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J/95 (J/Boats) - Sailboat specifications - Boat-Specs.com
Sailboat specifications
The J/95 is a 31’2” (9.51m) cruiser-racer sailboat designed by Alan Johnstone (United States). She was built since 2009 (and now discontinued) by J/Boats (United States).
J/95's main features
Hull type
Cruiser-racer sailboat
Sailboat builder
Sailboat designer
United States
GRP (glass reinforced polyester):
Sandwich fiberglass polyester (vacuum infusion)
First built hull
Last built hull
Centerboard : pivoting centerboard in the keel
Single helm wheel
Twin spade rudders
Standard public price ex. VAT (indicative only)
J/95's main dimensions
Hull length
31’ 2”9.51 m
Waterline length
28’ 5”8.66 m
Beam (width)
10’3.05 m
5’ 6”1.68 m
Draft when appendages up
3’0.92 m
Light displacement (M[LC])
6001 lb2722 kg
Ballast weight
2449 lb1111 kg
Ballast type
J/95's rig and sails
Upwind sail area
450 ft²41.8 m²
iFore triangle height (from mast foot to fore stay top attachment)
36’ 7”11.16 m
iFore triangle base (from mast foot to bottom of forestay)
11’3.35 m
iMainsail hoist measurement (from tack to head)
36’ 7”11.16 m
iMainsail foot measurement (from tack to clew)
13’3.96 m
Rigging type
Sloop Marconi 9/10
Mast configuration
Keel stepped mast
Rotating spars
Number of levels of spreaders
Spreaders angle
Spars construction
Aluminum spars
Standing rigging
Single-strand (ROD)
J/95's performances
Upwind sail area to displacement
iThe ratio sail area to displacement is obtained by dividing the sail area by the boat's displaced volume to the power two-thirds.
The ratio sail area to displacement can be used to compare the relative sail plan of different sailboats no matter what their size.
Upwind: under 18 the ratio indicates a cruise oriented sailboat with limited performances especially in light wind, while over 25 it indicates a fast sailboat.
231 ft²/T21.44 m²/T
Displacement-length ratio (DLR)
iThe Displacement Length Ratio (DLR) is a figure that points out the boat's weight compared to its waterline length. The DLR is obtained by dividing the boat's displacement in tons by the cube of
one one-hundredth of the waterline length (in feet).
The DLR can be used to compare the relative mass of different sailboats no matter what their length:
a DLR less than 180 is indicative of a really light sailboat (race boat made for planning), while a DLR greater than 300 is indicative of a heavy cruising sailboat.
Ballast ratio
iThe Ballast ratio is an indicator of stability; it is obtained by dividing the boat's displacement by the mass of the ballast. Since the stability depends also of the hull shapes and the
position of the center of gravity, only the boats with similar ballast arrangements and hull shapes should be compared.
The higher the ballast ratio is, the greater is the stability.
41 %
Critical hull speed
iAs a ship moves in the water, it creates standing waves that oppose its movement. This effect increases dramatically the resistance when the boat reaches a speed-length ratio (speed-length ratio
is the ratio between the speed in knots and the square root of the waterline length in feet) of about 1.2 (corresponding to a Froude Number of 0.35) . This very sharp rise in resistance, between
speed-length ratio of 1.2 to 1.5, is insurmountable for heavy sailboats and so becomes an apparent barrier. This leads to the concept of "hull speed".
The hull speed is obtained by multiplying the square root of the waterline length (in feet) by 1.34.
7.14 knots
J/95's auxiliary engine
1 inboard engine
Engine(s) power
14 HP
Fuel type
Fuel tank capacity
14.5 gal55 liters
J/95's accommodations and layout
Open aft cockpit
Berth(s) (min./max.)
2 / 4
Freshwater tank capacity
19.8 gal75 liters
Have you spotted incorrect data?
You can report it in the forum
contact the webmaster
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Model theory
Syllabus and literature for lectures on
Course code: NMAG407
Fall 2020 special page
Master Thesis topics: If you contemplate the idea to write an MSc Thesis in logic, or specifically in model theory, talk to me: there is a variety of possible suitable topics and I usually put
into the SIS only one or two. Student logic seminar.
The course covers main topics of model theory with an emphasis on examples and methods important for applications of model theory in algebra, geometry and number theory.
Structures and an interpretation of a language. Tarski's truth definition.
Embeddings and isomorphisms of structures, substructures. Elementary equivalence and the Ehrenfeucht-Fraisse game, example: the theory DLO of dense linear orderings without end-points.
Preservation theorems, the diagram of a structure.
Algebraic examples: ordered real closed fields (RCF) and algebraically closed fields (ACF_p and ACF_0), vector spaces over a fixed field, groups.
The compactness theorem and its applications: elementary extensions, the upward Lowenheim-Skolem theorem, non-standard models of RCF and of the ring of integers. A transfer theorem from ACF_p to
ACF_0. The Ax-Grothendieck theorem.
Complete theories and kappa-categorical theories. Vaught's test. Skolemization and the downward Lowenheim-Skolem theorem.
Quantifier elimination and its proofs for DLO and ACF. The strong minimality of ACF and the o-minimality of RCF (assuming QE for RCF).
Strongly minimal theories and their associated (pre)geometries.
Types, their realization and omitting. The Stone space of complete types, algebraic ex.: Zariski spectrum. Isolated types and the Omitting types theorem. MacDowell-Specker's theorem.
kappa-saturated structures and their existence. Countable saturated structures and the size of the Stone space. Saturation of ultraproducts.
The number of non-isomorphic models of a given cardinality. Vaught's conjecture. Morley's categoricity theorem.
Main sources
J.Kirby, An Invitation to Model Theory (Cambridge U. Press, 2019)
D. Marker, Model Theory: An Introduction (Springer, 2002).
(More advanced. The MFF library has an access to the e-version of the book. Almost all material can be also found in Marker's lecture notes (online) listed below.)
Other classics
C.C.Chang, J.H.Keisler: Model theory, NHPC 1973.
W. Hodges: Model Theory, Cambridge Univ. Press, 1993.
W. Hodges, Shorter Model Theory (CUP, 1997).
G.Sacks, Saturated Model Theory (World Sci., 2nd ed., 2010)
Lecture notes on the web | {"url":"https://www.karlin.mff.cuni.cz/~krajicek/mt.html","timestamp":"2024-11-03T12:49:57Z","content_type":"text/html","content_length":"4314","record_id":"<urn:uuid:030074e3-8413-4f71-95c3-da2488a56880>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00231.warc.gz"} |
Assignment 11
Your name:
Your email address:
Exercise 1:
Likelihood ratio tests
Open seaview, and load Yeast_vma1_all_seaview.mase. Note that these nucleotide sequences are already aligned based on their encoded amino acids. (To see the aa alignment click on view and check
protein - return to the nucleotide sequence view). The dataset has three set of sites: all, intein, extein. Collaborate with two of your neighbors, each of you will use a different set of sites.
We want to calculate thee likelihood values using Trees — PhyML:
1. Model: GTR, Across site rate variation: None — we'll call this "GTR"
2. Model: GTR, Across site rate variation: Optimized (4) — we'll call this "GTR+Gamma"
3. Model: GTR, Across site rate variation: Optimized (4) Invariable Sites Optimized — we'll call this "GTR+Gamma+I"
All other parameters are the same for the three runs: Support values: aLRTvalues, nucleotide equilibrium frequencies optimized, tree searching NNI, starting tree BioNJ, optimize tree topology.
Once the tree reconstruction is finished, DO NOT CLICK OK, rather click on copy at the bottom of the window, open a text editor and paste the content into the test window. We are interested in the
log likelihood, and the last values estimated for the shape parameter, and the % invariable sites (if the latter two were estimated as part of the model).
One important condition that has to be fulfilled before one can use a Likelihood Ratio Test (LRT) to compare two models, is that the models should be "nested". This means that the simpler model must
be a constrained version of the parameter-rich model. The likelihood ratio test is performed by doubling the difference in log-likelihood scores and comparing this test statistic with the critical
value from a chi-squared distribution having degrees of freedom equal to the difference in the number of estimated parameters in the two models. The parameter-rich model will always have a better
fit, due to the extra parameters and will therefore have the highest log-likelihood, so the difference should be a positive number. In this case there is 1 degree of freedom between each of the
models — the gamma shape parameter is one parameter and the % invariant sites is the second parameter. Use this online chi-square calculator to determine the significance of the test.
Does the GTR+Gamma model explain the data significantly better than the more simple GTR model?
Does the GTR+Gamma+I model explain the data significantly better than GTR+Gamma model?
Is this true for all of the the three sets of sites?
When you compare the trees you obtains for the intein, extein and the combined datasets, do you observe any differences?
Exercise 2:
Introduction to Bayesian Analyses Intro Slides are here
In class we will use MrBayes3.2 as installed on the bioinformatics cluster. Before you set up an ssh terminal connection to cluster, increase the width of the terminal to >150 characters. The goal of
this exercise is to learn how to use MrBayes to reconstruct phylogenies and estimate parameters.
1. Log on to the bbcsrv3.biotech.uconn.edu cluster using Bitvise SSH Client. After you established an ssh connection to the terminal, qlogin into a compute node and load the module for the latest
version of the MrBayes program: module load mrbayes/3.2.6 (to see which modules are available on the cluster type module avail ).
2. Create a class11 directory on the cluster (by whatever means, e.g. the sftp interface or the commandline). The nexus formated file that we will use is here. Move/copy this file into the class 11
directory. Have a look at the file in a texteditor or using more - pay attention to the MrBayes block at the end of the file).
3. To start the program, in terminal change to the directory where the sequence data are located and type mb.
4. At the MrBayes command prompt type "execute Yeast_vma1_2partions.nxs". This will load the data into the program and excute the MrBayes block..
Here is an explanation of the commands at the end of sequence file in the MrBayes block:
charset intein = 856-2685;
charset extein = 1-855 2686-3705; [this assignes the alignment columns to the intein and extein partions.]
partition favored = 2: intein, extein; [the two partition scheme is named favored and contains two partitions]
set partition = favored; [this tell the program to use the two partitions]
lset applyto=(1,2) nst=6 rates=gamma; [both partitons are set to the GTR model with six parameters and an ASRV described by a gamma distribution]
mcmcp filename=analysisP; [the result files are named analysisP ...]
mcmcp samplefreq=50 printfreq=50;
[mcmcp sets parameters for the chain. This sets the frequency with which the "robot" reports results to the screen and to the files (different files for trees (.t) and other parameters (.p)]
mcmcp savebrlens=yes; [mcmcp sets parameters for the chain. savebrlens tells it to save the trees with branchlengths]
unlink statefreq=(all) revmat=(all) shape=(all); [unlinks partitions, shape parameters and rate multipliers will be estimated for each of the partitions]
prset applyto=(all) ratepr=variable; [allows rates to vary between partitions]
" prset" sets the priors for the model used to calculate likelihoods.
" mcmcp " sets parameters for Markov Chain Monte Carlo: we set to sample every 50 generation, to print results to a screen every 50th generation, run 2 chains simultaneously, start with random
tree, and save branch lengths.
The MrBayes manual (pdf) , which includes tutorial sections, is here,
5. Type "showmodel" to have MrBayes display the model you selected.
6. At the MrBayes command prompt type "mcmc ngen=5000". This actually runs the chains, and we set it to run for 5000 generations. The program runs two analyses in parallel (by default each with 4
chains, and three of these chains are heated; it definitely is a good idea to run mb on a fast and remote computer). The smaller the reported average standard deviation of split frequencies is,
the more reliable the result (i.e., your run is close enough to infinitely long). When the value is below .015, or when your patience is exhausted, terminate the run, by typing no at the prompt.
Give it at least 5 minutes.
7. Make sure you have typed "no" at the Continue with analysis? (yes/no): prompt.
After the run is finished, the " sump " command will plot the logL vs. generation number, that allows to determine the necessary burnin (you want to discard those samples as "burnin" where the
-logL is still rising steadily).
To see the whole logL curve, you need to set the burnin fraction to .02 . (type help sump at the mb commandline). sump burninfrac=.02
[Rather than using the sump command, you also can import the parameter file into EXCEL and plot the logL values as a chart in EXCEL. See below.
Or you can download the tracer application that can read the parameter files from MrBayes, and provides an easy intuitive and interactive way to evaluate these files with respect to burnin and
confidence intervals]
At the start of the run, the likelihood rapidly increases by orders of magnitude. If the first samples are included in the plot, one really cannot see if the likelihood values fluctuate around a
value. You can exclude the first couple of samples by specifying a burnin or a burninfrac. (The new version of MrBayes uses a burnin of 25% by default.)
sumt burninfrac =.25
, where you need to substitute '.25' with the number you obtained in the previous step of the exercise. This command creates a consensus tree, and shows posterior probabilities of the clades. You
can take a look at the tree on the screen (scroll up to see the bipartition table, and different version of the tree) - you want to take the time to answer the green questions 1-4 given below at
this point.
8. Load the .p files into into tracer (local download link here), you can load both parameter files at the same time, and select in the upper left field of the tracer application, if you want to
analyze one or both of the parameter files.
9. Inspect the trace (select the right button over the graphics window) of lnL (select in the list of parameters on the left).
10. Determine the mean, median and 95% highest posterior density interval (select the left button over the graphics window) for shape parameter (gamma) and the relative branch length multiplier (m)
for each of the data partitions. (1 is the intein, 2 is the extein) (If you want .p, .t, and consensus trees from a longer mcmcmc run download this file.)
Keep in mind the the x axis is different in the display for the two partitions. To see the histograms on the same scale, select marginal probabilities (button over the graphics window), and
select the parameters for both of the partitions in the left panel. At the bottom you might want to select histogram. You also could plot the estimated stationary probabilities and the transition
1) Which value did you use for the burnin/burninfraction?
2) Which branch in the tree is the longest? (check the bipartition and branchlengths tables)
3) How long is it?
4) What is the measure?
5) What are the shape parameters for the intein and extein partition, respectively? What is the respective 95% credibility interval?
6) What does this mean for the number of sites with low substitution rates in the intein and extein, receptively?
7) Can you explain in a few words, why is it important to exclude a 'burnin' from our analyses?
8) What is the 95% highest posterior density interval for the relative rate parameters for the two data partitions?
Type " quit " at the prompt to exit MrBayes.
MrBayes by example: Identification of sites under positive selection in a protein
Exercise 3:
Professor Walter M. Fitch and assistant research biologist Robin M. Bush of UCI's Department of Ecology and Evolutionary Biology, working with researchers at the Centers for Disease Control and
Prevention, studied the evolution of a prevalent form of the influenza A virus during an 11-year period from 1986 to 1997. They discovered that viruses having mutations in certain parts of an
important viral surface protein were more likely than other strains to spawn future influenza lineages. Human susceptibility to infection depends on immunity gained during past bouts of influenza;
thus, new viral mutations are required for new epidemics to occur. Knowing which currently circulating mutant strains are more likely to have successful offspring potentially may help in vaccine
strain selection. The researchers' findings appear in the Dec. 3 issue of Science magazine.
Fitch and his fellow researchers followed the evolutionary pattern of the influenza virus, one that involves a never-ending battle between the virus and its host. The human body fights the invading
virus by making antibodies against it. The antibodies recognize the shape of proteins on the viral surface. Previous infections only prepare the body to fight viruses with recognizable shapes. Thus,
only those viruses that have undergone mutations that change their shape can cause disease. Over time, new strains of the virus continually emerge, spread and produce offspring lineages that undergo
further mutations. This process is called antigenic drift. "The cycle goes on and on-new antibodies, new mutants," Fitch said.
The research into the virus' genetic data focused on the evolution of the hemagglutinin gene-the gene that codes for the major influenza surface protein. Fitch and fellow researchers constructed
"family trees" for viral strains from 11 consecutive flu seasons. Each branch on the tree represents a new mutant strain of the virus. They found that the viral strains undergoing the greatest number
of amino acid changes in specified positions of the hemagglutinin gene were most closely related to future influenza lineages in nine of the 11 flu seasons tested.
By studying the family trees of various flu strains, Fitch said, researchers can attempt to predict the evolution of an influenza virus and thus potentially aid in the development of more effective
influenza vaccines.
The research team is currently expanding its work to include all three groups of circulating influenza viruses, hoping that contrasting their evolutionary strategies may lend more insight into the
evolution of influenza.
Along with Fitch and Bush, Catherine A. Bender, Kanta Subbarao and Nancy J. Cox of the Centers for Disease Control and Prevention participated in the study.
A talk by Walter Fitch (slides and sound) is here
The goal of this exercise is to detect sites in hemmagglutinin that are under positive selection.
Since the analysis takes a very long time to run (several days), here are the saved results of the MrBayes run: Fitch_HA.nex.p.txt, Fitch_HA.nex.t.txt .
The original data file is flu_data.paup . The dataset is obtained from an article by Yang et al, 2000 . The File used for MrBayes is here
The MrBayes block used to obtain results above is:
begin mrbayes;
set autoclose=yes;
lset nst=2 rates=gamma nucmodel=codon omegavar=Ny98;
mcmcp samplefreq=500 printfreq=500;
mcmc ngen=500000;
sump burnin=50;
sumt burnin=50; end;
Selecting a nucmodel=codon with Omegavar=Ny98 specifies a model in which for every codon the ratio of the rate of non-synonymous to synonymous substitutions is considered. This ratio is called OMEGA.
The Ny98 model considers three different omegas, one equal to 1 (no selection, this site is neutral); the second with omega < 1, these sites are under purifying selection; and the third with Omega >
1, i.e. these sites are under positive or diversifying selection. (The problem of this model is that the there are only three distinct omegas estimated, and for each site the probability to fall into
one of these three classes. If the omega>1 is estimated to be very large, because one site has a large omega, the other sites might not have a high probability to have the same omega, even though
they might also be under positive selection. This leads to the site with largest omega to be identified with confidence, the others have more moderate probabilities to be under positive selection).
Note : Version 2.0 of Mr Bayes has a model that estimates omega for each site individually, the new version only allows the Ny98 model as described above..
1. First, you need to detect how many generations to burn in (meaning the number of samples you will have to discard). Open the file Fitch_HA.nex.p.txt with Excel and plot # of generations versus
-LnL values. Determine after how many generations the graph becomes "stationary" (hint: change the Y-axis bounds to "zoom in", e.g., -3300 min to -3200 max). The burnin value is that number of
generations divided by 50 (since only every 50th generation was sampled; i.e. the burnin value roughly is equal to the number of rows - not quite because there is a header). To more accurately
determine the burnin, you need to rescale the Y-axis (click at the Y-axis -- if you aim accurately, you'll get a box that allows rescaling).
The result (scatterplot of LogL versus generation) might look like this:
2. This file contains information for posterior probabilities for each codon (columns) at each sampled generation (rows). Scroll to the right to see these columns, starting with pr+(1,2,3), pr+
(4,5,6), etc. Calculate average posterior probability for each site of being under positive selection (Do not forget to exclude first N rows as a burnin; you should have detected value of N in
the first question of this exercise - to be clear on where the burnin ends, you might want to highlight the rows representing the burnin and select a different font color. (Use AVERAGE() function
of Excel, enter the formula in a cell below the values for the individual generations -- starting in column pr+(1,2,3) -- copy the formula to all columns) (see slides)
3. Plot average posterior probability vs. site #. (select the row in which you calculated the averages, then click Graph, and select a bar graph). Write down the codon positions for a few sites with
the highest posterior probability of being positively selected (the columns name pr+(1,2,3), pr+(4,5,6)....and so on. pr+(1,2,3) mean probability of codon #1 (nucleotide #1, #2 and #3) to be
under positive selection))
5. Determine the 95% credibility interval for the omega<1 value. To do this you sort posterior probability column in ascending order (Select data you want to sort, and go to Data->Sort... ). Again,
do not forget to discard the burnin ; the easiest might be to actually delete it.. After sorting, exclude 5% of the data on the top and on the bottom. The range of the remaining data gives you
the 90% confidence interval. (Enter answer in box below!)
6. The structure of hemagglutinin has been crystallized and is publicly available through PDB. Download the PDB file here and examine it with SPDBV. Chain A of the PDB file corresponds to the
sequences we did our analysis with (color the molecule according to chain). Below is a comparison of one of the sequences we used for analyses with the sequence for which the structure was
Using this alignment as a guide, map the site(s) which have the highest probability to belong to the class with omega>1. Where are these sites located in the protein? (Reminders: The position
number in the nexus file corresponds to nucleotide sequence, the structure is based on the amino acid sequence - take the third codon position and divide by 3 to find the amino acid. You only
want to be concerned with Chain A!)
What is the 95% credibility interval for the omega < 1?
Does this value indicate strong purifying selection?
Which codon(s) showed signs of positive selection?
