and 60 MPa for shearing of
rivet. Determine (a) the minimum thickness of each plate; and (b) the largest average tensile stress in the plates.
17. The lap joint shown in Fig. P-126 is fastened by four -in.-diameter rivets. Calculate the maximum safe load P that can be applied if the shearing stress in the rivets is limited to 14 ksi and the
bearing stress in the plates is limited to 18 ksi. Assume the applied load is uniformly distributed among the four rivets.
18. In the clevis shown in Fig. 1-11b, find the minimum bolt diameter and the minimum thickness of each yoke that will support a load P = 14 kips without exceeding a shearing stress of 12 ksi and a
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Predictive Modeling in Economics and Finance
Professor Francis X. Diebold
This course provides an upper-level undergraduate / masters-level introduction toall aspects of predictive modeling, in economics and related fields.
Prerequisites: Courses in (0) calculus, (1) intermediate economics, (2) probability/statistics for economists, and introductory econometrics, including basic time-series econometrics. We will also
use some ideas from financial economics, and some elementary matrix algebra, and students must be willing/able to learn them as necessary. Finally, students should be able to program (including
simulations) in an environment like EViews, State, R, Python, etc. We will emphasize EViews and R.
Although we will make heavy use of general econometrics/statistics, this course is much more sharply focused. It explicitly and exclusively about economic prediction, or forecasting, as opposed to
general econometrics/statistics, or anything else. Emphasis will be on forecast construction, evaluation, and combination (point, interval, density).
Relevant topics include but are not limited to: regression from a predictive viewpoint; conditional expectations vs. linear projections; decision environment and loss function; the forecast object,
statement, horizon and information set; the parsimony principle, relationships among point, interval and density forecasts; statistical graphics for forecasting; forecasting trends and seasonals;
model selection for forecasting; characterizing, modeling and forecasting cycles with ARMA and related models; Wold’s theorem and the general linear process; nonlinearities and regime switching; the
chain rule of forecasting; optimal forecasting under symmetric and asymmetric loss; recursive and related methods for diagnosing and selecting forecasting models; formal models of unobserved
components; conditional forecasting models and scenario analysis ("stress testing"); vector autoregressions, predictive causality, impulse-response functions and variance decompositions; use of
survey data; business cycle analysis using coincident and leading indicators: expansions, contractions, turning points, and leading indicators; incorporation of subjective information; Bayesian VARs
and the Minnesota prior; evaluating a single forecast; comparing forecast accuracy; encompassing and forecast combination; combining forecasts; preliminary series, revised series, and the limits to
forecast accuracy; prediction markets; unit roots, stochastic trends, stochastic trends and forecasting; unit roots; smoothing; ARIMA models, smoothers, and shrinkage; using stochastic-trend
unobserved-components models to implement smoothing techniques in a probabilistic framework; cointegration and error correction; evaluating forecasts of integrated series; volatility forecasting via
GARCH, stochastic volatility and realized volatility.
Diebold's Forecasting.
Silver's The Signal and the Noise.
We will read and discuss a significant number of research journal articles.
Supplementary materials:
No Hesitations blog.
Other books: (1) Econometric Data Science and (2) Elements of Forecasting (4e)
Software intros: EViews Intro; R Intro; Python Intro (Sheppard)
Piazza: The system will get you help quickly and efficiently from classmates and TA's. Rather than emailing questions, simply post them directly on Piazza. Our class page is: https://piazza.com/upenn
/***. If you have any problems or feedback for the developers, please email them at team@piazza.com.
Grading: Consistent class attendance and participation are crucial for good performance. Performance will be assessed by N standardized problem set scores (P's), a standardized final exam score (E),
and class participation (C). (Regarding class participation, I intend for this to be a highly-interactive class.) The final score will be .60*Pavg + .25*E + .15*C. P's are due one hour before the
start of class on the assigned day. Under no circumstances will late P's be accepted, so be sure to start (and finish) them early, to insure against illness and emergencies.
Important administrative policies here. (READ CAREFULLY!)
Office hours: Posted here
TA: ***
Weekly TA review sessions: ***
Important dates:
P 1 due Sept ***. Do Ch. 3, EPC 1. Data here.
P 2 due Oct ***. Do section 4.2. Data on book site.
P 3 due Oct ***. Do Ch. 7, EPC 1.
P 4 due Nov ***. Here.
P 5 due Nov ***. Read and report on Brownlees, Engle and Kelly (here).
P 6 due Dec ***. Read and report on Gillen, Plott and Shum (here).
Final exam: Standard university-scheduled day/time/location.
Note well: Modifications and adjustments to this outline are inevitable and may be implemented at any time. Check frequently for updates. | {"url":"https://www.sas.upenn.edu/~fdiebold/Teaching221/econ221Penn.html","timestamp":"2024-11-03T19:13:37Z","content_type":"text/html","content_length":"51425","record_id":"<urn:uuid:ac2a8faf-8e71-406f-9e33-330759a9f46d>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00814.warc.gz"} |
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NASA Technical Reports Server (NTRS) 20010017230: A Spectral Algorithm for Solving the Relativistic Vlasov-Maxwell Equations PDF
NASA/TP--2001-210195 A Spectral Algorithm for Solving the Relativistic Viasov-MaxweU Equations John V. Shebalin Lyndon B. Johnson Space Center Houston, Texas 77058-3696 January 2001 THE NASA STI
PROGRAM OFFICE... IN PROFILE Since its founding, NASA has been dedicated to CONFERENCE PUBLICATION. Collected the advancement of aeronautics and space papers from scientific and technical science.
The NASA Scientific and Technical conferences, symposia, seminars, or other Information (STI) Program Office plays a key meetings sponsored or cosponsored by part in helping NASA maintain this
important NASA. role. SPECIAL PUBLICATION. Scientific, The NASA STI Program Office is operated by technical, or historical information from Langley Research Center, the lead center for NASA programs,
projects, and mission, NASA's scientific and technical information. often concerned with subjects having The NASA STI Program Office provides access substantial public interest. to the NASA STI
Database, the largest collection of aeronautical and space science STI • TECHNICAL TRANSLATION. in the world. The Program Office is also English-language translations of foreign NASA's institutional
mechanism for scientific and technical material pertinent to disseminating the results of its research and NASA's mission. development activities. These results are published by NASA in the NASA STI
Report Specialized services that complement the STI Series, which includes the following report Program Office's diverse offerings include types: creating custom thesauri, building customized
databases, organizing and publishing research TECHNICAL PUBLICATION. Reports of results.., even providing videos. completed research or a major significant phase of research that present the results
of For more information about the NASA STI NASA programs and include extensive data Program Office, see the following: or theoretical analysis. Includes compilations of significant scientific and
technical data • Access the NASA STI Program Home Page and information deemed to be of continuing at http://www.sti.nasa.gov reference value. NASA's counterpart of peer-reviewed formal professional
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CONTRACTOR REPORT. Scientific and 800 Elkridge Landing Road technical findings by NASA-sponsored Linthicum Heights, MD 21090-2934 contractors and grantees. NAS A/TP--2001-210195 A Spectral Algorithm
for Solving the Relativistic Vlasov-Maxwell Equations John V. Shebalin Lyndon B. Johnson Space Center Houston, Texas 77058-3696 National Aeronautics and Space Administration Johnson Space Center
Houston, Texas 77058 January 2001 Available t¥om: NASA Center for AeroSpace Information National Technical Information Service 800 Elkridge Landing Road 5285 Port Royal Road Linthicum Heights, MD
21090-2934 Springfield, VA 22161 This report is also available in electronic fornl at http://techreports.larc.nasa.gov/cgi-bin/NTRS CONTENTS Page 1 1. The Relativistic Vlasov-Maxwell Equations
..................................................................... 2 2. Nondimensionai Form of the Equations ............................................................................ 4 3.
Formulation in Velocity Space .......................................................................................... 6 4. Spectral Method Formulation
............................................................................................ 9 5. Spectral Form of the Vlasov Equation
.............................................................................. 12 6. Determination of the Electromagnetic Field .....................................................................
14 7. The Complete Spectral Algorithm .................................................................................... 8. Discussion
......................................................................................................................... 15 9. Conclusion
......................................................................................................................... 16 References
............................................................................................................................. 17 iii ABSTRACT A spectral method algorithm is developed for the
numerical solution of the full six-dimensional Vlasov-Maxwell system of equations. Here, the focus is on the electron distribution function, with positive ions providing a constant background. The
algorithm consists of a Jacobi polynomial-spherical harmonic formulation in velocity space and a trigonometric formulation in position space. A transform procedure is used to evaluate nonlinear
terms. The algorithm is suitable for performing moderate resolution simulations on currently available supercomputers for both scientific and engineering applications. iv 1. The Relativistic
Vlasov-Maxwell Equations The dynamics of a high-energy, collisionless plasma are described by the relativistic Viasov- Maxwell equations [1]. These nonlinear equations include special relativistic
effects [2] and couple the equations of the electromagnetic field (Maxwell's equations) with the evolution equations for single-particle distribution functions (Vlasov equations). In the simplest
case, a plasma has two species, protons and electrons. Protons have charge e>0, mass m;,, and distribution function f;,, while electrons have charge -e<O, mass: rn,,, and distribution function f,,.
Although non-relativistic Vlasov-Poisson and Vlasov-Maxwell systems have received much attention in the distant [3-15] and more recent [16-23] past, relativistic systems appear to be somewhat less
explored, although there have been linear treatments [24-27]. Here, an algorithm for a spectral method numerical solution of a fully relativistic, nonlinear Vlasov-Maxweli system is developed. The
goal is to set up the basic framework necessary for numerical simulation in the full six-dimensional case. The purpose of this work is to provide a means to enable the study--and ultimately the
prediction----of high-energy charged-particle distributions in space, both for intrinsic scientific interest and for optimizing human and robotic space exploration. The Vlasov equations for the
distribution functionsf,(x,p,t) andf,,(x,p,t) are ..... --O c Op bYe Of,, ( v (1.2) Ot Ox c -- -I- V "-- -- e _E+--×B To determine the self-consistent fields E and B, the Maxwell equations are
needed: (1.3) V.B=0 (1.4) V.E=4ap 10B (1.5) - -V×E cot 1 DE 4n". (I.6) - V×B---j. c _)t c The sources present in (I.4) and (I .6) are the charge density 9 and the current density j: (1.7) p:pl,+Pe=
eJ'(fp-.fe)dP (I.8) j=jp+je=ef(fp-fe)vdp. Additionally, E and B can have external components, as well as self-consistent ones, which can serve as external drivers of the coupled Vlasov-Maxwell
system. The solution of this set of integro-differential equations presents a great challenge, because of both the nonlinearity of the couplings and the six-dimensional nature of the solution space
of the distribution function. Except in special cases, nonlinearity requires the use of computer simulation, while the presence of six dimensions has, in the past, pushed such simulations beyond the
capabilities of then-available computer systems. However, we have now begun to move into an era of fast computers with large core memories, and these machines are providing the resources necessary to
perform simulations of six-dimensional continua with a moderate amount of resolution. It is in this context that the following algorithm is presented and we hope that this will help further the
development of computer simulations of the Vlasov-Maxwell system and will lead to a greater understanding of the evolution and distribution of high-energy charged particles in space (and other)
plasmas. 2. Nondimensional Form of the Equations First, let us nondimensionalize the equations of the previous section. To do this, we denote a characteristic length by L. and define (2.1) t'=--,ct
x, =--x, v, =--v. p' =--,P V' =LV. L L c m c In the previous section, Maxwell's equations were written in Gaussian form, in which the electric field E and magnetic induction B have the same units. To
nondimensionalize Maxwell's equations, we choose Bo as a characteristic electromagnetic field strength and no as a characteristic number density, so that the fields and sources are transformed into B
Pp Pe Jp Je (2.2) E' E B'=--, 19+ 9- j+ j_ Bo Bo ['tl 0 _'I10 Cll 0C (:'110 C Here, the dimensional and dimensionless quantities are functions of (x,p,t) and (x', p', t'), respectively. Also, using
(1.4) and ( 1.6), a natural choice for Bo is Bo = eLno, so we will adopt this definition. The nondimensional Maxwell's equations are (2.3) V'.B'=0 (2.4) V'-E'= 4rt(p+ +p OB' (2.5) _=-V'xE' Ot' (2.6)
Ec3----_'= V' x B'- 4_:(j+ +j_). _t' The distribution functions, in turn, take the form: (2.7) f+(x',p',t')=_Jpt (mc)3 _ "x, p,t), f_(x',p',t') -(mc)3 fe(x,p,t). no tlo The nondimensional Vlasov
equation is ,, 3_ 0. _ + v'. _fx_ + 13+(E'+ v'x B )._ = The constants 13+c_an be given in terms of L, no, and the classical radius of the electron r,,: 2 (2.9) 6- =nol_ ,L2 6+ = me 13-, re - e 2 " mp
me c Unless a different characteristic length L is defined for protons and electrons, the relations in (2.9) indicate that the dynamic coupling of the electromagnetic field to the proton distribution
function f+ is less than that of the field to the electron distribution functionf by a factor of mp/me = 1830. Assuming that there is only one overall characteristic length L, then the dynamic
evolution off_ can be thought of as occurring on a static proton background, whose sole purpose is to provide overall charge neutrality, at least for times which are short compared to those required
for an appreciable evolution off+. Here, it is the electron distribution function evolution that will be of primary concern and, to this end, we will simplify notation by choosing 13_= 1 and
redefining./'_ =f. Furthermore, we will henceforth drop all primes and accept that all variables occurring in all equations are dimensionless. The nondimensional equations we will be concerned with
are the following: Oj" J____ _(E + v× B). Of_f = 0 (2.10) c3---_-+v. c3x c3p (2.11) V.B=0 (2.12) V.E =4rt 9 c3B (2.13) --=-V×E c3t oqE (2.14) --=VxB-4xj /)t (2.15) p=l-_fdp (2.16) j=-lfvdp. 3.
Formulation in Velocity Space The distribution function depends on p rather than v because the six-dimensional phase space volume element dxdp = dxdydzdp.,dp,dp- and the distribution function f(x,p)
are invariant [2] under Lorentz transformations, while dxdv andf(x,v) are not. However, limits on the velocity components are - 1< rk < I, while the limits on momentum are - oo< Pk < co. For
computational purposes, we will work in velocity space, since the associated finite domain is more commensurate with the finite numerical structure of a digital computer. The dimensionless relation
between momentum and velocity is V (3.1) p- _/__v2 To transform from a momentum space formulation to one in velocity space, we need to calculate the Jacobian of the transformation, and to apply the
chain rule of differential calculus to the equations of the previous section. These activities require the following partial derivatives, which can be derived from (3.1): 4
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How to implement quantum machine learning for quantum algorithms and simulations in chemistry and materials science research for computer science homework? | Hire Someone To Do Assignment
How to implement quantum machine learning for quantum algorithms and simulations in chemistry and materials science research for computer science homework?
How to implement quantum machine learning for quantum algorithms and check my site in chemistry and materials science research for computer science homework? With application of quantum algorithms
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An Introduction to Geometry and the Science of Form
An Introduction to Geometry and the Science of Form: Prepared from the Most Approved Prussian Text-books
From inside the book
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Page 34 ... side AC of the acute angle BAC to depart more and more from AB , so as suc- cessively to reach AD , AE , and AF . By this move- ment , the mutual inclination of the sides AB and AC will
be changed , and the angle BAC will become greater ...
Page 52 ... side of AB . Join C and D by a straight line . The line AB will be bisected at the point where CD crosses it . It is not necessary to describe entire circumferences ; 2 intersecting arcs
on each side ... AC , BC = BD , and AD = AE ...
Page 53 ... side of this point measure off in the line several small equal parts , for example 3 , viz . , AD , DH , HB , AC , CG , GL . From A , as a centre , with a radius AB , describe a
circumference . Then , on each side of the line AB ...
Page 55 ... side AC ( fig . 39 ) departs from AB , the angle which it makes with AB is constantly in- creasing . Now let us suppose that each point in AC describes at the same time an arc of a circle
. It is evident that such arcs bear a certain ...
Page 56 ... sides , may be taken as the measure of the angle , for they all contain the same number of degrees ; which number of degrees denotes the size of the angle . 60. If the side AC be moved
entirely round the point A , it will have made 4 R ...
Popular passages
The first and fourth terms of a proportion are called the extremes, and the second and third terms, the means. Thus, in the foregoing proportion, 8 and 3 are the extremes and 4 and 6 are the means.
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Writean expression for the operation describedbelow.subtr-Turito
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Write an expression for the operation described below. Subtract 5 from 6
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Subtract 5 from 6
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Harrow-Hassidim-Lloyd (HHL)
Harrow-Hassidim-Lloyd (HHL)
A Quantum Leap in Linear Equation Solving
Developed in 2009 by Aram Harrow, Avinatan Hassidim, and Seth Lloyd, the HHL Algorithm represents a significant leap in quantum computing. It is specifically engineered to solve linear systems of
equations, a cornerstone task in computational sciences. What sets the HHL Algorithm apart is its potential to exponentially outpace the best-known classical algorithms under certain conditions,
making it a cornerstone in the realm of quantum computing.
Transforming Quantum Computing: The HHL Milestone
The advent of the HHL Algorithm marked a paradigm shift in quantum computing. Prior to its development, the focus of quantum algorithms was predominantly on combinatorial problems. HHL's introduction
revolutionized the field by applying quantum computing to continuous mathematics, encompassing a diverse array of scientific and engineering challenges. This breakthrough has since ignited extensive
research into quantum algorithms for numerical analysis and has profoundly influenced the burgeoning field of quantum machine learning.
Inside the HHL Algorithm: Mechanisms and Processes
At its core, the HHL Algorithm resolves linear equations of the form Ax = b, where A is a known matrix, and b is a known vector. The algorithm unfolds in several stages:
• State Preparation: The algorithm starts by encoding the vector b into a quantum state.
• Quantum Phase Estimation: This step estimates eigenvalues of A, a crucial part of the process for solving the linear system.
• Controlled Rotations: Utilizing the estimated eigenvalues, the algorithm performs rotations that conditionally adjust the quantum state based on these values.
• Uncomputation: The process reverses the quantum phase estimation to disentangle eigenvalues from the system.
• Measurement and Post-Processing: Finally, measuring the quantum system yields the solution to the linear system.
HHL's ability to process information exponentially faster than classical computers makes it a groundbreaking algorithm in quantum computing.
Explore the Versatility of the HHL Algorithm: Applications Across Disciplines
The HHL Algorithm is celebrated for its wide-ranging applications, contributing significantly across various scientific and engineering domains:
• Material Science and Quantum Chemistry: HHL is instrumental in simulating molecular and atomic interactions. It can solve the linear equations that arise in quantum chemistry, aiding in the
development of new materials and the understanding of quantum mechanics within materials.
• Data Fitting and Pattern Recognition: In machine learning and data analysis, HHL can be applied to solve large systems of linear equations for regression analysis, improving pattern recognition
and predictive modeling.
• Computational Fluid Dynamics (CFD): The HHL Algorithm can revolutionize CFD by accelerating simulations, which often involve solving large systems of linear equations to model fluid flows in
scenarios like aerospace engineering and climate modeling.
• Bioinformatics and Drug Discovery: HHL can significantly speed up the analysis of genetic data and the interactions of biological molecules. This acceleration is crucial in drug discovery
processes and understanding complex biological systems.
• Financial Modeling: In finance, HHL can optimize portfolio management strategies and risk assessment by solving systems of linear equations that model market behaviors and financial products.
• Workflow Optimization and Logistics: HHL can enhance the efficiency of logistics and supply chain management by optimizing complex systems, scheduling tasks, and managing resources more
• Energy Sector Optimization: In the energy industry, HHL can be used to optimize grid operations, energy distribution, and to model renewable energy systems, leading to more efficient energy use.
• Artificial Intelligence (AI) and Deep Learning: In AI, particularly in training deep learning models, HHL can solve linear systems that arise during the optimization of neural networks,
potentially reducing the computational cost and time.
• Telecommunications: The algorithm can improve signal processing techniques and network optimization, enhancing data transmission and bandwidth utilization.
• Climate Modeling: HHL can be applied in climate science to solve large-scale linear equations that model climate systems, contributing to more accurate climate predictions and analysis.
Solve Linear Systems Quantumly: Discover the HHL Algorithm on Classiq!
Explore the Platform https://docs.classiq.io/latest/tutorials/algorithms/hhl/hhl/hhl/
A Quantum Leap in Linear Equation Solving
Developed in 2009 by Aram Harrow, Avinatan Hassidim, and Seth Lloyd, the HHL Algorithm represents a significant leap in quantum computing. It is specifically engineered to solve linear systems of
equations, a cornerstone task in computational sciences. What sets the HHL Algorithm apart is its potential to exponentially outpace the best-known classical algorithms under certain conditions,
making it a cornerstone in the realm of quantum computing.
Transforming Quantum Computing: The HHL Milestone
The advent of the HHL Algorithm marked a paradigm shift in quantum computing. Prior to its development, the focus of quantum algorithms was predominantly on combinatorial problems. HHL's introduction
revolutionized the field by applying quantum computing to continuous mathematics, encompassing a diverse array of scientific and engineering challenges. This breakthrough has since ignited extensive
research into quantum algorithms for numerical analysis and has profoundly influenced the burgeoning field of quantum machine learning.
Inside the HHL Algorithm: Mechanisms and Processes
At its core, the HHL Algorithm resolves linear equations of the form Ax = b, where A is a known matrix, and b is a known vector. The algorithm unfolds in several stages:
• State Preparation: The algorithm starts by encoding the vector b into a quantum state.
• Quantum Phase Estimation: This step estimates eigenvalues of A, a crucial part of the process for solving the linear system.
• Controlled Rotations: Utilizing the estimated eigenvalues, the algorithm performs rotations that conditionally adjust the quantum state based on these values.
• Uncomputation: The process reverses the quantum phase estimation to disentangle eigenvalues from the system.
• Measurement and Post-Processing: Finally, measuring the quantum system yields the solution to the linear system.
HHL's ability to process information exponentially faster than classical computers makes it a groundbreaking algorithm in quantum computing.
Explore the Versatility of the HHL Algorithm: Applications Across Disciplines
The HHL Algorithm is celebrated for its wide-ranging applications, contributing significantly across various scientific and engineering domains:
• Material Science and Quantum Chemistry: HHL is instrumental in simulating molecular and atomic interactions. It can solve the linear equations that arise in quantum chemistry, aiding in the
development of new materials and the understanding of quantum mechanics within materials.
• Data Fitting and Pattern Recognition: In machine learning and data analysis, HHL can be applied to solve large systems of linear equations for regression analysis, improving pattern recognition
and predictive modeling.
• Computational Fluid Dynamics (CFD): The HHL Algorithm can revolutionize CFD by accelerating simulations, which often involve solving large systems of linear equations to model fluid flows in
scenarios like aerospace engineering and climate modeling.
• Bioinformatics and Drug Discovery: HHL can significantly speed up the analysis of genetic data and the interactions of biological molecules. This acceleration is crucial in drug discovery
processes and understanding complex biological systems.
• Financial Modeling: In finance, HHL can optimize portfolio management strategies and risk assessment by solving systems of linear equations that model market behaviors and financial products.
• Workflow Optimization and Logistics: HHL can enhance the efficiency of logistics and supply chain management by optimizing complex systems, scheduling tasks, and managing resources more
• Energy Sector Optimization: In the energy industry, HHL can be used to optimize grid operations, energy distribution, and to model renewable energy systems, leading to more efficient energy use.
• Artificial Intelligence (AI) and Deep Learning: In AI, particularly in training deep learning models, HHL can solve linear systems that arise during the optimization of neural networks,
potentially reducing the computational cost and time.
• Telecommunications: The algorithm can improve signal processing techniques and network optimization, enhancing data transmission and bandwidth utilization.
• Climate Modeling: HHL can be applied in climate science to solve large-scale linear equations that model climate systems, contributing to more accurate climate predictions and analysis.
Solve Linear Systems Quantumly: Discover the HHL Algorithm on Classiq!