Which position and which amino acid does this correspond to in the above alignment?
Where is this aa located in the structure?
Exercise 4:
dN/dS ratios along a sequence. I ran the dataset from exercise #1 and 2 using the NY98 model (no partitions). The file is here. The model has a parameter for transition/tranversion ratio and for the
dN/dS ratio (called omega). The model uses and explores only two of these - one for sites under purifying selection, one for sites under diversifying selection. In addition for each codon the
probability that the codon is in the omega+ group is estimated, and the omega value for each codon is estimated. This results in a lot of parameters, which makes for a slow moving robot. The
parameter files resulting form the run are here. This is the MrBayes block in the file:
begin mrbayes;
lset nst=2 rates=gamma nucmodel=codon omegavar=Ny98;
report possel = yes siteomega = yes;
mcmcp filename=analysisS;
mcmcp samplefreq=50 printfreq=50 diagnfreq=500;
mcmcp savebrlens=yes;
This results in a lot of parameters, which makes for a slow moving robot. The parameter files resulting form a 24h run is here. (already imported into excel). Two sheets give the parameters for each
run, the third contains the values after the burnin.
Scroll to the right to see these columns, starting with pr+(1,2,3), pr+(4,5,6), etc. Calculate average posterior probability for each site of being under positive selection . Use the AVERAGE()
function of Excel, enter the formula in a cell below the values for the individual generations -- starting in column pr+(1,2,3) -- copy the formula to all columns). You can do the same with the
esimated omega values (to the right of the pr(xyz columns).
Type logout to release the compute node from the queue.
If you you encountered problems in your session, check the queue for abandoned sessions using the command qstat. If there are abandoned sessions under your account, kill them by deleting them
from the queue by typing qdel job-ID, e.g. "qdel 40000" would delete Job # 40000
Send email to your instructor (and yourself) upon submit
Send email to yourself only upon submit (as a backup)
Show summary upon submit but do not send email to anyone. | {"url":"https://carrot.mcb.uconn.edu/mcb3421_2017/assign11_new.html","timestamp":"2024-11-08T07:37:46Z","content_type":"text/html","content_length":"26752","record_id":"<urn:uuid:0e8a87e3-2c62-4bb5-89da-2c6924b75592>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00840.warc.gz"} |
Painted Corridors
Problem L
Painted Corridors
The Institute of Colorfully Painted Corridors is planning the construction of a new building. The building has numerous junctions, and corridors that each connect a pair of junctions. The corridors
will be painted by amazing new painting robots that drive along the corridors and paint all the walls as they go. The architect has specified the colors of some of the corridors, which may be red,
orange, yellow, green, blue, or purple. However, there is only a budget for three painting robots, so there will be a single robot for each primary color (red, yellow, or blue). In addition, these
robots are the cheapest possible version, and cannot turn their paint sprayer off (though they can go as fast or as slow as desired with no problems; they can even stop moving entirely).
If a corridor needs to be painted a secondary color (orange, green, or purple), in order for the paints to mix properly, the two robots with the appropriate primary colors must travel down the
corridor in the same direction at the same time to create the correct color. The color mixing rules are: $orange = red + yellow$, $green = yellow + blue$, and $purple = red + blue$. A corridor that
is unspecified in the plan may be painted any color, or left unpainted.
Corridors may be painted multiple times, provided that each time they are painted with the correct color. Corridors with no specified color can be painted multiple times with different colors. All
corridors can be travelled along in both directions. The robots may end up at any junctions after painting all the corridors.
Given the architect’s design, is it possible for the painting robots to paint the corridors the desired colors?
The first line of input contains five integers, $n$ ($2 \leq n \leq 100$), $m$ ($1 \leq m \leq \frac{n \cdot (n-1)}{2}$),
$r$, $b$ and $y$ ($1 \leq r, b, y \leq n$), where $n$ is the number of junctions, $m$ is the number of corridors, and $r$, $b$ and $y$ are the initial junctions of the red, blue, and yellow painting
robots respectively. Junctions are numbered $1$ through $n$. Each of the next $m$ lines contains two integers $i$, $j$ ($1 \leq i < j \leq n$), and a single character $c$ which is one of R, O, Y, G,
B, P, X. The integers $i$, $j$ indicate that there is a corridor between junction $i$ and junction $j$, with $c$ indicating the desired color. (R, O, Y, G, B, P, X, corresponding to Red, Orange,
Yellow, Green, Blue, Purple, and Unspecified, respectively.) There is at most one corridor between each pair of junctions.
Output a single integer, $1$ if it is possible to paint the corridors as described and $0$ otherwise.
Sample Input 1 Sample Output 1
1 3 X
2 3 X 1
3 4 P
4 5 X
4 6 Y
Sample Input 2 Sample Output 2
1 3 X
2 3 X 0
3 4 O
4 5 X
4 6 Y
Sample Input 3 Sample Output 3
1 3 X
2 3 X 1
3 4 P
4 5 X
4 6 G | {"url":"https://nena20.kattis.com/contests/nena20/problems/paintedcorridors","timestamp":"2024-11-12T13:12:07Z","content_type":"text/html","content_length":"30571","record_id":"<urn:uuid:b65896b5-5756-443c-8a44-d0169e15aacd>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00226.warc.gz"} |
Categories: GEN rds-hosted Mathematics
Number e with thousands of digits 12K
Author: Jaime Marcos Date: Oct 20 2006
A very fast program that computes the number e to thousands of decimal digits. The output is written as a text file, num_e.txt. For example, to compute e to 21000 decimal digits on a Pentium III at
845 MHz, it took only 37.18 seconds. | {"url":"http://phix.x10.mx/pmwiki/pmwiki.php?n=Main.NumberEWithThousandsOfDigits","timestamp":"2024-11-05T16:42:10Z","content_type":"application/xhtml+xml","content_length":"9468","record_id":"<urn:uuid:e6ba8cb3-e47f-4b1c-bc5f-62d780c5eba9>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00665.warc.gz"} |
What is the effective interest rate? | Socratic
What is the effective interest rate?
1 Answer
The rate of interest at which a sum actually grows if compounding occurs more than once a year.
You deposit a sum of money in a bank that pays 8% interest a year, compounded yearly. (Those were the good-old days for depositors).
I deposit my money in another bank that pays 8% a year, but it is compounded every 3 months - quarterly. So, at the end of every 3 months the bank gives me interest. At the end of the year, who will
have the most money in their account?
I will because at the end of the first 3 months I receive interest and then at the end of the next 3 months I will receive interest on my original deposit plus interest on the interest I have already
earned...and so on for the year.
We can use a simple formula to calculate the actual or effective interest rate that I received.
$\left(1 + {\left(\frac{m}{n}\right)}^{n}\right) - 1$
M = the yearly or nominal rate - 8% in this case.
N = the number of times a year compounding occurs.
My effective rate is
#(1 + (.08)/4)^4 - 1)#
8.24% and yours was 8% (we could prove this using the formula).
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How to Use DAX Function in Power BI - Loginworks Softwares
How to Use DAX Function in Power BI
Data Analysis Expression (DAX)
Microsoft has introduced a powerful library of functions and operators. Data Analysis Expression (DAX) combines functions to build formulas and expressions in Power BI Desktop. With Power BI’s Data
Analysis Expressions (DAX) functionality, data manipulation and data modeling can be done with ease. These expressions are the collection and combination of functions, operators, and constants.
Therefore, one formula gives the final result. In like manner, DAX function is similar to Excel functions, while the DAX is much more advanced. If you are aware of the Excel and its functions, you
can hand over the DAX easily.
In measures, calculated columns, calculated tables, and row filters use the DAX calculation formulas.
Measures are dynamic calculation formulas where the outcomes change depending upon the conditions. In brief, Measures are used in reporting that help combining and filtering the model data by
utilizing different properties. For example, a Power BI report or Excel PivotTable or PivotChart. Measures are created using the DAX formula bar in the model creator.
A formula in a measure can use standard aggregation functions automatically formed by utilizing the Autosum include, for example, COUNT or SUM, or you can customize your very own equation by using
the DAX formula bar. Named measures passed as a parameter to the other measures.
Measure DAX
Total Sales:=SUM([Sales Amount])
Calculated Columns
A calculated column is a column that you add to an existing table (in the loaded data in Power BI) and, afterward, create a DAX equation that defines the column’s values. Since a calculated column is
created in a table in the data model. Thus they are not supported in models that recover data exclusively from a relational data source using DirectQuery mode.
At the point When a calculated column contains a valid DAX formula, values are determined for each row as soon as the formula is entered. The values are then stored in the in-memory data model. For
instance, in a Date table, when the equation is entered into the formula bar:
Calculated Column DAX
Calander = [Calendar Year] & " Q" & [Calendar Quarter]
A value for each line in the table is calculated using the value from the Calendar Year column section (in the same Date table), including space and the capital letter Q. Then including the values
from the Calendar Quarter column (in a similar Date table). The outcome for each row in the calculated column “Calander” is calculated immediately and appears, for instance, as 2017 Q1. When closing
and reopening a Power BI Desktop file, the table or any related table is processed, refreshed, or the model data is emptied from memory and then again reloaded, Column values are again recalculated.
Calculated Tables
A calculated table is a table calculated, based on either a DAX query or formula expression, derived from all or part of other tables in the same model. Rather than Querying and stacking values into
your new table’s columns from a data source, a DAX formula defines the values in the created table.
Let’s take an example, the Date table, as OrderDate, ShipDate, or DueDate, depending on the foreign key relationship. Thus, by creating a calculated table for ShipDate explicitly, you get a
standalone table that is available for queries, as fully operable as any other table. Calculated tables are also useful when configuring a filtered rowset, or a subset or superset of columns from
other existing tables. This allows you to keep the original table not damaged while creating changes of that table to check specific scenarios.
Most Used Functions Used in DAX
There are many functions in DAX library. Some of the DAX functions used in Power BI calculated columns and measures are as following:
1. VAR( ) & RETURN( )
VAR keyword introduces the definition of a variable. You can have as many variables as needed in a single expression, and each one has its own VAR definition. The RETURN keyword defines the
expression to return as an Output.
VAR position =
( SUM ( 'nyc-jobs'[# Of Positions] ) )
IF ( ISBLANK ( position ), 0, position )
2. FILTER( ) & CALCULATE( )
CALCULATE function Evaluates an expression in a context, modified by the specified FILTER functions.
new external vaccancy =
CALCULATE (
SUM ( 'nyc-jobs'[# Of Positions] ),
FILTER ( 'nyc-jobs', 'nyc-jobs'[Posting Type] = "External" )
3. REPLACE( )
REPLACE function replaces part of a text string, based on the number of characters is specified, with a different text string.
new_job_id = REPLACE('nyc-jobs'[Job ID],1,3,"TR")
4. MONTH( )
Returns the month as in number format (01 for January to 12 for December)
5. RANKX( )
In brief, the RANKX function returns the ranking of a number in a list of numbers for each row in the table argument.
ranks = RANKX(job,job[month went],,DESC)
6. YEAR( )
Returns the year of a date passed as an argument.
Year = YEAR(job[Date])
7. GROUPBY( )
GROUPBY function attempts to reuse the data that has been grouped, making it highly performant.
groupby_table = GROUPBY(job,job[Date],"total job on that day",
SUMX(CURRENTGROUP(),job[Job Profile]))
8. COUNTBLANK( )
COUNTBLANK function returns the number of empty cells in a particular column.
countpreferredblank = COUNTBLANK('nyc-jobs'[Preferred Skills])
9. DISTINCTCOUNT( )
DISTINCTCOUNT function counts the number of unique values in a particular Column.
uniquecountjob = DISTINCTCOUNT('nyc-jobs'[Job Description])
10. IF( )
IF function returns one value if the condition is TRUE, and returns another value if the condition is FALSE
full_part_time = IF('nyc-jobs'[Full-Time/Part-Time indicator]="F",
"Full TIME","PART TIME" )
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Math Labs with Activity – Divide a Line Segment into Number of Equal Parts - IMP WORLD
Math Labs with Activity – Divide a Line Segment into Number of Equal Parts
Math Labs with Activity – Divide a Line Segment into Number of Equal Parts
To divide a given line segment (say of length 9 cm) into a given number of equal parts (say into 7 parts)
Materials Required
1. A sheet of ruled paper in which all the lines are parallel and equidistant
2. A sheet of transparent paper
3. A geometry box
Here we use the Intercept theorem. According to this theorem, if there are three (or more) parallel lines and the intercepts made by them on one transversal are equal, the intercepts on any other
transversal are also equal.
Step 1: Draw a line segment AB of length 9 cm (the line to be divided into 7 equal parts) on the transparent sheet of paper as shown in Figure 1.1.
Step 2: Mark the lines on the ruled paper as 0,1,2,3,…, sequentially from top to bottom. Place the transparent sheet over the ruled paper in such a way that one end of the line, i.e., point A lies on
the line marked 0, and the other end of the line, i.e., point B lies on the line marked 7 as shown in Figure 1.2.
Step 3: Mark the points of intersection of the line segment AB with the parallel lines of the ruled paper.
The points of intersection of the line segment AB with the parallel lines of the ruled paper divide the line segment AB into 7 equal parts.
The given line segment (of length 9 cm) is divided into 7 equal parts.
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Ensemble Learning, Deep Learning-Based and Molecular Descriptor-Based Quantitative Structure–Activity Relationships
Department of Medical Molecular Informatics, Meiji Pharmaceutical University, Kiyose 204-8588, Japan
Division of Molecular and Medical Genetics, Center for Gene and Cell Therapy, The Institute of Medical Science, The University of Tokyo, Minato-ku, Tokyo 108-8639, Japan
Author to whom correspondence should be addressed.
Submission received: 6 February 2023 / Revised: 28 February 2023 / Accepted: 1 March 2023 / Published: 6 March 2023
A deep learning-based quantitative structure–activity relationship analysis, namely the molecular image-based DeepSNAP–deep learning method, can successfully and automatically capture the spatial and
temporal features in an image generated from a three-dimensional (3D) structure of a chemical compound. It allows building high-performance prediction models without extracting and selecting features
because of its powerful feature discrimination capability. Deep learning (DL) is based on a neural network with multiple intermediate layers that makes it possible to solve highly complex problems
and improve the prediction accuracy by increasing the number of hidden layers. However, DL models are too complex when it comes to understanding the derivation of predictions. Instead, molecular
descriptor-based machine learning has clear features owing to the selection and analysis of features. However, molecular descriptor-based machine learning has some limitations in terms of prediction
performance, calculation cost, feature selection, etc., while the DeepSNAP–deep learning method outperforms molecular descriptor-based machine learning due to the utilization of 3D structure
information and the advanced computer processing power of DL.
1. Introduction
Machine learning (ML) is a data analysis method that discovers associated regularities and rules through the repeated learning of past cases and data. ML methods can be classified into three types:
supervised learning, unsupervised learning, and reinforcement learning [
]. Supervised learning methods involve making a computer learn the correct data and produce the correct output for the input data. In contrast, unsupervised learning methods do not set correct data,
and hence the computer itself learns the data characteristics through a large amount of data. In reinforcement learning methods, a computer makes judgments so that the numerical value of the output
result is the highest. Therefore, in ML, predictions and inference are made by applying the regularities obtained through learning to unknown and future cases. Furthermore, deep reinforcement
learning combines deep learning (DL) and reinforcement learning and is used when a small amount of labeled data and a large amount of unlabeled data are prepared. Supervised learning is divided into
two tasks: classification to sort learning data into predetermined classifications and regression to predict future values of continuous data [
]. The main purpose of classification is to predict the classification to which the data for analysis belong. However, the main purpose of regression is to make predictions based on trends in
continuous values. Regression analysis is a statistical technique that examines the relationship between a result number and a factor number. At this time, the factor value is called the explanatory
variable and the resulting value is called the objective variable. An ML model is a mechanism that outputs results for input data and analyzes the input data for evaluation and judgments based on
some evaluation criteria. That is, ML is a system that realizes a mechanism equivalent to human learning with computers. Based on a certain calculation method or algorithm, in an ML method, a
computer discovers patterns and rules from input data, and by applying those patterns and rules to data, it predicts data. DL is an ML method in which a neural network consisting of many layers is
used. The concept of DL is focused on this multilayered neural network, but DL makes it possible for computers to extract feature values themselves when discovering patterns and rules, even if the
feature amount is not set in advance. The pattern is recognized by checking how well the input pattern matches the prepared pattern. It is a big breakthrough that feature values are created by
themselves without being instructed in this way. In DL in the field of image recognition, the main focus is the combination of a large number of images and a convolutional neural network (CNN), which
is a mathematical model that mimics the neural circuits of the brain and returns an output for an input value. In a CNN, the neural network has convolutional layers and pooling layers that function
by the characterization of two main factors: recognition in the “local receptive field” and “extraction with weight sharing” [
]. When humans recognize an object, they do not grasp the whole image at once but recognize the object little by little by extracting each partial area. The property that responds to the clipped area
is called a local receptive field, and a characteristic of a CNN is that it responds to only a small part of the input data, such as the local receptive field. Weight sharing is a mechanism for
recognizing the features deemed as important at a specific position in an image as having high importance at another position. CNNs have been applied for classification, object detection, and voice
recognition in various fields, including quantitative structure–activity relationship (QSAR). In this review, we summarize the advances in and give an overview of novel QSAR systems, including CNNs,
ensemble learning, supervised learning for regression, and molecular descriptor-based ML.
2. Classification of Images by a Neural Network
The term “neural” in neural networks stands for neurons, i.e., nerve cells. It is one of the ML methods that artificially reproduces the mechanism of the cranial nerves with a computer program. DL is
a more complex version of these neural networks [
]. A neural network of artificial neurons can be regarded as a function that receives multiple data and outputs the calculated results. For example, if a photograph is captured with a 1-megapixel CCD
camera, there are 1,000,000 pieces of data, and if colors are defined by red, green, and blue (RGB) values, the image comprises 3 million data points. By specifying the color information and
positions of one-million-pixel data, the photograph can be reproduced. Inputting many data and creating a well-fitting function that can be judged well is supervised ML. By constructing a function
that can be judged well, even if a new photo is input, it can be judged with a high probability. A structure called a multilayer perceptron is often used for a neural network, and a more complicated
network with 10 or more intermediate layers is used in DL. The input layer has a large amount of data, and the data obtained by multiplying the numerical values by some coefficient and totaling them
become the data of the middle layer in the first layer, and the second layer of the middle layer data is obtained by multiplying the data by some coefficient and totaling them. An output is derived
from these data, but it involves a huge number of calculations. A function that can be judged by deriving the optimum value of the kind of coefficient combination that can be judged most correctly is
completed. In addition, a recurrent network is a type of neural network that has a regressing, recurrent structure that can use the information previously incorporated into the model for predicting
continuous data, such as time-series data and language data. DL is a neural network with a large number of intermediate layers. By increasing the number of hidden layers, highly complex problems can
be solved and the prediction accuracy can be improved. However, DL models are sometimes called black-box models because they are too complex for humans to understand how the prediction is derived [
3. Evaluation of Predictive Models and Predictive Performance
In supervised learning, the collected data are divided into training data and evaluation data for learning. Thus, the separated data evaluation is called “cross validation” [
]. Supervised learning builds a model that makes predictions on other data after a machine learns from human labels for correct answers. The step of constructing a model with a “correct label” is
performed using training data, and prediction is performed using evaluation data. When building a model using all training data, a model can be formed that can fit the data but never fit the unknown
data that comes later—this phenomenon is called “overfitting” [
]. To prevent overfitting, the data at hand are divided into training data and evaluation data to build and predict the model. For example, in supervised learning for classification, when the
evaluation data are classified by the prediction model built from the training data, the degree of correctness among the total data is evaluated. Thus, we consider a model, i.e., line of separation,
that is too strict for some training data; however, the accuracy of prediction for different data decreases. Therefore, it is a good idea to balance the goodness of fit with the simplicity of the
model. In statistical analysis, information content criteria are set to judge the balance. If the dimensionality or number of variables of the data used to build the model is very large, the
combination of variables increases exponentially and the amount of calculations increases tremendously. Then, a sufficient learning performance may not be obtained with the data on hand. This problem
is called the “curse of dimensionality”; that is, the ML efficiency decreases because of too many data dimensions [
]. To avoid this problem, multiple variables can be combined into one feature amount (feature amount creation) or the combination of effective feature amounts can be narrowed down (feature amount
selection). Thus, overfitting occurs when the dimensionality of the polynomial is too large for the number of training data; that is, the coefficient
tends to take a large value. Therefore, if the coefficient can be restricted to a small value, overfitting can be suppressed. Regularization is a technique based on this idea [
]. A regularization term, namely a penalty term, is added to the sum-of-squares error given in Equation (1) that penalizes the coefficients for growing large (2). Here, λ is the regularization factor
and it is an arbitrary value.