Explore the Platform https://docs.classiq.io/latest/tutorials/algorithms/hhl/hhl/hhl/
About "The Qubit Guy's Podcast"
Hosted by The Qubit Guy (Yuval Boger, our Chief Marketing Officer), the podcast hosts thought leaders in quantum computing to discuss business and technical questions that impact the quantum
computing ecosystem. Our guests provide interesting insights about quantum computer software and algorithm, quantum computer hardware, key applications for quantum computing, market studies of the
quantum industry and more.
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Using Computational Thinking Activity #2: Wind Power
Introduce learners to wind power using websites such as the following:
Learners can explore background information about wind generated power and use the online simulations to create multiple scenarios of variables that affect wind power. Learners should be encouraged
to continually ask questions, record findings, and engage in argumentation when sharing results.
Related Crosscutting Concepts:
Related Disciplinary Core Ideas:
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zla_gbrcond_x - Linux Manuals (3)
zla_gbrcond_x (3) - Linux Manuals
zla_gbrcond_x.f -
DOUBLE PRECISION function zla_gbrcond_x (TRANS, N, KL, KU, AB, LDAB, AFB, LDAFB, IPIV, X, INFO, WORK, RWORK)
ZLA_GBRCOND_X computes the infinity norm condition number of op(A)*diag(x) for general banded matrices.
Function/Subroutine Documentation
DOUBLE PRECISION function zla_gbrcond_x (characterTRANS, integerN, integerKL, integerKU, complex*16, dimension( ldab, * )AB, integerLDAB, complex*16, dimension( ldafb, * )AFB, integerLDAFB, integer,
dimension( * )IPIV, complex*16, dimension( * )X, integerINFO, complex*16, dimension( * )WORK, double precision, dimension( * )RWORK)
ZLA_GBRCOND_X computes the infinity norm condition number of op(A)*diag(x) for general banded matrices.
ZLA_GBRCOND_X Computes the infinity norm condition number of
op(A) * diag(X) where X is a COMPLEX*16 vector.
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate Transpose = Transpose)
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
KL is INTEGER
The number of subdiagonals within the band of A. KL >= 0.
KU is INTEGER
The number of superdiagonals within the band of A. KU >= 0.
AB is COMPLEX*16 array, dimension (LDAB,N)
On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
The j-th column of A is stored in the j-th column of the
array AB as follows:
AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KL+KU+1.
AFB is COMPLEX*16 array, dimension (LDAFB,N)
Details of the LU factorization of the band matrix A, as
computed by ZGBTRF. U is stored as an upper triangular
band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
and the multipliers used during the factorization are stored
in rows KL+KU+2 to 2*KL+KU+1.
LDAFB is INTEGER
The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by ZGBTRF; row i of the matrix was interchanged
with row IPIV(i).
X is COMPLEX*16 array, dimension (N)
The vector X in the formula op(A) * diag(X).
INFO is INTEGER
= 0: Successful exit.
i > 0: The ith argument is invalid.
WORK is COMPLEX*16 array, dimension (2*N).
RWORK is DOUBLE PRECISION array, dimension (N).
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Definition at line 154 of file zla_gbrcond_x.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-zla_gbrcond_x/","timestamp":"2024-11-13T19:11:12Z","content_type":"text/html","content_length":"10403","record_id":"<urn:uuid:c1512360-9d05-4feb-8894-61b0ca9016ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00354.warc.gz"} |
How Deep Does Veritasium's Bullet Go?
After a couple of very productive days where I closed my Twitter tab because it was too freakin' annoying to read, I checked in briefly Wednesday morning, and found Rhett Allain and Frank Noschese
discussing this Veritasium bullet-in-block experiment:
Tom at Swans On Tea offers some analysis, and Rhett offers a video response doing out some of the math:
I basically agree with their explanations, and that should have been that, except that in his discussion Rhett mentioned that the bullet probably doesn't go as far into the spinning block, which
prompted Frank to ask whether you could measure and/or calculate the difference. Which sucked me in, and rather than working on the book chapter I'd planned to do this morning, I found myself
scribbling equations. Which just goes to show that Twitter is the work of the devil.
Having gone through the analysis, though, I might as well get a blog post out of the deal. So, how much of a difference would you see in the depth of the bullet for the spinning and non-spinning
cases? The full calculation would be very messy, but we can at least make a crude estimate.
First and foremost, let's work out the bits that we can do exactly: as Rhett explains in his video, we know that momentum needs to be conserved, here, so we can relate the initial velocity of the
bullet to the final velocity of the bullet-and-block:
$latex mv_i = (m+M)v_f $
I'm using m for the mass of the bullet and M for the mass of the bullet. Since the thing we can measure from the video is the final velocity of the bullet-and-block, v[f] let's solve this for the
thing we don't measure, namely the initial speed of the bullet v[i]:
$latex v_i = \frac{(m+M)}{m}v_f $
(As a check: this is clearly going to be larger than the final speed, which makes sense, since the bullet is going too fast to see.)
So far, so good. But what we really care about is the energy, and specifically the change in the energy, which is different in the rotating-block case than the straight-up case. In the straight-up
case, you only have two kinds of energy to worry about, kinetic and "internal," which is a catch-all for the energy that goes into sound, deformation of the wood, heating of the block, etc. The final
energy is just the kinetic energy of bullet-and-block plus this internal energy:
$latex E_{final} = \frac{1}{2}(m+M)v_f^2 + E_{int}$
This has to be equal to the initial energy, which is just the kinetic energy of the bullet:
$latex E_{initial} = \frac{1}{2}mv_i^2 = \frac{1}{2}(m+M)v_f^2 \frac{m+M}{m} $
where I've used the value for v[i] from up above to put this in terms of measurable quantities. If you set these two equal, and do a bit of algebra, you can get a simple expression for the internal
energy after the bullet buries itself into the wood:
$latex E_{int} = \frac{1}{2}(m+M) v_f^2 \frac{M}{m} $
(Do the algebra for homework, and send it to Rhett to be graded...). As a sanity check, this suggests that the vast majority of the initial kinetic energy is converted to internal energy-- M is
presumably considerably larger than m. Which makes sense-- this is doing quite a bit of violence to the block.
In the rotating case, the initial energy is the same, but you pick up some additional terms in the final energy, due to the rotation. In very general terms, this looks like:
$latex E_{final, rot}=\frac{1}{2}(m+M)v_f^2 + E_{int} + K_{rot,block} + K_{rot,bull} $
Those two K terms represent the rotational kinetic energy of the block spinning about its center of mass, and the bullet embedded in the block. I have these separated because Frank asked on Twitter
about the effect of the "impact parameter," namely the distance off-center at which the bullet is fired, and this let me calculate it. These kinetic energies will depend on the "moment of inertia"
for the block and bullet, which is going to be equal to the mass of the rotating object multiplied by some distance squared-- in the case of the bullet, it's just the "impact parameter" which
traditionally gets the symbol b. In the case of the block, it depends on the size and shape of the block, and I'm just going to use a single number for the size, which I'll call R, and stick in a
fudge factor β to cover the effect of the shape, because I'm too lazy to look it up. Doing that, we get:
$latex E_{final, rot}=\frac{1}{2}(m+M)v_f^2 + E_{int} + \frac{1}{2}\beta M R^2 \omega^2 + \frac{1}{2}mb^2 \omega^2 $
where we've been forced to introduce another new variable, the "angular velocity" ω. This is something we can measure from the video, though, so it's just a new parameter like v[f].
The internal energy in the rotating-block case, then, is:
$latex E_{int,rot} = \frac{1}{2}(m+M) v_f^2 \frac{M}{m} - \frac{1}{2}\beta M R^2 \omega^2 - \frac{1}{2}mb^2 \omega^2 $
As you would expect, this is lower than the energy in the non-rotating case, because some of the initial kinetic energy got turned into rotational energy of the block and bullet.
Up to this point, everything is exact, except for my lazy β thing. This doesn't let us do anything about the penetration depth, though. To get that information, we need a crude approximation. This is
not a thing that will make a lot of physics education research types happy, but we're going to say that this internal energy is associated with some work done by the frictional-type forces that
resist the bullet's entry into the block. "Work done by friction" is a Bad Phrase to a lot of folks these days, but it's the only way to get the penetration depth, so we'll assume that the internal
energy change comes from this work, which is equal to some force times some distance:
$latex E_{int} = F_{avg} \Delta x $
This isn't really a constant force, of course, but you can finesse that by calling it some "average" force, which isn't too awful an approximation. What's less justifiable, but absolutely necessary
to make this work, is the assumption that the "average" force is the same for both rotating and non-rotating cases. That's probably not a very good approximation, but there's no other simple way to
attack it, so we'll boldly forge ahead.
Now, we've got two different values for the internal energy, each involving an unknown average force and an unknown penetration depth, so this might seem hopeless. But all we really care about is the
change in the penetration depth-- that is, if we take the depth of the bullet in the straight-up case as another empirical parameter, we can find the depth of the bullet in the rotating case in terms
of that, and see how much they differ.
To do this, we first solve for the average force in terms of the internal energy in the straight-up case:
$latex F_{avg} = \frac{E_{int}}{\Delta x_s} = \frac{1}{2}\frac{M}{m}\frac{(m+M)v_f^2}{\Delta x_s} $
Then we solve for the penetration depth in the rotating case using that internal energy:
$latex \Delta x_{rot} = \frac{E_{int,rot}}{F_{avg}} = \frac{\frac{1}{2}(m+M) v_f^2 \frac{M}{m} - \frac{1}{2}\beta M R^2 \omega^2 - \frac{1}{2}mb^2 \omega^2}{F_avg} $
That looks pretty awful, but we can plug in the value we found for F[avg] in the straight-up case, and do a bunch of algebra to get:
$latex \Delta x_{rot} = \Delta x_s(1-\beta\frac{R^2 \omega^2}{v_f^2}\frac{m}{m+M} - \frac{b^2 \omega^2}{v_f^2} \frac{m^2}{M(m+M)})$
(Again, work this out on your own for homework, and send it to Rhett to be graded.)
Take a deep breath, the worst of the math is over. Let's look at what we've got, here: we have an expression for the penetration depth in the rotating case that looks like the penetration depth in
the straight-up case minus two factors, one related to the rotation of the block, the other related to the bullet within the block. These are kind of complicated, but the basic form makes sense: the
faster the block spins after the collision, the larger these correction terms get, and the smaller the penetration depth. that's exactly what we expect.
Now, you'd have a major problem if those corrections get too big, but there's good reason to suspect that they'll be small-- β is going to be less than 1, and both terms contain a ratio of masses
that is guaranteed to be less than 1. As long as the rotation rate isn't absurdly large, we're probably fine.
So, does this give a measurable difference between the two cases? (If you remember back before all the math, that's the question that got me sucked into this whole mess...) Well, to estimate that, we
need to know the speeds. If I were Rhett, I would download the YouTube video, crank it into Tracker Video, and measure those exactly, but that would require installing software, and anyway, I'm tired
from all that algebra. So I'm going to do a really crude back-of-the-envelope approximation, and see what that suggests.
To get the final speed of the block, we can look at how long it spends in the air. Since we know the acceleration due to gravity, the time in the air depends simply on the velocity-- looking at the
video, a bit less than two seconds pass between the gun going off and the block falling back to its original height; if we call it two seconds, that's an initial speed of 10 m/s, a nice round number.
To get the angular speed, we need to know how fast the block spins. In the high-speed video, it looks like maybe 10 revolutions on the way up, which, again, is a nice round number. That would be an
angular speed of 20π radians per second.
But really, the velocities only come in as a ratio of some length times the angular speed divided by the linear speed. Which means we need an estimate of the size of the block-- in the spirit of
picking nice round numbers, I'm going to say that both R and b are 10cm, or 0.1m. Which makes that ratio equal to 2π/10, and it's squared, which my calculator tells me is about 0.4.
That leaves only the masses to worry about, which is hard to figure. Because I'm just doing order-of-magnitude type things here, I'll say that the block is about a kilogram and the bullet about ten
grams. Those are wild guesses, but probably not off by more than an order of magnitude, which is good enough for my purposes. These really come in as a ratio, which is about 100 to 1 for the numbers
I picked.
So, then, plugging these crude numbers in, we have:
$latex \Delta x_{rot} = \Delta x_s (1 - \beta (0.4) \frac{1}{100} - (0.4)\frac{1}{100^2}) $
This requires some kind of estimate for the mystery fudge factor β-- if I call it 1/4, that cancels the 4 in the 0.4, and leaves us with a difference between the two penetration depths of about one
part in a thousand. If the bullet made it ten centimeters into the block (which would probably be just about all the way through), that'd be a difference of a tenth of a millimeter, which would be
extremely difficult to measure.
Now, I could be wrong about the masses-- if it's a 50g bullet and a half-kilogram block, you'd be looking at a difference of order one percent of the total depth. And you could maybe pick up another
bit by more accurately estimating the speeds. But this is going to be a small number, any way you look at it.
Does that make sense? Well, yes, or I wouldn't be posting this. See, in the crude work-by-friction approximation, a 1% change in the penetration depth means that 1% of the internal energy gets turned
into rotational kinetic energy. And the internal energy was something like 100 times the kinetic energy of the bullet-and-block after the collision (for the 100 to 1 mass ratio I used above), which
would require a really fast rotation of the block to achieve-- it's hard to pack much rotational kinetic energy into a small object.
So, that's my overthought analysis of the bullet-and-block video, and where the energy goes. Which is way more than was required, but that's why I'm a physicist, I guess...
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The rifle used almost certainly is firing .22 short ammunition. Wikipedia ( http://en.wikipedia.org/wiki/.22_Short ) says
The standard velocity .22 Short launches a 29-grain (1.9 g) bullet at 1,045 feet per second (319 m/s) with 70 ft·lbf (95 J) of energy from a 22 in (559 mm) rifle barrel and can penetrate 2 inches
(51 mm) of soft pine.
The remark that a .22 short can penetrate 2 inches of soft pine confirms that it is unlikely that anything more powerful would be used.
I should've noted that, despite growing up in a somewhat redneck-y area, I know basically nothing about guns...
Another video response to this estimated the mass of the block at more like 200g, so it sounds like I might've had the 100:1 ratio right but the wrong absolute value.
Thinking about energy is a poor way to determine the motion of this system. The collision is far too inelastic. (though it can be interesting to ask about the energy budget)
Think instead about impulse and momentum. The block's linear momentum is that of the initial bullet plus any vertical impulse it may have received from the nail. The nail cannot apply any downward
impulse, so it is right away clear that the block cannot rise less high than originally. The nail does not apply an upward impulse either; it would only do so if there had been an initial tendency
for its contact point to jump downwards. But its contact point is essentially the same as the center of mass, which we have already decided is going up. Hence no impulse from the nail and no
difference in final height.
If instead the nail was offset to the left of the center of mass ( and the block supported maybe by some tissue paper ) then the initial tendency of the contact point would have been to go DOWN, and
the nail would supply an upward impulse, to counter that, and the final height of the block would be higher.
I absolutely agree that energy is the wrong way to attack the original question. Of course, it's the obvious wrong way to attack it, which is why it works as a trick-question video...
The impulse-from-the-nail thing is an interesting approach, and seems like something that ought to be testable-- just drive in a second nail, off to the left. I'm still trying to think of a way to do
this with PASCO stuff (as I don't own a gun, and doubt the college would be all that wild about me shooting one off on campus even if I did...), and I'll keep that in mind as an add-on...
I like the cartoon bullets. That's a nice touch.
I should note that I didn't intend this post to really be an answer to the Veritasium video per se-- it's working out a different, more difficult question, hence all the equations.
As Veritasium is in Australia, which is not at all friendly towards guns, and .22 Short being pretty uncommon these days, I'd be surprised if he was running anything other than standard velocity .22
LR. With that kind of barrel length, you can generally count on ~320-340 m/s with a standard 40 gr (2.6 g) bullet. If shot at soft pine it'll blow through, yes, but it'll only get so far into treated
hardwood blocks. I run rimfire shooting matches at my club and while they're certainly not pretty afterwards, the pine backers we use to hold up targets hold up just fine even to the high velocity
I don't think penetration depth really makes much of a difference here. Consider a linear approximation of the bullet's velocity in the block, where it goes from 340 m/s to 0 m/s over a distance of
about 50 mm (obviously it won't be linear, but hey, I'm trying). This happens over about 0.3 ms, which is far faster than anything on the time scales of this video, especially when compared to the
rate of rotation of the block. Also, consider that the speed of sound in the block is probably some 3500 m/s or so, so the pressure from the bullet will equalize effectively instantly on these time
scales as well.
I'd look at this in terms of a momentum transfer of the bullet into a stationary block as an initial condition and then just go from there. The dynamics of the bullet in the block seems like a
Also the velocity profile of the bullet is much more likely an exponential-ish decay, so the momentum transfer will happen somewhat more rapidly than the linear approximation above.
An easy way to test this on an energy vs. momentum front, on that note, would be to fire high-velocity 22 LR, like a CCI Mini-Mag or the like, into the block and test that. The bullet weight is the
same, but you get an extra 20-30% on the velocity (44-69% energy boost). That's significant enough of a difference that you should be able to see what the relationship is between the two tests and
decide from that what is the driving factor. | {"url":"https://scienceblogs.com/principles/2013/08/21/how-deep-does-veritasiums-bullet-go","timestamp":"2024-11-05T06:03:31Z","content_type":"text/html","content_length":"64616","record_id":"<urn:uuid:54235c3e-39d9-42bf-b152-20e881dd87de>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00621.warc.gz"} |
How do you prove the statement lim as x approaches 9 for root4(9-x) = 0 using the epsilon and delta definition? | Socratic
How do you prove the statement lim as x approaches 9 for #root4(9-x) = 0# using the epsilon and delta definition?
1 Answer
The $\delta \text{-} \epsilon$ definition of a limit states that:
the limit of a function $f \left(x\right)$, as $x$ approaches some value $c$, is $L$if, for every possible $\epsilon > 0$, we can find a $\delta > 0$that depends on $\epsilon$, such that $\left\
mid f \left(x\right) - L \right\mid < \epsilon$ whenever $\left\mid x - c \right\mid < \delta$.
It's like a game. Player 1 picks an $\epsilon > 0$, and Player 2 is trying to find a $\delta > 0$ such that every $x$ within $\pm \delta$ of $c$ (that is, $\textcolor{n a v y}{x \in \left(c - \delta
, c + \delta\right)}$) is guaranteed to get mapped to an $f \left(x\right)$ within $\pm \epsilon$ of $L$ (that is, $\textcolor{g r e e n}{f \left(x\right) \in \left(L - \epsilon , L + \epsilon\
right)}$). Player 1 keeps picking smaller and smaller $\epsilon$, and Player 2 keeps having to find smaller and smaller $\delta$.
If we can prove that, for every $\epsilon$ Player 1 picks, Player 2 can find a suitable $\delta$, then we've proven the limit. In other words:
$\forall \text{ "epsilon > 0" "EE" "delta>0}$
such that
$\left\mid f \left(x\right) - L \right\mid < \epsilon$ when $\left\mid x - c \right\mid < \delta$
${\lim}_{x \to c} f \left(x\right) = L$.
Okay, so that's a lot of mumble jumble. Let's put it to use.
We need to find a $\delta$ that depends on $\epsilon$. That means you can think of $\delta$ as a function of $\epsilon$. We also want to have $' x \text{ is within "delta" of } c '$ imply $' f \left
(x\right) \text{ is within " epsilon " of } L '$, or in math lingo:
$\left\mid x - c \right\mid < \delta \implies \left\mid f \left(x\right) - L \right\mid < \epsilon$
So we start with $\left\mid x - c \right\mid < \delta$ and $\textcolor{b l u e}{\text{try to get it to look like } \left\mid f \left(x\right) - L \right\mid < \epsilon}$.
In this example, $c = 9$, $L = 0$, and $f \left(x\right) = \sqrt[4]{9 - x}$.
$\left\mid x - c \right\mid < \delta \implies \left\mid x - 9 \right\mid < \delta$
$\textcolor{w h i t e}{\left\mid x - c \right\mid < \delta} \implies \left\mid 9 - x \right\mid < \delta$
$\textcolor{w h i t e}{\left\mid x - c \right\mid < \delta} \implies \sqrt[4]{\left\mid 9 - x \right\mid} < \sqrt[4]{\delta}$
$\textcolor{w h i t e}{\left\mid x - c \right\mid < \delta} \implies \left\mid \sqrt[4]{9 - x} \right\mid < \sqrt[4]{\delta}$
$\textcolor{w h i t e}{\left\mid x - c \right\mid < \delta} \implies \left\mid \sqrt[4]{9 - x} - 0 \right\mid < \sqrt[4]{\delta}$
$\textcolor{w h i t e}{\left\mid x - c \right\mid < \delta} \implies \left\mid f \left(x\right) - 0 \right\mid < \sqrt[4]{\delta}$
Hey—looks like we may have found our connection between $\delta$ and $\epsilon$! If we let $\sqrt[4]{\delta} = \epsilon$, then we have
$\textcolor{w h i t e}{\left\mid x - c \right\mid < \delta} \implies \left\mid f \left(x\right) - 0 \right\mid < \epsilon$
and so we've shown that $\left\mid x - 9 \right\mid < \delta \implies \left\mid f \left(x\right) - 0 \right\mid < \epsilon$.
The last thing to do is to solve $\sqrt[4]{\delta} = \epsilon$ for $\delta$:
$\text{ } \sqrt[4]{\delta} = \epsilon$
$\implies \delta = {\epsilon}^{4}$
All the work we've done here simply means that no matter how small an $\epsilon$ Player 1 may pick, Player 2 can always just choose their $\delta$ to be ${\epsilon}^{4}$, and they'll win every time.
That is, as long as $x$ is within ${\epsilon}^{4}$ of $9$, $\sqrt[4]{9 - x}$ will be within $\epsilon$ of $0$. Thus,
${\lim}_{x \to 9} \sqrt[4]{9 - x} = 0$
has been proven.
$Q E D .$
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Deploying UAV Base Stations in Communication Network Using Machine Learning
Xukai Zhong
B.ASC., Simon Fraser University, 2017
A Report Submitted in Partial Fulfillment of the Requirements for the Degree of
in the Department of Electrical and Computer Engineering
Xukai Zhong, 2019 University of Victoria
All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author.
Supervisory Committee
Dr. Xiaodai Dong, Supervisor
(Department of Electrical and Computer Engineering)
Dr. Wu-Sheng Lu, Departmental Member
Supervisory Committee
Dr. Xiaodai Dong, Supervisor
(Department of Electrical and Computer Engineering)
Dr. Wu-Sheng Lu, Departmental Member
(Department of Electrical and Computer Engineering)
Today has witnessed a constantly increasing demand for high-quality wireless communications services. Moreover, the quality of service (QoS) requirement of future 5G and beyond cellular networks
leads to the possible use of the unmanned aerial vehicle base station (UAV-BS). Deploying UAV-BSs to assist the communications network has become a research direction with great potential. In this
project, we focus on the problem of deploying UAV-BSs to provide satisfactory wireless communication services, with the aim that maximizes the total number of covered user equipment subject to user
data rate requirements and UAV-BS capacity limit. Then, the report extends to a reinforcement learning based method to adjust the locations of UAVs to maximize the sum data rate of the user equipment
(UE). Numerical experiments under practical settings provide supportive evidences to our design.