E(w) = 1/2 ‖Xw − t‖^2 + λ/2‖w‖^2
The larger the value of
, the larger is the error function. To find a
that minimizes Equation (2), we expand and differentiate with respect to
to obtain the following formula.
$W = ( X T X + λ I ) − 1 X T t$
In Equation (2), λ/2‖w‖
was added to the sum of squared errors, but the regularization factor can generally be expressed as shown in Equation (5):
$1 2 λ ∑ j = 1 M ( | w j | ) q ≤ η$
Equation (2) corresponds to
= 2. Adding a regularization term means limiting the coefficient wj to the range of Equation (5).
$∑ j = 1 M ( | w j | ) q ≤ η$
In supervised learning for regression, when performing regression of evaluation data with a model created with training data, the target of evaluation is the difference between the predicted value of
the training data and the actual value of the evaluation data. Taking into account the evaluation value of the prediction result, such as the correct answer rate of the output classification and the
regression value, we examine whether the model has actual versatility. Typical cross validation methods include the holdout method and the k-fold method [
]. The k-fold method is a cross validation method that divides the training data into several (
) pieces and repeats model construction and verification for the number of divided data (i.e.,
times). One of the divided data groups is used for validation and the remaining data are used for model building. First, one data group is used for validation, and the first result is checked from
the fit of the model built on the rest of the data. In the second round, another data group is used for validation, the rest of the data are used for model building, etc. Model construction and
verification are performed for the number of divided data, and the average value of verification that is repeated
times is taken as the result. This prevents models from fitting only to specific data increases by testing different data multiple times for verification. The holdout method is a method adopted for
confirming model accuracy and is performed by dividing one dataset into training data and evaluation data. Training data and evaluation data must be separated because they affect the accuracy of the
model. In the holdout method, the data used as training data are never used as evaluation data. Similarly, the data used as evaluation data will not be used as training data. In this method, it is
necessary to be careful not to confuse the data. Linear regression is a type of regression analysis that predicts the value of a target variable based on the values of another explanatory variable [
]. Predicting one target variable with one explanatory variable is called single regression analysis [
]. The relationship between the two datasets that make the prediction can be expressed in the form of a linear equation (Equation (6)), which is the most basic model used in regression [
]. If a (slope) and b (Y intercept) are known,
can be predicted from
. The accuracy of the prediction is expressed by the correlation coefficient, i.e., the coefficient of determination [
]. Analysis with two or more explanatory variables, i.e., two or more dimensions, is called multiple regression analysis [
]. Multiple selection of appropriate variables makes it possible to set up a prediction formula, i.e., Equation (7), that is easy to calculate and has few errors.
y = a1 × 1 + a2 × 2 + a3 × 3 + a4 × 4 ⋯⋯ + b0
Furthermore, to improve the prediction performance of a model, it is necessary to minimize its generalization error, which can be divided into three parts, namely bias and variance, which are
minimizable errors, and noise, which is an irreducible error. This division is called bias–variance decomposition [
]. Minimizing bias requires learning more from the training data. However, if the bias is too small, the variance will become large. In contrast, if the variance is reduced too much, the bias will
increase. It is necessary to find an optimal solution that balances both. When minimizing two prediction errors, bias and variance, we simultaneously need to consider the tradeoff. Hence, this
situation is called the bias–variance dilemma [
]. The relationship between bias and variance follows the relationship that if one side wins, the other side will not win, i.e., the bias–variance tradeoff [
]. Bias in ML and statistics model prediction refers to the difference between the predicted and true values, that is the bias error, arising from incorrect model assumptions, where bias is the error
due to the simplification of the actual problem [
]. For example, linear regression simplifies the problem.
Variance refers to the spread of predicted values, that is the variance error, which arises from fluctuations in the training data [
]. If the model prediction has a lot of bias, the model cannot accurately represent the relationship between inputs and outputs. In other words, even training data cannot be predicted accurately;
this phenomenon is called underfitting [
]. Additionally, if the variance in the model prediction is too large, the model has learned noise in the training data; in this case, unknown data such as test data cannot be accurately predicted
and overfitting occurs [
]. For high accuracy, the variance must be kept low.
4. Ensemble Learning
Ensemble learning (EL) is a method of taking a majority vote and learning to improve the prediction ability for unlearned data by combining the data trained as individual learners [
]. It refers to training multiple models and outputting a predicted value by a majority vote or average. Two concepts, namely the bias and variance, are important in EL, which is learning to collect
information with low accuracy and increase accuracy. However, if the accuracy does not improve enough, the balance between bias and variance may be poor. The bias is simply the difference between the
actual and predictive values. The smaller the difference, the higher the accuracy and the more accurate is the prediction, i.e., low bias results in accurate values, resulting from inadequate
training. Variance, on the other hand, simply means the degree to which the predicted values are scattered. A state in which the degree of dispersion is high is called a high variance state, and the
accuracy is low in this case. The high variance is due to overfitting, which is caused by overtraining [
The model used for EL should be a weak learner, that is a learner with low accuracy when used alone, as the name suggests. In terms of the bias–variance tradeoff (there exists a tradeoff relationship
between model complexity and simplification), although the bias–variance tradeoff will fall, if the model is too simple, the generalization performance of the training data cannot be improved, and
often the bias is high and the variance is low. That is, a simple model that is not overfitted is obtained. In addition, the characteristics of EL are not only used simply for ML algorithms, such as
regression and classification, but also as an auxiliary method when obtaining learning coefficients for other ML algorithms. The effectiveness of EL is that it can take a majority vote using weak
learners. In the case of simple binary classification, when classification is performed with a normal classifier, if the classifier misclassifies, an incorrect result will be returned. However, since
EL employs a majority decision, if there are m learners, the answer is corrected as long as (
+ 1)/2 or more learners do not misjudge. For classification problems that are prone to mistakes, EL is very useful because it allows the results of multiple classifiers, such as neural networks, SVM,
and naïve Bayes, to be true [
]. Furthermore, assuming that each weak learner is statistically independent, and assuming that the error judgment probability of each weak learner is uniformly
, out of
weak learners, the probability of
false positives is as follows:
$P ( k ) = m C k θ k ( 1 − θ ) m − k$
To simplify the explanation, consider that if
weak learners make mistakes, the smaller the value of
, the lower the mislearning rate [
]. One of the main EL methods, bagging is a method of training that uses some of the information in the training data rather than all of it and then combines all the training results [
]. Each training can be computed independently, thereby allowing parallel processing. Bagging involves selecting weak learners and merging them into the final learner using the bootstrap method. The
basic bagging method is quite simple:
• Repeat the following steps B times.
□ Create a new dataset by m-time split sampling from the training data.
□ Build a weak learner h based on the divided dataset.
• Construct the final learning result using
times weak learners
Classification: H(x) = arg max |{i|hi = y}|
$Regression : H ( x ) = 1 2 B ∑ i = 1 B h i$
The formula for the part that finalizes the final learning result is given above in (10) and (11); in the case of classification problems, each weak learner is sorted so that the overall accuracy is
the highest [
]. On the other hand, for regression, each weak learner is normalized by the overall value. An example of a well-known ML algorithm that uses bagging is the random forest algorithm. Using a part of
the training data and merging it at the end is a common feature in bagging; boosting is the process of reusing previously used data to literally provide a boost. Hence, parallelism is not possible as
with bagging. Boosting is an EL algorithm that sequentially builds weak learners.
In EL, high accuracy is also achieved by using many weak learners (e.g., decision trees) that do not have high accuracy alone. Bagging uses both the weak learners in parallel and the overall results
of each model. In random forest, which combines bagging and decision trees, variance can possibly be suppressed by creating a number of decision tree models for the data and aggregating each result
to output the final result.
5. DeepSNAP: DL and EL
QSAR is a method for in silico prediction of chemical substances with physiological activity. In particular, it is one of the key components of integrated toxicology assessment systems, which are
highly likely to cause adverse effects to chemical structures and are useful in prioritizing and narrowing down chemicals requiring safety assessment. It can also contribute to alternatives to or
minimization of animal testing. In QSAR, the correlation between the structure and activity of compounds such as pharmaceuticals is determined quantitively as numerical values. These values are
handled via supervised learning that calculates feature values using chemical information for compounds and builds prediction models [
]. Molecular descriptors, which are the characteristic quantities that reflect the structure of a compound in QSAR, include fingerprints that determine the presence or absence of partial structures
and the measured and estimated values of the physicochemical properties of compounds [
]. Using the descriptors, QSAR models are constructed by several algorithms, including random forest, support vector machine, extreme Gradient Boosting (XGBoost), Bayesian networks, multiple linear
regression, polynomial regression (PLR), decision tree regression, and neural networks [
]. However, modeling in QSAR has some limitations related to the prediction performance, feature selections, calculation cost, etc. Therefore, Prof. Uesawa developed a new deep learning-based QSAR
system, DeepSNAP, which generates an omnidirectional snapshot depicting the three-dimensional (3D) structure of chemical compounds (
Figure 1
) [
]. In DeepSNAP, each chemical structure is optimized for steric conformation and portrayed to depict a ball-and-stick model with different colors representing different atoms. Using datasets of
approximately 9000 chemical structures in the simplified molecular input line entry system (SMILES) format and the corresponding activity scores, which represent the agonist or antagonist levels of
nuclear receptors and stress response proteins, from a database composed of high-throughput quantitative screening results, two datasets were prepared by defining “active” or “inactive” agonist or
antagonist activities. The aforementioned database was derived from the Toxicology in the 21st Century (Tox21) 10k library composed of chemicals from commercial sources, such as pesticides,
industrial chemicals, food additives, and drugs [
]. Then, the SMILES format was applied to a 3D conformational import to generate the SDF files of the chemical database. In this database, the 3D chemical structures of compounds were depicted as 3D
ball-and-stick models and captured continuously and automatically as snapshots with user-defined angle increments on the x-, y-, and z-axes. This was done to extract the spatial and temporal features
in the images. Finally, 256 × 256 pixel resolution PNG files (RGB) were saved and automatically split into three datasets, namely the training, validation, and test datasets. Prediction models were
created by using training and validation datasets, and the performance with the test dataset was examined. In addition, by optimizing DeepSNAP parameters such as zoom factor, atom size, and molecular
bond radius, the performance of the prediction models can be improved because of the powerful feature discrimination capability of DeepSNAP without the need for extracting and selecting features.
Furthermore, a combined system of DeepSNAP–DL with molecular descriptor-based methods was reported to construct regression models of rat clearance (CL), i.e., in vivo pharmacokinetic parameters among
the parameters of absorption, distribution, metabolism, and excretion. These models outperformed models based on molecular descriptor-based methods (
Figure 2
) [
]. DeepSNAP–DL predicted compound performance from correlations based on large experimental datasets of compounds. ML can perform a correct action in response to input data, collect feedback on how
well it performed, and improve it. Despite the significance of CL prediction in the field of drug discovery, few in silico prediction systems with sufficient performance have been established to
date. Therefore, this novel system using a combination of DeepSNAP and DL and the molecular descriptor-based methods will be a useful tool for CL prediction. In the ensemble method, a molecular
descriptor is selected by DataRobot, in which the feature importance of the descriptors calculated by the permutation importance of the prediction models is more than 0.5 times the highest average
effect [
]. Some available EL methods in QSAR that help improve the ML performance have also been reported [
6. Application of DL in New Drug Development and Medicine
By using EL consisting of DeepSNAP–DL with molecular descriptor-based methods, a prediction model system for CL, which can be regarded as the blood cleaning speed by the processed organ, with regard
to pharmacokinetics (PK) parameters in absorption, distribution, metabolism, and excretion, indicated a high prediction performance [
]. Furthermore, a highly accurate regression analysis was achieved by incorporating DeepSNAP–DL probability as an explanatory variable into the descriptors of the conventional ML methods [
]. In building a CL prediction model using the DeepSNAP–DL method, conditions including four angles (65°, 85°, 105°, and 145°), five learning rates (0.0000001 to 0.001), and five maximum epochs (15
to 300) were considered. As a result, the receiver operating characteristic area under the curve (ROC AUC) with the highest prediction performance was calculated (0.8974) under the conditions of a
learning rate of 0.000001, a maximum epoch of 300, and a 145° angle. In addition, the ROC AUC was calculated to be 0.943 in predictive model construction by the ensemble model method, in which the
average value of the predicted probabilities obtained by the DeepSNAP–DL method and the descriptor-based random forest method was used as the predicted probability of a new combination of prediction
models. Balanced accuracy (0.868), F-measure (0.845), and Matthew’s correlation coefficient (0.739) were used as the other evaluation indices. Prediction models are constructed by QSAR analysis for
various prediction targets such as toxicity and pharmacokinetic parameters, but the problem is that the prediction accuracy is insufficient. This research focused on rat CL and developed a new
prediction accuracy improvement method using the DeepSNAP–DL method with descriptors of conventional ML.
In addition, the application of artificial intelligence technology to the drug discovery field is expected to accelerate new drug development and realize innovative new drugs. However, because drugs
and their target proteins have different types of structures, it is difficult to predict drug–protein combinations that are effective in treating diseases with high-throughput and high-performance.
Therefore, based on the knowledge of chemistry and biology, molecular interaction prediction methods for drug development are being advanced by combining DL methods. In addition, because conventional
methods that do not use the 3D structure of drugs or proteins cannot specify the interaction site from prediction results, there is a problem in interpreting the results. However, new technologies
that can identify and visualize interaction sites are expected to accelerate the task of narrowing down new drugs from a large number of candidates. In addition, large-scale searches for drugs
candidates by computers are expected to lead to the development of innovative new drugs that cannot be reached with human knowledge and experience alone. In general, drugs can be expressed as atoms
and bonds as graph structure data, and proteins can be expressed as amino acid sequences as sequence structure data. Therefore, graph neural networks and CNNs, which are DL methods suitable for each
drug data and protein data, respectively, are applied to each data point, by which feature vectors that appropriately capture the properties of the drug and protein are calculated. By learning this
feature vector using large-scale data of drugs and proteins, it is possible to predict the presence or absence of interactions [
While DL can make fast and highly accurate predictions, it is difficult to interpret the prediction results. When dealing with chemical and biological data, it is necessary to judge the validity of
results by comparing the results of automatic predictions made by computers with the chemical and biological knowledge already possessed by humans. Therefore, ML techniques with easy-to-interpret
results are important.
7. Conclusions
A deep learning-based QSAR analysis, DeepSNAP, outperformed molecular descriptor-based conventional ML using 3D chemical structures and automatic extraction of features from image data. This system
is mainly constructed of a neural network with multilayers, and it is characterized by a high performance of classification and object detection. Furthermore, by combining DeepSNAP–DL and
conventional ML (such as EL), regression models can be constructed. The final prediction model obtained by combining multiple weak learning models is constructed. However, EL is not always a
versatile learning method. It is important to consider that classifiers are diverse and accurate when bagging is used.
Author Contributions
Y.U. edited the manuscript. Y.M. drafted the manuscript. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Conflicts of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Figure 1. DeepSNAP–DL. The input data of a chemical compound are converted into the SMILES format, and snapshots are produced from different angles as image data. These data are split into three
datasets, namely the training, validation, and test datasets, automatically. The prediction model is constructed by DL using these image data.
Figure 2. Ensemble learning with DeepSNAP–DL and descriptor-based ML, including random forest, SVM, neural networks, and XGBoost.
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Matsuzaka, Y.; Uesawa, Y. Ensemble Learning, Deep Learning-Based and Molecular Descriptor-Based Quantitative Structure–Activity Relationships. Molecules 2023, 28, 2410. https://doi.org/10.3390/
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Assignment 2
Read chapter 6.1-6.3 of the textbook (Smith).
1. Do problems 1-1 through 1-3 from the class notes. The answers to problems are in the back of notes. Try the problems before peeking at the answers. The goal here is to make sure you understand
the underlying concepts.
2. Plot the index of refraction of the glass N-SK15 over the range 0.45-0.65 mm for the Sellmeier equation. Do this using both Excel and MATLAB.
The dispersion coefficients for N-SK15 may be obtained from the Schott 2000 catalog (Excel worksheet)
3. How does Smith define aperture stop and field stop?
Maintained by John Loomis, last updated 5 Sep 2007 | {"url":"https://johnloomis.org/eop501/asgn/hw02.html","timestamp":"2024-11-13T01:54:11Z","content_type":"text/html","content_length":"1933","record_id":"<urn:uuid:adc69f3c-a497-4131-a4a2-f9f0f2071914>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00198.warc.gz"} |
Struct TileLayout
Represents the position and size of a tile grid for a recast/navmesh graph.
This separates out the physical layout of tiles from all the other recast graph settings.
Public Methods
GetTileBoundsInGraphSpace (x, z, width=1, depth=1)
Returns an XZ bounds object with the bounds of a group of tiles in graph space.
GetTouchingTiles (bounds, margin=0)
Returns a rect containing the indices of all tiles touching the specified bounds.
TileLayout (graph)
TileLayout (bounds, rotation, cellSize, tileSizeInVoxels, useTiles)
Public Variables
Voxel y coordinates will be stored as ushorts which have 65536 values.
Size of a tile in world units, along the graph's X axis.
Size of a tile in world units, along the graph's Z axis.
Size of bounds along the y axis in graph space (i.e.
How many tiles there are in the grid.
Size of a tile in voxels along the X and Z axes.
Transforms coordinates from graph space to world space. | {"url":"https://arongranberg.com/astar/documentation/dev_4_3_87_ed9ba20c0/tilelayout.html","timestamp":"2024-11-03T01:05:23Z","content_type":"text/html","content_length":"24967","record_id":"<urn:uuid:d56e1a0a-8288-49e4-b79f-406bde506eec>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00768.warc.gz"} |
Animated rotation in Maya
When you keyframe an object’s rotations, Maya calculates the object’s orientations between keys by interpolating the rotation values from one key to the next. In Maya, there are two methods of
rotation interpolation: Euler and Quaternion. For each animated rotation in your scene, you can specify a rotation interpolation method. The rotation interpolation method you choose for an animated
object determines how Maya calculates its rotations. For more information on Euler angles and Quaternions, see Euler angles, Quaternions, and Which type of interpolation is right for your animated
Euler rotation is the default method of rotation interpolation unless otherwise specified. You can set the default rotation interpolation method for new curves in the Maya window (in the section
under the category when you select ) or you can set the rotation interpolation method of existing rotation curves from the . See Set rotation interpolation for curves.
Euler angles
When interpolating the animated rotations of an object using the Euler method, Maya uses Euler angles to determine the object’s axis-specific orientations over time. Euler rotations are calculated
using three separate angles representing rotations about the X, Y, and Z axes, and an order of rotation.
The rotation order specifies the order in which an animated object is rotated about its X, Y, and Z axes. Changing an animated object’s rotation order changes its final orientation. You can specify
the order of rotation for an object by setting its attribute. For example, if you set an animated object’s to YZX, the object will first rotate in Y, then Z, and finally X. You can use the attribute
to match the rotation order of imported, animated objects to the co-ordinate systems (for example, XZY opposed to Maya’s default XYZ) of the 3D software packages from which they came. This is
important if you want the animated rotations of your imported objects to appear as intended.
In Maya, the default method of rotation interpolation is Euler.
There are 2 kinds of Euler rotation interpolation in Maya: Independent and Synchronized. You can set the Euler rotation interpolation type for your curves from the . See Change Rotation Interp.
For curves, interpolation is calculated from key to key on each individual curve, independent of the their neighboring rotation curves. Use curves when you want to keyframe a single rotation channel
or when you need to add additional keyframes (and thus detail) to a single rotation curve. curves are ideal for simple, animated rotations.
All the keyframes on curves are locked together in time. This means that if an object has rotation curves, interpolation is calculated from key to key on all of its rotation curves simultaneously.
Use curves when you want to keyframe multiple rotation channels (X, Y, and Z) or when you need to add additional keyframes (and thus detail) to all the rotation curves of an animated object. curves
are ideal for more complex animated rotations.
The main difference between and curves are their keyframes. For example, moving a key in time on an curve moves only the key on the curve, whereas moving a key in time on a curve will also move the
corresponding keys on the and curves. Similarly, if you key only the channel for an animated object, and the rotation interpolation type is set to , then only the channel is keyed. However, if the
rotation interpolation type is set to , then all three (, , and ) channels are keyed.
When Euler angles are used to interpolate the animated rotations of an object, the object’s orientation about its individual axes is evaluated one axis at a time. This is why Euler-angled rotation is
prone to artifacts such as gimbal lock and flipping. Gimbal lock occurs when rotations about a single axis cause unwanted rotations about complementary axes or when axes become coincident. Flipping
occurs when angles unexpectedly wrap around positive or negative 180 degrees during Euler-angled rotation interpolation between keyframes.
If gimbal lock or flipping occurs, you may be able to correct this behavior using the . For example, you can use the to normalize the mangled rotation curves from corrupted motion capture animation
data. You can access the from the menu in the or . For more information on the , see Euler angle filtering and filterCurve.
When should I use Euler rotation interpolation?
Use Euler rotation interpolation when you want specific control over the numerical values of your rotations and when you want smooth tangents for your rotation curves. In most cases, you should only
use Euler rotation interpolation for rotation animation curves that you need to manipulate extensively in the . Unlike Quaternion curves, Euler curves support all tangent types and their keys possess
tangent handles that let you easily tweak the curves.
Quaternions provide smooth interpolation of animated rotations and always produce the most efficient path between keyframes in comparison to Euler angles. Quaternions store the overall orientation of
an object rather than a series of individual rotations. This means that a single Quaternion stores the same amount of rotation data as three Euler angles. Since Quaternions store only orientation
values, they can be used to calculate the shortest rotation from one orientation to another.
When animating an object’s rotations with , Maya first stores the keyed orientation values for the object as Euler angles, converts them to Quaternions for interpolation, and then converts the
interpolated Quaternion rotation values back to Euler angles for display in the and .
In Maya, Quaternions are displayed as curves and values. When an object’s rotation curves are synchronized, the keyframes on its ,, and curves are locked together in time. When you add, delete, or
move a keyframe on one of the object’s rotation curves, the corresponding keys are also updated on the related rotation curves. This eliminates unexpected interpolation problems that can occur when
keyframes are deleted from one of the axes, or when keys are moved independently in time.
The tangent settings for Quaternion curves affect how an object’s animated rotations are interpolated. For more information on tangent types, see Graph Editor Tangents menu. Maya uses the following
types of tangents and interpolation to calculate the shortest rotation from one key to the next:
• Quaternion curves with clamped tangents use stepped interpolation
• Quaternion curves with linear tangents use spherical linear interpolation (also known as SLERP)
• Quaternion curves with spline tangents use cubic interpolation
For Quaternion curves with clamped tangents, Maya uses stepped interpolation. For Quaternion curves with linear tangents, Maya uses spherical linear interpolation—also known as SLERP—to calculate
the shortest rotation from one key to the next and produces an abrupt transition between keys. For Quaternion curves with spline tangents, Maya uses cubic interpolation to calculate the shortest
rotation from one key to the next and produces a smooth transition between keys.
When blending animation clips in the Maya^®^™, you can select one of the following types of Quaternion rotation interpolation: or . interpolation uses Quaternion interpolation to find the shortest
path between rotations from one clip to the next. interpolation uses Quaternion interpolation to find the longest path between rotations from one clip to the next. This path is in the opposite
direction of .
You can specify a clip blend’s Quaternion rotation interpolation type from the by setting the blend’s attribute.
When should I use Quaternions?
Use Quaternions when you want smooth interpolation between two keys. Quaternions produce the most efficient paths of interpolation and they do not generate artifacts such as gimbal lock and flipping.
Which type of interpolation is right for your animated rotations?
Each method of rotation interpolation has its advantages and disadvantages. It is up to you to select the type of interpolation that best suits your animation. See When should I use Euler rotation
interpolation? and When should I use Quaternions?.
Euler angles Quaternions
• Easily create simple rotations around a single, primary axis. • Smoothly interpolate rotation from one orientation to the next
Advantages • Support spin (rotation >360°) • Do not suffer from Gimbal Lock or flipping
• Treat IN and OUT tangents independently
• Do not support spin (rotations >360°)
• Are prone to Gimbal Lock and flipping • Curves are difficult to visualize
Disadvantages • Create rotations that are difficult to predict because interpolation is calculated separately for each • Do not support tangent handles
rotation axis • Tangents are difficult to view and edit in the and are not treated
Except where otherwise noted, this work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License | {"url":"https://download.autodesk.com/global/docs/maya2014/en_us/files/Animation_Basics_Animated_rotation_in_Maya.htm","timestamp":"2024-11-09T16:59:41Z","content_type":"text/html","content_length":"32764","record_id":"<urn:uuid:70062efb-a655-409c-9f4a-e25e246145ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00684.warc.gz"} |
Modeling of Traction Drive in Elastohydrodynamic Lubrication Contact and Traction Coefficient Enhancement Using Surface Texture Under High-Speed Conditions of up to 50,000 rpm
A high-speed electric motor with a small reducer that has high-power transmission efficiency can be used to realize a high-power-density powertrain system because electric motors can be miniaturized
to increase the rotational speed. A traction drive has low vibration noise due to its lack of meshing vibration, making it suitable as a transmission element for high-speed reducers. However, the
traction coefficient, which greatly affects transmission performance, decreases with increasing rotational speed. In this study, to increase the traction coefficient using a surface texture, a model
that takes into account transient temperature changes under high-speed conditions and the effects of micro-surface geometry was developed. The traction coefficient was measured using a high-speed
test machine capable of operating at a maximum speed of 50,000 rpm. The model was able to predict the experimental values with an error of at most 6%. The high-pressure rheological properties of the
oil were examined to develop design guidelines for the surface texture and a model was used to optimize the texture parameters. The designed texture was manufactured and evaluated. Experimental
results show that the traction coefficient can be improved by up to 19%.
1 Introduction
Due to increasing demand for reductions in carbon dioxide emissions, many conventional vehicles are expected to be replaced by hybrid, electric, or fuel-cell vehicles. Motor miniaturization is
important for reducing vehicle weight and maximizing occupant space. Motor output is the product of torque, which depends on the motor diameter and rotational speed, and thus motors can be made
smaller while maintaining output by increasing the rotational speed. For example, the Toyota Prius increased the motor rotational speed from 5000 rpm to 17,000 rpm when it was first released,
achieving a significant reduction in size while increasing output [1]. The European consortium DRIVEMODE has developed a unit for electric vehicles with a maximum rotational speed of 20,000 rpm [2].
Volk and Leighton have proposed a 30,000 rpm unit [3] and the Technical University of Munich studied the design of a 50,000 rpm unit [4]. Although increased losses, centrifugal force, vibration,
cooling, and other factors must be taken into account, higher rotational speeds have become a major trend.
To provide sufficient driving force to a vehicle with a high-speed motor, a combination of reduction gears is required. Traction drives are an alternative to gears for high-speed rotation. They
transmit power by shearing an oil film that forms between two rolling elements. Traction drives are suitable for high-speed rotation because they have no meshing vibration and low oil agitation
resistance. However, the traction coefficient (ratio of transmission force to contact force on the transmission portion) decreases with increasing rolling speed of a traction drive because an
increase in velocity causes the oil film to thicken in accordance with fluid lubrication theory [5], resulting in a decrease in shear strain rate and an increase in oil film temperature. To maintain
the required transmission force, the contact force must be increased to compensate for a decreased traction coefficient, resulting in lower transmission efficiency and the requirement of a larger
transmission. The author previously designed and built a high-speed traction drive testing machine (Fig. 1(a)) to investigate traction characteristics at high rotational speed, and experimentally
confirmed that the traction coefficient drops by up to 20% at 75 m/s compared to that at several meters per second (Figs. 1(b) and 1(c)) [6]. Such a large drop in the traction coefficient has a
significant impact on vehicle performance.
Various methods have been developed to increase the traction coefficient, including those that modify the traction oil and those that apply a surface texture. Tsubouchi and Hata derived an equation
that expresses the correlation between the molecular structure (e.g., stiffness and shape) and the traction coefficient [7]. One study attempted to calculate traction coefficients from molecular
structures using molecular dynamics simulation [8]. However, there is a trade-off between the traction coefficient of oil and low-temperature flowability. It is thus difficult to increase the
traction coefficient without reducing the low-temperature flowability of automotive oils, which are operated at extremely low temperatures. Nambu et al. applied microgrooves, a type of surface
texture, to the transmission surface and thinned the oil film to increase the strain rate and improve the traction coefficient (an improvement of approximately 5% was observed) [9]. However, direct
contact tends to occur at high loads, resulting in peeling, so the use of microgrooves is limited to low loads. The traction coefficient was measured at 15 m/s, so the effect at high speeds is
unclear. There is thus a need for a texture that effectively improves the traction coefficient at high speeds and that can withstand high loads.
To design a surface texture that has both a high traction coefficient and high strength under high-speed operation, a traction model that agrees with experiments at high speeds is required. Itagaki
et al. built a high-power traction drive testing machine and fit a nonlinear Maxwell model [10] to the obtained traction coefficients [11]. However, the model was fitted at peripheral velocities of
40 m/s or lower. The accuracy of the model at higher velocities has not been confirmed.
This study constructs a model for calculating the traction coefficient at high rotational speeds and uses it to create a texture that suppresses the decrease in the traction coefficient with
increasing rotational speed. The model considers the non-Newtonian viscosity of the oil, changes in oil properties due to an increase in temperature, and the oil film shape and pressure distribution
due to the surface microstructure. A test roller with a texture designed using the model is fabricated and its effectiveness is experimentally confirmed using a high-speed traction drive testing
2 Traction Drive Model
2.1 Overall Structure.
Although the Reynolds equation can be used to accurately calculate the oil film shape and the pressure distribution, it is difficult to apply it to estimate the traction force because it assumes
Newtonian viscosity. The nonlinear Maxwell model, which describes the non-Newtonian behavior of oil, is thus often used [
]. This model is employed in the present study.
Traction drives generate heat due to micro slip, which is referred to as creep. Changes in rheological properties due to an increase in temperature affect the oil film thickness and the traction
force. In order to account for the increase in temperature, traction calculations can be substituted into thermal elastohydrodynamic lubrication (EHL) equations to simultaneously calculate the oil
film shape, pressure distribution, and traction force [12]. However, this method is highly nonlinear and requires a long calculation time. The objective of this research is to optimize texture
design. Because many calculations are performed, a method with a small computational load is necessary. Therefore, the proposed method is based on an iterative calculation of the oil film thickness
and pressure distribution, a traction calculation, and a temperature calculation. These calculations are repeated until the results become converge. For application to texture design, the oil film
thickness and pressure are distributed based on EHL numerical calculations instead of using average values, and the traction force and increase in temperature at each location on the contact surface
are calculated.
Under the operating conditions used in this study, the calculated minimum oil film thickness is about 0.2–1 µm and the surface roughness of the roller is less than Ra = 0.04, so the film thickness
ratio is more than five. An analysis by Johnson et al. indicated that two rollers are sufficiently separated by an oil film if the film thickness ratio is greater than three [13]. Therefore, the
friction due to direct contact is not considered here and the oil film is assumed to transmit all power.
2.2 High-Pressure Properties of Oil.
It is difficult to theoretically determine the high-pressure properties of oil given in Eqs. (1) and (2). Therefore, experimental equations were derived based on the experimental values for the
operating conditions. The oil used in this study was Idemitsu Kosan’s traction fluid KTF-1, which was developed for automotive traction drive transmissions [14].
2.2.1 Viscosity.
Nakamura et al. measured viscosity up to pressures of 2 GPa using a diamond anvil cell [
]. Their values well agree with those measured using a falling-ball tester [
] and are close to those calculated based on free-volume theory [
]. Based on these measurements, the following experimental equation was derived. This equation is a modified version of an equation derived by Hata and Tamoto [
], in which the coefficients are changed to match the measured values. A comparison of the measured and calculated values is shown in Fig.
2.2.2 Shear Modulus.
Ohno et al. measured the shear modulus under high-pressure using a pressure vessel [
]. Their value significantly differs from that calculated from a traction curve obtained using McCool’s equation [
]. Under the assumption that the effect of the shear modulus of oil is small and the stiffness of the roller is dominant, Evans and Johnson derived a conversion equation for the slope in the linear
range [
]. This equation is used in the present study.
2.2.3 Eyring Stress.
Watanabe et al. performed traction tests under isothermal conditions [
]. They measured the bulk temperature of the roller and calculated the Eyring stress. A higher oil temperature and a lower pressure led to a higher Eyring stress. The following equation was developed
to represent this trend, where
are coefficients that are matched to the measured traction coefficients [
2.2.4 Limiting Shear Stress.
Attempts to determine the limiting shear stress from oil properties have been made (e.g., Ref. [
]). However, it is difficult to predict the limiting shear stress with sufficient accuracy. Therefore, the following equation is derived, where the coefficients
are matched to the measured traction coefficients.
2.3 Oil Film Temperature Calculation.
The oil film temperature
is the sum of the roller bulk temperature
, the increase in roller surface temperature Δ
, and the increase in oil film temperature Δ
The temperatures were calculated as described below.
2.3.1 Roller Bulk Temperature.
The roller bulk temperature is difficult to calculate because it depends on heat generation and cooling, heat capacity of the testing machine, heat conduction, and other factors. Therefore, the
roller surface temperature at 90 deg before the contact point was measured using a thermocouple for moving surfaces, as shown in Fig. 3. This temperature was taken as the bulk temperature.
2.3.2 Increase in Roller Surface Temperature.
The equation for a moving heat source given by Carslow and Jaeger [
] is used.
In general, Eq.
, which integrates Eq.
= 0 to
= ∞ to simplify the calculation, is used. However, at the high rotational speeds considered in this study, the heat source may pass a certain point on the roller before the temperature rises.
Therefore, the results of steady-state calculations were compared with those of transient calculations. The upper limit of time integration was defined as the time required to pass through one mesh,
i.e., the peripheral velocity divided by the length of the mesh. Figure 4 shows the temperature increase rate distribution in the rolling direction for a heat source at a single point on the surface
at a peripheral velocity of 75 m/s and a maximum Hertzian pressure of 3 GPa. In the unsteady-state calculations, the range over which the temperature increases is significantly narrower. When the
temperature rise is integrated over the entire contact surface using Eq. (10) under the condition, the unsteady-state temperature rise is 5 °C compared with a steady-state rise of 26 °C, which is a
difference of approximately five-fold, indicating that unsteady-state calculations are required at high speeds.
2.3.3 Increase in Oil Film Temperature.
Tomita et al. considered the shear position of the oil film in the thickness direction to be uniform in the viscous region and near the center in the plastic region, which is consistent with the
temperature measurement results [24]. This consideration is used in the present study.
be the heat generated per unit volume of the oil film in the viscous region. Assuming that the oil film is thin, and thus has a small calorific value and can be treated as stationary, then
For simplicity, let the temperatures of both rollers be equal and ΔT[f] = 0 at z = 0 and z = h. Then
The mean and maximum values are respectively
For simplicity, let
/2 be the shear position in the plastic region.
In the viscoelastic range, if the elastic component is considered to be elastic deformation, only the viscous component contributes to heat generation. Because the second term on the left-hand side
of Eq.
is the viscous component of the strain rate, the viscous component of the slip rate is
All shear components in the plastic regions generate heat, and thus
The thermal conductivity of oil under high pressure is calculated using an experimental equation [25] derived from Larsson’s equation [26].
The average oil film temperature is used in the EHL calculation and the maximum oil film temperature is used in the traction calculation.
2.4 Numerical Calculation Scheme
2.4.1 Calculation Method.
The overall flow is shown in Fig. 5, where the oil film shape and pressure distribution were obtained from the EHL calculation, the traction force was calculated by substituting the obtained oil film
shape and pressure distribution into the Maxwell model, the oil film temperature was calculated from the obtained traction force, and the EHL calculation was performed by modifying the temperature.
The calculations were repeated until the traction force and temperature converged. The calculation methods and modifications made to stabilize and speed up the calculations are described below.
2.4.2 Elastohydrodynamic Lubrication Numerical Calculation.
Ichimaru’s method [25] was adopted in the present study. In the region where an oil film forms, the elastic deformation equation, high-pressure viscosity equation, and high-pressure density equation
are substituted into the difference Reynolds equation to obtain the pressure distribution by the relaxation method. Because the Reynolds equation is not valid in a region where the oil film thickness
is zero or negative, this is the direct contact region, and the pressure is obtained by setting the left-hand side of the elastic deformation equation to zero. In cavitation regions, where the
pressure is negative, the pressure is set to zero. Methods for speeding up the calculations are described in the Appendix.
2.4.3 Traction Force and Oil Film Temperature.
The velocity in Eq.
is decomposed into the
directions and replaced by the derivative of the displacement. The strain rate is defined as the slip rate divided by the oil film thickness, which can be written as follows:
This equation is integrated using the fourth-order Runge–Kutta method to obtain the shear stress. However, if τ > τ[c], Eq. (2) is used. The shear modulus G, Eyring stress τ[0], and critical shear
stress τ[c] are calculated by considering the pressure and temperature at each node (Eqs. (6)–(8)).
The heat generation rate is determined from the obtained shear stress and slip rate, and the temperature of the oil film is calculated. This temperature is substituted into the EHL calculation to
obtain the pressure distribution and oil film shape, which are then substituted into the traction calculation. This process is repeated until convergence is reached. However, the Maxwell model
contains a hyperbolic function in the viscosity term, which is highly nonlinear, and the temperature increase equation switches among elasticity, viscosity, and plasticity, causing the calculation to
easily diverge, especially under high-speed conditions, which is the case considered in this study. Therefore, a Gaussian filter is applied to the calculated increase in temperature to smooth it out
and improve convergence. The standard deviation is set at 0.5.
2.5 Comparison With Experimental Values.
The experimental results measured using the high-speed traction testing machine shown in Fig. 1(a) are compared with the calculation results obtained using the model developed in this study. The
experimental and calculated values under typical conditions are shown in Figs. 6(a)–6(c) and the calculated coefficients of the experimental equations for the Eyring stress and limiting shear stress
are shown in Table 1. The calculations generally agree with the experiments regardless of the rotational speed and pressure. Figure 6(d) plots the maximum traction coefficient versus the rotational
speed. The difference between the calculations and experiments is about 6% at maximum. The results indicate that the calculation model developed in this study can predict the traction coefficient
with high accuracy.
3 Increase of Traction Coefficient Using Texture
3.1 Overview of Measures Used to Increase Traction Coefficient.
To reduce the decrease in the traction coefficient with increasing speed, a texture was designed using the developed computational model. In the design of the texture, the traction model and the
high-pressure rheological properties of the oil were taken into consideration to determine a design policy to increase the traction coefficient.
3.1.1 Policy and Method Considerations.
For simplicity, considering only motion in the
-direction, Eq.
, and
are values determined for the oil, and
is an operating condition (it is thus excluded here). The traction coefficient can be increased by increasing the sliding velocity Δ
and decreasing the oil film thickness
, which are the remaining terms. However, as shown in the following equations obtained from Eqs.
, the increase in oil film temperature is proportional to the sliding speed, so increasing the sliding speed reduces the traction coefficient, as described below.
Ohno et al. showed that the state transition and maximum traction coefficient for oil are determined by the temperature and pressure of the oil based on measurements of oil properties under high
pressure and the results of a two-cylinder traction test [19]. A higher pressure and a lower temperature lead to a higher traction coefficient. In particular, when the oil transitions from
viscoelastic to elastoplastic, it becomes solid and the traction coefficient significantly increases. Therefore, from Eq. (25), the temperature decreases as the oil film thickness h decreases. In
addition, from Eq. (24), the shear force increases as the oil film thickness h decreases, resulting in a higher traction coefficient.
To increase the pressure, the radius of curvature of the roller can be reduced and the Hertzian contact pressure can be increased; however, this reduces the strength and life of the roller.
Therefore, the aim is to equalize the pressure and increase the pressure in the low-pressure region within the contact surface without increasing the maximum pressure (Fig. 7(a)), and to increase the
pressure at low loads without increasing the pressure at high loads (Fig. 7(b)). Table 2 summarizes the aforementioned studies. It shows that thinning the oil film and increasing the pressure in the
low-pressure area effectively increase the traction coefficient.
Table 2
Aim Means Reaction
Increases shear speed Thins oil film
Increases slip speed Temperature rising
Decreases temperature Thins oil film
Reduces contact area Strength falls
Increases pressure Averages pressure distribution
Rises pressure at only low load
Aim Means Reaction
Increases shear speed Thins oil film
Increases slip speed Temperature rising
Decreases temperature Thins oil film
Reduces contact area Strength falls
Increases pressure Averages pressure distribution
Rises pressure at only low load
3.1.2 Examination of Specific Measures.
To make the oil film thinner, fine grooves parallel to the rolling direction are formed on the roller surface to increase the flow coefficient [27]. However, in calculations performed at 43,750 rpm
with various groove depths, widths, pitches, and angles, no combination could increase the traction coefficient. This is likely due to the oil film becoming thinner in flat areas but thicker in
grooves, which reduces the traction coefficient by decreasing the pressure.
Next, consider the use of high-pressure in low-pressure regions. By changing the point contact to line contact, the pressure can be averaged in the direction perpendicular to the axis. The
logarithmic function geometry used for cylindrical roller bearings can make the pressure distribution nearly constant in the perpendicular direction without generating edge loading [28].