Supervisory Committee ii
Abstract iii
Table of Contents iv
List of Tables vi
List of Figures vii
Acknowledgements viii
Dedication ix
1 Introduction 1
1.1 Background . . . 1
1.2 Air to Ground (A2G) Channel Model . . . 2
1.3 Genetic Algorithm . . . 4 1.4 Related Work . . . 5 1.5 Report Outline . . . 6 2 Reinforcement Learning 7 2.1 Neural Network . . . 7 2.1.1 A Single Neuron . . . 8
2.1.2 Feedforward Neural Network . . . 9
2.1.3 Backpropagation . . . 9
2.1.4 Q-Learning . . . 10
2.1.5 Deep Q-Network . . . 10
3.1 System Model . . . 13
3.2 Problem Formulation of Finding Optimal 3D Location of UAV-BS . . 14
3.2.1 2D UAV-BS Deployment Problem . . . 14
3.2.2 Finding the Optimal Altitude for UAV-BS . . . 16
3.3 GA based UAV-BS Deployment Strategy . . . 16
3.4 Numerical Result . . . 18
4 Dynamic Movement Strategy in a UAV-Assisted Network 24 4.1 UAV-Assisted Network System Description . . . 24
4.2 UAV Dynamic Movement Problem Formulation . . . 25
4.3 Deep Q-Network based UAV Movement Strategy . . . 26
4.3.1 State Representation . . . 27
4.3.2 Action Space . . . 27
4.3.3 Reward Design . . . 28
4.3.4 Training Procedure . . . 28
4.4 Numerical Result . . . 29
5 Conclusion and future work 32 5.1 Optimal 3D Location of UAV-BS with Maximum Coverage . . . 32
5.2 Optimal UAV Dynamic Movement Strategy . . . 32
5.3 Future Work . . . 33
A Genetic Algorithm Python Implementation 34
B Deep Q-Network Python Implementation 39
Table 3.1 Coverage ratio comparison in urban environment. . . 20 Table 4.1 Comparisons of processing time of different algorithm . . . 29
List of Figures
Figure 1.1 Radius vs. altitude curve for different maximum path loss. . . . 3
Figure 1.2 GA workflow . . . 5
Figure 2.1 RL workflow . . . 7
Figure 2.2 A communication system model of multiple UAV-BSs serving ground users . . . 8
Figure 2.3 A single neuron . . . 8
Figure 2.4 Fully Connected Neural Network . . . 9
Figure 2.5 A simple Q-Table . . . 11
Figure 2.6 Deep Q-Network . . . 11
Figure 3.1 A communication system model of multiple UAV-BSs serving ground users . . . 14
Figure 3.2 Path loss vs. altitude for given radii in urban environment. . . 17
Figure 3.3 The 100% coverage ratio result of GA deployment with 80 UEs in a 5000 m × 5000 m square region with different data rate requirements. . . 21
Figure 3.4 The coverage ratio versus the number of UAV-BSs in four envi-ronments. . . 22
Figure 3.5 The UAV’s average transmit power comparison of altitude with maximum coverage, fixed altitude and random altitude in urban environments. . . 23
Figure 4.1 A communication system model of UAV-assisted Network . . . 25
Figure 4.2 The UE distribution and association with 500 UEs in a 5000 m × 5000 m area. . . 30
Figure 4.3 sum data rate comparison of different methods. . . 30
any conditions, providing help whenever I am in need. I’m also grateful to many of my friends who have brought me happiness, joy as well as generous help, especially Ahmed Elmoogy, Hoang Minh Tu,
Dr.Jinlong Zhan, Tong Zhu, Tianzhu Li, Ying Wang and Ji Shi.
Wireless communications systems which include unmanned aerial vehicles (UAV) are capable of providing cost-effective wireless connectivity for devices without fixed in-frastructure base stations.
Compared to terrestrial communications or those based on high-altitude platforms, on-demand wireless systems with low-altitude UAVs are in general more flexibly reconfigured, and likely to have
better communications chan-nels due to the presence of short-range line-of-sight links [15]. For example, in the extreme situations like natural disaster or battlefield where it is not cost-efficient
nor time-efficient to re-deploy onsite terrestrial base stations, the utilization of unmanned aerial vehicle base stations (UAV-BSs) becomes a valid solution since UAV-BSs can be deployed and
reconfigured rapidly. Also, the UAV can play an important role in practical applications of Internet of Things (IoT) where UAV collects data from IoT devices [12]. Moreover, UAVs have a great
potential to be used in many 5G and be-yond applications, for example, the authors in [7] propose a multi-layer UAV network model for UAV-enabled 5G and beyond applications.
With their high mobility and low cost, in the past few decades, UAVs have found a wide range of applications including wireless communications, rescue and agriculture. Historically, UAVs have been
primarily used in the military [15], mainly deployed in hostile territory to reduce pilot losses. With the continued reduction of the cost as well as the size of the devices, small UAVs are now
becoming more easily accessible to the general public. Therefore, lots of new applications in the civilian and com-mercial domains have emerged, with typical examples including weather monitoring,
communications relaying, and others.
For practical use of UAV in wireless communications, one promising solution to enhance the performance is by letting the UAVs learn the environment by various sensors and adapt their movement and
communications resource allocation in real time. Thus, the implementation of intelligent learning algorithms are common in designing UAV-networks for various purposes including navigation, deployment
and anti-jamming.
Despite of the benefits in enabling UAV-BSs, there are many remaining issues to be addressed. A significant one is to find suitable UAV-BSs’ positions when deploying the UAV-BSs network. Since the
life time of the battery powering one UAV-BS is limited and the number of available UAV-BSs is also constrained, UAV-BSs should be deployed in an energy-efficient method. Another critical challenge
is the design of the movement strategy for UAVs. Since in realistic situations, in order to take advantage of the high mobility of UAVs, it is important that a reasonable strategy needs to be
designed for UAVs to cope with various environments.
Air to Ground (A2G) Channel Model
The A2G channel adpoted follows that in [1] where line-of-sight (LoS) occurs with a certain probability. The probability of a LoS and non line-of-sight (NLoS) channel between UAV j at horizontal
position mj = (xj, yj) and user i at horizontal location
ui = (˜xi, ˜yi) are formulated as [1] PLoS = 1 1 + a exp(−b(180 π tan −1[(]Hj rij) − a)) , PN LoS =1 − PLoS, (1.1)
where Hj is the altitude of UAV-BS j; a and b are environment dependent variables;
rij = p(xj− ˜xi)2+ (yj − ˜yi)2 is the horizontal euclidean distance between the ith
user and jth [UAV. Then the path loss for LoS and NLoS can be written as]
P LLoS = 20 log( 4πfcdij c ) + ηLoS, P LN LoS = 20 log( 4πfcdij c ) + ηN LoS (1.2)
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 Altitude (H) (m) 0 1000 2000 3000 Radius (R) (m)
Figure 1.1: Radius vs. altitude curve for different maximum path loss.
where fc is the carrier frequency, c is the speed of light and dij denotes the distance
between between the UE and UAV-BS given by dij =
q H2
j + rij2. Moreover, ηLoS and
ηN LoS are the environment dependent average additional path loss for LoS and NLoS
condition respectively. According to (1.1), (1.2), the path loss (PL) can written as: P L =P LLoS× PLoS+ P LN LoS× PN LoS
= A
1 + a exp(−b(180[π] (H[r]) − a)) + 20 log r
cos(H[r]) + B
where A = ηLoS− ηN LoS and B = 20 log(4πf[c]c) + ηN LoS.
In order to show the effect of different P Lmax on the radius-altitude curve, we
have plotted this relation (2.2) in Fig. 1.1 where the coverage radius is a function of both, the altitude H and the P Lmax, by keeping a constant environment parameters
such as those of urban.
The path loss for UEs which are associated with the GBSs at distance rik can be
modeled by P Lik = ηBrα[ik]B where ηB is the additional PL over the free space PL and
αB is the PL exponent.
at a distance rij/rik from its associated UAV-BS j or GBS k can be expressed as
SIN Rij/ik =
Pj/kh0P L−1[ij/ik]
, (1.4)
I[i¯][j/i¯][k] = Pih0P L−1[i¯][j/i¯][k], (1.5)
represents interference from other UAV-BSs/GBSs, Pj/krepresents the transmit power
of the serving base station. h0 is the small fading gain assumed to be an independent
number following the exponential distribution and σ2 [is the variance of the additive]
white Gaussian noise component. Therefore, according to the Shannon Capacity The-orem, the data rate Ciof of the ith UE can be expressed as Ci = B log2(1+SIN Rij/ik)
where B is the bandwidth of the channel.
Genetic Algorithm
Genetic Algorithm (GA) works on a population which consists of some candidate solutions and the population size is the total number of solutions. Each solution is considered to be a chromosome and
each chromosome has a set of genes where each gene is represented by the features of the solutions. Then, each individual chromosome has a fitness value which is computed based on the fitness
function representing the quality of the chromosome. Moreover, a selection method called roulette wheel method where the chromosome with higher fitness value has a higher chance to survive the
However, the selection process can only generate the best candidate solution with no more change of the chromosome. In order to ensure the diversity of the solution to avoid falling into local
optimal solutions, crossover and selection are applied after selection process. In crossover procedure, two chromosome are selected in a proba-bility of crossover rate to exchange information so new
chromosomes are generated. Also, in mutation procedure, each chromosome has a probability of mutation rate to replace a set of genes with new random values. This process repeats for t iteration until
t reach a preset iteration limit. Fig. 1.2 illustrates the general work flow of a complete GA.
Figure 1.2: GA workflow
Related Work
Research on UAV-BSs development has focused on finding horizontal positioning [10]-[12] and altitude optimization [13]-[15]. In [9] and [11], an identical coverage radius is assumed for all UAV-BSs.
The work in [9] proposes an efficient spiral placement algorithm aiming to minimize the required number of UAVs, while [11] models the UAV deployment problem based on circle packing theory and study
the relationship between the number of deployed UAV-BSs and the coverage duration. In [6], the authors use a K-Means clustering method to partition the ground users to k subsets and users belonging
to the same subset are served by one UAV. All these works have a fixed altitude assumption. The relationship between the altitude of UAV-BSs and the coverage area is studied in [1] and [5]. In [1],
the method of finding the optimal altitude of a single UAV placement for maximizing the coverage is studied based on a channel model with probabilistic path loss (PL). Reference [5] formulates an
equivalent problem based on the same channel model as [1] and proposes an efficient solution. Moreover, [3] studies multiple UAV-BS 3D placements with a given radius taking into account energy
efficiency by decoupling the UAV-BS placement in the vertical dimension from the horizontal dimension. In recent year, artificial intelligence algorithms are growing in various research fields. The
authors in [2] applied GA which is a popular artificial intelligence algorithm to derive the optimal UAV locations in
5G applications with the consideration of energy consumption and coverage range. Moreover, machine learning techniques have begun to gain popularity to be uti-lized in deploying UAVs [7]-[9]. In
particular, in [16], a machine learning framework based on Gaussian mixture model (GMM) and a weighted expectation maximiza-tion (WEM) algorithm to predict the locamaximiza-tions of UAVs in which the
total power consumption is minimized are proposed. Also, the authors in [4] study a Q-learning based algorithm to find the optimal trajectory to maximize the sum rates of ground users for a single
UAV base station (UAV-BS). Reference [8] proposes a deep rein-forcement learning based movement design for multiple UAV-BSs.
Report Outline
The structure of the report is as followed:
Chapter 2 presents an introduction of reinforcement learning.
Chapter 3 presents a Optimal 3D Deployment method for deploying UAV Base Stations.
Chapter 4 describes a reinforcement learning based method to obtain the UAV movement strategy in UAV-assisted network.
Figure 2.1: RL workflow
Chapter 2
Reinforcement Learning
Fig. 2.1 illustrates the workflow of a basic reinforcement learning (RL) method. The RL task is to train an agent who interacts with the environment that provides feedback to each of its actions. The
agent arrives at different states by performing actions. Actions lead to rewards so we reinforce the agents to learn to choose the best actions based on the rewards. Therefore, the only objective of
the agent is to maximize its total reward across an episode. The way the agent chooses its actions is known as policy. The RL examples include Q-learning, deep Q-learning, policy gradient and etc.
Neural Network
Figure 2.2: A communication system model of multiple UAV-BSs serving ground users
Figure 2.3: A single neuron
A Single Neuron
The basic unit of a neural network is called neuron which receives numerical input from some other nodes, or from an external source and computes an output. Each input has an associated weights and a
bias and the nerual applies an activation function to the weighed sum of the inputs as shown in Fig. 2.3. The purpose to have an activation function is to have a non-linear representation of the
outputs. In neural network, the sigmoid function is used as activation function.
f (x) = sigmoid(x) = 1
Figure 2.4: Fully Connected Neural Network
Feedforward Neural Network
A feedforward neural network is a collection of neurons which are connected with each other in a particular way. A neuron takes inputs from other neurons and output the computation results to another
neuron. Fig. 2.4 shows a simple fully connected neural network. The layer1 is called input layer and layer 4 is called output layer. The layers in between are called hidden layer. The neurons in one
layer are connected with all the neurons from previous layer. In a feedforward neural network, the information moves in only one direction which is forward. It goes through the neurons in hidden
layers and to the neurons in output layers without any loop.
Initially all the weights in the neurons are randomly assigned. For the inputs from the training dataset, the neural network takes those inputs and the outputs can be derived. The outputs are
compared with the desired outputs so that the difference between the computed outputs and desired outputs can be observed. According to the difference which is also known as propagated, the values of
weights can be adjusted until the propagated is below a predetermined threshold.
Once the above algorithm terminates, we consider the nerual network is ready to take inputs which are not from training dataset to accurately predict the outputs.
Q-learning is an off policy reinforcement learning algorithm which finds the best action for a given state. It’s considered off-policy because the q-learning function learns from actions that are
outside the current policy. More specifically, q-learning learns a policy that maximizes the total reward.
• Q-Value: The Q-Value Q(s, a) represents the total rewards of agents being at state s and performing action a and the Q-Value for each state and action can be found in the Q-Table. It can be
computed by equation:
Q(s, a) = r(s, a) + γmaxaQ(s0, a) (2.2)
The above equation states that the Q-value which is derived from the agent being at state s and taking action a equals to the immediate reward r(s, a) plus the highest possible Q-value of the next
state s0. γ is the discount factor which represents the contribution of future rewards.
• Q-Table: Q-Table is a look up table which states the Q-Value that represents the future values for actions for each states. Fig. 2.5 illustrates the format of a Q-Table where the Q-Value for
actions to each states are stated.
To begin with, the Q-Table is initialized with all zeros. Then the agent chooses an action based on epsilon greedy strategy that 90% the agent chooses the action with highest Q-Value while 10% the
agent chooses a random action. After, based on the action the agent chooses, the reward of performing the action is observed. According to the outcome and the reward, Q-Value can be updated based on
Qnew(s, a) = Qold(s, a) + α(r(s, a) + γmaxQ(s0, a) − Qold(s, a)) (2.3)
Deep Q-Network
The traditional Q-Learning is a powerful algorithm to create a look up table for the agent so that the agent is capable for making rational action in each state. However, the drawback of Q-Learning
is when there are too many states in the environment, it requires a large amount of memory since we need a long Q-Table. Therefore, the
Figure 2.5: A simple Q-Table
Figure 2.6: Deep Q-Network
neural network is a powerful tool that can be utilized to estimate Q-value as shown in Fig. 2.6.
Therefore, the next action is determined by the maximum output of the neu-ral network. Refer to equation (2.3), if we make the loss function Loss = (r + γmaxaQ(s˜ 0, a; Θ) − Q(s, a; Θ))2 where Θ
represents the parameters of the Q-Network,
it becomes a simple regression problem.
However, in this loss function, Q(s, a; Θ) plays the role of a desired target in a regression problem which needs to be stationary in order to converge the network. Therefore a separate network is
used to calculate the target. This target network has the same architecture as the the network to predict Q-Value but with frozen parameters. The parameters of predicted network are copied to target
network in every C iterations and C is a predetermined value.
Also, another important factor in Deep Q-Network is experience replay. It stores a fixed size of samples from training data into a memory tuple. In each training step, a mini-batch of samples are
randomly selected from the memory to train the Q-Network. Experience replay breaks up the correlation in the training data by
sampling batch of experiences randomly from a large memory pool which also helps the network to converge.
QoS-Compliant Optimal 3D
Deployment Method
The 3D deployment of UAV-BSs can be decomposed into the 2D horizontal locations optimization and altitude determination. This is because the UAV altitude only impacts the cell radius and path loss
experienced in the cell, while the horizontal location and a radius determine which UEs are covered by the UAV. As clearly seen in Fig. 1.1, for a given P Lmax, there is a maximum radius Rmax and a
altitude Hmax. If the altitude is smaller or larger than Hmax, while maintaining the
same radius, the path loss on the cell edge will be larger than the given P Lmax. Since
the cell radius affects the total number of the covered UEs, we want the cell radius to be maximized in order to potentially cover more users. Hence the 3D deployment solution takes the procedure as
follows. First, a maximum cell radius upper bound Rmax that guarantees the desired P Lmax requirement is derived. Second, the 2D
placements of |Q| UAVs and their respective coverage radii bounded by Rmax that
maximize the total number of UEs supported while satisfying the individual data rate requirements and the UAV capacity constraint are formulated and solved. Finally, given the actual coverage radius
of each UAV obtained from the second step, the altitude that leads to the achieved minimum cell edge path loss is determined.
System Model
Fig. 4.1 shows a communication network model where many UEs are clustered to be served by multiple UAV-BSs. The objective is to find the optimal locations for
Figure 3.1: A communication system model of multiple UAV-BSs serving ground users
UAV-BSs so that the ground users’ coverage ratio and the coverage radii can be maximized. Let P be the set of all the UEs which are labelled as i = 1, 2, ... |P|. Each UE has a unique data rate
requirement ci and all UEs have a maximum tolerated
path loss P Lmaxthat serves the purpose to guarantee all the data rate requirements
from UEs are feasible, for QoS compliance. Q denotes the set of available UAV-BSs labelled as j = 1, 2, ... |Q| and each UAV-BS has a data rate capacity Cj. In our
system, we assume that no ground base station is available but the locations and data rate requirement of all users are known. Despite of the known interference issue in UAV cells, this work does not
take into account multi-cell interference, which may be mitigated by various techniques such as beamforming, frequency planning, etc.
Problem Formulation of Finding Optimal 3D
Location of UAV-BS
2D UAV-BS Deployment Problem
Since we model the 2D deployment problem via placing multiple circles of different sizes, unlike authors in [2] who investigate a problem of solving for the least number of UAVs to cover users in a
region, this problem is equivalent to finding the appropriate location and radius for each UAV-BS to cover as many UEs as possible while simul-taneously satisfying the data rate requirements and the
UAV capacity constraint.
within the serving area of a UAV-BS, the UAV-BS can allocate certain data channels to the user which has a unique data rate requirement ci. For simplicity, we assume
that for any UE, the allocated data rate equals what it requires. Then the data rate allocation problem can be expressed asP|Q|
j=1ciγij ≤ Cj, where Cj is the data capacity
of UAV j.
Now, the deployment problem becomes a rucksack-like problem which is a NP-hard problem. It can be expressed as
maximize Rj,mj |Q| X j=1 |P| X i=1 γij, s.t. C1 : kmj− γijuik ≤ Rj + M (1 − γij), i ∈ {1, 2, ... |P|} , j ∈ {1, 2, ... |Q|} , γij ∈ {0, 1} C2 : |P| X i=1 ciγij ≤ Cj, j ∈ {1, 2, ... |Q|} C3 : |Q| X j=
1 γij ≤ 1, i ∈ {1, 2, .. |P|} C4 :Rj ≤ Rmax, j ∈ {1, 2, ... |Q|} (3.1)
Our objective is to maximize the number of served users. First, C1 in (5), guarantees that a UE can be served by a UAV-BS, when the horizontal distance between the UE and the UAV-BS is less than
UAV-BS’s coverage radius. Then C2 regulates that the total data rate of all covered users served by one UAV-BS cannot exceed the data rate capacity of the UAV-BS. Furthermore, C3 ensures each user
should be served by at most one UAV-BS. Last, Fig. 1.1 shows that the function of coverage radius respective to altitude for a given P Lmax is a concave function so there exists a maximum radius
Rmax that any coverage radii R > Rmax does not have a feasible solution. Thus, C4
solve this optimization problem will be presented in the next section.
Finding the Optimal Altitude for UAV-BS
After, the horizontal locations and coverage radii of UAV-BSs have been determined and all the coverage radii are less than Rmax. Therefore, for each UAV-BS, the range
of altitude which results in the P L value less than P Lmax can be obtained from
Fig. 1.1. The objective for this step is to find the optimal altitude for each UAV-BS which requires least transmit energy, ie., the minimum path loss, to provide service for the coverage range
derived in step 1.
As observed from (2.2), the path loss between a UAV-BS and UE is a function of the horizontal distance r and the altitude H, that is, P L = f (r, H). Also, from Fig. 1.1, for a given P Lmax, defining
the elevation angle θ = H[R], there exists an
elevation angle θmax that maximizes the radius R by solving ∂R[H] = 0. As derived
in [1], θmax satisfies the following equation:
9 ln(10)tan(θmax) +
abA exp(−b(180[π] θmax− a))
(a exp(−b(180[π] θmax− a)) + 1)2
= 0 (3.2)
where θmax is environment dependent so it is a constant in a given environment. It
has been proven by [3] that this elevation angle provides the minimum P L of the users in the boundary which is equivalent to the P L of all the UEs within the covered range are minimized so the
required transmit power of the UAV-BS is minimized. Therefore, once the actual coverage radius R of each UAV-BS is obtained in Subsection III-A, the UAV-BS altitude Hopt is given by Hopt = R tan
(θmax). Fig. 3.2 shows the relationship
between P L and altitude for given radii. It can be observed that as long as the radius is fixed, a minimum value of P L always exists.
GA based UAV-BS Deployment Strategy
As illustrated in Algorithm 2, the horizontal location, and the coverage radius of each UAV-BS are treated as a gene in the GA model. Therefore, for UAV-BS j, the combination (xj, yj, Rj) is a gene.
Placing genes for all the available UAV-BSs
together, i.e., {xj, yj, Rj}[j∈Q] makes a chromosome. The required inputs include K,
D, P, Q, Rmax, {ci}[i∈P], {ui}[i∈P], θopt, pm, pc where K is the number of iterations for
500 1000 1500 2000 2500 3000 Altitude (H) (m) 32 34 36 38 40 42 44 Path Loss (db)
Figure 3.2: Path loss vs. altitude for given radii in urban environment.
crossover rate for GA respectively. The outputs are the horizontal locations, altitudes and coverage radii, denoted by Oj, j = 1, 2, ... |Q|, of all the UAV-BSs.
First, |Q| empty lists are created and each of them is to store the covered UEs of the corresponding UAV-BSs. Also, two arrays r, ˆr are created, respectively, to store the number of covered UEs in
each UAV-BS and the total number of covered UEs of all UAV-BSs known as the fitness score. In step 3, the first population ν1 is
generated by creating D chromosomes where the horizontal locations of all UAV-BSs are initialized by assigning each of them with the equidistant point of 3 random UEs’ locations, and the coverage
radius are initialized by generating random numbers in the range from 1 to Rmax.
Then, K iterations are executed to find the 2D deployment result from Step 4 to Step 20. In Step 5 and Step 6, if the horizontal distance between a UE and a UAV-BS is less than the coverage radius,
the UE can be served by the UAV-BS. Also, if a UE is within the coverage range of more than one UAV-BS, it is assigned to
the closest one. In the for loop from Step 7 to Step 16, calculate the sum data rate P
p∈Ojcpˆ of all covered UEs for each UAV-BS. If the sum data rate is smaller than
the data capacity Cj, the number of covered UEs |Oj| is stored to array r. Otherwise,
a negative number is stored to array ˆr and the algorithm breaks out the loop and goes back to Step 5, which means the fitness of this chromosome is negative. In Step 15, the fitness function of the
chromosome is the total number of covered UEs and it is saved into array ˆr.
In Step 17, the roulette wheel method is applied to update the current population νˆ[k]. A random chromosome is selected within the current population to be the
com-petitor. Comparing the fitness score of all the chromosomes with the competitor, the chromosomes with less fitness scores are replaced by the competitor. Afterward, in the crossover procedure, pc
of chromosomes are randomly selected and paired. Each
pair is considered to be the parent chromosomes. In each parent chromosomes, the first half genes of one chromosome and second half genes of the other chromosome are exchanged to produce children
chromosomes. In Step 19, all the chromosomes have a probability of pm to perform mutation process in which one gene of the mutated
chromosome is selected to be replaced by random horizontal location and coverage radius.
Finally, in Step 21 and Step 22, we can obtain the result of horizontal locations and coverage radii of UAV-BSs via choosing the chromosome with the maximum fitness score. Finally, the optimal
altitudes are obtained by Hopt = R tan(θmax).