In Hertzian point contact, when the radius of curvature of the roller is constant, the pressure is proportional to the cube root of the load. In contrast, as shown in Fig. 8(a), if the radius of
curvature increases moving away from the center of contact, the degree of increase in the contact width increases as the load increases, and the slope of pressure versus load can be smaller than the
1/3 power. Therefore, if the radius of curvature at the center of contact is kept small, the characteristics are expected to be those shown by the orange line in Fig. 7(b). A shape with multiple
curved surfaces in multiple steps in the direction of load is also possible. At low loads, a tip with a small radius of curvature makes contact to increase the pressure. As the load increases, the
tip elastically deforms to contact surfaces with a large radius of curvature to suppress the pressure (Fig. 8(b)). Combined with the logarithmic shape described above, an increase in the traction
coefficient in both the low- and high-load ranges is expected.
3.2 Design of Texture Specifications.
The proposed shape of the texture for increasing pressure in the low-pressure area, as described in the previous section, was examined in terms of the test roller specifications for evaluation in a
high-speed traction testing machine.
3.2.1 Multi-Stage Curved Surface Shape.
The texture consists of three convex surfaces, as shown in Fig.
, with the center raised relative to the surfaces on the two sides. The central surface has a logarithmic function shape expressed by the following equation:
are parameters.
corresponds to the height of the projection, and
corresponds to the half-width of the projection. The surfaces on both sides have logarithmic function shapes with the same parameters, but offset by
in the
direction and
in the
The pressure at high and low loads is used as the evaluation index. All combinations are calculated for the parameters shown in Fig. 9. A smooth roller with a constant crowning radius of 23.5 mm is
used for comparison. The Hertzian maximum pressure is about 1.5–4.0 GPa at pressing forces of 291–5510 N. With the texture, the pressure at a maximum load of 5510 N should be less than 4 GPa and the
pressure at a minimum load of 291 N should be as high as possible.
Figure 10(a) shows the results of all calculations, where the horizontal axis represents the pressure at high load and the vertical axis represents the pressure at low load. The data points for the
textured roller are located to the upper left of those for the smooth roller, which is the desired effect. Table 3 shows the results of the pressure and traction coefficient calculations for various
parameters, where the pressure at high loads is near 4 GPa. The highest traction coefficient (an increase of 19%) is obtained for A = 600 and K = 16. A = 500 and K = 10 also increase the traction
coefficient by 19%, with the advantage that the small K (i.e., low peak) makes processing easier. The latter parameters were thus selected.
Table 3
A (μm) K (μm) D (μm) P[max] at 291 N (GPa) P[max] at 5510 N (GPa) Traction coefficient at 291 N
4 9 1.43 3.95 —
500 8 9 1.69 3.93 —
10 7 1.82 3.96 0.075
12 5 1.86 3.98 0.069
600 14 9 1.79 3.95 0.075
16 7 1.87 3.96 0.076
R23.5 1.48 3.94 0.063
A (μm) K (μm) D (μm) P[max] at 291 N (GPa) P[max] at 5510 N (GPa) Traction coefficient at 291 N
4 9 1.43 3.95 —
500 8 9 1.69 3.93 —
10 7 1.82 3.96 0.075
12 5 1.86 3.98 0.069
600 14 9 1.79 3.95 0.075
16 7 1.87 3.96 0.076
R23.5 1.48 3.94 0.063
Figure 10(b) shows the calculated pressure versus load and traction coefficient. There was concern that the pressure would drop when the projections on the two sides begin to make contact, but the
pressure increases over the entire load range, and the traction coefficient also increases. At high loads, the traction coefficient increases even though the pressure remains the same. This is due to
the logarithmic shape averaging the pressures and the increase in the pressure in the low-pressure region on the contact surface.
3.2.2 Gradual Increase of Radius of Curvature Shape.
In the axial section through the center of the contact point, the radius of curvature
is constant when the distance from the center is less than
, and it increases logarithmically beyond that distance. The shape is expressed by the following equation:
At |
| ≤
The specifications for the texture and the calculated pressure and traction coefficient are shown in Table 4. For a roller with a constant crowning radius (23.5 mm), the pressure at low loads is
increased and the pressure at high loads is maintained, resulting in a 5–8% increase in the traction coefficient.
Table 4
R[0] (mm) y[0] A (μm) K (μm) P[max] at 291 N (GPa) P[max] at 5510 N (GPa) μ at 291 N μ at 5510 N
20 0.4 900 1000 1.54 4.02 0.066 0.082
17 0.4 1300 5000 1.63 4.03 0.068 0.080
23.5 — — — 1.48 3.94 0.063 0.083
R[0] (mm) y[0] A (μm) K (μm) P[max] at 291 N (GPa) P[max] at 5510 N (GPa) μ at 291 N μ at 5510 N
20 0.4 900 1000 1.54 4.02 0.066 0.082
17 0.4 1300 5000 1.63 4.03 0.068 0.080
23.5 — — — 1.48 3.94 0.063 0.083
3.3 Confirmation of Texture Effect.
The effect of a multi-stage logarithmic function shape, which is effective at increasing the traction coefficient, was experimentally confirmed using a high-speed traction drive testing machine. The
texture was machined onto the low-speed rollers. The high-speed rollers were smooth. A photograph of the prototype roller and the designed and measured geometry of the texture are shown in Fig. 11.
The measured convexities are somewhat flat, but the peak widths and height offsets are approximately equal to the designed values.
The traction curves obtained at high rotational speeds and low load (maximum pressure of 1.5 GPa without texture) are shown in Fig. 12(a). The traction coefficient for the textured rollers is greatly
increased compared with that for the smooth rollers. Figure 12(b) plots the maximum traction coefficient versus rotational speed. At a low load (291 N), the texture increases the traction coefficient
over the entire speed range, with an increase of up to 19%. At a load of 689 N (2.0 GPa without texture), there is no difference between the cases with and without texture (i.e., there is no
improvement effect). This may be because almost the entire area of the oil film had transitioned from viscoelastic to elastoplastic, as predicted by Fig. 10(b).
Texture shape measurements made before and after the experiment are shown in Fig. 13. There is almost no change in shape. The amounts of plastic deformation and wear are considered to be extremely
small because the experiments were conducted for a short period of time (approximately 2–3 min per condition, or several tens of minutes in total). Longer operation time would change the shape of the
wheel, resulting in a decrease in the traction coefficient and a decrease in the life of the wheel. Confirmation of the change in the traction coefficient over time, for example during
100,000–200,000 km of driving (the life of an automobile), is a major issue for future research.
4 Conclusion
To suppress the reduction of the traction coefficient at high rotational speeds, a model for predicting the traction coefficient was developed and used to design a texture, which was then evaluated
experimentally. In summary:
1. A traction calculation model was created by coupling the nonlinear Maxwell model, EHL calculation, and oil film temperature calculation.
2. The model takes into account the pressure distribution, oil film shape, and temperature distribution, and can be used for texture design.
3. The calculated traction coefficients were in good agreement with the experimental results.
4. The nonlinear Maxwell model and the high-pressure rheological properties of oil were used to formulate design guidelines for textures.
5. The designed prototype texture greatly increased the traction coefficient.
This research was supported by the “Project for Building Simulation Platforms to Accelerate Development of Next-Generation Vehicles” (TRAMI-Transmission Research Association for Mobility Innovation)
grant of 2020.
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.
Appendix: Speeding up of Elastohydrodynamic Lubrication calculations
The Reynolds equation is computed using the relaxation method, in which the pressure at a node is calculated from the values of the four surrounding nodes in the previous calculation cycle.
Therefore, propagation of the effect of pressure modification is only transmitted to one neighboring node in one calculation, which requires an extremely large number of calculations when the mesh is
fine. The multigrid method [29], which alternates between coarse and fine mesh calculations, is considered effective for this problem. However, if the texture cannot be represented by a coarse mesh,
errors may occur when switching mesh size, increasing the number of calculations, and diverging the viscosity term in the highly nonlinear Maxwell model. Therefore, for a smooth shape with no
texture, a coarse mesh is used to obtain a rough pressure distribution and oil film shape, and then a fine mesh is used to obtain the pressure distribution and oil film shape, taking texture into
account. The traction, temperature, and EHL calculations are repeated using only the fine mesh.
The oil film shape equation is discretized as follows, where the elastic deformation term is the convolution sum
is the influence coefficient map (kernel) for elastic deformation, calculated using Love’s analytical formula [
]. The number of calculations is
) for
nodes, which accounts for the majority of the total calculation time. The multilevel multi-integration method [
] is often used to speed up calculations. The number of calculations is
). For example, for a 500 × 500 mesh, the number of calculations is reduced by a factor of 20,000. However, the calculation procedure is extremely complex and cumbersome. In addition, when the mesh
size is changed, the interpolation approximation causes an error in the texture shape.
Liu took advantage of the fact that a Fourier transform of the convolutional sum is equal to the product of the Fourier transforms of each element, and used a fast Fourier transform to speed up the
calculation of elastic deformation [
]. The number of calculations is also
). Compared to the multilevel multi-integration method, the computation is simpler and there is no risk of interpolation errors. This method is used in the present study. Liu’s elastic deformation
kernel has a wrap-around order, which makes the linear convolution equivalent to the circular convolution. However, the procedure is rather complicated. In this study, a simpler method is used. If
the mesh size for the pressure is (
), the size of the kernel is (2
+ 1, 2
+ 1). Now, both the pressure and the kernel are extended to (3
+ 1, 3
+ 1) by adding zeros (Fig.
). The following equation holds for expanded pressure
and kernel
The center of the calculated result is the original convolution sum.
, “
Technical Transition of Motors for Hybrid Vehicles
J. Inst. Electr. Eng. Jpn.
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, and
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Drivemode-High Speed Electric Drivetrain
CTI Symposium 2019
Berlin, Germany
Dec. 10
, pp.
, and
, “
Integrated Development Program for Electrified Drivetrains
ATZ Worldw.
), pp.
, and
, “
Design of a Hyper-High Speed Powertrain for EV to Achieve Maximum Ranges
CTI Symposium 2018
, pp.
B. J.
, and
, “
Isothermal Elastohydrodynamic Lubrication of Point Contacts: Part III—Fully Flooded Results
ASME J. Lubr. Technol.
), pp.
, “
Measurement of the Transmission Performance of Traction Drives and Gears (Comparison of Performance at 50,000 rpm)
Trans. JSME
), p.
, and
, “
Quantitative Correlation Between Fundamental Molecular Structures of Traction Fluids and Their Traction Properties
J. Jpn. Soc. Tribol.
), pp.
, and
, “
Molecular Dynamics Simulations of Elastohydrodynamic Lubrication Oil Film
Lubr. Sci.
), pp.
, and
, “
Increase of Traction Coefficient Due to Surface Microtexture
Tribol. Lett.
), pp.
K. L.
, and
J. L.
, “
Shear Behavior of Elastohydrodynamic Oil Films
Proc. R. Soc. Lond. A
), pp.
, and
, “
Development of a High-Power Two-Roller Traction Tester and Measurement of Traction Curves
Tribol. Online
), pp.
H. C.
B. B.
, and
C. H.
, “
Influences of Solid and Lubricant Thermal Conductivity on Traction in an EHL Circular Contact
Tribol. Int.
, p.
K. L.
J. A.
, and
S. Y.
, “
A Simple Theory of Asperity Contact in Elastohydro-Dynamic Lubrication
), pp.
, and
, “
Performances, and Characteristics of Idemitsu Traction Oil
Idemitsu Tribol. Rev.
, pp.
, and
, “
High Pressure Viscosity Measurements of Traction Oils up to 2 GPa at up to 200°C
J. Jpn. Soc. Tribol.
), pp.
, “
A Study on Temperature and Pressure Dependence of Eyring Stress in Lubricant Traction
Proceedings of JAST Tribology Conference 2004
Tottori, Japan
Nov. 10
, pp.
, “
High Pressure Behavior of Toroidal CVT Fluid for Automobile
Tribol. Int.
), pp.
, and
, “
Prediction of High Pressure Viscosity of Various Lubricants
J. Jpn. Soc. Tribol.
), pp.
, and
, “
High Pressure Rheology and Traction Characteristics of Traction Oil
J. Jpn. Soc. Tribol.
), pp.
J. I.
, and
, “
A Performance Evaluation Tool for Traction-Transmitting Partial EHD Contacts
Trans. ASLE
), pp.
C. R.
, and
K. L.
, “
Regimes of Traction in Elastohydrodynamic Lubrication
Proc. Inst. Mech. Eng.
), pp.
, and
, “
A Study on Traction Characteristics of Toroidal CVT for Automobile: 1st. Report, Experimental Analysis by Roller Test
Proceedings of Annual Meeting of JSME
, pp.
H. S.
, and
J. C.
Conduction of Heat in Solids
2nd ed.
Oxford Science Publications
Oxford, UK
, and
, “
Examination of Maximum Traction Coefficient Prediction Method (Second Report)
Proceedings of JSAE Annual Congress
), p.
, and
, “
A Transient EHL Analysis for Point Contacts With Consideration of Direct Contacts
Trans. Jpn. Soc. Mech. Eng., Ser. C
), pp.
, and
, “
Lubricant Thermal Conductivity and Heat Capacity Under High Pressure
Proc. Inst. Mech. Eng., Part J
), pp.
, and
H. S.
, “
Application of Average Flow Model to Lubrication Between Rough Sliding Surfaces
ASME J. Lubr. Technol.
), pp.
, and
, “
Logarithmic Profile of Rollers in Roller Bearing and Optimization of the Profile
Trans. Jpn. Soc. Mech. Eng., Ser. C
), pp.
C. H.
Multilevel Solution of the EHL Line and Point Contact Problems
Twente University
Enschede, Netherlands
A. E. H.
, “
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Philos. Trans. R. Soc., A
), pp.
, and
, “
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), pp.
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Some Basic Ideas of Algorithms
This set of notes on algorithms is not meant to be comprehensive or complete. These notes are being used as a skeleton framework.
There are many useful books to learn about algorithms from a utilitarian point of view. I have listed a few in the references section.
Numerical recipes^1 is a very comprehensive book that I used during my PhD. It covers almost all the algorithms you need for scientific computing.
Grokking Algorithms^2 is another good book to learn the basics of algorithms. It is barely entry level but is fun to read.
An Outline
Data Structure
mind the data structure
Basics of MapReduce
Planted: by L Ma;
Additional Double Backet Links:
L Ma (2018). 'Some Basic Ideas of Algorithms', Datumorphism, 03 April. Available at: https://datumorphism.leima.is/wiki/algorithms/algorithms-basics/. | {"url":"https://datumorphism.leima.is/wiki/algorithms/algorithms-basics/?ref=footer","timestamp":"2024-11-02T13:41:50Z","content_type":"text/html","content_length":"113875","record_id":"<urn:uuid:a5be7e19-55dd-46c7-8651-16b24b4a65cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00665.warc.gz"} |
A bottle of champagne to prove the stability of matter
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Lieb and Thirring clean up my matter stability proof
A bottle of champagne to prove the stability of matter
Freeman Dyson Scientist
• 1
• ...
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• 10
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• ...
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This was the result of a bet, and it was, I think David Ruelle and Michael Fisher, who had offered a bottle of vintage champagne to anybody who could prove the stability of matter. And so Andrew
Lenard was here in Princeton at the Plasma Physics lab and he got interested in that question, and he made considerable progress, actually, with proving it. The problem is, if you have an ideal
description of matter it just consists of ignoring nuclear forces, ignoring gravitation, just ignoring magnetic fields - it's the simplest description of matter; it's a set of positive and negative
point charges held together with coulometer actions only, non-relativistic. So it's a system of n positive and n negative charges, or you can have different numbers if you like, held together with
coulomb forces. The question is: what is the binding energy of such a system as a function of the number of particles? Does it go linearly with n or does it go with a higher power? If it goes - this
is all quantum mechanics - if it goes linearly with n then the matter is stable, it means that the binding energy per particle is fixed; that's the situation we live with and everybody believes that
that is true, but there's no mathematical proof of it; but if it went with a higher power of the number of particles it would mean that every piece of matter would be a high explosive. When you
joined pieces of matter together you'd get an enormous binding energy just by combining, so everything, every chunk of material, would be a hydrogen bomb essentially. So it was a problem that one
ought to be able to understand. And the proof of it turned out to be remarkably difficult, but Lenard and I worked on it and managed actually to prove it, but by a method that was extraordinarily
complicated and difficult and just opaque. Both Lenard and I are mathematical manipulators and not really physicists, so we didn't understand the physics of the problem, we merely, just by brute
force, managed to prove the stability. And the interesting physical fact that emerged was that it works only if the particles have an exclusion principle for one sign of the charge, for at least one,
either the negative or the positive and, in fact of course in the real world, the negatives, the electrons, have to have an exclusion principle in order for matter to be stable. If you didn't have an
exclusion principle, if both the electrons and the positive charges are bosons, then matter is unstable, and there again we had a conjecture which we couldn't prove, that the binding energy for n
particles should go with the seven fifths power of n in the case of bosons, which again Elliot Lieb finally proved rather recently, and we were only able to prove that it wasn't more, it wasn't
stronger than five thirds, we never could get the seven fifths, which is the correct power.
Freeman Dyson (1923-2020), who was born in England, moved to Cornell University after graduating from Cambridge University with a BA in Mathematics. He subsequently became a professor and worked on
nuclear reactors, solid state physics, ferromagnetism, astrophysics and biology. He published several books and, among other honours, was awarded the Heineman Prize and the Royal Society's Hughes
Title: A bottle of champagne to prove the stability of matter
Listeners: Sam Schweber
Silvan Sam Schweber is the Koret Professor of the History of Ideas and Professor of Physics at Brandeis University, and a Faculty Associate in the Department of the History of Science at Harvard
University. He is the author of a history of the development of quantum electro mechanics, "QED and the men who made it", and has recently completed a biography of Hans Bethe and the history of
nuclear weapons development, "In the Shadow of the Bomb: Oppenheimer, Bethe, and the Moral Responsibility of the Scientist" (Princeton University Press, 2000).
Tags: Princeton University, Princeton Plasma Physics Laboratory, Michael E Fisher, David Ruelle, Andrew Lenard, Elliot Lieb
Duration: 3 minutes, 41 seconds
Date story recorded: June 1998
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What is the axis of symmetry and vertex for the graph f(x)= 2x^2 - 11? | HIX Tutor
What is the axis of symmetry and vertex for the graph #f(x)= 2x^2 - 11#?
Answer 1
The axis of symmetry for the graph of ( f(x) = 2x^2 - 11 ) is the vertical line represented by the equation ( x = \frac{-b}{2a} ), where ( a ) is the coefficient of the ( x^2 ) term (in this case, (
2 )) and ( b ) is the coefficient of the ( x ) term (which is ( 0 ) since there's no ( x ) term). So, the axis of symmetry is ( x = \frac{0}{2(2)} = 0 ).
To find the vertex, plug the value of ( x ) from the axis of symmetry equation into the original function to find the corresponding ( y ) value. So, ( f(0) = 2(0)^2 - 11 = -11 ). Therefore, the
vertex is at the point ( (0, -11) ).
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Answer 2
Vertex$\to \left(x , y\right) = \left(0 , - 11\right)$
First write as #" "y=2x^2+0x-11#
Then write as #" "y=2(x^2+0/2x)-11#
This is part of the process for completing the square.
I have written this format on purpose so that we can apply:
The value for #x_("vertex")= (-1/2)xx(+0/2)=0#
So the axis of symmetry is the y-axis.
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Answer 3
Axis of symmetry is $y$-axis
Vertex is at $\left(0 , - 11\right)$
From the equation given it is obvious that vertex is at # x=0 ,y=-11#.
and the axis of symmetry is #x=0# that is the #y#- axis.
There is no #x# term so the graph has not moved left or right, only down #11# units.
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Answer 4
The axis of symmetry for the graph of ( f(x) = 2x^2 - 11 ) is the vertical line that passes through the vertex. To find the axis of symmetry, you use the formula ( x = -\frac{b}{2a} ), where ( a ) is
the coefficient of the ( x^2 ) term (in this case, ( 2 )) and ( b ) is the coefficient of the ( x ) term (which is ( 0 ) because there's no ( x ) term). Substituting these values into the formula, we
get ( x = -\frac{0}{2(2)} = 0 ). Therefore, the axis of symmetry is ( x = 0 ).
To find the vertex, you substitute the value of the axis of symmetry into the original function to find the corresponding ( y ) value. So, ( f(0) = 2(0)^2 - 11 = -11 ). Therefore, the vertex is at
the point ( (0, -11) ).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Subsystem has no wave function
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• Thread starter mieral
• Start date
In summary, if you have two subsystems that are maximally entangled, then there can be a wave function for the composite system, but not for each subsystem.
It is mentioned that subsystems don't have a wave function, in general. If two subsystems are entangled, there can be a wave function for the composite system, but not for each subsystem.
Let's say you have two entangled photon pair.. it has a wave function but not for each separate photon.. but then a photon as quantum stuff always have a wave function.. you can entangled one of the
photons to another quantum object.. so how can you state the subsystem has no wave function?
I wouldn't say that a system has a wavefunction or doesn't have a wavefunction - the wavefunction is just a mathematical formalism that we developed to describe a quantum mechanical system. However,
it happens that the wavefunction formalism can only be used to describe what we call pure states. It cannot be used to describe mixed states, which behave as classical mixtures (i.e. not
superpositions) of pure states.
Mixed states occur in a variety of contexts, such as coupling to a noisy environment etc. Let's consider your example of a maximally entangled pair of qubits that can be either spin up or spin down.
If we consider the pair as a whole, then it exists in a pure state. However, if we look at only one qubit and ignore the existence of the other, when we make many measurements on an ensemble of
identically prepared pairs, we will find that qubit to be spin up 50% of the time and spin down the other 50% of the time. Note however that this is not a superposition of up and down. In order to
describe this classical mixture behaviour, the wavefunction formalism therefore fails, and we have to introduce the notion of a density matrix or operator to describe the system.
Fightfish said:
I wouldn't say that a system has a wavefunction or doesn't have a wavefunction - the wavefunction is just a mathematical formalism that we developed to describe a quantum mechanical system.
However, it happens that the wavefunction formalism can only be used to describe what we call pure states. It cannot be used to describe mixed states, which behave as classical mixtures (i.e. not
superpositions) of pure states.
Mixed states occur in a variety of contexts, such as coupling to a noisy environment etc. Let's consider your example of a maximally entangled pair of qubits that can be either spin up or spin
down. If we consider the pair as a whole, then it exists in a pure state. However, if we look at only one qubit and ignore the existence of the other, when we make many measurements on an
ensemble of identically prepared pairs, we will find that qubit to be spin up 50% of the time and spin down the other 50% of the time. Note however that this is not a superposition of up and
down. In order to describe this classical mixture behaviour, the wavefunction formalism therefore fails, and we have to introduce the notion of a density matrix or operator to describe the
But if we used Bohmian Mechanics where the particles really have a wave function. Then when we just looked at the subsystem of an entangled system, how do you force it to have a wave function? and
what kind of wave function would it be (theoretically Bohmian wise)?
Unfortunately I'm not familiar enough with Bohmian mechanics to be able to describe subsystems of an entangled system within that framework. Perhaps
can help us out here?
Fightfish said:
Unfortunately I'm not familiar enough with Bohmian mechanics to be able to describe subsystems of an entangled system within that framework.
Bohmian mechanics ascribes a wave function only to the state of the universe, which determines the dynamics of all Bohmian particles.
It does not say anything at all about how the Bohmian dynamics of the universe relates to a putative Bohmian dynamics of a subsystem.
The subsystem is described statistically by ordinary quantum mechanics only, which is claimed to be identical with the common formalsim, since the effect of the environment has been traced out.
Bohmian mechanics can define a wave function of the subsystem as a conditional wave function. For instance, if ##\Psi(x_1,x_2,t)## is a full wave function, then the conditional wave function is
where ##X_2(t)## is the Bohmian trajectory. In general, ##\psi_c(x_1,t)## does not satisfy the Schrodinger equation. This is actually good, because it can explain the effective "collapse" of the
conditional wave function.
Furthermore, when the entanglement is absent, i.e. when
where ##f(t)=\psi_2(X_2(t),t)##. The factor ##f(t)## does not influence the trajectory ##X_1(t)##, so the physical content of ##\psi_c(x_1,t)## does not differ from the physical content of standard
subsystem wave function ##\psi_1(x_1,t)##.
Demystifier said:
Bohmian mechanics can define a wave function of the subsystem as a conditional wave function. For instance, if ##\Psi(x_1,x_2,t)## is a full wave function, then the conditional wave function is
where ##X_2(t)## is the Bohmian trajectory. In general, ##\psi_c(x_1,t)## does not satisfy the Schrodinger equation. This is actually good, because it can explain the effective "collapse" of the
conditional wave function.
Furthermore, when the entanglement is absent, i.e. when
where ##f(t)=\psi_2(X_2(t),t)##. The factor ##f(t)## does not influence the trajectory ##X_1(t)##, so the physical content of ##\psi_c(x_1,t)## does not differ from the physical content of
standard subsystem wave function ##\psi_1(x_1,t)##.
For a system consisting of two noninteracting subsystems that are entangled with each other, do the subsystems propagate under the conditional Bohmian dynamics in the same way as under the full
Bohmian dynamics?
Last edited:
A. Neumaier said:
For a system consisting of two noninteracting subsystems that are entangled with each other, do the subsystems propagate under the conditional Bohmian dynamics as under the full Bohmian dynamics?
I'm not sure I understand the question. Did you perhaps mean
rather than
Or if you meant what you wrote, the answer is a trivial yes.
Last edited:
Demystifier said:
I'm not sure I understand the question. Did you perhaps mean or rather than as?
''in the same way as''
Demystifier said:
This is actually good, because it can explain the effective "collapse" of the conditional wave function.
How does it explain the effective "collapse"?
mieral said:
but then a photon as quantum stuff always have a wave function
This is not correct. A quantum
always has a wave function, but a single photon is only a quantum system (instead of just a subsystem) if it is isolated, i.e., if it does not interact with anything else. But if it does not interact
with anything else, it can't be entangled with anything else.
A. Neumaier said:
How does it explain the effective "collapse"?
I will explain if you promise that you will not complain that "it is not a rigorous proof".
Demystifier said:
I will explain if you promise that you will not complain that "it is not a rigorous proof".
I don't expect any rigorous proof, so I won't complain about that.
A. Neumaier said:
I don't expect any rigorous proof, so I won't complain about that.
Fine. Let ##x_2## represent the collective coordinate for positions of the macroscopic pointer of the measuring apparatus. Then, as I explained to you several times, the total wave function at some
fixed time ##t## after the measurement is of the form
$$\Psi(x_1,x_2)=\sum_A c_A \psi_A(x_1) \varphi_A(x_2)$$
$$\varphi_A(x_2)\varphi_B(x_2)\simeq 0...(1)$$
for ##A\neq B##. The corresponding conditional wave function is
$$\psi_c(x_1)=\sum_A c_A \psi_A(x_1) \varphi_A(X_2)...(2)$$
where ##X_2## represents the actual particle positions at time ##t##. But from (1) we see that if ##\varphi_A(X_2)## is non-negligible for one value of ##A##, then it is negligible for all other
values. Hence (2) reduces to
$$\psi_c(x_1) \simeq c_A \psi_A(x_1) \varphi_A(X_2)\equiv \tilde{c}_A \psi_A(x_1)$$
which corresponds to an effective collapse.
FAQ: Subsystem has no wave function
What does it mean when a subsystem has no wave function?
When a subsystem has no wave function, it means that the state of the subsystem cannot be described using quantum mechanics. This could be due to a lack of information or understanding about the
system, or it could be an indication that classical mechanics is a more appropriate framework for describing the subsystem.
Can a subsystem have no wave function in a quantum system?
Yes, a subsystem can have no wave function in a quantum system. This usually occurs when the subsystem is not isolated and is interacting with other subsystems or the environment, making it difficult
to accurately describe its state using quantum mechanics.
What are the implications of a subsystem having no wave function?
The implications of a subsystem having no wave function can vary depending on the specific system and context. In some cases, it may indicate a breakdown of our understanding of the system, while in
others it may suggest the need for a different approach or theoretical framework.
How can we determine if a subsystem has no wave function?
Determining if a subsystem has no wave function requires careful analysis of the system and its interactions. This may involve conducting experiments, collecting data, and performing calculations to
assess the system's behavior and determine if it can be described using quantum mechanics.
Is it possible for a subsystem to eventually develop a wave function?
Yes, it is possible for a subsystem to eventually develop a wave function. This could happen if the subsystem becomes isolated from its environment or if new information is obtained about the system
that allows for a more accurate description using quantum mechanics. | {"url":"https://www.physicsforums.com/threads/subsystem-has-no-wave-function.901686/","timestamp":"2024-11-03T12:43:29Z","content_type":"text/html","content_length":"150983","record_id":"<urn:uuid:b73f8cb6-ae71-4a6c-9757-c22835e477a8>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00056.warc.gz"} |
Square Root of 69 - How to Find the Square Root of 69?
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Square Root of 69
The number 69 is a number with only two prime factors 3 and 23. The square root of a number implies a number whose product with itself gives the initial number. We will now find the value of the
square root of 69 using various methods and a few interesting facts and problems as well.
• Square root of 69: √69 = 8.30662
• Square of 69: 69^2 = 4761
1. What Is the Square Root of 69?
2. Is Square Root of 69 Rational or Irrational?
3. How to Find the Square Root of 69?
4. FAQs on Square Root of 69
5. Important Notes on Square Root of 69
What is The Square Root of 69?
• The square root of 69 is written as √69 in radical form which is equal to 8.30662 (approximately).
• The number 69 only has two prime factors that are, 3 and 23. So, its square root cannot be simplified further using prime factorization.
• The square root of 69 can be written as (69)^1/2 in exponential form.
Is Square Root of 69 Rational or Irrational?
The square root of 69 is a non-repeating and non-terminating number so, it cannot be expressed in the form of p/q where q ≠ 0.
Hence, the square root of 69 is an irrational number.
How to Find the Square Root of 69?
We will now find the square root of 69 using the below-given methods:
Square Root of 69 Using Approximation Method
• Firstly, find two consecutive perfect squares such that 69 lies between them. The two numbers are 64 (8^2) and 81 (9^2). So, the whole number part of the square root of 69 will be 8.
• Now, for determining the decimal part we use the formula: (Given number – Smaller perfect square) / (Greater perfect square – smaller perfect square) = (69 – 64)/(81 – 64) = 5/17 = 0.294
• Hence, the approx. the square root of 69 by the approximation method is 8.294.
Square Root of 69 By Long Division
Now we will calculate the square root of 69 by the long division method.
• Start pairing the digits from the unit’s place in pairs of two by putting a bar on top of them. We will have one pair in this case (69).
• Find a number(a) such that a × a ≤ 69. So, a will be 8 as 8 × 8 = 64.
• We get the remainder as 5 (69-64) and the quotient as 8. Now, we will add the divisor a with itself and get the new divisor (16).
• Put a decimal in the dividend and quotient simultaneously. Also, place 3 pairs of zero in the dividend part after the decimal.
• Bring down one pair of zero. So, our new dividend is 500. Find a number(n) such that 16n × n ≤ 500. The number n will be 3 as 163 × 3 = 489 ≤ 500.
• Repeat the above step for the remaining pairs of zero.
So, we get the square root of √69 = 8.306 by the long division method.
Explore square roots using illustrations and interactive examples
• The number 69 is not a perfect square.
• The square root of 69 is an irrational number.
• The square root of -69 is an imaginary number.
Square Root of 69 Solved Examples
1. Example 1: Mike wants to find out the square root of -69. Can you help Mike?
The square root of all negative numbers is an imaginary number. So, the square of -69 is represented as √-69 = ±8.306i. (where i = √-1).
Example 2: How much minimum area in square feet should be added in a field of surface area 69 square feet, such that it can be converted into a field of square shape with its length as an
To convert the field into square-shaped its surface area should be a perfect square. Then the length of the field will be an integer. The nearest perfect square greater than 69 is 81. Therefore,
the area of the field that should be added is 81 – 69 = 12 square feet.
Show Solution >
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Become a problem-solving champ using logic, not rules. Learn the why behind math with our certified experts
Interactive Questions
Show Answer >
FAQs on Square Roots of 69
What is the negative square root of 69?
The negative square root of 69 is -8.3066.
What is the square root of 69 up to 10 decimal places?
The square root of 69 is √69 = 8.3066238629.
Can we find the square root of 69 using the prime factorization method?
No, we can’t find the square root of 69 via the prime factorization method. Because the number 69 has only two prime factors 3 and 23. So, we can’t reduce it further.
Is the square root of 69 is a rational number?
No, the square root of 69 is not a rational number. Because it cannot be expressed in the form of p/q where q ≠ 0.
Is the number 69 a perfect square?
No, the number 69 is not a perfect square. The square root of 69 is an irrational number and all perfect squares have integers as their square root.
Math worksheets and
visual curriculum | {"url":"https://www.cuemath.com/algebra/square-root-of-69/","timestamp":"2024-11-02T20:14:50Z","content_type":"text/html","content_length":"217165","record_id":"<urn:uuid:1d47c658-4cb8-4df0-b2bc-7f5dfc92452e>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00584.warc.gz"} |
Understanding Gravity: Probing Deeper Into Einstein's Theory Of Relativity
Understanding Gravity: Probing Deeper into Einstein’s Theory of Relativity
Understanding Gravity: Scrutinizing The Framework of Spacetime: A Glance at the Past
Albert Einstein’s general theory of relativity is a groundbreaking scientific proposition that deciphers the peculiar concept of spacetime – how it bends and contorts due to mass. To illustrate, our
Sun’s gravitational force distorts the spacetime around it, causing the Earth to revolve around the Sun, much like a marble spiralling in a funnel. It’s this intricate dance between mass, gravity,
and spacetime that prevents the Earth from being sucked into the Sun, primarily due to Earth’s sideways momentum.
The general theory of relativity, introduced in 1915, revolutionized our perception of gravity by portraying it as a warp in spacetime. Despite being an indispensable cornerstone in understanding the
space that envelops us, physicists speculate that it may not encapsulate the entire cosmic picture. They postulate that the theories of quantum gravity, which strive to bridge the gap between general
relativity and quantum physics, may hold the key to unfurling the universe’s deepest secrets.
Unveiling Quantum Gravity: A Collision Course
The quest for quantum gravity’s footprints often leads scientists to the cataclysmic collisions between black holes. Black holes are the densest entities in the universe, possessing a gravitational
force so immense that it elongates any object that falls into it into a noodle-like structure. When two black holes collide and amalgamate into a larger entity, they agitate spacetime around them,
transmitting gravitational waves that radiate in all directions.
An Ongoing Investigation: LIGO and the Test of Time
Since 2015, the National Science Foundation-funded LIGO (Laser Interferometer Gravitational-Wave Observatory), supervised by Caltech and MIT, has been regularly detecting gravitational waves produced
by black hole mergers. Joined by its partner observatories, Virgo and KAGRA, in 2017 and 2020 respectively, this initiative has yet to find any evidence contradicting the general theory of
A Step Further: Probing Quantum Gravity via Black Holes
Two recent studies led by Caltech, featured in Physical Review X and Physical Review Letters, propose novel methodologies to conduct more rigorous tests of general relativity. By scrutinizing the
structures of black holes and the spacetime ripples they generate, researchers aim to find minuscule deviations from general relativity that could indicate the existence of quantum gravity.
“When two black holes amalgamate to form a larger black hole, the resultant black hole resonates like a bell,” expounds Yanbei Chen (Ph.D. ’03), a physics professor at Caltech and co-author of both
studies. “The resonance quality, or its timbre, may differ from general relativity’s predictions if certain quantum gravity theories hold true. Our methodologies aim to detect differences in this
ringdown phase, such as the harmonics and overtones.”
A Ground-Breaking Equation: Paving the Way for Quantum Gravity Exploration
The first paper, spearheaded by Caltech graduate student Dongjun Li, introduces a unique equation that portrays how black holes would resonate within the realm of certain quantum gravity theories or
beyond the general relativity regime.
The research builds upon a trailblazing equation formulated 50 years ago by Saul Teukolsky (Ph.D. ’73), the Robinson Professor of Theoretical Astrophysics at Caltech. Contrary to numerical relativity
methodologies that require supercomputers to solve a multitude of differential equations pertaining to general relativity, the Teukolsky equation provides direct physical insight into the problem,
simplifying the process significantly.
Li has adapted Teukolsky’s equation to suit black holes within the beyond-general-relativity regime, a pioneering move. “Our novel equation enables us to model and comprehend gravitational waves
radiating around black holes that are more exotic than Einstein predicted,” he says.
Applying the New Equation: Unmasking Gravity’s Mysteries
The second paper, published in Physical Review Letters and led by Caltech graduate student Sizheng Ma, outlines a fresh approach to applying Li’s equation to actual data collected by LIGO and its
partners in their upcoming observational run. This data analysis methodology uses a series of filters to eliminate black hole ringing features predicted by general relativity, thereby potentially
revealing beyond-general-relativity signatures.
“We can search for features as described by Dongjun’s equation in the data that LIGO, Virgo, and KAGRA will gather,” Ma says. “Dongjun has found a way to condense a large set of complex equations
into just one equation, and this is incredibly beneficial. This equation is more efficient and easier to use than previous methods.”
Li and Ma’s studies mutually enhance each other. “I was initially concerned that the signatures my equation predicts would be obscured under the multiple overtones and harmonics; fortunately,
Sizheng’s filters can eliminate all these known features, which allows us to focus solely on the differences,” Li says.
Chen concludes, “Together, Li and Ma’s discoveries can significantly enhance our community’s ability to probe gravity.”
Keywords: Einstein’s theory of relativity, LIGO data, Quantum gravity, Black holes, Spacetime, Gravitational waves, Teukolsky equation.
Read Orginal Article: https://journals.aps.org/prx/abstract/10.1103/PhysRevX.13.021029
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Question 11: I want to plot the following piecewise defined function in Maple but I always seem to get an error. Let
f := proc(x,y) if x>y then erf(x-y) else erf(y-x) fi end;
But when I try to plot this I get
> plot3d( f(x,y), x=-2..2, y=-2..2 );
Error, (in f) cannot evaluate boolean | {"url":"http://thproxy.jinr.ru/Documents/MapleV/qa/tableofcontents3_1.html","timestamp":"2024-11-03T06:14:23Z","content_type":"text/html","content_length":"5420","record_id":"<urn:uuid:ce49e1a4-32d0-43b2-b163-8158980dd22c>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00061.warc.gz"} |
Problem with unsuspected coordinate values when creating polygons with arcpy.
01-24-2020 04:37 AM
I have a problem when creating polygon features with arcpy. I want to save rounded X and Y vertices for example to two decimal places. Example:
I have coordinates given on two decimal places. I am expecting rounding error due to limitations in precision of float data type.
Lets say we are working with projected coordinate system with base units of 1 Meter.
Example script (using dummy coordinate values):
import arcpy
point_coord = arcpy.Point(1.10, 2.13)
point = arcpy.PointGeometry(point_coord)
linestring_coords = [arcpy.Point(1.10, 2.13), arcpy.Point(2.44, 5.12)]
ls_array = arcpy.Array(linestring_coords)
linestring = arcpy.Polyline(ls_array)
poly_coords = [arcpy.Point(1.10, 2.13), arcpy.Point(2.44, 5.12), arcpy.Point(0.32, 4.2), arcpy.Point(1.10, 2.13)]
p_array = arcpy.Array(arcpy.Array(poly_coords))
poly = arcpy.Polygon(p_array)
print("Constructed point: {}".format(point.WKT))
print("Constructed polyline: {}".format(linestring.WKT))
print("Constructed polygon: {}".format(poly.WKT))
Constructed point: POINT (1.1000000000000001 2.1299999999999999)
Constructed polyline: MULTILINESTRING ((1.1000000000000001 2.1299999999999999, 2.4399999999999999 5.1200000000000001))
Constructed polygon: MULTIPOLYGON (((1.10009765625 2.130126953125, 2.44012451171875 5.1201171875, 0.32012939453125 4.2000732421875, 1.10009765625 2.130126953125)))
As said I am expecting rounding errors on for the values in point and polyline. But error on polygon coordinates is a lot bigger than 10e-10. If we say that base projection units are in meters that
brings an error of 0.1 mm which is a lot in some surveying applications. I know that this error can be virtually rounded to 2 decimal places in arcmap or some other, but if I want to take these
coordinates programically and perform some calculations the error grows and stays in data.
Is there a way to enforce rounding to values of coordinates for polygon features. How can this be the case in software meant for managing acurrate and high quality data.
I know for a fact, that some libraries outside arcgis ecosystem round coordinates normally for polygons (like point and polyline above).
How can I achieve same 'precision' as when creating polylines but with Polygons.
Thanks in advance.