Numerical Result
In our simulations, we consider the UEs are uniformly distributed in a 5000 m × 5000 m area. Referring to [1], the environment parameters are set up as followed: fc
= 2 GHz, P Lmax = 110 dB, (a, b, ηLoS, ηN LoS) is configured to be (4.88, 0.43, 0.1,
21), (9.61, 0.43, 0.1, 20), (12.08, 0.11, 1.6, 23), (27.23, 0.08, 2.3, 34) corresponding to suburban, urban, dense urban and high-rise urban environments, respectively. The GA parameters set (K, D,
pm, pc) is configured to be (10000, 100, 0.01, 0.8). Also,
we assume there are three different data rate requirements of all UEs, c1 [= 5 × 10]6
bps, c2 [= 2 × 10]6 [bps and c]3 [= 1 × 10]6 [bps, and each UE has one of these three]
data rate requirements. Moreover, all the UAV-BSs have the same data rate capacity C = 1 × 108 [bps. Fig. 3.3 illustrates the UE distribution and the GA deployment]
{Oj}[j∈Q]. 7: for j = 1; j ≤ |Q|; j + + do 8: if P ˆ p∈Ojcpˆ≤ Cj then 9: r [j] ← |Oj| 10: else 11: ˆr[ˆi] ← −100
12: Continue and go back to step 5
13: end if
14: end for
15: Fitness Function: ˆr[ˆi] = sum(r)
16: end for
17: Selection: update ν[k]ˆ using roulette wheel method to select chromosomes
18: Crossover: Based on pc, update νˆ[k] by exchanging information of parent chromosomes to
produce children chromosomes
19: Mutation: Based on pm, gene are selected randomly to be replaced new random values
20: end for
21: Find the chromosome with maximum value in ˆr and obtain {mj}[j∈Q] and {Rj}[j∈Q] from the
22: Obtain {Hj}[j∈Q] by solving Hopt= R tan(θopt)
23: return {Hj}j∈Q, {Rj}j∈Q, {mj}j∈Q
Fig. 3.4 shows the average coverage ratios of 80 UEs by 8 available UAV-BSs with 10 realizations in four different environments when increasing the number of UAV-BSs. The UEs are arbitrarily
distributed. As seen from Fig. 3.4, the coverage ratio varies significantly in four deployment scenarios, particularly with high-rise urban one much more challenging than others.
By applying Shannon Capacity Theorem, the required SN R of each UAV-BS can be calculated through C = B log[2](1 + Pr
Pn), where B is the bandwidth of the
channel, Pr and Pn denote the required received power and average noise power,
respectively. In our model, we assume that B = 1 × 107 Hz, Pr = −74 dBm and
Pn = −100 dBm. Thus, we can obtain the minimum required power for each UAV-BS
by Pt = Pr+P L(Rj, Hj). Fig. 3.5 shows that in urban environments with 10 available
UAV-BSs and 3 different approaches to determine altitudes, the average minimum required transmit power of all UAV-BSs when increasing the number of UEs. In
the fixed altitude approach, all the UAV-BSs are deployed in the same altitude. In random altitude approach, each UAV-BS is deployed into a random altitudes. The altitudes from both fixed altitude
and random altitudes are selected from the range where P Lmax requirement is met. As we can see, if the UAV-BSs are deployed in the
altitude in the way we proposed, less average transmit power is required to provide wireless service.
For further performance comparison, we test 3 algorithms to obtain the coverage percentage of UEs given 10 available UAV-BSs fixing environment parameters (Ur-ban). In each test, we generated 10
times of arbitrary UE distributions of 80, 200 and 450 UEs respectively. Besides the GA deployment strategy proposed, we have two other schemes for comparison. The first one is random placement which
ran-domly selects a location within the square region and a coverage radius. The second one is K-means algorithm which partitions the UEs into ˆK clusters to be covered by ˆK UAV-BSs. The results are
presented in TABLE 4.1. Compared with these two other algorithms, GA has the significant advantage of solving the optimization problem with many variables involved. It is observed that the result of
GA based deployment has higher coverage percentage and this advantage is more pronounced when the number of UEs increases.
Table 3.1: Coverage ratio comparison in urban environment.
|P| 80 200 450
GA Deployment Method 99.2% 88.6% 75.3%
K-Means 98.6% 82.3% 69.4%
-1000 0 1000 2000 3000 4000 5000 6000 x-dimension (m) -1000 0 1000 2000 3000 4000 5000 6000 y-dimension (m) UE c1 UE c2 UE c3 UAV-BS location
Figure 3.3: The 100% coverage ratio result of GA deployment with 80 UEs in a 5000 m × 5000 m square region with different data rate requirements.
1 2 3 4 5 6 7 8 9 10 Number of UAV-BSs 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Coverage Ratio Dense Urban Urban Suburban Highrise Urban
80 90 100 110 120 130 140 150 160 Number of UEs 43.5 44 44.5 45 45.5 46 46.5 47 47.5
Average Received Power(dbm)
Altitude with Maximum Coverage Fixed Altitude
Random Altitude
Figure 3.5: The UAV’s average transmit power comparison of altitude with maximum coverage, fixed altitude and random altitude in urban environments.
Chapter 4
Dynamic Movement Strategy in a
UAV-Assisted Network
In this chapter, we investigate a real-time dynamic UAV movement strategy design on a deep reinforcement learning framework called Deep Q-Network (DQN) [14] to maximize the sum data rate. Our
contribution lies in the formulating the design problem of the UAVs’ movement strategy to find the optimal locations of UAVs in every single time instant, in response to the ground users’ movement.
UAV-Assisted Network System Description
Fig. 4.1 shows the framework of UAV-assisted wireless communications system model where UAVs serve as aerial base stations and provide wireless communications to the ground UEs. Also, the traditional
terrestrial infrastructures are capable of serving the UEs which are not covered by UAV-BSs. Let P be the set of all the UEs which are labelled as i = 1, 2, ..., |P|. Q denotes the set of available
UAV-BSs labelled as j = 1, 2, ..., |Q| and O denotes the set of ground base stations (GBSs) labelled as k = 1, 2, ..., |O|. In our system, we assume that the UEs are assigned to the closest base
station to receive wireless communication service and all the UAV-BSs cells are deployed at the same altitude H. Also, considering the existing interference mitigation technologies, both the
interference between BSs and interference between UAV-BSs and GUAV-BSs cells are assumed to be negligible. Moreover, ground users are assumed to move at each time instant so the locations of each UE
at each time instant can be expressed as mi(t) = [xi(t), yi(t)] , t ∈ T where T is the time window considered..
Figure 4.1: A communication system model of UAV-assisted Network
Similarly, the locations of UAV-BS j can be written as nj(t) = [˜xj(t), ˜yj(t)]. Also,
uk = [ˇxk, ˇyk] denotes the the location of the k-th GBS, which is a known parameter
in the study.
UAV Dynamic Movement Problem
The dynamic UAV-BS movement strategy problem can be treated as a design of determining the positions of the UAV-BSs at each time instant. The objective is to find the optimal positions for all
UAV-BSs in each time-slot, to maximize the sum data rates of users. γij/ik(t) is a binary variable indicating whether the user i is
associated with UAV-BS j or GBS k at time instant t, with 1 for service and 0 for no association. Thus, the optimization problem at each time instant t can be formulated as:
maximize nj(t),j∈Q |P| X i=1 Ci(t), s.t. C1 : knj(t) − γij(t)mi(t)k ≤ n¯[j](t) − mi(t) + M |1 − γij(t)| , ∀j ∈ Q, ∀¯j ∈ {O, Q \ j} C2 : kuk− αik(t)mi(t)k ≤ ku¯[k]− mi(t)k + M |1 − αik(t)| , ∀k ∈ O,
∀¯k ∈ {Q, O \ k} C3 :X j γij(t) + X k αik(t) = 1, ∀i, j, k. (4.1)
Constraints C1 and C2 in (6) guarante all the UEs are associated with the nearest UAV-BSs/GBSs. Also, M is a large number to ensure the constraints hold in any UE association conditions. Then C3
guarantees all the UEs are associated with only single base station. Therefore, the objective of the optimization problem is to find the optimal positions of UAV-BSs in each instant over time
duration T so that the sum data rates of the users can be maximized.
Deep Q-Network based UAV Movement
In this section, given the real-time locations of a set of UEs, we present a rein-forcement learning based UAV-BS movement strategy to obtain the optimal real-time locations of UAV-BSs. Before
discussing the movement of UAV-BSs, the mobility model of UEs needs to be discussed first. The random walk model [13] is chosen as the UE mobility model in this letter, but other models can be easily
included. The moving direction of UEs are uniformly distributed among left, right, forward, backward and staying still. Moreover, the initial positions of the ground users are assumed to be fixed. At
each instant t ∈ T when ground users move, all UAV-BSs take action in response to the movement of the ground users.
The objective is to train a neural network to represent the action-value function which takes the local observations of the positions of both UEs and UAV-BSs in any instant as inputs and derives the
action-value functions of the UAV-BSs movement. The Deep Network consists of four parts: states, actions, rewards and the Q-Network. At each time slot t, each agent observes a state st, from the
state space S
6: Observe s[t]
7: Choose the action aj[t] which maximizes the ¯Q(sj[t], aj[t]; ¯Θj)
8: end for
9: All agents take actions, observe rewards rjt, update state s
j t → s
j t+1
10: for each UAV-BS agent j do
11: Observe sj[t+1]
12: Store (sj[t],aj[t], r[t+1]j , sj[t+1]) into replay memory Dj
13: Uniformly sample mini batch from replay memory Dj
14: Perform a gradient descent on Loss = (rj[t]+ βmaxa0Q(s˜ j[t+1], a0; ˜Θ[j]) − ¯Q(sj[t], aj[t]; ¯Θ[j]))2with
respect to network parameters ¯Θj.
15: Update ˜Θj= ¯Θj every C time steps
16: end for
17: end for
18: end for
and takes an action atin the action space A based on the decision from Q-Network ¯Q.
The principle of the Q-Network is to obtain the maximum Q-value which maximizes the sum data rates of UEs. Following the action, the state of each agent transits to a new state st+1 and the agents
receive a reward rt which is determined by the
instantaneous sum data rates of ground users.
State Representation
All agents’ states are defined as: s = (xuav, yuav) which is the horizontal position
of the UAVs. Assuming that the initial states of all UAV-BSs are at the optimal positions where the sum data rates of ground users are maximized at time instant t0.
The optimal positions can be derived by conducting exhaust search.
Action Space
At each time step, all the UAV-BSs take an action at ∈ A which includes choosing
a direction for UAV-BSs to move according to the current state st, based on the
same speed in any time step therefore the moving distances for any UAV-BSs from any time instant t to t + 1 are assumed to be the same. More specifically, since we assume that all the UAV-BSs are at
the same altitude H, there are 5 different actions in A: (1,0) means the UAV-BS will turn right, (-1,0) means the UAV-BS will turn left, (0,1) means the UAV-BS will move forward, (0,-1) means the
UAV-BS will move backward and (0,0) means the UAV-BS will stay still.
Reward Design
After performing an action, the UAV-BS has a different location so the UEs need to change the association based on problem ??. Therefore, the new association comes with a new instantaneous sum data
rates of the ground UEs. The principle of de-signing the reward function is to improve the UEs’ instantaneous data rates, which enables the agent to receive a positive reward. When the action results
in a reduction of the sum data rates of the UEs, the UAV-BS receives a negative reward. Thus, the reward function can be expressed as
1, if sum rates increase
−0.2, if sum rates remain the same −1, if sum rates decrease
Training Procedure
The training procedure requires a learning rate α and a discount factor β. The learning procedure is divided into several episodes, and at the beginning of each episode, the positions of UAV-BSs will
be reset to the initial values. We leverage a DQN with experience replay to train the agents [14]. Each agent j has a DQN ¯Q that takes an input of the observation of the current state sj[t] and
generate the output of the value functions corresponding to all the actions. At each training step t, each agent chooses the action aj[t] which leads to the maximum estimated Q value. Based on the
action taken by the agent, the transition tuple (sj[t], aj[t], rj[t], sj[t+1]) is collected and stored into the replay memory D with a size of N . Then, in each episode, a predetermined size of
mini-batch experiences E are uniformly sampled to update Θ using gradient
stabilize the training.
Numerical Result
In our simulation, we consider UAV-assisted model in a 5000 m × 5000 m area and uni-formly divide the area into 4 sections, that is, Section 1 : 0 < x ≤ 2500, 0 < y ≤ 2500, Section 2 : 2500 < x ≤
5000, 0 < y ≤ 2500, Section 3 : 0 < x ≤ 2500, 2500 < y ≤ 5000, Section 4 : 2500 < x ≤ 5000, 2500 < y ≤ 5000. We assume that initially all of the UEs are distributed in the whole area, and then in the
middle of the time duration, the majority (90%) of the UEs converge to Section 1. At the end of the time duration, all the UEs go back to the uniformly distributed in the whole area. The UEs follow
random walk mobility model inside the section area. There is one GBS available located at u0 = [2500, 2500]. Moreover, referring to [1], the environment parameters
are set up as follows: fc= 2 GHz, P Lmax = 103 dB, (a, b, ηLoS, ηN LoS) is configured
to be (9.61, 0.43, 0.1, 20) corresponding to the urban environment. The transmit powers of UAV-BSs and GBS are set to be 37 dBm and 40 dBm, respectively. Also, the Deep Q-Network parameter set (α, β,
N, B, C) is configured to be (0.01, 0.9, 2000, 50, 200). Fig. 4.2 shows the UEs distribution and their association in a time instant. The UEs and base stations with same color represent the
association and all the UEs are associated with the closest base stations.
Table 4.1: Comparisons of processing time of different algorithm NA Processing Time (ms)
Deep Q-Network 210
Exhaust Search 4117
K-Means 387
Fixed 0
Fig. 4.3 shows the comparison of the sum data rates in all the time instants with different algorithms. It can be observed that the overall performance in 500
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 x-dimension(m) 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 y-dimension(m) UAV[1] UAV[2] UAV 3 UAV[4] Base Station UE[1] UE 2 UE[3] UE[4] UE
Figure 4.2: The UE distribution and association with 500 UEs in a 5000 m × 5000 m area. 0 50 100 150 200 250 300 350 400 450 500 Time Instance 1.94 1.95 1.96 1.97 1.98 1.99 2 2.01 2.02
Sum Data Rate (bps)
Exhaust Search Deep Q Network K-Means Fixed
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Number of episodes 104 0.6 0.8 1 1.2 1.4
Sum Data Rate (bps)
Figure 4.4: Sum data rate versus number of training episodes.
time instant of Deep Q-Network outperforms the fixed location or K-Means deploy-ment strategy and closely follows the performance of the Exhaust Search. However, considering the computation cost
analysis in Table 4.1, which is generated via using Intel CoreR TMi5- 4430 Processor to run the algorithm 10 times and take the average
processing time, exhaustive search as expected achieves the highest performance but the computation complexity can be too high for real-time processing. The Deep Q-Network performs close to the
exhaustive search but with significantly less processing resource and time, which is particularly critical for low-latency communications and mission execution involving UAVs.
Fig. 4.4 further plots the sum data rates against the number of training episodes. It can be observed that the UAV-BSs are capable of carrying out their actions via iterative learning from their past
experience to improve the performance.
Chapter 5
Conclusion and future work
Optimal 3D Location of UAV-BS with
Maxi-mum Coverage
In this report, we have proposed and evaluated a cost-efficient 3D UAV-BS deploy-ment algorithm for providing real-life wireless communication services when all the UEs are randomly distributed with
various data rate requirements. A GA-based deployment algorithm has been proposed to maximize the number of covered UEs, and simultaneously meet the UEs’ individual data rate requirements and UAV-BS
capacity limit. The proposed algorithm outperforms other algorithms in requiring less UAV-BSs to provide full coverage of all UEs.
Optimal UAV Dynamic Movement Strategy
In this report, we have also proposed and evaluated a dynamic UAV-BS deployment algorithm for optimizing the real-time performance of wireless communication ser-vices when all the UEs are moving. A
Deep Q-Network based algorithm has been proposed to maximize the sum data rates of ground UEs in a dynamic UAV-assisted network. Results show that the proposed algorithm outperforms other existing
dy-namic deployment algorithms.
into consideration. Moreover, another critical factor of UAV network is the energy consumption. A more effective energy model of UAV network is another research field with great potential.
Appendix A
Genetic Algorithm Python
# F i t n e s s f u n c t i o n
d e f e v a l u a t e ( chromosome , UEpoint , uav number ,
d a t a r a t e c a p a c i t y , d a t a r e q u i r e m e n t ) :
s c o r e = 0
UAV assig = d i s t r i b u t e U E ( chromosome , UEpoint , uav number )
U A V a s s i g a r r a y = np . a s a r r a y ( UAV assig )
#p r i n t ( U A V a s s i g a r r a y )
f o r i i n r a n g e( uav number ) :
UAV index = np . where ( U A V a s s i g a r r a y==i ) [ 0 ]
#p r i n t ( UAV index . s h a p e ) s u m d a t a r a t e = 0 f o r i t e m i n UAV index : s u m d a t a r a t e += d a t a r e q u i r e m e n t [ i t e m ] i f s u m d a t a r a t e > d a t a r a
t e c a p a c i t y : p r i n t( ” Data r a t e e x c c e d s c a p a c i t y ” ) s c o r e = −1 b r e a k e l s e : s c o r e += l e n( UAV index )
: param k : UAV number
: param p o p u l a t i o n S i z e : p o p u l a t i o n number : param d a t a i n : U E l o c a t i o n : r e t u r n : ””” chromosomes = np . z e r o s ( ( p o p u l a t i o n S i z e , k ∗ 3 ) ,
dtype=np . f l o a t) u a v p o s i t i o n f i n a l = np . z e r o s ( ( p o p u l a t i o n S i z e , k ∗ 2 ) , dtype=np . f l o a t) #UEPoints = np . random . r a n d i n t ( 1 0 0 , s i z e =(UE
number , 2 ) ) f o r i i n r a n g e( p o p u l a t i o n S i z e ) : u a v p o s i t i o n = np . z e r o s ( ( k , 3 ) , dtype=np . f l o a t) u a v l o c a t i o n = np . z e r o s ( ( k , 2 ) ,
dtype=np . f l o a t) f o r j i n r a n g e( k ) :
random = np . random . rand ( 2 ) ∗5000
r a n d o m r a d i u s = np . random . rand ( ) ∗1300 u a v p o s i t i o n [ j , 0 ] = random [ 0 ] u a v p o s i t i o n [ j , 1 ] = random [ 1 ] u a v p o s i t i o n [ j , 2 ] = r a n d o m r a
d i u s u a v l o c a t i o n [ j , 0 ] = random [ 0 ] u a v l o c a t i o n [ j , 1 ] = random [ 1 ] chromosomes [ i , : ] = u a v p o s i t i o n . f l a t t e n ( )
u a v p o s i t i o n f i n a l [ i , : ] = u a v l o c a t i o n . f l a t t e n ( ) r e t u r n chromosomes , u a v p o s i t i o n f i n a l # c r o s s o v e r t o g e n e r a t e new d e s c e n
d a n t d e f c r o s s o v e r ( p o p u l a t i o n , pc , uav number ) : ””” : param p o p u l a t i o n : chromosomes : param pc : p r o b a b i l i t y o f c r o s s o v e r i s 0 . 8 : r e t u
r n : new p o p u l a t i o n ”””
# The number o f t h e chromosomes need t o be c o n s i d e r e d b a s e d on pc
m, n = p o p u l a t i o n . s h a p e numbers = np . u i n t 8 (m ∗ pc )
# Make s u r e t h e number i s even
i f numbers % 2 != 0 : numbers += 1 # G e n e r a t e an empty s t r u c t u r e u p d a t e p o p u l a t i o n = np . z e r o s ( (m, n ) , dtype=np . f l o a t) # G e n e r a t e random i n d e x
i n d e x = rd . sample (r a n g e(m) , numbers ) # Copy t h e unused o n e s f o r i i n r a n g e(m) : i f not i n d e x . c o n t a i n s ( i ) : u p d a t e p o p u l a t i o n [ i , : ] = p o p
u l a t i o n [ i , : ] # C r o s s o v e r w h i l e l e n( i n d e x ) > 0 : a = i n d e x . pop ( ) b = i n d e x . pop ( ) # G e n e r a t e a c r o s s o v e r p o i n t c r o s s o v e r P o i
n t i n d e x =
u p d a t e p o p u l a t i o n [ b , 0 : c r o s s o v e r P o i n t ] = p o p u l a t i o n [ b , 0 : c r o s s o v e r P o i n t ] u p d a t e p o p u l a t i o n [ b , c r o s s o v e r P o i n t
: ] = p o p u l a t i o n [ a , c r o s s o v e r P o i n t : ] r e t u r n u p d a t e p o p u l a t i o n d e f s e l e c t ( chromosomes , f i t , t o u r n a m e n t s i z e ) :#Tournament s e l
e c t i o n ””” : param t o u r n a m e n t s i z e : tournament s i z e : param chromosomes : chromosomes
: param f i t : f i t n e s s r e s u l t : r e t u r n : ””” m, n = chromosomes . s h a p e n e w p o p u l a t i o n = np . z e r o s ( (m, n ) , dtype=np . f l o a t) # Check v a l i d i t y o f
tournament s i z e . i f t o u r n a m e n t s i z e >= m: msg = ’ Tournament s i z e ( { } ) i s l a r g e r than p o p u l a t i o n s i z e ( { } ) ’ r a i s e V a l u e E r r o r ( msg .f o r m a
t( t o u r n a m e n t s i z e , m) ) # S e l e c t a f a t h e r and a mother . f o r i i n r a n g e(m) :
c o m p e t i t o r s = rd . sample (r a n g e(m) , t o u r n a m e n t s i z e )
n e w p o p u l a t i o n [ i , : ] =
chromosomes [max( c o m p e t i t o r s , key=lambda x : f i t [ x ] ) , : ]
r e t u r n n e w p o p u l a t i o n
d e f mutation ( chromosomes , pm, uav number ) :
Deep Q-Network Python
i m p o r t numpy a s np d e f b u i l d n e t ( s e l f ) : s e l f . s = t f . p l a c e h o l d e r ( t f . f l o a t 3 2 , [ None , s e l f . n f e a t u r e s ] ) s e l f . q t a r g e t = t f .
p l a c e h o l d e r ( t f . f l o a t 3 2 , [ None , s e l f . n a c t i o n s ] ) w i t h t f . v a r i a b l e s c o p e ( ’ Q net ’ ) : c names , n l 1 , w i n i t i a l i z e r , b i n i t i a
l i z e r = \ [ ’ Q net params ’ ,
t f . GraphKeys . GLOBAL VARIABLES ] , 1 0 , \ t f . r a n d o m n o r m a l i n i t i a l i z e r ( 0 . , 0 . 3 ) , t f . c o n s t a n t i n i t i a l i z e r ( 0 . 1 ) w i t h t f . v a r i a b l
e s c o p e ( ’ l 1 ’ ) : w1 = t f . g e t v a r i a b l e ( ’ w1 ’ , [ s e l f . n f e a t u r e s , n l 1 ] , i n i t i a l i z e r = w i n i t i a l i z e r )
b1 = t f . g e t v a r i a b l e ( ’ b1 ’ , [ 1 , n l 1 ] , i n i t i a l i z e r = b i n i t i a l i z e r ) l 1 = t f . nn . r e l u ( t f . matmul ( s e l f . s , w1 ) + b1 ) # s e c o n d l a y e
r . c o l l e c t i o n s i s use d l a t e r when a s s i g n t o t a r g e t n e t w i t h t f . v a r i a b l e s c o p e ( ’ l 2 ’ ) : w2 = t f . g e t v a r i a b l e ( ’ w2 ’ , [ n l 1 , s e l
f . n a c t i o n s ] , i n i t i a l i z e r =w i n i t i a l i z e r , c o l l e c t i o n s=c names ) b2 = t f . g e t v a r i a b l e ( ’ b2 ’ , [ 1 , s e l f . n a c t i o n s ] , i n i t i a l
i z e r = b i n i t i a l i z e r , c o l l e c t i o n s=c names ) s e l f . q e v a l = t f . matmul ( l 1 , w2 ) + b2 w i t h t f . v a r i a b l e s c o p e ( ’ l o s s ’ ) : s e l f . l o s s =
t f . reduce mean ( t f . s q u a r e d d i f f e r e n c e ( s e l f . q t a r g e t , s e l f . q e v a l ) ) w i t h t f . v a r i a b l e s c o p e ( ’ t r a i n ’ ) : s e l f . t r a i n o p = t
f . t r a i n . RMSPropOptimizer ( s e l f . l r ) . m i n i m i z e ( s e l f . l o s s ) s e l f . s = t f . p l a c e h o l d e r ( t f . f l o a t 3 2 , [ None , s e l f . n f e a t u r e s ] ) #
i n p u t w i t h t f . v a r i a b l e s c o p e ( ’ t a r g e t n e t ’ ) : c names = [ ’ t a r g e t n e t p a r a m s ’ , t f ] w i t h t f . v a r i a b l e s c o p e ( ’ l 1 ’ ) : w1 = t f . g
e t v a r i a b l e ( ’ w1 ’ , [ s e l f . n f e a t u r e s , n l 1 ] , i n i t i a l i z e r = w i n i t i a l i z e r )
w i t h t f . v a r i a b l e s c o p e ( ’ l 2 ’ ) : w2 = t f . g e t v a r i a b l e ( ’ w2 ’ , [ n l 1 , s e l f . n a c t i o n s ] , i n i t i a l i z e r = w i n i t i a l i z e r ) b2 = t f .
g e t v a r i a b l e ( ’ b2 ’ , [ 1 , s e l f . n a c t i o n s ] , i n i t i a l i z e r = b i n i t i a l i z e r ) s e l f . q n e x t = t f . matmul ( l 1 , w2 ) + b2 d e f s t o r e t r a n s i
t i o n ( s e l f , s , a , r , s ) : t r a n s i t i o n = np . h s t a c k ( ( s , [ a , r ] , s ) ) i n d e x = s e l f . memory counter % s e l f . m e m o r y s i z e s e l f . memory [ i n d e
x , : ] = t r a n s i t i o n s e l f . memory counter += 1 d e f c h o o s e a c t i o n ( s e l f , o b s e r v a t i o n ) : o b s e r v a t i o n = o b s e r v a t i o n [ np . newaxis , : ] i f
np . random . u n i f o r m ( ) < 0 . 1 : a c t i o n s v a l u e = s e l f . s e s s . run ( s e l f . q e v a l , f e e d d i c t ={ s e l f . s : o b s e r v a t i o n } ) a c t i o n c h o s e n
= np . argmax ( a c t i o n s v a l u e ) e l s e :
a c t i o n c h o s e n = np . random . r a n d i n t ( 0 , s e l f . n a c t i o n s )
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[5] R. I. Bor-Yaliniz, A. El-Keyi, and H. Yanikomeroglu. Efficient 3-D placement of an aerial base station in next generation cellular networks. In Proc. IEEE Int. Conf. Communications (ICC), pages
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[6] B. Galkin, J. Kibilda, and L. A. DaSilva. Deployment of UAV-mounted access points according to spatial user locations in two-tier cellular networks. In Proc. Wireless Days (WD), pages 1–6, March
[7] Yiming Huo, Xiaodai Dong, Tao Lu, Wei Xu, and Marvin Yuen. Distributed and multi-layer uav networks for next-generation wireless communication and power transfer: A feasibility study. IEEE
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[10] Donald Michie, David J Spiegelhalter, CC Taylor, et al. Machine learning. Neural and Statistical Classification, 13, 1994.
[11] M. Mozaffari, W. Saad, M. Bennis, and M. Debbah. Efficient deployment of multiple unmanned aerial vehicles for optimal wireless coverage. IEEE Commu-nications Letters, 20(8):1647–1650, August
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(GLOBECOM), pages 1–6, December 2018. | {"url":"https://5dok.net/document/wyevjx0z-deploying-base-stations-communication-network-using-machine-learning.html","timestamp":"2024-11-02T23:26:39Z","content_type":"text/html","content_length":"207387","record_id":"<urn:uuid:1df8d624-7b99-46b7-90a3-10397f0abf4e>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00883.warc.gz"} |
WINETASTER ON 10/02/06 WITH 8 JUDGES AND 5 WINES BASED ON RANKS, IDENT=N
Copyright (c) 1995-2006 Richard E. Quandt, V. 1.65
FLIGHT 1:
Number of Judges = 8
Number of Wines = 5
Identification of the Wine: The judges' overall ranking:
Wine A is Grace 2002 ........ 4th place
Wine B is Cardinale 2002 ........ 3rd place
Wine C is Rudd 2002 ........ 5th place
Wine D is Heitz 2001 ........ 2nd place
Wine E is Poetry 2001 ........ 1st place
The Judges's Rankings
Judge Wine -> A B C D E
Ed 5. 3. 4. 2. 1.
Bob 5. 1. 4. 2. 3.
Mike 5. 4. 3. 2. 1.
Frank 2. 1. 5. 4. 3.
Burt 3. 4. 5. 2. 1.
Orley 5. 1. 4. 2. 3.
John 3. 4. 5. 2. 1.
Dick 1. 5. 3. 4. 2.
Table of Votes Against
Wine -> A B C D E
Group Ranking -> 4 3 5 2 1
Votes Against -> 29 23 33 20 15
( 8 is the best possible, 40 is the worst)
Here is a measure of the correlation in the preferences of the judges which
ranges between 1.0 (perfect correlation) and 0.0 (no correlation):
W = 0.3187
The probability that random chance could be responsible for this correlation
is quite small, 0.0372. Most analysts would say that unless this
probability is less than 0.1, the judges' preferences are not strongly
We now analyze how each taster's preferences are correlated with the group
preference. A correlation of 1.0 means that the taster's preferences are a
perfect predictor of the group's preferences. A 0.0 means no correlation,
while a -1.0 means that the taster has the reverse ranking of the group.
This is measured by the correlation R.
Correlation Between the Ranks of
Each Person With the Average Ranking of Others
Name of Person Correlation R Correlation Price
Ed 0.9000 -0.9000
John 0.9000 -0.9000
Burt 0.9000 -0.9000
Mike 0.7000 -0.7000
Bob 0.5000 -0.5000
Orley 0.5000 -0.5000
Frank 0.2000 -0.2000
Dick -0.1000 0.1000
The wines were preferred by the judges in the following order. When the
preferences of the judges are strong enough to permit meaningful differentiation
among the wines, they are separated by -------------------- and are judged to be
significantly different.
1. ........ 1st place Wine E is Poetry 2001
2. ........ 2nd place Wine D is Heitz 2001
3. ........ 3rd place Wine B is Cardinale 2002
4. ........ 4th place Wine A is Grace 2002
5. ........ 5th place Wine C is Rudd 2002
We now test whether the ranksums AS A WHOLE provide a significant ordering.
The Friedman Chi-square value is 10.2000. The probability that this could
happen by chance is 0.0372
We now test whether the group ranking of wines is correlated with the
prices of the wines. The rank correlation between them is -1.0000. At the
10% level of significance this would have to exceed the critical value of
0.8000 to be significant.
We now undertake a more detailed examination of the pair-wise rank correla-
tions that exist between pairs of judges. First, we present a table in which you
can find the correlation for any pair of judges, by finding one of the names in the
left hand margin and the other name on top of a column. A second table arranges
these correlations in descending order and marks which is significantly positive
significantly negative, or not significant. This may allow you to find clusters
of judges whose rankings were particularly similar or particularly dissimilar.
Pairwise Rank Correlations
Correlations must exceed in absolute value 1.00 for significance at the 0.05
level and must exceed 0.90 for significance at the 0.1 level
Ed Bob Mike
Ed 1.000 0.600 0.900
Bob 0.600 1.000 0.300
Mike 0.900 0.300 1.000
Frank -0.100 0.300 -0.500
Burt 0.700 0.100 0.600
Orley 0.600 1.000 0.300
John 0.700 0.100 0.600
Dick -0.300 -0.900 -0.100
Frank Burt Orley
Ed -0.100 0.700 0.600
Bob 0.300 0.100 1.000
Mike -0.500 0.600 0.300
Frank 1.000 0.100 0.300
Burt 0.100 1.000 0.100
Orley 0.300 0.100 1.000
John 0.100 1.000 0.100
Dick -0.100 0.300 -0.900
John Dick
Ed 0.700 -0.300
Bob 0.100 -0.900
Mike 0.600 -0.100
Frank 0.100 -0.100
Burt 1.000 0.300
Orley 0.100 -0.900
John 1.000 0.300
Dick 0.300 1.000
Pairwise correlations in descending order
1.000 Bob and Orley Significantly positive
1.000 Burt and John Significantly positive
0.900 Ed and Mike Significantly positive
0.700 Ed and Burt Not significant
0.700 Ed and John Not significant
0.600 Ed and Bob Not significant
0.600 Mike and John Not significant
0.600 Ed and Orley Not significant
0.600 Mike and Burt Not significant
0.300 Bob and Frank Not significant
0.300 John and Dick Not significant
0.300 Bob and Mike Not significant
0.300 Frank and Orley Not significant
0.300 Burt and Dick Not significant
0.300 Mike and Orley Not significant
0.100 Frank and John Not significant
0.100 Bob and Burt Not significant
0.100 Bob and John Not significant
0.100 Frank and Burt Not significant
0.100 Burt and Orley Not significant
0.100 Orley and John Not significant
-0.100 Ed and Frank Not significant
-0.100 Frank and Dick Not significant
-0.100 Mike and Dick Not significant
-0.300 Ed and Dick Not significant
-0.500 Mike and Frank Not significant
-0.900 Bob and Dick Significantly negative
-0.900 Orley and Dick Significantly negative
All the wines were quite amazing. They had a substantially similar
bouquet but did not taste identical by any means. Nevertheless, the
tasters claimed that they had difficulty in distinguishing among the
wines. On the whole, the Poetry was deemed to be significantly good and
the Rudd was thought to be significantly bad. One taster judged the
wine that was second worst in the aggregate as being first. The real
question was whether the tasters could differentiate among the very
expensive wines (ranging from $105 to $159) and the relatively inexpensive
Heitz costing only $45. In fact, the Heitz was the second highest ranked wine,
which suggests that the higher priced wines are substantially overpriced.
The tasters were asked to identify the Heitz in a
secret ballot, and only one out of eight tasters succeded in identifying
this wine. Every taster's preferences among the wines was negative
correlated with the wine prices except for the one contrarian taster who
ranked the Grace first.
It is worth mentioning that this is a lanmdmark tasting in that it is the
100th tasting since we have started to record the tastings and the statistical
results in a systematic way. The only noteworthy observation we can make is that
a statistical analysis of the results of the tastings suggests on the basis of the
Kendall W-coefficients (or rather, of the p-values corresponding to these coefficients)
that we have not increased over time the degree of agreement among the tasters---that
is to say, we have not learned from each other and have not adopted over time the
tasting standards of other tasters.
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QP as LP: cutting planes
In (1) I described a simple linearization scheme for a QP model we we can solve it as a straight LP. For a simple (convex) quadratic function \(f(x)\), instead of solving:
we solve
\[\begin{align} \min\>&z\\
&z \ge a_i x + b_i\end{align}\]
In this post I do things slightly different: instead of adding the linear inequalities ahead of time, we add them one at the time based on the previously found optimal point. This approach is called
a cutting plane technique (2).
Example: portfolio model
We consider again the simple portfolio model:
\[\bbox[lightcyan,10px,border:3px solid darkblue]{
\begin{align} \min \>& \sum_t w_t^2\\
& w_t = \sum_i r’_{i,t} x_i\\
& \sum_i \mu_i x_i \ge R\\
& \sum_i x_i = 1\\
&x_i\ge 0\end{align}}
The model is linearized as an LP model:
\[\bbox[lightcyan,10px,border:3px solid darkblue]
{\begin{align}\min \>& \sum_t z_t\\
&z_t \ge a_{t,k} w_t + b_{t,k} & \text{Added $K\times T$ cuts}\\
& w_t = \sum_i r’_{i,t} x_i\\
& \sum_i \mu_i x_i \ge R\\
& \sum_i x_i = 1\\
&x_i, z_t\ge 0\end{align}}
Initially we start without cuts. Later on, during the Cutting Plane algorithm, we will add linear cuts in each iteration. The algorithm would look like:
1. \(k:=1\)
2. Solve model LP, let \(w^*_t\) be the optimal values for \(w_t\).
3. if \(k=\)MAXIT: STOP
4. Add the constraint \(z_t \ge a_{t,k} w_t + b_{t,k}\) where \(a_{t,k}=2 w^*_t\) and \(b_{t,k}=-(w^*_t)^2\). Note that we add one cut for each \(w_t\) here (our dataset has \(T=717\) time periods).
5. \(k:=k+1\)
6. go to step 2
Here we stop simply when a certain number of iterations MAXIT has been exceeded. That can be refined by stopping when the objective does not seem to change much anymore. Another optimization could be
to only add cuts that are different from the ones added before (for some \(t\) we may converge quicker than for others).
The algorithm converges very fast:
---- 118 PARAMETER report2
obj(LP) w'w
iter1 1.02907546
iter2 0.21577087 0.46879016
iter3 0.33210537 0.48432099
iter4 0.37143835 0.41397603
iter5 0.40990988 0.41362293
iter6 0.41109694 0.41222051
iter7 0.41183602 0.41212331
iter8 0.41192720 0.41204267
iter9 0.41196991 0.41204112
iter10 0.41197620 0.41203782
Note that the optimal QP solution has an objective: 0.41202816.
This is pretty good performance especially because the dataset is not very small (it has 717 time periods and 83 stocks).
Here is a picture of the cuts introduced for the first element \(z_1=w_1^2\):
A combined approach
We can even combine the two methods:
1. Start with a coarse-grained (i.e. cheap) initial set of cuts. In (1) we used \(n=10\) inequalities per \(w_t\). For this experiment I reduced this to \(n=5\).
2. Then apply our cutting plane algorithm. Instead of MAXIT=10 iterations we now do 5 iterations.
This yields even faster convergence:
---- 142 PARAMETER report3
obj(LP) w'w
iter1 0.32921509 0.41401219
iter2 0.40812966 0.41471624
iter3 0.41017228 0.41211351
iter4 0.41188869 0.41208158
iter5 0.41195624 0.41203588
1. QP as LP: piecewise linear functions, http://yetanothermathprogrammingconsultant.blogspot.com/2017/04/qp-as-lp-piecewise-linear-functions.html
2. J. E. Kelley, Jr. "The Cutting-Plane Method for Solving Convex Programs." J. Soc. Indust. Appl. Math. Vol. 8, No. 4, pp. 703-712, December, 1960.
3. Cardinality Constrained Portfolio Optimization: MIQP without MIQP Solver, http://yetanothermathprogrammingconsultant.blogspot.com/2016/02/cardinality-constrained-portfolio.html. Here this cutting
plane method is applied to a MIQP model (not strictly “allowed” as this is no longer convex, but useful as a heuristic).
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4.4 Newton’s Third Law of Motion: Symmetry in Forces
4 Dynamics: Force and Newton’s Laws of Motion
26 4.4 Newton’s Third Law of Motion: Symmetry in Forces
• Understand Newton’s third law of motion.
• Apply Newton’s third law to define systems and solve problems of motion.
There is a passage in the musical Man of la Mancha that relates to Newton’s third law of motion. Sancho, in describing a fight with his wife to Don Quixote, says, “Of course I hit her back, Your
Grace, but she’s a lot harder than me and you know what they say, ‘Whether the stone hits the pitcher or the pitcher hits the stone, it’s going to be bad for the pitcher.’” This is exactly what
happens whenever one body exerts a force on another—the first also experiences a force (equal in magnitude and opposite in direction). Numerous common experiences, such as stubbing a toe or throwing
a ball, confirm this. It is precisely stated in Newton’s third law of motion.
Whenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that it exerts.
This law represents a certain symmetry in nature: Forces always occur in pairs, and one body cannot exert a force on another without experiencing a force itself. We sometimes refer to this law
loosely as “action-reaction,” where the force exerted is the action and the force experienced as a consequence is the reaction. Newton’s third law has practical uses in analyzing the origin of forces
and understanding which forces are external to a system.
We can readily see Newton’s third law at work by taking a look at how people move about. Consider a swimmer pushing off from the side of a pool, as illustrated in Figure 1. She pushes against the
pool wall with her feet and accelerates in the direction opposite to that of her push. The wall has exerted an equal and opposite force back on the swimmer. You might think that two equal and
opposite forces would cancel, but they do not because they act on different systems. In this case, there are two systems that we could investigate: the swimmer or the wall. If we select the swimmer
to be the system of interest, as in the figure, then[latex]\textbf{F}_{\textbf{wall on feet}}[/latex]is an external force on this system and affects its motion. The swimmer moves in the direction of
[latex]\textbf{F}_{\textbf{wall on feet}}.[/latex]In contrast, the force[latex]\textbf{F}_{\textbf{feet on wall}}[/latex]acts on the wall and not on our system of interest. Thus[latex]\textbf{F}_{\
textbf{feet on wall}}[/latex]does not directly affect the motion of the system and does not cancel[latex]\textbf{F}_{\textbf{wall on feet}}.[/latex]Note that the swimmer pushes in the direction
opposite to that in which she wishes to move. The reaction to her push is thus in the desired direction.
Figure 1. When the swimmer exerts a force F[feet on wall] on the wall, she accelerates in the direction opposite to that of her push. This means the net external force on her is in the direction
opposite to F[feet on wall]. This opposition occurs because, in accordance with Newton’s third law of motion, the wall exerts a force F[wall on feet] on her, equal in magnitude but in the direction
opposite to the one she exerts on it. The line around the swimmer indicates the system of interest. Note that F[feet on wall] does not act on this system (the swimmer) and, thus, does not cancel F
[wall on feet]. Thus the free-body diagram shows only F[wall on feet], w, the gravitational force, and BF, the buoyant force of the water supporting the swimmer’s weight. The vertical forces w and BF
cancel since there is no vertical motion.
Other examples of Newton’s third law are easy to find. As a professor paces in front of a whiteboard, she exerts a force backward on the floor. The floor exerts a reaction force forward on the
professor that causes her to accelerate forward. Similarly, a car accelerates because the ground pushes forward on the drive wheels in reaction to the drive wheels pushing backward on the ground. You
can see evidence of the wheels pushing backward when tires spin on a gravel road and throw rocks backward. In another example, rockets move forward by expelling gas backward at high velocity. This
means the rocket exerts a large backward force on the gas in the rocket combustion chamber, and the gas therefore exerts a large reaction force forward on the rocket. This reaction force is called
thrust. It is a common misconception that rockets propel themselves by pushing on the ground or on the air behind them. They actually work better in a vacuum, where they can more readily expel the
exhaust gases. Helicopters similarly create lift by pushing air down, thereby experiencing an upward reaction force. Birds and airplanes also fly by exerting force on air in a direction opposite to
that of whatever force they need. For example, the wings of a bird force air downward and backward in order to get lift and move forward. An octopus propels itself in the water by ejecting water
through a funnel from its body, similar to a jet ski. In a situation similar to Sancho’s, professional cage fighters experience reaction forces when they punch, sometimes breaking their hand by
hitting an opponent’s body.
Example 1: Getting Up To Speed: Choosing the Correct System
A physics professor pushes a cart of demonstration equipment to a lecture hall, as seen in Figure 2. Her mass is 65.0 kg, the cart’s is 12.0 kg, and the equipment’s is 7.0 kg. Calculate the
acceleration produced when the professor exerts a backward force of 150 N on the floor. All forces opposing the motion, such as friction on the cart’s wheels and air resistance, total 24.0 N.
Figure 2. A professor pushes a cart of demonstration equipment. The lengths of the arrows are proportional to the magnitudes of the forces (except for f, since it is too small to draw to scale).
Different questions are asked in each example; thus, the system of interest must be defined differently for each. System 1 is appropriate for Example 2, since it asks for the acceleration of the
entire group of objects. Only F[floor] and f are external forces acting on System 1 along the line of motion. All other forces either cancel or act on the outside world. System 2 is chosen for this
example so that F[prof] will be an external force and enter into Newton’s second law. Note that the free-body diagrams, which allow us to apply Newton’s second law, vary with the system chosen.
Since they accelerate as a unit, we define the system to be the professor, cart, and equipment. This is System 1 in Figure 2. The professor pushes backward with a force[latex]\textbf{F}_{\textbf
{foot}}[/latex]of 150 N. According to Newton’s third law, the floor exerts a forward reaction force[latex]\textbf{F}_{\textbf{floor}}[/latex]of 150 N on System 1. Because all motion is horizontal, we
can assume there is no net force in the vertical direction. The problem is therefore one-dimensional along the horizontal direction. As noted, [latex]\textbf{f}[/latex]opposes the motion and is thus
in the opposite direction of[latex]\textbf{F}_{\textbf{floor}}.[/latex]Note that we do not include the forces[latex]\textbf{F}_{\textbf{prof}}[/latex]or[latex]\textbf{F}_{\textbf{cart}}[/latex]
because these are internal forces, and we do not include[latex]\textbf{F}_{\textbf{foot}}[/latex]because it acts on the floor, not on the system. There are no other significant forces acting on
System 1. If the net external force can be found from all this information, we can use Newton’s second law to find the acceleration as requested. See the free-body diagram in the figure.
Newton’s second law is given by
The net external force on System 1 is deduced from Figure 2 and the discussion above to be
[latex]\boldsymbol{\textbf{F}_{\textbf{net}}=\textbf{F}_{\textbf{floor}}-\textbf{f}=150\textbf{ N}-24.0\textbf{ N}=126\textbf{ N}.}[/latex]
The mass of System 1 is
[latex]\boldsymbol{m=(65.0 + 12.0 + 7.0)\textbf{ kg} = 84\textbf{ kg}.}[/latex]
These values of[latex]\boldsymbol{F_{\textbf{net}}}[/latex]and[latex]\boldsymbol{m}[/latex]produce an acceleration of
[latex]\begin{array}{r @{{}={}}l} \boldsymbol{a} & \boldsymbol{\frac{F_{\textbf{net}}}{m}} \\[1em] \boldsymbol{a} & \boldsymbol{\frac{126 \;\textbf{N}}{84 \;\textbf{kg}} = 1.5 \;\textbf{m} / \;\
textbf{s}^2} \end{array}[/latex]
None of the forces between components of System 1, such as between the professor’s hands and the cart, contribute to the net external force because they are internal to System 1. Another way to look
at this is to note that forces between components of a system cancel because they are equal in magnitude and opposite in direction. For example, the force exerted by the professor on the cart results
in an equal and opposite force back on her. In this case both forces act on the same system and, therefore, cancel. Thus internal forces (between components of a system) cancel. Choosing System 1 was
crucial to solving this problem.
Example 2: Force of the Cart—Choosing a New System
Calculate the force the professor exerts on the cart in Figure 2 using data from the previous example if needed.
If we now define the system of interest to be the cart plus equipment (System 2 in Figure 2), then the net external force on System 2 is the force the professor exerts on the cart minus friction. The
force she exerts on the cart,[latex]\textbf{F}_{\textbf{prof}},[/latex]is an external force acting on System 2.[latex]\textbf{F}_{\textbf{prof}}[/latex]was internal to System 1, but it is external to
System 2 and will enter Newton’s second law for System 2.
Newton’s second law can be used to find[latex]\textbf{F}_{\textbf{prof}}.[/latex]Starting with
and noting that the magnitude of the net external force on System 2 is
we solve for[latex]\boldsymbol{F_{\textbf{prof}}},[/latex]the desired quantity:
The value of[latex]\boldsymbol{f}[/latex]is given, so we must calculate net[latex]\boldsymbol{F_{\textbf{net}}}.[/latex]That can be done since both the acceleration and mass of System 2 are known.