01-24-2020 06:23 AM
01-24-2020 06:43 AM
01-24-2020 07:29 AM | {"url":"https://community.esri.com/t5/python-questions/problem-with-unsuspected-coordinate-values-when/td-p/160361","timestamp":"2024-11-03T17:21:52Z","content_type":"text/html","content_length":"249652","record_id":"<urn:uuid:af22292e-8d7f-41b4-9d36-b495a87a16b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00654.warc.gz"} |
Minkowski Distance: A Comprehensive Guide
Minkowski distance is a way of measuring the straight or curved path between two points, depending on a chosen parameter that affects the shape. Keep reading to learn about the fundamentals,
applications, and comparisons of Minkowski distance in various fields.
Oct 9, 2024 · 11 min read
Become an ML Scientist
Upskill in Python to become a machine learning scientist.
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How does Minkowski distance relate to Euclidean and Manhattan distances?
What happens when `p` approaches infinity in Minkowski distance?
Can Minkowski distance be used with categorical data?
Is Minkowski distance affected by the scale of features?
Are there any limitations to using Minkowski distance?
Discover machine learning with Python and work towards becoming a machine learning scientist. Explore supervised, unsupervised, and deep learning.Learn how to build advanced and effective machine
learning models in Python using ensemble techniques such as bagging, boosting, and stacking.Explore data structures such as linked lists, stacks, queues, hash tables, and graphs; and search and sort
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Understanding Euclidean Distance: From Theory to Practice
Explore how Euclidean distance bridges ancient geometry and modern algorithms, with coding examples in Python and R, and learn about its applications in data science, machine learning, and spatial
Understanding Chebyshev Distance: A Comprehensive Guide
Learn how Chebyshev distance offers a unique approach to spatial problems. Uncover its applications in robotics, GIS, and game development with coding examples in Python and R.
What is Cosine Distance?
Explore cosine distance and cosine similarity. Discover calculations, applications, and comparisons with other metrics. Learn to implement in R and Python using numpy.
What is Manhattan Distance?
Learn how to calculate and apply Manhattan Distance with coding examples in Python and R, and explore its use in machine learning and pathfinding.
Mean Shift Clustering: A Comprehensive Guide
Discover the mean shift clustering algorithm, its advantages, real-world applications, and step-by-step Python implementation. Compare it with K-means to understand key differences.
See MoreSee More | {"url":"https://www.datacamp.com/tutorial/minkowski-distance","timestamp":"2024-11-09T17:24:27Z","content_type":"text/html","content_length":"407434","record_id":"<urn:uuid:bd147aef-f584-4fd6-88c0-5f58cc218594>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00381.warc.gz"} |
Google CodeJam 2020 - Qualification Round - Problem 1 - Vestigium
In the first shot at the problem on paper, it becomes apparent that in a Natural Latin Square, the trace always turns out to be n*(n+1)/2
The elements in such a main diagonal may be either completely distinct from 1 to n or all same as the midpoint n/2 or (n+1)/2
Attempt 1: Time Complexity O(n^2)
By the end of this attempt, we arrive at no. of repeated rows
# Read rows and sum them
## If the sum doesn't add up to n(n+1)/2, call it a repeated row
### n^2 visits
# Read columns and sum them
## If the sum doesn't add up to n(n+1)/2, call it a repeated column
### n^2 visits again. total = 2*(n^2) visits
# if repeated_row and repeated_column are zeroes, directly print that
# the trace is n*(n+1)/2
# else sum the diagonals
# n visits again. Total = 2*(n^2)+n visits
def execute():
test_case_count = int(next(lines))
for case_num in range(test_case_count):
n = int(next(lines))
expected_sum = n*(n+1)/2
repeated_row_count = 0
repeated_col_count = 0
row_arr = []
for row_id in range(n):
row = next(lines)
row_sum = sum(map(int, row.split()))
if row_sum != expected_sum:
repeated_row_count += 1
print("repeated row count: " + str(repeated_row_count))
lines = open('1_in', 'r') | {"url":"https://blog.fossterer.com/Technical/google-codejam-2020-1-vestigium-attempt-1.html","timestamp":"2024-11-12T20:15:16Z","content_type":"text/html","content_length":"7362","record_id":"<urn:uuid:eec4f195-0f47-4e79-a4f1-3c98e4cda01d>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00084.warc.gz"} |
MATH 151 MCC Mathematics Algebra Worksheet - Custom Scholars
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Nima Rasekh's Academic Home Page
I am currently a postdoctoral researcher at the Universität Greifswald, in the research group of Konrad Waldorf and Matthias Ludewig, and a Quantum Formalism (QF) Fellow at the Zaiku Group. I am
currently on the academic job market, seeking tenure track positions.
For more information about me check out:
I am a homotopy theorist, meaning I like to understand how two things are equal. Homotopical thinking has already found many applications throughout mathematics, such as algebra, geometry and
mathematical physics, and I am always excited about finding new applications and connections. If you want get a better sense of homotopical thinking you can check out this excellent article by Emily
Riehl discussing the rise of higher category theory or this other general science article about the nature of equalities. Finally, if you want to know more about me beyond mathematics, I recently had
the pleasure of being interviewed by my former Master student, Qi Zhu, the transcript of which can be found here.
In my passion to make homotopical thinking more accessible to a wider range of mathematicians and computer scientists I am also pursuing formalization of mathematics and particularly homotopical
structures. You can find my work on my Github page.
I have also taught a course focused on homotopy theory and higher category theory that I would like to expand upon in the future. In the meantime you can find the lecture notes for my course.
Before coming to Greifswald I was a postdoctoral fellow at the Max-Planck-Institut für Mathematik and spent some time as a collaborateur scientifique (postdoctoral researcher) at the École
Polytechnique Fédérale de Lausanne working in the research group of Kathryn Hess. I was a PhD student at the University of Illinois at Urbana-Champaign, where I worked with my advisor Charles Rezk.
Email: nima.rasekh [at] uni-greifswald.de
Office: 5.16
Address:Walther-Rathenau-Straße 47, 17489 Greifswald
Telephone: +49 3834 420 4663
Here is a summary of some of my ongoing research projects and formalizations that I am pursuing:
A chronological list of my papers can be found on my ArXiv page or my Google Scholar page. Thematically, my work breaks down into three broad themes:
Formalization of Homotopy Theory:
• Insights From Univalent Foundations: A Case Study Using Double Categories - Formalization in Coq Unimath - Talk Recording - Talk Slides,
joint with Benedikt Ahrens, Paige North and Niels van der Weide,
To appear in Computer Science Logic 2025:
We continue our study of univalence properties of various notions of double categories. As part of that effort we more generally introduce the "Univalence Maxim" for categorical structures,
establishing a correspondence between categorical notions and equivalences. With this method we in particular obtain strict double setcategories (invariant under isomorphisms), pseudo double
setcategories (invariant under isomorphisms), univalent pseudo double categories (invariant under vertical equivalences), Verity double bisetcategories (invariant under isomorphisms) and
univalent Verity double bicategories (invariant under gregarious equivalences).
• Univalent Double Categories - Formalization in Coq Unimath - Talk Slides 1 - Talk Slides 2 - Talk Notes,
joint with Benedikt Ahrens, Paige North and Niels van der Weide,
Published in Certified Programs and Proofs 2024, doi:
We introduce a notion of univalent double category motivated by pseudo double categories. In particular, we consider several equivalent definitions, construct non-trivial examples, and construct
the univalent bicategory of univalent double categories.
• Constructing Coproducts in locally Cartesian closed ∞-Categories - Talk Slides - Talk Recording,
joint with Jonas Frey,
Published in Homology, Homotopy and Applications, doi:
We show that locally Cartesian closed ∞-categories with subobject classifier have an initial object and coproducts generalizing a well-known result in elementary topos theory.
• Univalence in Higher Category Theory:
We study univalence in locally Cartesian closed ∞-categories using internal ∞-categories (as complete Segal objects), generalizing a definition by Gepner and Kock in the presentable setting.
• Filter Quotients and Non-Presentable (∞,1)-Toposes - Talk Notes - Slides 1 - Slides 2,
Published in Journal of Pure and Applied Algebra, doi:
We generalize the filter quotient construction from elementary toposes to (∞,1)-categories to construct new non-presentable (∞,1)-toposes.
• Truncations and Blakers-Massey in an Elementary Higher Topos - Talk Notes - Slides 1 - Slides 2 - Slides 3 - Talk Recording:
We prove that every elementary higher topos has a universal truncation functor using the join construction. Moreover, we show it satisfies the Blakers-Massey theorem.
• Yoneda Lemma for Elementary Higher Toposes:
We prove that every object in an elementary higher topos embeds into its ``object of maps''. This corresponds to the Yoneda lemma for spaces.
• Every Elementary Higher Topos has a Natural Number Object - Talk Notes - Slides,
Published in Theory and Applications of Categories:
We compute the loop space of the circle and use that to show that every elementary higher topos as a natural number object and internal countable limits and colimits.
• A Theory of Elementary Higher Toposes - Talk Notes:
We define an elementary higher topos and show it generalizes elementary toposes and higher toposes.
• Complete Segal Objects - Talk Notes - Talk Recording (from Workshop in the Fields Institute):
We define an internal version of a higher category and show it has the same characteristics as a higher category (such as objects, morphisms, composition, ...). Then we use it to define
Limits in (∞,n)-Categories:
Homotopy Coherent Hochschild Homology:
• Shadows are Bicategorical Traces-Talk Recordings-Other Talk Recordings-Talk Slides-Other Talk Slides
joint with Kathryn Hess:
We characterize shadows, originally due to Ponto, as enriched THH and use that to give an alternative proof for Morita invariance.
• The cotangent complex and Thom spectra,
joint with Bruno Stonek
Published in Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, doi:
We use the results in the previous paper to compute the cotangent complex of Thom spectra, generalizing a result by Basterra and Mandell. As part of the work we also compare various definitions
of cotangent complexes that can be found in the literature using methods from Goodwillie calculus.
• Thom spectra, higher THH and tensors in ∞-categories-Talk Notes-Slides,
joint with Bruno Stonek and Gabriel Valenzuela,
Published in Algebraic & Geometric Topology, doi:
We compute tensor and in particular THH of Thom spectra using tensors of presentable ∞-categories, building on work of Gepner, Groth and Nikolaus and generalizing and streamlining a result by
Other Work:
• Analyzing RGB Images using Topology with Ruth Davidson, Chuan Du, Rosemary Guzman, Adarsh Manawa and Christopher Szul:
In this talk we discuss how to use a code developed at Australian National University to do image analysis with discrete Morse theory. We show how to use the code in two different scenarios:
water scarcity and crime data.
• RGB image-based data analysis via discrete Morse theory and persistent homology with Ruth Davidson, Chuan Du, Rosemary Guzman, Adarsh Manawa and Christopher Szul:
We use a code developed at ANU that can detect fundamental topological features of a grayscale image and enhance it so that it can also analyze RGB images. As a result we can perform data
analysis directly on RGB images representing water scarcity variability as well as crime variability.
• An Introduction to TFTs:
This is a talk I gave in the graduate student homotopy seminar. I introduce the basic notions of topological field theories and show that even simple computations necessitate using higher
categorical tools.
• A New Approach to Straightening:
These are my slides for the talk I gave in GSTGC (Graduate Student Geometry Topology Conference) 2016. I show a method to introduce the unstraightening construction to a larger mathematical
• I took my preliminary exam March 3rd, 2015. Here is my prelim syllabus and the slides of my prelim talk. | {"url":"https://nimarasekh.github.io/","timestamp":"2024-11-04T18:20:33Z","content_type":"text/html","content_length":"27677","record_id":"<urn:uuid:52d765ce-202e-4b10-ac44-23b18511932e>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00091.warc.gz"} |
1. INTRODUCTION NMR spectroscopy has been applied to edible oils and fats (2-9), but little is known about relaxivity properties of olive oil. NMRD could become an interesting tool to characterize
olive oil quality from a novel point of view, and to acquire information on its composition in a rapid and non destructive way. 2. METHOD AND MATERIAL NMRD profiles of Tuscan olive oil samples were
obtained by means of a Fast Field Cycling Relaxometer, in the 10kHz-35MHz range and processed (numerically and graphically) by means of programs written in the SAS (Statistical Analysis System)
software environment. 3. RESULTS AND DISCUSSION 3.1 A mono-exponential model was tentatively fitted to the experimental data: The line linking the experimental points regularly crosses the graph of
the corresponding monoexponential model three times. Thus a systematic error is apparent, either in the Pre-Polarized (PP, 0.001- 15MHz) or in the Not-Polarized (NP, 15-35MHz) experiments and the
multi-exponential fitting is mandatory. 3.2 The bi-exponential model reduces the residuals of two-three orders of magnitude. No further reduction was achieved incrementing the number of exponential
components. On the beginning, the bi-exponential models were fitted to experimental data allowing all the parameters to change freely, even the between-component ratio (q1 in the graphs, kk1bi in the
tables). Then a unique value was estimated. 3.3 The between-component ratio is a very critical parameter, not only for its physical meaning, but also because it is not easy to estimate, as shown by
the table and by the graph below. An ascending trend can be seen and the ratio seems to be field-dependent because it increases together with the frequency of the relaxing field (BRLX). The
between-component ratio was estimated here on the whole BRLX range (FREE estimate in tables and graphs). Then the model has been re-fitted forcing the ratio to be equal to the estimated value (REFIT
estimate in tables and graphs). In the same way a tri-exponential model (not reported) has been tested, but without further decrease of the error, neither systematic nor random, and it has been
discarded. 3.4 The mono- and multi-exponential models have been fitted to NMRD data by means of Non-Linear Regression Analysis. In the following graphs the Sums of Squares of Errors (SSEs) are
plotted and compared, for each relaxation field (BRLX) and for the three cases: 1) SSEmono (mono-exponential model): MAGNITUDES = a1mono+ b1mono*(exp(-TAU1/t1mono)) 2) SSEbiFREE (bi-exponential
model; all parameters allowed to change) MAGNITUDES = a1bi+ b1bi*(exp(-TAU1/t1bi) + kk1bi*(exp(-TAU1/t2bi))) 3) SSEbiREFIT (bi-exponential model; all parameters allowed to change except the
between-component ratio) MAGNITUDES = a1bi+ b1bi*(exp(-TAU1/t1bi) + kk1bi*(exp(-TAU1/t2bi))) with the bound kk1bi = 0.7380 The SSEs of the bi- and tri-exponential (not reported) FREE models are
equivalent and better than the mono-exp.. The SSE of the bi-exp. model, REFITTED including the kk1bi value estimated on the whole BRLX range, is less than the corresponding one of the mono-exp, for
all BRLX values. The FREE and the REFIT SSE of the bi-exp. model are alike. 3.5 R1 (t1) pre-analysis The following graphs show the trend of R1. R1 is derived from the mono-exp. model (diamonds) and
from the first (circles) and the second (stars) component of the bi-exp. one. The first graph is the result of the FREE kk1bi fitting (the between-component ratio of the bi-exponential model was
allowed to change with BRLX). In the second graph R1 has been calculated considering the bound: kk1bi = 0.7380 as explained above. No outlier is detectable and the trend of R1 is more smoothed if the
kk1bi unique estimated value is included in the model and the between components ratio is not allowed to change freely. 3.6 Fitting lorentzian models to R1 data Multi-Lorentzian, “model-free”
equations (see the table below) were fitted to the two components, by means of Non-Linear Regression, testing n values from 1 to 5. 4. CONCLUSIONS NMRD profiles of Tuscan olive oil samples have been
acquired for their relaxometric characterization. The magnetization decays detected by the relaxometer could be well fitted by means of a bi-exponential function. Inclusion of a third exponential
term did not reduce the sum of the differences between experimental and back-calculated values. Two relaxation rate profiles for each set of data were determined as a function of the magnetic field.
The profiles were then analysed as a sum of Lorentzian functions. Comparative analysis of the NMRD profiles of different types of oils is in progress, to investigate whether the quality of the
different oils may be correlated to the parameters related to relaxometric properties. The fitted mono- and multi-Lorentzian models were compared evaluated and selected within the Non- Linear
Regression Analysis framework, as shown by the following tables and graphs. In the first table the selected models are summarized and the parameters are all normalized (norm). In the tables below,
the parameters are not normalized and differently labeled: qq replaces x and the constant value 1 replaces (1-x). The models were evaluated taking in account their significance, the analysis of the
residuals, and the confidence intervals of the parameters. Each selected model is illustrated by two graphs. In the former the fitting is plotted, together with the residuals. In the latter the
selected Lorentzian is drawn with its components.
MULTI-EXPONENTIAL FIT OF NMRD DATA FOR OLIVE OIL ANALYSIS / S. ALESSANDRI; C. LUCHINAT; G. PARIGI; A. CIMATO. - STAMPA. - (2005), pp. P13-P13. (Intervento presentato al convegno 4TH CONFERENCE ON
FIELD CYCLING NMR RELAXOMETRY tenutosi a TORINO).
1. INTRODUCTION NMR spectroscopy has been applied to edible oils and fats (2-9), but little is known about relaxivity properties of olive oil. NMRD could become an interesting tool to characterize
olive oil quality from a novel point of view, and to acquire information on its composition in a rapid and non destructive way. 2. METHOD AND MATERIAL NMRD profiles of Tuscan olive oil samples were
obtained by means of a Fast Field Cycling Relaxometer, in the 10kHz-35MHz range and processed (numerically and graphically) by means of programs written in the SAS (Statistical Analysis System)
software environment. 3. RESULTS AND DISCUSSION 3.1 A mono-exponential model was tentatively fitted to the experimental data: The line linking the experimental points regularly crosses the graph of
the corresponding monoexponential model three times. Thus a systematic error is apparent, either in the Pre-Polarized (PP, 0.001- 15MHz) or in the Not-Polarized (NP, 15-35MHz) experiments and the
multi-exponential fitting is mandatory. 3.2 The bi-exponential model reduces the residuals of two-three orders of magnitude. No further reduction was achieved incrementing the number of exponential
components. On the beginning, the bi-exponential models were fitted to experimental data allowing all the parameters to change freely, even the between-component ratio (q1 in the graphs, kk1bi in the
tables). Then a unique value was estimated. 3.3 The between-component ratio is a very critical parameter, not only for its physical meaning, but also because it is not easy to estimate, as shown by
the table and by the graph below. An ascending trend can be seen and the ratio seems to be field-dependent because it increases together with the frequency of the relaxing field (BRLX). The
between-component ratio was estimated here on the whole BRLX range (FREE estimate in tables and graphs). Then the model has been re-fitted forcing the ratio to be equal to the estimated value (REFIT
estimate in tables and graphs). In the same way a tri-exponential model (not reported) has been tested, but without further decrease of the error, neither systematic nor random, and it has been
discarded. 3.4 The mono- and multi-exponential models have been fitted to NMRD data by means of Non-Linear Regression Analysis. In the following graphs the Sums of Squares of Errors (SSEs) are
plotted and compared, for each relaxation field (BRLX) and for the three cases: 1) SSEmono (mono-exponential model): MAGNITUDES = a1mono+ b1mono*(exp(-TAU1/t1mono)) 2) SSEbiFREE (bi-exponential
model; all parameters allowed to change) MAGNITUDES = a1bi+ b1bi*(exp(-TAU1/t1bi) + kk1bi*(exp(-TAU1/t2bi))) 3) SSEbiREFIT (bi-exponential model; all parameters allowed to change except the
between-component ratio) MAGNITUDES = a1bi+ b1bi*(exp(-TAU1/t1bi) + kk1bi*(exp(-TAU1/t2bi))) with the bound kk1bi = 0.7380 The SSEs of the bi- and tri-exponential (not reported) FREE models are
equivalent and better than the mono-exp.. The SSE of the bi-exp. model, REFITTED including the kk1bi value estimated on the whole BRLX range, is less than the corresponding one of the mono-exp, for
all BRLX values. The FREE and the REFIT SSE of the bi-exp. model are alike. 3.5 R1 (t1) pre-analysis The following graphs show the trend of R1. R1 is derived from the mono-exp. model (diamonds) and
from the first (circles) and the second (stars) component of the bi-exp. one. The first graph is the result of the FREE kk1bi fitting (the between-component ratio of the bi-exponential model was
allowed to change with BRLX). In the second graph R1 has been calculated considering the bound: kk1bi = 0.7380 as explained above. No outlier is detectable and the trend of R1 is more smoothed if the
kk1bi unique estimated value is included in the model and the between components ratio is not allowed to change freely. 3.6 Fitting lorentzian models to R1 data Multi-Lorentzian, “model-free”
equations (see the table below) were fitted to the two components, by means of Non-Linear Regression, testing n values from 1 to 5. 4. CONCLUSIONS NMRD profiles of Tuscan olive oil samples have been
acquired for their relaxometric characterization. The magnetization decays detected by the relaxometer could be well fitted by means of a bi-exponential function. Inclusion of a third exponential
term did not reduce the sum of the differences between experimental and back-calculated values. Two relaxation rate profiles for each set of data were determined as a function of the magnetic field.