Using Newton’s second law we see that
where the mass of System 2 is 19.0 kg ([latex]\boldsymbol{m}[/latex]= 12.0 kg + 7.0 kg) and its acceleration was found to be[latex]\boldsymbol{a=1.5\textbf{ m/s}^2}[/latex]in the previous example.
[latex]\boldsymbol{F_{\textbf{net}}=(19.0\textbf{ kg})(1.5\textbf{ m/s}^2)=29\textbf{ N}.}[/latex]
Now we can find the desired force:
[latex]\boldsymbol{F_{\textbf{prof}}=29\textbf{ N}+24.0\textbf{ N}=53\textbf{ N}.}[/latex]
It is interesting that this force is significantly less than the 150-N force the professor exerted backward on the floor. Not all of that 150-N force is transmitted to the cart; some of it
accelerates the professor.
The choice of a system is an important analytical step both in solving problems and in thoroughly understanding the physics of the situation (which is not necessarily the same thing).
Visualize the gravitational force that two objects exert on each other. Change properties of the objects in order to see how it changes the gravity force.
Figure 3. Gravity Force Lab
Section Summary
• Newton’s third law of motion represents a basic symmetry in nature. It states: Whenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and
opposite in direction to the force that the first body exerts.
• A thrust is a reaction force that pushes a body forward in response to a backward force. Rockets, airplanes, and cars are pushed forward by a thrust reaction force.
Conceptual Questions
1: When you take off in a jet aircraft, there is a sensation of being pushed back into the seat. Explain why you move backward in the seat—is there really a force backward on you? (The same reasoning
explains whiplash injuries, in which the head is apparently thrown backward.)
2: A device used since the 1940s to measure the kick or recoil of the body due to heart beats is the “ballistocardiograph.” What physics principle(s) are involved here to measure the force of cardiac
contraction? How might we construct such a device?
3: Describe a situation in which one system exerts a force on another and, as a consequence, experiences a force that is equal in magnitude and opposite in direction. Which of Newton’s laws of motion
4: Why does an ordinary rifle recoil (kick backward) when fired? The barrel of a recoilless rifle is open at both ends. Describe how Newton’s third law applies when one is fired. Can you safely stand
close behind one when it is fired?
5: An American football lineman reasons that it is senseless to try to out-push the opposing player, since no matter how hard he pushes he will experience an equal and opposite force from the other
player. Use Newton’s laws and draw a free-body diagram of an appropriate system to explain how he can still out-push the opposition if he is strong enough.
6: Newton’s third law of motion tells us that forces always occur in pairs of equal and opposite magnitude. Explain how the choice of the “system of interest” affects whether one such pair of forces
Problems & Exercises
1: What net external force is exerted on a 1100-kg artillery shell fired from a battleship if the shell is accelerated at[latex]\boldsymbol{2.40\times10^4\textbf{ m/s}^2}?[/latex]What is the
magnitude of the force exerted on the ship by the artillery shell?
2: A brave but inadequate rugby player is being pushed backward by an opposing player who is exerting a force of 800 N on him. The mass of the losing player plus equipment is 90.0 kg, and he is
accelerating at[latex]\boldsymbol{1.20\textbf{ m/s}^2}[/latex]backward. (a) What is the force of friction between the losing player’s feet and the grass? (b) What force does the winning player exert
on the ground to move forward if his mass plus equipment is 110 kg? (c) Draw a sketch of the situation showing the system of interest used to solve each part. For this situation, draw a free-body
diagram and write the net force equation.
Newton’s third law of motion
whenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that the first body exerts
a reaction force that pushes a body forward in response to a backward force; rockets, airplanes, and cars are pushed forward by a thrust reaction force
Problems & Exercises
Force on shell:[latex]\boldsymbol{2.64\times10^7\textbf{ N}}[/latex]
Force exerted on ship =[latex]\boldsymbol{-2.64\times10^7\textbf{ N}},[/latex]by Newton’s third law | {"url":"http://pressbooks-dev.oer.hawaii.edu/collegephysics/chapter/4-4-newtons-third-law-of-motion-symmetry-in-forces/","timestamp":"2024-11-07T14:01:43Z","content_type":"text/html","content_length":"174345","record_id":"<urn:uuid:78256423-d32f-491c-b481-70e3db626a9c>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00386.warc.gz"} |
Advanced Medium Combat Aircraft (AMCA) News and Discussion
Our forces must be having a hearty laugh every time these gamechanging articles come out, must be the highlight of their evening when they retire to their barracks etc 🤣 At least they n we can go
to bed smiling. Thank you admin for your sense of humour 🙏
Like they had a hearty laugh when they embarassed all of us by shooting off a cruise missile, shoot down their own colleagues mistaking it for a missile, or got shot down in enemy territory?
The forces are not ealiens. They come from our own and has got the same flaws.
How exactly do you detect things behind mountains??
AGENDA - AMCA's SUPERCRUISE
Different people speculate AMCA to Supercruise b/w Mach 1.2-1.4 with F414 engines & Mach 1.5-1.6 with JV engine producing 75 KN dry thust.
> On one side we have Mother Nature's unbeatable laws of PCM putting limits of performance -
higher drag, higher KE required, higher complexity design
> On the other side we have
global engineers pushing for Speed (both cruising & maximum)
Turboprop -> Turbojet/fan -> Ramjet -> Turbo-Ramjet -> Variable cycle adaptive engine
> KE required increases as square of velocity, looks like panic🙀, but comes from Calorific value of researched fuels with secret sauce 🍛 & ingredients - small volume but big kick👢, especially after
> Currently SuCr is attached to Turbo-jet/fan, considered an "overkill", inefficient, gimmick, etc by many as per Performance studies on engine types. Some would say it is war-time mode/feature which
it is.
But if nations're already prepared to do it in war-time since 3 decades & will continue in future also then what can civillians do?
Supercruise provides ability to -
- launch weapons to have higher range w/o increasing IRS of jet.
- Intercept targets better.
- Evade enemy's weapon.
In peace time, fighter jets fly subsonic due to multiple reasons -
- Sonic booms disturbs residential areas.
- Fuel efficiency. Typically, less/more throttle means less/more fuel flow means less/more thrust/speed/distance flown.
Jet engines like Turbo-jet/fan have their efficiency boundaries but still since decades scientists & engineers are working on
better airframe design & engine to use same amount of fuel but achieve higher thrust/speed/distance travelled.
> Given any engine with an inlet diameter, it is upto designer how much thrust can be squeezed out. Engineers either do not know that limit or it is above top secret.
> 2 same jets with different wing & fuselage design but with same # & type of engine(s) will have different performance.
If we take 3 Supercruising jets -
F-22 (SuCr M 1.8), Rafale (SuCr M 1.4), EF-2000 (SuCr M 1.5)
& their engines
F119, M-88-2, EJ-200
& compare with
then it is very difficult to find governing reason resulting in max dry thrust bcoz there are many permutations & combinations of individual engine parts design & performance.
I created a graph, manipulating the values up/down to bring the graph lines closer to visually compare better:
We see that -
> Turbine inlet temp. is a very low slope line. It takes a dip with EJ-200.
> Inlet diameter, inlet area, engine weight, volume, air mass flow show identical increasing trend.
> But, Engine length, dry thrust, dry T/W ratio, dry T/Vol ratio, Bypass ratio take a dip with F414.
So the
big dip in Bypass ratio might have impacted dry thrust & then dry T/W ratio, dry T/Vol. ratio
I wonder if engine length also influenced it.
> # of compressor & turbine stages take a dip with EJ-200. This could have affected compression ratio also.
> F119's # length, inlet dia/area, body volume, weight, air mass flow jumps obviously.
But # of stages, compression ratio, fuel SFC, take a BIG dip but impacting its dry T/W & T/Vol ratios
STILL its dry thrust is like DOUBLE.
Fuel consumption
is measured in units like
g/KN/s or lb/lbf/hr
, called
SFC or Specific Fuel Consumption
. But different people can use different metrics like fuel used as per airframe weight, distance travelled, etc.
F-22's F119 engine's SFC with
inlet dia. 100cm at 100% power (116-120.3 KN) is around 17 g/KN/s.
2 engines, so
F-22 SFC is 34 g/KN/s
at 100% power &
Sup.Cr. Mach 1.5-1.8 (514.5-617.4 m/s)
So 3.94-4Kg/s fuel for covering 514.5-617.4 m/s or
128.6-156.7 m/Kg or 6.38-7.77 gm/m.
Empty weight 19.7 T + 50% fuel 4.1 T + full IWB 8 AAMs 1.1 T = 24.9 tons
Airframe T/W ratio at 100% power = 2x(116 to 120.3)/9.8 /24.9 =
0.94 to 0.98
Fuel per ton = (3,940-4,000)/24.9 =
158.23-160.64 gm/s/T
50% fuel 4.1 tons while supercruise will be depleted in 1,025-1040 seconds or 17-18 minutes covering 527-642 Kms.
GE F-414 engine's SFC with
inlet dia. 79cm at 100% power (57.8-61.83 KN) is 20.5-23.25 g/KN/s
depending upon model. 75 KN JV engine is planned.
2 engines, so
AMCA SFC will be 41-46.5 g/KN/s
at 100% power.
2.37-2.87Kg/s fuel
will be used.
empty weight 12 T + 50% fuel 3.25 T + 4 Astr MK3 SFDR 0.88 T =
16.13 tons
T/W ratio at 100% power = 2x58/9.8 /16.13 =
Fuel per ton = (2,370-2,870)/16.13 =
146.93-177.92 gm/s/T
let's assume that with 0.73 T/W AMCA can also supercruise at M 1.2 (411.6 m/s).
50% fuel 3.25 tons while on supercuise will be depleted in 1,132-1,371 seconds or 18-23 minutes covering 466-564 Kms.
new engine with 75 KN dry thrust
will be available then
hopefully 6 AAMs
will be carried.
T/W ratio at 100% power = 2x75/9.8 / (16.13 + 0.44) =
Then hopefully AMCA will supercruise around M 1.5
Rafale's M-88-2 engine's SFC with
inlet dia. 70cm at 100% power (50KN) is 22.14 g/KN/s
2 engines, so
Rafale SFC is 44.28 g/KN/s
at 100% power &
Sup.Cr. Mach 1.4 (480.2 m/s)
So 2.21 Kg/s fuel for covering 480.2 m/s or
217.28 m/Kg or 4.6 gm/m
To go this extra 59 m/Kg-fuel Vs F-35, the SFC is increased from 20.3 to 22.14 g/KN/s.
EF-2000's EJ-200 engine's SFC with
inlet dia. 74cm at 100% power (60 KN) is 21-23 g/KN/s
2 engines so
EF-2000 SFC is 42-46 g/N/s
at 100% power &
Sup.Cr. Mach 1.5 (514.5 m/s)
so 2.52-2.76 Kg/s fuel for covering 514.5 m/s or
186.41-204.16 m/Kg or 4.9-5.36 gm/m
So we see that
Rafale with empty design weight 8.5 T, 492 sqft clipped delta wing & 50KN engine can supercruise at M 1.4
but F-18E/F with empty design weight 14.5 T, 500 sqft. trapezoidal wing & 58 KN engine cannot due to 6T weight increase due to carrier-ops MLG & other things & higher drag wing.
I mentioned about DRAG where people panic a lot. We should dive little more into it.
Drag are of many types
Some drag increase with speed & some decrease, but total drag increase.
That's why most people panic even before calculating. Why the world is pushing for increasing cruise & max speed?
The propulsion performace of Turbo-fan is limited around Mach 1.6 aspergraph below. Yet we see F-22 SuCr at M 1.8 with F119 engnes whose SFC is lowest 17 gm/N/s at 100% throttle. So there is
definitely something(s) classified.
File:Specific-impulse-kk-20090105.png - Wikimedia Commons
File:Gas turbine efficiency.png - Wikimedia Commons
That means if military is persistent on SuCr then we civillians are stuck with something somewhere, perhaps with engine efficiency & drag graphs are bothering us too much, while there are structural
factors also.
We should keep in mind that objective, priorities of military & civil jets are different.
MoD & Air force also have budget & SOP for peace time Ops incl. pre-planned routes, responses,flight altitude, speed kkeping in mind min. fuel expenses, maintenance & spares charges, etc.
But design focuses on war time performance also.
Let's look at the collage of drag, the highlighted part of graph in green color.
Real world is not ideal but full of resistance, losses, still as wing sweep angle increases, the drag decreases drastically.
Coefficient of drag Cd & Fd Force of drag are different, just like (Cf=u) coefficient of ground friction & (F=u.M.g) ground friction force.
So just like
ground force equation (F - Mg = Ma)
, we need
Flight equation of motion
. As per the scope of forums, we common people enthusiasts don't need complex 3-axis equation including roll, pitch, yaw, like Navier-Stokes equation, etc. But this kind of forum has to go on for 1-2
decades at least.
Let's take a
basic example of level flight
Make corrections/alterations where you like.
But for our low IQ minds, we need a simplified formula for overall drag - The Drag equation
Fd increases as square of Velocity🙀, but
the Cd of swept wing jet is 0.02 +/-
Air density at cruise altitudes is < 1 Kg/M^3. At 30Kft it is 0.458, at 50kft it is 0.186
NOTE - Make corrections/alterations as required.
Drag Force Equation Fd
(1/2) (Air density X Cd X Cross Section Area X Velocity^2)
Air density @ 40,000 feet
= 0.3 Kg/m^3
Coefficient of drag Cd
wing sweel angle around 50 degrees
= 0.02
let's consider
Mach 1.2
(411.6 m/s, round down to 410 m/s) which is considered bad for SuCr
Cross Section Area
of AMCA at wingtip level, let's say = 8 m^2
= (0.3 X 0.02 X 8 X 410 X 410)/2 = 4,034.4 N =
4.034 KN
2 F414 engines
together produce 2x58 KN =
116 KN dry thrust
net thrust = T - Fd = 112 KN, it is like an engine with 56 KN dry thrust
It is analogous to 116 people are pulling something forward & 4 people are trying pull behind.
Net result is 112 pulling forward. This is simple theoretical level-flight example. I am curious to know actual values.
Those who want deeper dive can include laws like conservation of momentum/energy/mass; equations of Navier-Stokes, Bernouli, Laplace, Euler, etc; Reynold's number, Critical Mach number, Stagnation
pressure, etc, etc.
Practically the
avionics computer
of modern jet fighter is equivalent of compacted
average Super-computer
calculating many 3D equations every millisecond.
Computing power is measured in units like
MIPS - Millions Instructions/Second & FLOPS - Floating Point Operations/Second
So we see that real world physics will always have resistance but overall effect matters & as per that solutions or work-arounds are developed. Supercruise is war time feature & it will be used for
reasons mentioned. The variable cycle engine will extend its usage.
Even my grandchildren will not see the AMCA in action. It will be admired forever. While other nations will have 8th-generation fighter jets, we will proudly showcase our 5th-generation aircraft.
incapable DRDO and inefficient HAL would kill this project forever
Even my grandchildren will not see the AMCA in action. It will be admired forever. While other nations will have 8th-generation fighter jets, we will proudly showcase our 5th-generation aircraft.
incapable DRDO and inefficient HAL would kill this project forever
Other than GTRE, it is difficult to find out where the bottleneck is - Govt., IAF, ADA/NAL, etc.
DoD + IITs have made some appreciable progress like RAM, radar, EW sensors, etc but lagging in other design aspects & components like engine, DAS, etc.
May be there are some type of problems everywhere, multiple bottlenecks.
AGENDA - RAM (Radar Absorbent Material) for AMCA.
This is old news now. Our DoD organisations with some IITs have developed RAM paints, sheets named "Adrishya", "NiRaLa", etc, composite materials & working on geometric shaping starting with AMCA.
The RCS results would obviously be top secret.
There were some sheets also, somewhere on social media.
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Before we talk further on shape, structure, RCS, etc of AMCA further, let's have a look of 3D CAD designs made by 5 people i have spotted so far :
1- Murli Yadav (social media ID not available)
2- Ankur Singh Chauhan (x.com/Anx450z)
(DFI - defenceforumindia.com/members/wahmanrespector.37183/)
3- Kuntal Biswas (x.com/Kuntal__biswas)
4- Satwik Sadhukhan (x.com/i_m_satwikk)
5- Harshal Pal (x.com/HarshalPal5)
If anyone of you know them & other artists including international ones, please invite them here.
I will post only selected pics, rest can be checked on their Twitter, DFI, etc posts. Some are also present on 3D sites like Turbosquid, Artstation, Sketchfab, Behance, etc.
Murli Yadav
Satwik Sadhukhan
Harshal Pal
Kuntal Biswas
Older design
Revised design
Ankur Singh Chauhan
In war time, Fighter jets might plan a sortie waypoints as per fixed assets like airbases, SAMs, terrain, etc. So the jet can maintain flight at certain altitude & heading to have minimum RCS towards
certain areas.
But the dynamic assets like moving ground SAMs, AWACS, enemy fighter jets can force to tactically alter the plan, waypoints & maneuver in roll, pitch, yaw axis which increases RCS towards certain
The 5gen jets still use rudders but canted at angle matching the fuselage side wall. from the diagrams above, on rolling & banking, the surface area at that angle increases a lot for few seconds. The
entire body is reflecting some RF energy.
This may compel to apply RAM on entire ventral/bottom side.
Earlier in capitalist country like USA, private companies developed their version of RAS & RAM whose quality would differ & cost of application & maintenance would be very high. Special machines
would be needed to wrap the jet with RAM tapes, attach RAm panels, or paint the RAM.
Today multiple nations have developed their own RAS, RAM with easier application & reduced cost.
But bcoz of nature of RF radiation is not simple, & ultimately a fighter jet has to do so much maneuvering, sometimes to evade enemy jets & missiles, that RAM may have to be applied almost
everywhere. So people usually prioritise only front RCS but side, top, back RCS now would become equal priority.
Will AMCA have Towed Decoy?
A collage of diagrams of
EW antennas, GPS, SATCOM, Radar altimeter, TACAN, RWR, IFF, VHF/UHF, L-band, data link (IFDL/MADL)
The diagrams say "preliminary" so final positions may change.
The following is collage of F-22's & F-35's sensors & antennas:
AMCA Vs TFX Kaan Vs KF-21, top view, side view, front view, isometric/corner view, as per present state of designs.
Good AMCA diagrams are not yet available, even by CAD artists.
Turkey was given F110-GE-129 engine. India was offered F-16IN with F110-GE-132A engine. We can't go for older airframe designs but if the business was done for the engine then we could have designed
a jet better than AMCA.
I don't have an official refined infographic or static model yet. Possible locations & coverage sectors for DAS/MAWS:
It came with 5gen jets helping pilot to focus on 1 picture of battle space, coming from multiple sensors as part of the jet or from wingman, other friendly jets, AWACS, satellite, ground asset, etc.
RWR was a standard among 4gen jets, but with analog wide sector indicators. I guess no jet had spherical sensor coverage & narrow direction indicator of incoming missile or enemy jet locked on to us.
Thereafter moresensors were added - RF/EW/ESM/IFF, IR/MAWS, LWR.
It became important even for MLUed 4.5gen jets to have spherical coverage & some degree of sensor fusion with digital display.
Demo cockpit of AMCA has been shown at Aero-India expo. The static model has 1 wide primary MFD & 2nd MFD below it b/w legs.
The actual inducted jet will have a sensor fused view.
But this demo cockpit may not be showing it yet. In lower right coner we see RWR & stores display.
The main 4 bigger sections, from left to right -
- Digital gyroscope/attitude indicator
- Navigation display with map
- Radar/Attack display
- Multiple systems - Fuel, Hydraulics, Electrical, Nozzle position, Anti-ice, Engine RPM % bar,
Another pic:
The RWR has been enlarged on the right.
Navigation display remains at 2nd from left.
Radar/Attack display has been moved below to bottom row.
Navigation & RWR displays
2nd from left is navigation+map display.
Below it are RWR & Radar/Attack displays.
AFAIK the 5gen jets F-22, F-35 & others too, do not care anymore about individual displays like RWR, IRST sweeping, radar sweeps & AESA beams, passive ESM finds, etc. All those things become
processing overhead for pilot & for display GPU, hence fused into 1 situation display.
As the AMCA project progresses, we hope to see better version of demo cockpits, more precise, showing sensor fused display.
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It is very difficult to decipher some elements in the demo cockpit display. Those who play DCS, MSFS & other simulators might be able to guess better.
Top row:
> AP - Auto Pilot
> AHLD - Altitude Hold?
> ASEL - Altitude Select?
> FD - Flight Director?
> L | G ?
> AT - Auto Thrust?
> NAV - Navigation map/mode active?
> 0.35 206, 0.25 151 ?
> FUEL 2931, 2350 - remaining fuel.
> 027 degree ?
> 6090, 5080 - Altitude?
> SPOO1? > RT1, RT2?
> VOR - VHF Omni-directional Range?
> TAC - Tactical air navigation?
> IFF M3 - Interogate Friend or Foe frequency select?
> M? > DISP - Display options?
> 50X TR?
> 02 PKTS BULLS?
> 068 / 102 NM, 273 / X88 NM - may be navigation beacons bearing, distance.
> AMCA TAKE EASTERN PKT ??
Multiple systems status
> 2 circles at top corners with 52, 17 - could be nozzle position open %.
> A/ICE - Engine Anti-ice heating OFF / AUTO.
> Vertical white scale & green bar, range 1-10, AB (After Burner), value 82%, 88% - Engine RPM %.
> Vertical yellow scale & greenbar, range 2-10, value 610, 671 - could be engine temperature.
> Small vertical white scale & green bar, range 0-200, FF value 31, 83 - could be Fuel Flow.
> REMN 2931 - Remaining Fuel?
> INT 2350 - Internal Fuel?
> BINGO 400 - Bingo Fuel mark.
But INT should be total & REMN should be less than that, right?
> HYD1, HYD2 280 BAR - Hydraulic pressure.
> DC 28.0 V, AC 114 V - Electricity.
> OIL 6.6, 6.9 BAR - Engine oil pressure.
> LPL, LPR ON - LP no idea, but on Left & right are ON. > BPL, BPR - BP no idea, but on left & right.
May be LP, BP are pumps.
Navigation, Map display
> LOC - Localizer?
> DCN? - Display Contrast?
> DCL? - Display Color? > SCL? - Symbols Color?
> DAN?
> FD?
> FPI?
> OVR? - Overlay?
> OBL? > Lower left corner, blue color : ETA 11:30:55 - Estimated Time of Arrival at waypoint?
> Lower center, blue color : EF with some number - no idea
> Lower right corner, 096/3.42 NM, 058/2.4 NM - Waypoint bearing/distance?
> Top right corner, 6100, 4550 - Altitude?
Engine tech should've been done like China, un abashed copying which then took them to 5th gen quicker, where today they have squadrons of j-20s on the sikkim border and are saying they will get to
1000 aircraft very soon. Agreed their ws series isn't very advanced but they are getting there quickly. We should've followed them in copying everything.
AMCA Vs TFX Kaan Vs KF-21, top view, side view, front view, isometric/corner view, as per present state of designs.
Good AMCA diagrams are not yet available, even by CAD artists.
Turkey was given F110-GE-129 engine. India was offered F-16IN with F110-GE-132A engine. We can't go for older airframe designs but if the business was done for the engine then we could have
designed a jet better than AMCA.
View attachment 763 View attachment 764
There is Chinese in the picture, you stole the picture of Chinese people, shameful!
There is Chinese in the picture, you stole the picture of Chinese people, shameful!
"Stole"?? ROFL!
That's a fan art, not some copyrighted thing with Patent.
This is a casual chat forum for discussion.
BTW, there is Indian AMCA in the picture. The creator stole AMCA depiction & didn't take permission.
, do we have a thread for comedy replies?😁
Engine tech should've been done like China, un abashed copying which then took them to 5th gen quicker, where today they have squadrons of j-20s on the sikkim border and are saying they will get
to 1000 aircraft very soon. Agreed their ws series isn't very advanced but they are getting there quickly. We should've followed them in copying everything.
Our people just assume that diplomatic talks will avoid war forever, but in this century there will be a big multi-front war & our people will learn lesson very bad way.
National importance? What is that? We are happy with upgrading Tejas to mk2. We will put huge fund in mk2 and enjoy by cheating government.
in order for the country to progress It's important to prioritize long-term goals over shortcuts. With real determination and appropriate funding, the development of Tejas Mk2 can greatly enhance
national security and innovation.
sso id
Our people just assume that diplomatic talks will avoid war forever, but in this century there will be a big multi-front war & our people will learn lesson very bad way.
Also why not use Russian engines? I doubt they will say no for the latest Saturn engines for our AMCA and Tejas mk-1 & 2. I know they arnt as reliable as western engines but their thrust is more than
enough for our fighter requirements. | {"url":"https://defence.in/threads/advanced-medium-combat-aircraft-amca-news-and-discussion.4440/page-43#post-41308","timestamp":"2024-11-14T23:18:21Z","content_type":"text/html","content_length":"271584","record_id":"<urn:uuid:244e595a-26a9-4105-8412-c93afb7783bd>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00378.warc.gz"} |
Dice Notation
What our group sees as the essentials of dice notation.
Here are examples of dice notation: d4, d6, d8, d10, d12, and d20.
The notation refers to a die, but more often a roll of a die. The rules might say that a weapon causes d8 of damage, which is equivalent to saying it does 1–8 hit points of damage.
The notation has been extended in a couple of ways: 3d6 means to roll three 6-sided dice and sum them, generating a number in the range 3–18. 1d4+1 means to roll one 4-sided die and add one to it,
generating a number in the range 2–5.
The Introduction of Dice Notation
The notation isn't used in the original box set or the Monster Manual.
The Players Handbook (June 1978) was the first TSR publication to use it.
Jon Peterson
suggests the PHB was written as if players were already familiar with the notation, but the occurrences I've found are used in a parenthetical manner. For example, consider this spell description:
A fireball is an explosive burst of flame, which detonates with a low roar, and delivers damage proportionate to the level of the magic-user who cast it, i.e. 1 six-sided die (d6) for each level
of experience of the spell caster. Exception: Magic fireball wands deliver 6 die fireballs (6d6), magic staves with this capability deliver 8 die fireballs, and scroll spells of this type deliver
a fireball of from 5 to 10 dice (d6 + 4) of damage.
The November 1978 printing of the Holmes rulebook appends the following text:
In some places the reader will note an abbreviated notation for the type of die has been used. The first number is the number of dice used, the letter "d" appears, and the last number is the type
of dice used. Thus, "2d4" would mean that two 4-sided dice would be thrown (or one 4-sided would be thrown twice); "3d12" would indicate that three 12-sided dice are used, and so on.
The blurb suggests that dice notation is used elsewhere in the Holmes rulebook, but it isn't!
The Dungeon Masters Guide (August 1979) also explains the notation:
Before any further discussion takes place, let us define the accepted abbreviations for the various dice. A die is symbolized by "d", and its number of sides is shown immediately thereafter. A
six-sided die is therefore "d6", d8 is an eight-sided die, and so on. Two four-sided dice are expressed by 2d4, five eight-side dice are 5d8, etc. Any additions to or subtractions from the die or
dice are expressed after the identification, thus: d8 + 8 means a linear number grouping between 9 and 16, while 3d6 - 2 means a bell-shaped progression from 1 to 16, with the greatest
probability group in the middle (8, 9). This latter progression has the same median numbers as 2d6, but it has higher and lower ends and a greater probability of a median number than if 2d12 were
used. When percentage dice are to be used, this is indicated by d%.
As Jon Peterson has discussed, essentially the same notation, albeit with a capital D, was being used in the fanzines for several years before TSR embraced it. It appears, amazingly, in the first
issue of Alarums & Excursions from 1975.
The Old Notation Isn't Good Enough
Looking through the older texts, you can see a couple of different ways for specifying dice rolls: "5 + 1", "3-8 sided", and "2–24".
This notation is inferior in various ways. The first doesn't make clear which die to use: a 5d6+1 roll is intended. Of course, ambiguity can sometimes be advantageous. The way hit dice are specified
in the Monster Manual might have made the book more appealing to players still using d6 hit dice for monsters.
The third example, range notation, looks like a concise way to specify rolls, but it also can be ambiguous. For example, 3–12 can be either d10+2 or 3d4. The first method is uniform, whereas the
second starts to approximate a bell curve. If you roll it the first way, the chance of getting a 3 is 10%; if you roll it the second way the chance of getting a 3 is 1 in 64 or about 1.6%.
3–12 is the smallest range which can be ambiguous, and it is used in the PHB! A bardiche inflicts 3–12 hit points of damage on large opponents.
The New Notation Isn't Good Enough
On p. 10 the DMG explains dice notation and on the following page it describes a method for rolling attribute scores which can't be expressed with that dice notation: rolling 4d6 and dropping the
lowest die!
The site
has some notation for this. The roll can be written as 4d6d1 to indicate the lowest die is dropped, or 4d6k3 to indicate the highest three dice are kept.
There is alternate notation which make it explicit that the lowest die is dropped: 4d6dl1 and the highest three dice are kept: 4d6kh3. One could call for the highest die to be dropped: 4d6dh1 or the
lowest three dice are kept: 4d6kl3.
The lowercase L seems like an opportunity for confusion with a numeral one, so we just use 4d6kh3 for a 3d6 with negative skew and 4d6dh1 for a 3d6 with positive skew.
We don't like laptops, ipads, or even phones at the table. Nevertheless it was convenient to implement a command line tool which understands dice notation—including the "keep high" and "drop high"
$ roll 6d6
$ for i in $(seq 1 6); do roll 4d6kh3; done
The code is on
Factor Rolls
One can use the dice to generate other ranges of integers:
1–2 ⌈d6/3⌉
1–3 ⌈d6/2⌉
1–5 ⌈d20/4⌉
1–10 ⌈d20/2⌉
In case the notation on the right is not clear, one rolls the indicated die, divides by the following number, and then rounds up. The most practical notation is d2, d3, d5, d10. I'm not aware of a
standard term for this type of roll; we've been calling them factor rolls.
Product Rolls
Percentile dice are an example of what we've been calling a product roll. We could use two d6 to create a d36, for example. This is not multiplication, but more like working with base 6 numbers. The
percentile dice make the process easier by using zeros and distinguishing the tens die from the ones die. One formula for getting a range 1–36 is
6×(d6 - 1) + d6
Dice of different colors are needed for a product roll. Our convention is to use a white die for the ones die. If dice of different colors are not available, a single roll is not possible; roll the
most significant die first.
The dice do not have to have the same number of faces. If you wanted the use the 30 sided dice gaming tables published by the armory, you could generate d30 rolls with a d6 and a d10.
If factor rolls and product rolls are allowed, then the only numbers we cannot generate are ones containing a prime larger than 5 as a factor.
Thus there is no way to generate d7 uniformly using a single roll. One could roll a d8 and re-roll if the result is 8. Simply writing d7 is the best notation.
Seven sided dice have been manufactured. One design is a pentagonal prism, and another "is based on spacing points as equally as possible on a sphere and then cutting planar slices perpendicular to
those directions." It would be interesting to test these dice and see whether the distribution is uniform. | {"url":"https://www.athenopolis.net/2016/12/dice-notation.html","timestamp":"2024-11-07T12:18:10Z","content_type":"application/xhtml+xml","content_length":"63650","record_id":"<urn:uuid:c6bd2754-1ff1-436b-ad99-0b4cf6b2e8f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00300.warc.gz"} |
Rolling interface friction dynamics of hot strip continuous rolling and its effect on mill chatter
According to the fluid mechanics and rolling theory, the rolling interface friction dynamic characteristics and their influence on rolling speed, rolling load, roller material, rolling temperature
and oil properties were studied respectively with lubrication or without lubrication in hot rolling. The two-dimensional Reynolds equation was established for steady and unsteady rolling friction
interface. Through the research of rolling interface friction characteristics influence on the mill vibration, we found that when the lubrication film thickness in the roll gap was thicker, the
friction coefficient was lower, its damping effect and system stability was worse. And if the rolling speed is greater than a certain value, the rolling interface friction coefficient will fall
sharply with the increase of rolling speed and produces the self-excited vibration caused by negative damping. So the contrast test with different rolling interface has been carried out, such as the
emulsion is open or close, high chromium cast iron/high speed steel roller is put to use, high chromium cast iron roller is grinded finely or coarsely, rolling speed is normal/reduced. And the
results showed that it had an obvious effect on the mill vibration suppression by the adoption of the emulsion close and high speed steel roller.
1. Introduction
The CSP (Compact Strip Production) hot rolling process has a high temperature and big reduction. Its metal organization and properties have changed, and its surface has a strong oxidation, sticking
or sliding. The physical and chemical property of the rolling interface is very complex. In order to reduce the rolling force and roller wear in the rolling production, a lubricant is widely used in
hot rolling, which makes the interface friction to be more complex. In rolling mill vibration, people find that the bad lubrication condition is an important factor. When the emulsifier stability
performance is poor, oil film strength is insufficient or the film is broken, the oil film thickness is instable, the friction condition and friction type in the roll gap will change and make the
rolling process unstable. So, adopting proper lubrication conditions to improve the friction behavior and eliminate the adverse effect is an important aspect for the rolling process stability and
rolling mill vibration inhibition study.
Furumoto designed a chamber in the Mill Stabilizing Device and optimized its size for suppressing the mill vibration [1]. KIM modeled a rolling mill by multibody dynamics to investigate the cause and
characteristics of the mill chatter, and they found the chatter frequency was equal to 1190 Hz and was caused by rolling force. The amplitude of chatter vibration could be reduced by controlling the
rolling speed and rolling force of the static and dynamic components [2]. He also proposed a mathematical model of a cold rolling mill including the driving system and a novel combination of the
direct integration method and quasistatic analysis to solve the model efficiently, he found the horizontal chatter vibration had a strong effect on the dynamic characteristics [3]. Świątoniowski
presented a probabilistic model of the friction phenomena on the work-back up rolls contact surface and found that such a character of the disturbance in the distribution of zones with static and
kinetic friction could be regarded as one of the sources of self-excited vibrations appearing in the system consisting of a rolling mill and a strip [4]. Y. A. Amer studied the torsional vibration
reduction for a rolling mill’s main drive system via a negative velocity feedback under the parametric excitation and found the resonance condition was the first natural frequency vibration as one of
worse resonance cases [5]. Fujita proposed a new actuator for controlling the friction coefficient balance between the final stand and preceding stand as an intelligent hybrid lubrication control
system for preventing from chatter. The results showed that the hybrid lubrication system could prevent from chatter efficiently in high speed cold rolling region [6]. Kijima investigated the
influence of lubrication on elongation and roughness transfer in skin-pass rolling by experimental rolling tests in which the relationship between lubrication behavior and roll radius was clarified.
It was found that operational size rolls could be explained convincingly by height characterization parameters and were considered to be reasonable, some characteristics of skin-pass rolling related
to lubrication were not properly simulated using a small radius, laboratory size rolls due to the insufficient contact length between the rolls and the workpiece [7]. Although these researchers have
conducted a lot of researches, but they have hardly studied the effect of the rolling interface friction on the rolling mill vibration, meanwhile the hot rolling mill vibration problem has not been
solved perfectly.
This paper studied the influence characteristic of rolling lubrication on hot rolling interfacial friction dynamics through theoretical analysis and experimental research, these influencing factors
include rolling speed, rolling load, roller material, rolling temperature and lubricating oil influences on the mill vibration. Rolling friction and lubrication dynamic equations were established for
steady and non-steady state. The influence of rolling process and force parameters on the mill vibration from the rolling interface and vibration suppression measures was studied.
2. Rolling interface friction dynamics
Rolling interface friction is divided into four basic forms, such as dry friction, boundary friction, fluid friction and mixed friction state. A friction surface without any lubricant or defiled
objects is dry friction. A contact surface with very thin oil film (film thickness is around 0.1-0.01 microns) is boundary friction. The friction body surface has a thick oil layer and does not have
direct occlusion of surface roughness, the friction under this condition is liquid friction. The rolling interface friction dynamic characteristics are described by the sliding friction and shear
friction law, such as the Kalman curve obeying the dry friction law, Siebel curve obeying the local friction law, Nadai curve obeying the liquid friction law, Uermxo curve obeying the dry friction
law in the sliding zone and liquid law in the adhesion area [8].
2.1. Rolling interface friction properties without lubrication
The major influence factors on friction performance under the unlubrication condition are rolling speed, rolling load, roll surface roughness, tin oxide thickness, rolling temperature, etc. W. L.
Roberts deduced the friction coefficient $\mu$ experiential formula with the roll surface roughness ${R}_{a}$, tin oxide thickness ${h}_{s}$ and temperature $T$ according to the test data as follows
$\mu =3.6\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{-4810}{T+459}\right)+0.063\mathrm{l}\mathrm{n}\left(\frac{{R}_{a}}{{h}_{s}}\right),$
where, the unit of temperature $T$ is ℉. It can be seen the interface friction coefficient increases with the increment of temperature and roll surface roughness, and decreases with increment of tin
oxide thickness. And the production weight $M$ of iron oxide can be described by Eq. (2):
$M=a\rho t\mathrm{e}\mathrm{x}\mathrm{p}\left[-b/\left(T+460\right)\right]\left(1/h+1/w+1/l\right),$
where, $h$, $w$ and $l$ are strip thickness, width and length respectively, $t$ is the exposure time, $T$ is the strip temperature, $\rho$ is the steel strip density, $a$ and $b$ are constants. From
Eq. (2), with the extension of exposure time and the improvement of the strip steel temperature, the thickness of the tin oxide is also gradually thickening. The iron oxide producing speed is
different for steel types. Carbon, silicon, nickel, and copper can promote the iron oxide formation. But manganese, aluminum and chromium can slow tin oxide formation.
Under large reduction, rolling interface does not meet the coulomb friction model, but shear friction model based on the shear stress can be used. That is, the friction stress is a part of the
materials’ equivalent shear stress [10]. So, Wanheim and Bay deduced the relationship between friction coefficient $\mu$ and friction factor $m$, the pressure $p$ and rolled strip shear strength $k$
as $\mu =mk{A}_{r}/p{A}_{a}$. According to experiments, they gave out the regression polynomial between friction coefficient $\mu$ and friction factor $m$ as $m=$4.3923$\mu$-4.1402${\mu }^{2}$-1.1522
${\mu }^{3}$. In a word, the friction coefficient under non-lubrication rolling can be enlarged by raising temperature and roll surface roughness or reducing the speed (reflecting $t$ value).
2.2. Rolling interface friction properties with lubricant
From a field test, we found the process lubrication can reduce the contact arc surface friction in a deformation area, total rolling pressure and energy consumption, and increase the reduction per
pass, reduce the minimum rolling thickness and roll wear, prevent metal stick on roller [11]. Liquid lubricant is commonly used in hot rolling, it is divided into pure oil, oil-water mixture and
emulsion. Because the emulsion has an excellent performance (oil content stability, etc.), it has been widely used in rolling production, as this CSP production line.
Rolling interface lubrication schematic for strip rolling is shown in Fig. 1. For different rolling speed, the relationship between friction force and speed can be divided into as follows: I region,
that is a non-sliding zone, it has only elastic deformation; II boundary lubrication region, the speed is very low and has not enough lubricating oil to be sucked in, there is a solid and solid
contact; III segmental lubrication region, although it has certain speed and lubricating oil can be inhaled, but it is not enough to separate the contact area; IV full lubrication area, the contact
surface is separated by the oil film completely, its characteristic curve is shown in Fig. 2. Among them, the negative slope in part III is caused by the oil film gradual increase with speed
increasing and friction decreasing. There has the Stribeck effect, namely there is negative damping phenomenon [12].
Fig. 1Rolling interface lubrication schematic
Fig. 2Interface friction characteristic curve
When the interface has a full fluid lubrication state, the lubricant fills cuneate cracks on the surface of the roller and rolled strip, and forms a thin oil film with a certain bearing capacity
[13], so we set up the following two-dimensional Reynolds equation for steady and unsteady rolling process:
$\frac{\partial }{\partial x}\left(\frac{{h}^{3}}{12{\eta }_{0}}\frac{\partial p}{\partial x}\right)+\frac{\partial }{\partial y}\left(\frac{{h}^{3}}{12{\eta }_{0}}\frac{\partial p}{\partial y}\
right)=\frac{1}{2}\frac{\partial }{\partial x}\left(\stackrel{-}{U}h\right)+\frac{1}{2}\frac{\partial }{\partial y}\left(\stackrel{-}{V}h\right)+\frac{\partial h}{\partial t}.$
For wide strip ($l/\stackrel{-}{h}>5$, $l$ is contact arc length, $\stackrel{-}{h}$ is the average thickness of strip entrance and exit), Eq. (3) can be simplified as one dimensional Reynolds
equation as follows:
$\frac{\partial }{\partial x}\left(\frac{{h}^{3}}{12{\eta }_{0}}\frac{\partial p}{\partial x}\right)=\frac{1}{2}\frac{\partial }{\partial x}\left(\stackrel{-}{U}h\right)+\frac{\partial h}{\partial
where, $x$ is the distance along the rolling direction, $h$ is the oil film thickness, $p$ is lubricating oil film pressure on the length of the wedge front the entrance of deformation area, $\
stackrel{-}{U}$ is the average of the work roller surface circumferential velocity and strip steel velocity, ${\eta }_{0}$ is the dynamic viscosity of lubricant. For unsteady rolling process, the
lubricant dynamic viscosity ${\eta }_{0}$ is not a constant, and it can be represented as ${\eta }_{0}\left(T,p\right)={\eta }_{A}{e}^{\theta p-\delta \left(T-{T}_{s}\right)}$ (where, ${\eta }_{A}$
is the viscosity at temperature ${T}_{s}$ and normal pressure, $\theta$ is viscous pressure coefficient, $\delta$ is viscosity-temperature coefficient).
It can be seen from Eq. (4), the oil film pressure variation is composed by squeezing effect $\partial h/\partial t$ and dynamic pressure effect $\partial \left(\stackrel{-}{U}h\right)/\partial x$,
and oil film thickness has no fluctuation in the steady rolling process. That is, the latter effect can only be considered but not the former. Then oil film pressure $p\left(x\right)$ at any location
$x$ is deduced as follows:
$p\left(x\right)=\frac{12{\eta }_{0}\left({v}_{r}+{v}_{1}\right){R}^{2}}{{A}^{2}}\left\{\begin{array}{c}\frac{x}{{x}^{2}-{A}^{2}}\left[1-R{\xi }_{0}\left(\frac{1}{{x}^{2}-{A}^{2}}-\frac{3}{2{A}^{2}}\
right)\right]\\ -\frac{1}{x}\left(1+\frac{3R{\xi }_{0}}{2{A}^{2}}\right)\left(1+\frac{{A}^{2}}{3{x}^{2}}\right)\end{array}\right\},$
where, ${v}_{r}$ is work roller surface circumferential velocity, ${v}_{1}$ is the strip speed, ${A}^{2}=R\left(\mathrm{\Delta }h-2{\xi }_{0}\right)$, $R$ is the roll radius, the entrance lubrication
layer thickness is ${\xi }_{0}=3\theta {\eta }_{0}\left({v}_{r}-{v}_{1}\right)/\alpha \left[1-{e}^{-\theta \left(K-{\sigma }_{0}\right)}\right]$ (where, $\theta$ is viscosity compression coefficient,
$\alpha$ is the nip angle, $K$ is the rolled strip yield strength, ${\sigma }_{0}$ is the strip steel back tension); The oil film shear stress $\tau \left(x\right)$ at any location $x$ in deformation
area is as follows:
$\tau \left(x\right)=J\left[\frac{1}{3}\left(1+2\frac{1+{\epsilon }^{2}{z}^{2}}{1+\frac{{\epsilon }^{2}{x}_{\varphi }^{2}}{{l}_{d}^{2}}}\right)\sqrt{1+\frac{{l}_{d}-x}{I}}-\frac{1+\frac{{l}_{d}-x}
{I}}{\sqrt{1+\frac{{l}_{d}-{x}_{\varphi }}{I}}}\right],$
where, $J=6{v}_{r}{\eta }_{0}/{\xi }_{0}$ is the fluid resistance coefficient, $I=\left({\eta }_{0}{l}_{d}^{2}{v}_{r}\right)/2{p}_{m}{\xi }_{0}^{2}$ is the fluid lubrication coefficient, $z={x}_{\
varphi }/{l}_{d}$, ${x}_{\varphi }$ are the neutral point coordinates, ${l}_{d}$ is the contact arc length, $\epsilon$ is the pass reduction rate. From the friction factor $\mu \left(x\right)=p\left
(x\right)/\tau \left(x\right)$, and according to the Eq. (5) and Eq. (6), the average friction coefficient $\mu "$ of the deformation zone can be approximately expressed as the follows:
${\mu }^{"}\approx \frac{\epsilon \left({e}^{{\eta }_{0}\gamma }-1\right)}{6{\eta }_{0}K\left(2-\epsilon \right)}\sqrt{\frac{{h}_{1}\epsilon }{D}},$
where, ${h}_{1}$ is the rolled strip entry thickness, $D$ is the roller diameter. According to the rolling theory, only when $\mu "$ is greater than the smallest friction coefficient ${\mu }_{\mathrm
{m}\mathrm{i}\mathrm{n}}$, the rolling process can proceed steadily and there is Eq. (8):
${\mu }^{"}>{\mu }_{\mathrm{m}\mathrm{i}\mathrm{n}}=\sqrt{\frac{H\epsilon }{2D}}\approx \frac{\alpha }{2},\frac{\epsilon \left({e}^{{\eta }_{0}\gamma }-1\right)}{3{\eta }_{0}K\left(2-\epsilon \
So, in order to guarantee the rolling stability for a certain pass reduction rate $\epsilon$, dynamic viscosity ${\eta }_{0}$ must have the maximum permission. Because the interface friction
characteristics change is caused by the lubrication film thickness in the deformation area affected by the rolling speed and reduction rate, we deduce the average oil film thickness of deformation
area $h=6\stackrel{-}{U}{\eta }_{0}\left(1-2\epsilon /3\right)/K\mathrm{t}\mathrm{a}\mathrm{n}\alpha$. It can be seen the lube film thickness is proportional to the rolling speed and lubricant
viscosity within a certain scope, it is inversely proportional to the contact angle and strip yield limit. At the same time, the thickness of the lubricating oil is also related to the roller
material, such as cast iron roll is 30-40 % more than steel roll.
When the rolling interface is in the boundary or mixed friction state, there is ups and downs organization between a roller and strip, the lubrication film thickness of deformation area is very
uneven, the entire contact area is composed by alternate boundary friction, liquid friction and dry friction area. And the total friction force $T$ can be represented as $T={\tau }_{c}{F}_{c}+{\tau }
_{l}{F}_{l}+{\tau }_{dr}{F}_{dr}+{\tau }_{s}{F}_{s}$. where ${\tau }_{c}$ is the shear resistance within a very thin boundary lubrication layer, ${F}_{c}$ is the boundary friction zone area, ${\tau }
_{l}$ is the shear resistance in large lubricating oil thickness, ${F}_{l}$ is the liquid friction zone area, ${\tau }_{dr}$ is the shear resistance of direct contact with the surface area, ${F}_{dr}
$ is the dry friction area, ${\tau }_{s}$ is the sculpture resistance per unit, ${F}_{s}$ is the area to produce a sculpture function. Here, the segmental film elastohydrodynamic Reynolds equation
should be adopted as Eq. (9) to the boundary friction [14]:
$\frac{\partial }{\partial x}\left({\varphi }_{x}\frac{{h}_{W}^{3}}{12{\eta }_{0}}\frac{\partial p}{\partial x}\right)+\frac{\partial }{\partial y}\left({\varphi }_{y}\frac{{h}_{W}^{3}}{12{\eta }_
{0}}\frac{\partial p}{\partial y}\right)=\stackrel{-}{U}\frac{\partial {h}_{TW}}{\partial x}+\frac{\left({v}_{r}-{v}_{1}\right)\sigma }{2}\frac{\partial {\varphi }_{s}}{\partial x}+\frac{\partial {h}
_{TW}}{\partial t},$
where, ${h}_{W}$ and ${h}_{TW}$ are the nominal oil film thickness and the actual oil film thickness considering the effects of surface waviness. ${\varphi }_{x}$, ${\varphi }_{y}$ and ${\varphi }_
{s}$ are the roughness effect coefficient respectively, the comprehensive roughness $\sigma =\sqrt{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}}$ (${\sigma }_{1}$ and ${\sigma }_{2}$ are root mean square
deviation of two surface roughness).
2.3. Influence analysis of rolling interface friction characteristics
The influencing factors of rolling interface friction characteristics are available very distinctly. In different extents of rolling deformation zone (such as the back-slip zone, adhesion area,
stagnation zone, adhesive area and forward slip zone), the friction mechanism is different, and brings great difficulties to the theoretical calculation, so we often need to use experiments to study.
In general, the influence factors of the rolling friction interface can be divided into: main influence factors, such as rolling temperature, lubricant viscosity, rolling speed and pass reduction
rate, etc.; Secondary influence factors, such as the roll surface roughness, chemical composition, contact surface unit pressure, work roller diameter and vibration, etc.
2.3.1. Influence of temperature and speed
When the temperature is high, the microstructures and properties of metal will change, there will be surface bonding and the oxide film formation, so the rolled strip temperature influence on the
friction coefficient is the largest. For low carbon steel, when the rolling temperature is above 700 ℃, the friction coefficient $\mu$ decreases with the increase of the rolling temperature $t$ and
speed $v$. And we often calculate it by empirical formula as follows:
$\mu =a-0.0005t-0.056v,a=\text{const},$
$\mu =\left(0.7935-0.000356t+0.012\sqrt[3]{{R}_{z}^{2}}\right)\left[1-\left(0.348+0.00017t\right)C\right]\phi \left(v\right),$
where, ${R}_{z}$ is the roller surface roughness, $C$ is the carbon content of rolled strip, $\phi \left(v\right)=1-0.1v$ (0 $\le v\le$ 2 m/s) or $\phi \left(v\right)=1.44-0.28v$ (2 $\le v\le$ 3 m/
s), friction coefficient variety test curve with the change of temperature is shown in Fig. 3.
From the calculation of the experimental data, the F3 mill rolling friction coefficient is 0.01-0.1, which is at a mixture of fluid friction and boundary friction state. Based on field test data
fitting, the relationship between the friction coefficient and speed is shown in Fig. 4 (1.6 mm and 2.0 mm are strip finishing thickness respectively). It can be seen that the curve was consistent
with the Stribeck curve, namely, there is instability regions which friction fell sharply with the increase of rolling speed. And the friction coefficient is small, the regression equation between
the friction coefficient $\mu$ and rolling speed $v$ is obtained as follows (corresponding to 2.0 mm strip curve):
$\mu =0.35-0.06v+0.0024{v}^{3}.$
Fig. 3Relationship between friction coefficient and strip temperature
Fig. 4Relationship between friction coefficient and rolling speed
2.3.2. Oxide skin of roller and strip material influence analysis
The major influence of roller material on the friction coefficient is a carbon content, and each roller is different by carbide materials, such as the carbides in a high chromium cast iron roller is
M7C3 (Hv2500), carbide in an infinite chilled cast iron roller is Fe3C (Hv1300), a high-speed steel roller has MC (Hv3000). Namely, the wear resistance of the high-speed steel roller is better than
the one of the high chromium cast iron and infinite chilled cast iron roller, and it also has good red hardness to ensure it have high anti-wear ability under a high temperature. Their mechanical
properties comparison is shown in Table 1. We also find it because the high-speed steel has the special microstructure, high thermal conductivity and high hardness carbides on the roller surface, it
makes the friction coefficient increase by 3 %-5 % or greater. Its thermal cracking and anti-skid performance have obvious advantages.
Table 1Roller outer mechanical properties
Project High-speed steel High-chromium iron High alloy chilled cast iron
Carbide / % 10-15 5-10
Hardness / hsc 70-90 70-80 75-85
Strength of extension / MPa 700-1000 700-900 400-600
Compressive strength / MPa 2500-3200 1700-2200 1900-2500
Fracture toughness / MPa·m^-0.5 25-28 21-34 18–25
The oxide skin in hot rolling can be divided into: the thick primary oxide skin, the furnace life oxide skin, refractory oxide skin, secondary oxidation skin and red oxide, etc. Basic part of the
iron oxide close to the metal surface is FeO, which accounts about 60-80 %. Fe[3]O[4] and thin Fe[2]O[3] in turn are outward. Due to the primary oxide skin, FeO is in a softening state, the friction
coefficient is small, so it can cause slipping easily. The secondary oxidation skin is thin and hard, it can't completely fill the bumpy texture on the roller surface in the deformation zone, roller
surface bump will be pressed into the surface of rolled piece to form the mechanical occlusion bond and even make the friction coefficient increase. In addition, the friction coefficient is affected
by the carbon content in the roller or rolled piece, with the increase of carbon content in rolled piece, the friction coefficient has a significant reduction, such as the friction coefficient in
rolling stainless steel is 1.3 times smaller than rolling carbon steel. According to the field test data and Eq. (7), the friction coefficient influences the relation curve with rolled piece type and
roller material (roller temperature is also considered) fitted out for ordinary Carbon Steel (CS) and Steel Plate Atmospheric-Hot (SPA-H), and they are shown in Fig. 5 and Fig. 6 respectively. It can
be seen from Fig. 5, with strip finishing, the thickness becomes smaller, the average friction coefficient is smaller, the rolling stability becomes worse, the mill vibration is easy to be produced.
It can be seen from Fig. 6, the friction coefficient of high speed steel roller is the largest, that is, the use of a high-speed steel roll in certain conditions is beneficial to suppress the rolling
mill vibration.
Fig. 5Rolled piece material influence on friction coefficient
Fig. 6Roller material influence on friction coefficient
2.3.3. Lubricating oil influence
Hot rolling oil lubrication has a great influence on the rolling friction interface. The hot rolling friction coefficient of all kinds of additives is shown in Table 2. The kinematic viscosity of
mineral oil 1 is 40 mm^2·s^-1 at 40 ℃. It can be seen that the friction dynamics of rolling interface can be improved if the emulsion liquid containing mineral oil is used. The kinematic viscosity of
mineral oil 2 is 150mm^2·s^-1 at 40 ℃. In addition, the rolling oil lubrication performance will be different for a different roller material. Such as when the high alkaline organic acid salt rolling
oil is used in a high-speed steel roller, due to the high alkaline organic metal salts has high alkaline Ca sulfonate (with reducing) as the main ingredients of hot rolling oil, it can prevent from
the oxidation FeO of rolled piece transfer to the roller surface and turn into Fe[3]O[4]. So, it can inhibit the roller surface black oxidation generation, make the roller and surface oxidation film
to modify, and can improve the roller surface friction coefficient.
Under the same lubrication condition, rolling force will increase with the reduction rate increase. The increase of rolling force will lead to roller flattening, the distortion of surface contact arc
shape, curvature radius increase, elastic deformation in the contact region extension, reduction of hydrodynamic wedge angle of lubrication wedge forward area. It is advantageous for the lubricant
into the deformation zone and increases the lubrication film thickness. So, with the increase of reduction rate, the deformation area lubrication dose increases and the friction coefficient reduces.
For instance, when there is a vibration in the F3 mill, there is a sharp sound in the rolling interface sometimes, because with the increase of rolling speed, pumping oil increases, and the friction
coefficient decreases. With the workpiece deformation increasing and deformation zone temperature rising, lubricating oil viscosity declines, oil film relative strength declines or breaks, the sharp
noise sends out from the gap. Experiments showed that the roll with transverse cut has more than double lubricating layer thickness, because the transverse microscopic convex roughness has a “fence”
role and can block lubricant extrusion.
Table 2Hot rolling friction coefficient of all kinds of additives
Additive Friction coefficient Additive Friction coefficient
Mineral oil 1 0.314 P-c key compounds 0.11
Mineral oil 2 0.219 Vulcanized ester 0.106
Rapeseed oil 0.14 Alkaline acid calcium 0.1
Oleic acid 0.126 Calcium sulfonate 0.161
Dimer acids 0.1 Oil soluble boric acid ester 0.252
High viscosity resin 0.106
3. Relation between interface friction and mill vibration
3.1. Influence of interface friction on mill vibration
Interface friction mainly affects rolling force, rolling process stability and mill structure stability. And then strengthens or weakens the rolling mill vibration strength [15]. According to
experiment, the per-unit rolling force $p$ and its parameter effects (such as the friction coefficient) can be calculated as follows:
$p=14.5\left({h}_{1}-{h}_{2}\right){\sigma }_{fm}\xi {\left(\frac{{h}_{1}}{R}\right)}^{-0.97}\frac{{X}^{0.1}}{\sqrt{\frac{L}{{h}_{m}}}}{\mu }^{0.281},$
where, $X=hT/{\sigma }_{m}\mathrm{\Delta }v$ ($h$ is the heat transfer coefficient, $T$ is the inlet temperature, ${\sigma }_{fm}$ is the average flow stress, $\mathrm{\Delta }v$ is the interface
relative speed), $\xi$ is the constant, ${h}_{m}$ is the strip steel average thickness. By Eq. (13), rolling force increases with the friction coefficient increase.
The rolling process stability is poorer in partial or full lubrication, and when the vertical vibration occurs in the roller, convex bodies have a slight contact and separation, the oil film
thickness is thickening and thinning, the vertical contact stiffness of the whole system is also changing. When the work roller has a downward movement, the system stiffness increases; Conversely, it
has soft spring characteristic. The lubrication state has great changes, and even causes bearing oil film rupture, then the influence of the squeezing effect $\partial h/\partial t$ in the Reynolds
Eq. (4) cannot be ignored. That is, the vertical vibration is a typical nonlinear system caused by the friction force, the system is easy to produce the abnormal phenomenon such as parametric
resonance. Under the full hydraulic lubrication, the oil film thickness $h$ (reflecting roll gap friction coefficient) has the greatest influence on rolling force $P$. And the oil film thickness $h$
is mainly affected by the rolling speed $v$, reduction rate $\epsilon$ and lubricant viscosity $\eta$, then we can write $P$ as follows:
$P=f\left(h,\mu \right)=f\left[\varphi \left(v,\epsilon ,\eta \right),\phi \left(v,\epsilon ,\eta \right)\right].$
Rolling force fluctuation quantity can be expressed as:
$dP=\left(\frac{\partial P}{\partial h}\right)\left[\left(\frac{\partial h}{\partial v}dv\right)+\left(\frac{\partial h}{\partial \epsilon }d\epsilon \right)+\left(\frac{\partial h}{\partial \eta }d\
eta \right)\right]+\left(\frac{\partial P}{\partial \mu }\right)\left[\left(\frac{\partial \mu }{\partial v}dv\right)+\left(\frac{\partial \mu }{\partial \epsilon }d\epsilon \right)+\left(\frac{\
partial \mu }{\partial \eta }d\eta \right)\right]$$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}=\mathrm{\Delta }{P}_{1}+\mathrm{\Delta }{P}_{2}+\mathrm{\Delta }{P}_{3}+\mathrm{\Delta }{P}_
{4}+\mathrm{\Delta }{P}_{5}+\mathrm{\Delta }{P}_{6}.$
Lubricant viscosity $\eta$ is the biggest influence factor on the oil film thickness, so its impact on rolling force can be represented as:
$dP=\mathrm{\Delta }{P}_{3}=\left(\frac{\partial P}{\partial h}\right)\left(\frac{\partial h}{\partial \eta }d\eta \right).$
According to the rolling theory, rolling force $P$ is expressed as $P=3\eta B\left({v}_{r}+{v}_{1}\right)\sqrt{R"r}/\alpha h$, and there is:
$\frac{\partial P}{\partial h}=-\frac{3\eta B\left({v}_{r}+{v}_{1}\right)\sqrt{{R}^{"}r}}{\alpha {h}^{2}},\frac{\partial h}{\partial \eta }=\frac{6\stackrel{-}{U}}{K\mathrm{t}\mathrm{a}\mathrm{n}\
alpha }\left(1-\frac{2\epsilon }{3}\right).$
When reducing the reduction rate $\epsilon$ and increasing the lubricant viscosity $\eta$, we have the same effect on the oil film thickness and lubrication state. Setting roll vertical vibration
displacement equation as $X={A}_{0}\mathrm{c}\mathrm{o}\mathrm{s}\omega t$, there is:
$\epsilon =\frac{{h}_{1}-{h}_{2}}{{h}_{1}}=\frac{{h}_{1}-{\stackrel{-}{h}}_{2}-2{A}_{0}\mathrm{c}\mathrm{o}\mathrm{s}\omega t}{{h}_{1}}=\stackrel{-}{\epsilon }+\frac{-2{A}_{0}\mathrm{c}\mathrm{o}\
mathrm{s}\omega t}{{h}_{1}}=\stackrel{-}{\epsilon }+\mathrm{\Delta }\epsilon ,$
$d\eta =-d\epsilon =-\frac{-2{A}_{0}\mathrm{c}\mathrm{o}\mathrm{s}\omega t}{{h}_{1}}=\frac{2}{{h}_{1}}X.$
After substituting Eq. (17) and Eq. (18) into Eq. (16), we find $\mathrm{\Delta }{P}_{3}$ is negative. Namely, it increases stand vertical vibration system stiffness. Lubricating oil film in the roll
gap for the vertical motion of the rolling mill system also play a role in damping, the greater is the lubrication film thickness, the lower is the friction coefficient, worse damping effect,
reducing the friction influence on the rolling pressure, the closer is the relation between tension and rolling force, the worse system stability. When the rolling speed is greater than a certain
value, the relation between the speed of rolling mill and the friction coefficient can be approximately expressed as the following formula:
$\mu =a\mathrm{e}\mathrm{x}\mathrm{p}\left(-bv+c\right),$
where, $a$, $b$ and $c$ are constants. From the Eq. (19), friction coefficient falls sharply with the increase of rolling speed, this friction force down can cause rolling mill negative damping and
self-excited vibration easily, such as the Stribeck effect section in Fig. 2. Supposing the roll movement differential equation is expressed as $m\stackrel{¨}{x}+\left(c+c"\right)\stackrel{˙}{x}+kx=
F$, where, $c$ is the system damping, $c"$ is a friction change slope with the speed (equivalent to the negative damping). When in negative damping critical conditions, i.e. $\left(c+c"\right)=0$, a
roller likely produces self-excited chatter and is not stable. And when negative damping exceeds the critical condition $\left(c+c"\right)<0$, the rolling instability is divergent, rolling will not
be able to maintain and even cause the equipment destruction.
When there is torsional vibration or horizontal vibration, the tangential and horizontal slide of mill roller and rolling piece will play a major role. Namely, the contact region with adhesion and
sliding coexistence transits to full sliding. At this time, a neutral point will escape from the deformation zone periodically, rollers and rolled piece will produce the “slip-glue” skid phenomenon.
It can make the interface friction coefficient to be changed at a dramatic and periodical state; The stability of the rolling process will decrease and induce vibration. By the Kaller theory, when
the entire contact area from adhesion and sliding coexistence shift to full sliding, the friction (tangential force) of contact area reduces the maximum static friction force to dynamic friction, and
the friction varies. Namely, the system produces a critical speed, and stick slip has a typical form of instability caused by friction. Namely, in the mixed film, lubrication is larger because of the
effect of plastic fluid hydrodynamic lubrication, the interface shear stress decreases with the speed increase, and it can produce sudden acceleration and start an oscillation process, that is the
self-excited vibration process. So, in order to control the rolling mill vibration, it is best to control the rolling lubrication thickness with or without lubrication.
3.2. Contrast test with rolling interface friction changing
From the foregoing analysis, the friction coefficient increases within a certain range and will contribute to the rolling stability, so we took the following measures for vibration comparison test by
improving the interface friction state, such as open/close emulsion, high chromium cast iron/high speed steel roll and fine grinding/coarse grinding high chromium cast iron respectively,
The CSP production line accepts the HYG-1 lubricating oil and in proportion by mixing device, lubricating oil and water are mixed evenly, we should prevent from oil and water emulsification
completely. The horizontal acceleration time domain and Wigner-Ville distribution diagrams of F2 mill work roller were shown in Fig. 7 for rolling container plate at 2 mm finishing thickness when
emulsion open (Fig. 7(a)) and close (Fig. 7(b)). It can be seen that the roller horizontal acceleration root mean square value is 87.30 mV (the unit is not converted) in emulsion open and 65.74 mV in
emulsion closed, the amplitude decreased by 25 %. The rolling process and force parameter comparison is shown in Table 3 by changing the interface friction condition. It can be seen that although two
strength parameters changed a little, but roller horizontal vibration acceleration amplitude decreased obviously after the emulsion close. From the frequency domain curve of emulsion close, work
roller vibration energy is relatively concentrated, and shifts to a high frequency, which is beneficial to avoid system low order natural frequency and reduce the rolling mill vibration possibility.
That is, the rolling interface has an improvement after the emulsion shut off, and there is a suppression effect on the rolling mill vibration.
Table 3Rolling process and force parameters comparison after interface friction condition change
F2 high speed steel
Normal rolling F3 high speed steel F3 roller coarse grinding
F2, F4 emulsion open F2, F4 emulsion close
F3 entrance width mm 1143.07 1199.21 1199.4 1143.22 1143.25
F3 entry thickness mm 12.099 12.354 12.372 11.318 11.537
Reduction rate % 46.8 46.9 46.9 46.9 46.6
F3 inlet temperature ℃ 982.84 991.73 996.53 995.86 1000.3
F3 exit temperature ℃ 981.61 989.06 992.83 992.12 996.5
F3 contact arc length mm 48.3 49.9 49.9 48.1 47.4
F3 work roller temperature ℃ 52.84 38.02 53.36 47.86 46.83
F3 rolling force kn 23994 26066 25368 22060 21341
F3 rolling torque kn∙m 814.9 919.2 902.2 742.6 713.9
F3 largest rolling torque kn∙m 14315.5 14725.2 14538.9 12963.6 12064.8
F3 rolling power kw 5024 5508 5476 5056 5222
F3 backward tension N/mm^2 69.15 74.08 74.19 64.69 65.95
F3 speed m/s 2.32 2.37 2.4 2.73 2.83
F3 bending force kn 1399.9 1400.2 970.3 1403.1 1400
F3 axial float mm –38.55 –33.13 –100 –88.27 –72.12
F3 roller angular velocity round/sec 0.96 0.9480 0.9600 1.0920 1.1320
Fig. 7Horizontal acceleration time domain and Wigner-Ville distribution diagrams of F2 work roller
The horizontal acceleration of time domain and its Wigner-Ville distribution diagrams of F3 mill work roller were shown in Fig. 8 for rolling container plate at 2 mm finishing thickness with high
chromium cast iron roller (Fig. 8(a)) and high speed steel roller (Fig. 8(b)). It can be seen the mean square root value is 519.9 mV and 116.21 mV respectively. That is, the roller acceleration
amplitude decreased about 76 % after high speed steel adoption and the rolling force is also reduced. From their frequency domain graph, we can also see the energy distribution after using a
high-speed steel roller, there is no obvious advantage frequency and vibration, it also can be felt at the scene obviously. So, there are obvious effects on the rolling mill vibration suppression by
using high speed steel rollers.
Fig. 8Horizontal acceleration time domain and Wigner-Ville distribution diagrams of F3 work roller
We also made a contrast test by normal and reduced rolling speed at field, the horizontal acceleration time domain and its Wigner-Ville distribution diagrams of F3 mill work roller were shown in Fig.
9 for rolling container plate at 1.6 mm finishing thickness with normal speed (Fig. 9(a)) and reduced (Fig. 9(b)) rolling speed. Its corresponding RMS values were 1283.7 mV and 1607 mV. Namely, the
work roller acceleration amplitude increases about 20 % after reducing speed, vibration intensity has a tendency to increase after speed reducing. So, while it will be helpful to improve the friction
coefficient, reduce strip strain rate and the rolling force with rolling speed reducing, but due to CSP technology restrictions, the pass rolling temperature decreases and the rolling force increases
(which can be seen from Table 4) after reducing rolling speed. So, it is not desirable to suppress mill vibration by reducing the speed. Then the main factor affecting the rolling mill vibration
intensity is the rolled piece material, such as the harder are materials under the same process, the greater is the rolling force and the more intense is vibration. The noticeable influence on the
rolling mill vibration is the roll gap lubrication friction conditions for the same material conditions.
Fig. 9Horizontal acceleration time domain and Wigner-Ville distribution diagrams of F3 work roller
Table 4Rolling process force parameters at normal and slow speed
Normal rolling Slow speed Normal rolling Slow speed
Entrance width, mm 1142.54 1142.84 Rolling torque, KN∙m 861.8 849.6
Entry thickness, mm 10.546 10.543 Maximal rolling torque, KN∙m 11544.2 14095.7
Inlet temperature, ℃ 971.53 956.52 Rolling power, kw 6567 5303
Outlet temperature, ℃ 972.67 956.12 Backward tension, N/mm^2 60.24 60.25
Contact arc length, mm 46.7 46.9 Rolling speed, m/s 2.87 2.35
Work roll temperature, ℃ 38.66 46.56 Roll bending force, KN 1400 1401.4
Rolling force, KN 25165 25494 Axial float, mm -35.01 -28.58
Experiments of work roller surface coarse grinding were also carried out, the time domain horizontal acceleration and its Wigner-Ville distribution diagrams of F3 mill work roller were shown in Fig.
10 for rolling container plate at 2.0 mm finishing thickness with work roller coarse grinding. It can be found from the time-domain diagram, roller acceleration root mean square value is 1655.4 mV
and has gained more than two times, so we think there is no suppression benefit for coarse grinding roller. And it can be seen from the scene feeling, F3 vibration feeling is more intense after
coarse grinding.
Fig. 10Horizontal acceleration time domain and Wigner-Ville distribution diagrams of F3 work roller
4. Conclusions
Based on the fluid mechanics theory and rolling theory, we studied hot rolling friction interface dynamic characteristics under lubricated and non-lubricated condition respectively, the factors
influence such as rolling speed, rolling load, roll material, rolling temperature and oil properties. The two-dimensional Reynolds equation was set up for steady and unsteady rolling process. Through
the study of the interface friction influence on the rolling mill vibration, we found if the lubrication film thickness in the roll gap is greater, the friction coefficient is lower, its damping
effect is worse, this will reduce the influence of friction on the rolling force, the closer is the relation between tension and rolling force, the worse is the system stability. When the rolling
speed is greater than a certain value, the rolling interface friction coefficient falls sharply with the increase of rolling speed, this produces self-excited vibration by negative damping. The field
contrast tests were carried out by changing rolling interface state, such as open/close emulsion, high chromium cast iron/high speed steel roller, fine/coarse grinding high chromium cast iron roller
and normal/reduced speed. It can be found that the emulsion close and high speed steel roller adoption have the obvious effectiveness on rolling mill vibration suppression.
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About this article
Mechanical vibrations and applications
four-high mill
roller system
rolling interface
This research was supported by the Key Scientific Research Project of the Henan Province (No. 17A580003), Henan Polytechnic University Education Teaching Reform Research Projects (No. 2015JG034) and
Colleges and Universities Focus on Soft Science Research Project Plan (No. 16A630049).
Copyright © 2017 JVE International Ltd.
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Solved: Jake has knitted 5 scarves. 2 of them are green. If he
chooses one of the scarves at rand [Others]
Jake has knitted 5 scarves. 2 of them are green. If he chooses one of the scarves at random, what is the probability that it is green? Write your answer as a fraction and a decimal.
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Expert Verified Solution
Super Gauth AI
The probability that Jake chooses a green scarf at random is $$\frac{2}{5}$$ as a fraction and $$0.4$$ as a decimal
1. Identify the total number of scarves
Jake has knitted 5 scarves
2. Determine the number of green scarves
The number of green scarves is 2
3. Calculate the probability as a fraction
The probability is $$\frac{2}{5}$$
4. Convert the fraction to a decimal
$$\frac{2}{5} = 0.4$$
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