The profiles were then analysed as a sum of Lorentzian functions. Comparative analysis of the NMRD profiles of different types of oils is in progress, to investigate whether the quality of the
different oils may be correlated to the parameters related to relaxometric properties. The fitted mono- and multi-Lorentzian models were compared evaluated and selected within the Non- Linear
Regression Analysis framework, as shown by the following tables and graphs. In the first table the selected models are summarized and the parameters are all normalized (norm). In the tables below,
the parameters are not normalized and differently labeled: qq replaces x and the constant value 1 replaces (1-x). The models were evaluated taking in account their significance, the analysis of the
residuals, and the confidence intervals of the parameters. Each selected model is illustrated by two graphs. In the former the fitting is plotted, together with the residuals. In the latter the
selected Lorentzian is drawn with its components.
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Attention Getters
Is it Interesting?
Attention keeping methods in the body of a speech.
Objective: By the end of class the student will have an understanding of how to keep interest in the main section of any speech.
Materials: Only pen and paper are required.
1. Instructor will list clarity, inclusion, and emotion on the board.
2. The three terms will be explained to the class.
A. clarity-descriptive words to add “color” to ideas. ex. – dog> cute and fluffy or hairy and smelly
B. inclusion-creating a connection with listeners. ex. – death> “Something we all have to face.” NOT “Something you will face.”
C. emotion-using emotional words to inspire listeners. ex. – “The crowd felt pride and joy at the end of the victorious game.”
3. The class will be split into pairs and entire class will be given a number from 1 to 3.
A. The number that each pair receives will be the language style they are to use for their assignment.
4. Each pair will then write a brief speech on their favorite pet (alive or deceased) using the language style they’ve been assigned.
5. Upon completion, the pair will then have to improvise their own sentences separately in front of the class. The instructor will inform the two which style they are to use.
6. The written speeches will be handed in to the instructor.
Result: The student will have practiced giving a speech using the tools given to keep an audience attentive in the middle of a speech.
Fall 2007
Attention Getting Methods
Objective: The students will write effective attention getting methods and incorporate them into the introductions of their speeches. The students will also list at least four attention getting
methods from those discussed in class.
Materials: Rough draft copy of speech, or at least a possible speech topic.
1. Call the class to attention. Use one of the attention getting methods to introduce the topic. You might tell a short story about a speaker that didn’t get your attention, or you might ask a
question or pose a challenge. It is up to you! Tell them that today you will be talking about the introduction of their speeches. Explain that there are many important things that go in the
introduction, but today you are going to talk about attention getters.
2. Ask the students what you did to get their attention. Answers will vary based on what option you chose. Ask the students if this method was effective and if they wanted to pay attention.
3. Discuss the different types of attention getters. Following are some of the most common, but these can be developed and there are variations of each:
• Tell a story or anecdote: You might explain what led you to your topic. What interests you is likely to interest your audience too. Personal stories work well, as do stories or events from the news
that you can draw comparisons with your topic.
• Begin with a startling statement: An abrupt statement that is contrary to what is happening around you will get attention because it is so distinct. Even though it can easily be overused, shock
value can be an effective tool.
• Pose a question: When a question is asked, it is hard not to automatically think about an answer to the issue the question has raised. You can ask a dialectical question where you expect a
response, like what happens in class, but most speeches before an audience often use rhetorical questions, which don’t require an audible answer.
• Pose a challenge: Even though an audience may not like it if you have different views on a topic than they do, posing a challenge can help to earn their attention and respect if you have sufficient
support to back up your argument.
• Explore an analogy or draw comparisons to something else: Sometimes this indirect approach is very effective. An analogy serves to immediately gain your audience’s attention while gradually leading
them to the point of your talk.
• Introduce a quotation: This is another attention getter that can be effective if it is not overused. Quotations are useful if they are relevant and can give credibility to your position, especially
if they are from a well known figure. Try to avoid using dictionary definitions, they are overused and detract from your speech.
• Interject some humor: If you have something to say that is relevant to your topic and genuinely funny, then say it. However, if it is offensive or contrived, don’t do it! Humor is easy to overdo,
and it isn’t fun to listen to someone who thinks they are funny when they really aren’t. A good rule of thumb is to remember to laugh at yourself, and then you won’t be offending someone else!
4. Let the students brainstorm ideas of what methods of attention getters they might use in their speeches. List their speech topics on the board and discuss what attention getting methods might be
appropriate for each topic. For instance, if you are talking about a very serious subject, joking about it may not be appropriate.
5. Give the students a few minutes to write two different attention getting introductions for their speeches.
6. Have each student share their attention getters with a partner, then work together to choose the best one.
7. Ask each student to share his or her attention getting method with another partner.
8. Instruct each student to take out a piece of paper and write his or her name on it. Tell the students to list as many attention getting methods as they can. They will need to list at least four in
order to earn full points.
9. Collect the quizzes and tell the students that their homework assignment is to continue working on the attention getters in their introductions.
Results: At the end of this assignment, students should know the different methods of attention getters and should be able to incorporate them into their speeches. They will write introductions for
their speeches that have effective attention getters.
Homework: Continue working on speeches.
Elissa Martin
Summer 2007
Wait, Wait! I’m Not Done Yet!!
Attention Keeping Methods in the Body of a Speech.
Objective: The students will be able to discuss within a small and large group the notes they took on a specific passage from their text.
Materials: Your class’s textbook
1. Have the students read and take their own notes on the section in their book that talks about attention getters and motivating the audience to listen.
2. Split the students into groups of 3-4. In their groups they must discuss with each other what notes they took and add anything they missed to their won notes.
a. The Teacher should be actively monitoring the room and if there are questions encourage the students to write them down on a separate sheet of paper.
3. Give the students about 15 minutes to discuss.
4. Bring the groups back together and have them discuss their notes as a whole.
a. What did you find most interesting?
b. What did you learn?
c. How can you take “attention getters” and turn them into “attention keepers”
d. How can you apply this to your own speech.
5. During this time, allow students to ask their questions. Ask the class if anyone can explain. If someone can, let them, if no one volunteers, answer the question yourself. (you should probably
brush up on some attention grabbing techniques yourself)
a. Try to point out where the question is answered in the book.
Results: For homework, the student must proofread their speech and if they need to, add statements to maintain audience attention. On a sheet of paper, they must record what they changed and any
examples they already had in their speech of Maintaining Audience Attention.
Attention Getters
Startling Statements
OBJECTIVE: The students will be able to identity and write a startling statement for their speech topics.
MATERIALS: Topic worksheet, computer/internet usage
1. Give an overview to the students of what a startling statement/attention getter is. (It is statement or fact that grabs the audience’s attention in order to make them want to listen to your
2. Hand out the topic worksheet.
3. Pair students into groups of two.
4. Explain to students the instructions on the top of the sheet.
5. Give students a maximum of 20 minutes to search the internet to find or come up with a startling statement or attention getter for each of the topics on the worksheet. *BE SURE TO CONSTATNLY
MONITOR THE STUDENT’S INTERNET USAGE TO BE SURE THEY ARE STAYING ON TASK!*
6. After students have found a startling statement for each topic, bring them back together into one group and have them share a few what they found with the class.
7. Explain to the class which suggestions/findings were good ones and which were not, and explain why.
RESULTS: Have students write down 5 startling statements that work with their speech topic. They will be expected to share them with the teacher the next day.
Name:___________________________________ Date:________________
Directions: Using the internet, find startling statements that would work as an attention getter for each of the following speech topics. Record the website you obtained your information from. Get
creative! Don’t just use boring facts for each one! Remember: They are supposed to grab the audience’s attention and pull them into wanting to listen to the rest of your speech!
1. The movie Ghost Busters
Source: ___________________________________________________________
2. Football
Source: ___________________________________________________________
3. Volleyball
Source: ___________________________________________________________
4. Paper
Source: ___________________________________________________________
5. Spiders (any type, but PLEASE SPECIFY)
Source: ___________________________________________________________
6. Vehicle Manufacturing
Source: ___________________________________________________________
7. Abortion
Source: ___________________________________________________________
8. Alcoholism
Source: ___________________________________________________________
9. Your favorite music group.
Source: ___________________________________________________________
10. A topic of your choice (PLEASE SPECIFY): _____________________________
Source: ___________________________________________________________
Name: TEACHER’S COPY__(Examples)____ Date:________________
Directions: Using the internet, find startling statements that would work as an attention getter for each of the following speech topics. Get creative! Don’t just use boring facts for each one!
Remember: They are supposed to grab the audience’s attention and pull them into wanting to listen to the rest of your speech!
1. The movie Ghost Busters
Start speech by singing the theme song.
2. Football
The first girl to play tackle football was Luverne “Toad” Wise who played in 1939 for the Escambia County (Alabama) High School football team. (http://www.fortunecity.com/wembley/mueller/641/
3. Volleyball
Volleyball was created in 1895 for middle aged men who were looking for a sport to play that was less strenuous than basketball. (http://mountolivevolleyball.tripod.com/id6.html)
4. Paper
Recycling one ton of paper saves about 17 trees. (http://paperproject.org/paperfacts.html)
5. Spiders (any type, but PLEASE SPECIFY)
Daddy Long Legs are the most poisonous spider in the world, but their mouths are too small to bite anyone.
6. Vehicle Manufacturing
Over 10,000 people were laid-off by Chevy in 2009.
7. Abortion
About 3,700 abortions take place everyday in the U.S. (http://www.abortionno.org/Resources/fastfacts.html)
8. Alcoholism
Signs of alcohol show up in 10 minutes and last over an hour. (http://www.safemenopausesolutions.com/alcoholism-facts.html)
9. Your favorite music group.
Start by singing one of their songs
10. A topic of your choice (PLEASE SPECIFY): _____________________________
Mandy Brabec | {"url":"https://mcluhan.unk.edu/lawson/?page_id=52","timestamp":"2024-11-11T03:29:43Z","content_type":"text/html","content_length":"55016","record_id":"<urn:uuid:8da1ec81-c9e3-4b1b-a07b-102af31ed9a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00072.warc.gz"} |
C Program To Find Maximum Even Number In Row In Matrix
Problem :- C Program To Find Row In An Array That Include The Greatest Amount Of Even Number
Input Is Below
Solution :-
#include<iostream> using namespace std; int main() { int a[10][10],n,l,i,j,count=0; cout<<"\nEnter The Matrix Size (l*l)\n"; cin>>l; cout<<"\nEnter The Row Number Of Matrix\n"; cin>>n; if(n<0||n<=l)
{ } else { cout<<"\nRow Is Exceed The Limit Enter Value >0 And Less Than Size Of Matrix \n"; exit(0); } cout<<"Enter The Matrix Value\n"; for(i=1;i<=l;i++) for(j=1;j<=l;j++) cin>>a[i][j]; cout<<"\n\
nMatrix Row Is Given Below\n\n"; for(i=1;i<=l;i++) { cout<<a[n][i]<<" "; if(a[n][i]%2==0) count++; } cout<<"\n\nNo. Of Even Number In Row IS = "<<count; return 0; }
1 comment:
1. Anonymous05/01/2017, 07:55
amount - of uncountable
number - of countable
amount of sand/salt/flour
number of characters/flowers/numbers | {"url":"https://www.programmingwithbasics.com/2016/05/c-program-to-find-row-in-array-that.html","timestamp":"2024-11-06T04:50:11Z","content_type":"application/xhtml+xml","content_length":"130080","record_id":"<urn:uuid:bbedaf1e-60f8-414b-9b15-a179c2e10f44>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00191.warc.gz"} |
PROC CALIS: Direct Covariance Structures Analysis :: SAS/STAT(R) 9.22 User's Guide
Direct Covariance Structures Analysis
Previous examples are concerned with the implied covariance structures from the functional relationships among manifest and latent variables. In some cases, direct modeling of the covariance
structures is not only possible, but indeed more convenient. The MSTRUCT modeling language in PROC CALIS is designed for this purpose. Consider the following four variables from the Wheaton et al. (
1977) data:
Anomie 1967
Powerlessness 1967
Anomie 1971
Powerlessness 1971
The covariance structures are hypothesized as follows:
Variance of Anomie
Variance of Powerlessness
Covariance between Anomie and Powerlessness
Covariance between Anomie measures
Covariance between Powerlessness measures
In the hypothesized covariance structures, the variances of Anomie and Powerlessness measures are assumed to stay constant over the two time points. Their covariances are also independent of the time
of measurements. To test the tenability of this covariance structure model, you can use the following statements of the MSTRUCT modeling language:
proc calis nobs=932 data=Wheaton;
var = Anomie67 Powerless67 Anomie71 Powerless71;
matrix _COV_ [1,1] = phi1,
[2,2] = phi2,
[3,3] = phi1,
[4,4] = phi2,
[2,1] = theta1,
[3,1] = theta2,
[3,2] = theta1,
[4,1] = theta1,
[4,2] = theta3,
[4,3] = theta1;
In the MSTRUCT statement, you specify the list of variables of interest with the VAR= option. The order of the variables in the list will be the order in the hypothesized covariance matrix. Next, you
use the MATRIX _COV_ statement to specify the parameters in the covariance matrix. The specification is a direct translation from the hypothesized covariance matrix. For example, the [1,1] element of
the covariance matrix is fitted by the free parameter phi1. Depending on the hypothesized model, you can also specify fixed constants for the elements in the covariance matrix. If an element in the
covariance matrix is not specified by either a parameter name or a constant, it is assumed to be a fixed zero.
The analysis of this model is carried out in Example 25.17.
The MSTRUCT modeling language appears to be more restrictive than any of the other modeling languages discussed, in regard to the following limitations:
• It does not explicitly support latent variables in modeling.
• It does not explicitly support modeling of linear functional relations among variables (for example, paths).
However, these limitations are more apparent than real. In PROC CALIS, the parameters defined in models can be dependent. These dependent parameters can be defined further as functions of other
parameters in the PARAMETERS and the SAS programming statements. With these capabilities, it is possible to fit structural models with latent variables and with linear functional relations by using
the MSTRUCT modeling language. However, this requires a certain level of sophistication in statistical knowledge and in programming. Therefore, it is recommended that the MSTRUCT modeling language be
used only when the covariance and mean structures are modeled directly.
For more information about the MSTRUCT modeling language, see the section The MSTRUCT Model and the MSTRUCT statement. | {"url":"http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/statug_calis_sect015.htm","timestamp":"2024-11-13T03:12:56Z","content_type":"application/xhtml+xml","content_length":"15004","record_id":"<urn:uuid:7e6bbe34-11c6-45e2-a494-5bfc86848852>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00533.warc.gz"} |
ClimaCoreVTK.jl · ClimaCore.jl
ClimaCoreVTK.jl provides functionality for writing ClimaCore fields to VTK files, using the WriteVTK.jl package.
Write fields to as an unstructured mesh VTK file named basename.vtu.
fields can be either:
• a ClimaCore Field object,
• a FieldVector object,
• a NamedTuple of Fields.
The basis keyword option determines the type of cells used to write.:
• :cell (default): output values at cell centers (interpolating where necessary).
• :point: output values at cell vertices.
• :lagrange: output values at Lagrange nodes (valid only for spectral element spaces), using Use VTK Lagrange cells to accurately represent high-order elements.
The latlong=true keyword option will output a spherical or spherical shell domain using the Mercator projection, with longitude along the x-axis, latitude along the y-axis, and altitude along the
z-axis (if applicable). Note this currently only displays correctly if the number of elements across the cubed sphere face is even.
Any additional keyword arguments are passed to WriteVTK.vtk_grid.
Write a sequence of fields at times as a Paraview collection (.pvd) file, along with VTK files.
fields can be either be an iterable collection of fields, or a NamedTuple of collections.
vtk_grid(basename, gridspace::ClimaCore.Spaces.AbstractSpace;
basis=:cell, vtkargs...)
Construct a VTK grid from a ClimaCore.Spaces.AbstractSpace. If basis=:lagrange, it will construct a mesh made of Lagrange cells (valid only for spectral element spaces), otherwise it will it
subdivide the space into quads, with vertices at nodal points.
Construct a vector of MeshCell objects representing the elements of space as an unstuctured mesh of Lagrange polynomial cells, suitable for passing to vtk_grid.
Construct a vector of MeshCell objects representing the elements of space as an unstuctured mesh of linear cells, suitable for passing to vtk_grid.
The space for the grid used by VTK, for any field on space.
This generally does two things:
• Modifies the horizontal space to use a ClosedUniform quadrature rule, which will use equispaced nodal points in the reference element. This is required for using VTK Lagrange elements (see 1).
• Modifies the vertical space to be on the faces.
Construct a space for outputting cell data, when using outputting a grid gridspace. be stored.
This generally does two things:
• Modifies the horizontal space to use a Uniform quadrature rule, which will use equispaced nodal points in the reference element (excluding the boundary).
• Modifies the vertical space to be on the centers.
addfield!(vtkfile, prefix::Union{String,Nothing}, f, dataspace)
Add a field or fields f, optionally prefixing the name with prefix to the VTK file vtkfile, interpolating to dataspace.
f can be any of the following:
• a scalar or vector field (if no prefix is provided, then the field will be named "data")
• a composite field, which will be named accordingly
• a NamedTuple of fields | {"url":"https://clima.github.io/ClimaCore.jl/dev/lib/ClimaCoreVTK/","timestamp":"2024-11-06T17:52:51Z","content_type":"text/html","content_length":"19497","record_id":"<urn:uuid:32da1712-5af4-4e68-bd40-6d5460c16a61>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00127.warc.gz"} |
5 Best Ways to Minimum Removals from Array to Make GCD Greater in Python
π ‘ Problem Formulation: The challenge is to find methods that will minimize the number of elements we must remove from an array in Python to increase the greatest common divisor (GCD) of the
remaining elements. If we are given an array like [4, 6, 24, 36], we might look for a way to remove a minimum number of elements to make the GCD of the array larger than a certain threshold, such as
Method 1: Brute Force Approach
Method 1 involves checking all possible combinations of elements to find the minimum removals needed to increase the GCD. This method is straightforward but can be highly inefficient for large arrays
due to its exponential time complexity.
Here’s an example:
import math
from itertools import combinations
def min_removals_brute_force(arr, target_gcd):
for i in range(len(arr)):
for comb in combinations(arr, len(arr) - i):
if math.gcd(*comb) > target_gcd:
return i
return -1
# Test the function
array = [4, 6, 24, 36]
target = 4
removals = min_removals_brute_force(array, target)
print(f"Minimum removals: {removals}")
Output: Minimum removals: 1
This code snippet calculates the minimum number of removals required to increase the GCD of the given array above the specified target. By using the combinations method from Python’s itertools
module, it iteratively checks smaller subsets of the array until it finds a combination that meets the condition.
Method 2: Greedy Removal of Non-Multiples
Method 2 uses a greedy strategy to remove elements from the array that are not multiples of the target GCD. This method is not always optimal but offers a much faster execution time than the brute
force approach.
Here’s an example:
def min_removals_greedy(arr, target_gcd):
removals = 0
for num in arr:
if num % target_gcd != 0:
removals += 1
return removals
# Test the function
array = [4, 6, 24, 36]
target = 4
removals = min_removals_greedy(array, target)
print(f"Minimum removals: {removals}")
Output: Minimum removals: 1
The provided code iteratively checks each number in the array and increments a counter for every element that is not a multiple of the target GCD, effectively counting the number of elements to be
removed to increase the GCD.
Method 3: Dynamic Programming
Method 3 involves using a dynamic programming technique to determine the minimum removals. This method is more complex but can be efficient for larger arrays as it avoids recalculating solutions for
previously seen subproblems.
Here’s an example:
# Dynamic programming approach is too complex to be showcased in few lines of code
# and is beyond the scope of this simple example.
This method is mentioned here as a conceptual strategy and would require a more in-depth explanation and implementation, which may vary greatly depending on the specifics of the problem and the
constraints imposed by the array size and composition.
Method 4: Mathematical Insight and Optimization
Method 4 applies mathematical insights to optimize the removal process. By analyzing the properties of numbers and the GCD, certain shortcuts can be taken to speed up the calculation without having
to check every combination of elements.
Here’s an example:
# Mathematical optimization is highly specific and depends on properties of GCD
# and array elements which are beyond this simple example.
Even without specific code, we recognize that a mathematically-optimized solution would derive from properties like prime factorization or recognizing certain divisibility rules that might quickly
flag elements for removal.
Bonus One-Liner Method 5: Functional Python Approach
Method 5 utilizes Python’s functional programming capabilities to create a concise one-liner that removes elements lower than or equal to the target GCD, aiming to potentially increase the overall
Here’s an example:
min_removals_one_liner = lambda arr, target_gcd: len([x for x in arr if x <= target_gcd])
# Test the function
array = [4, 6, 24, 36]
target = 4
removals = min_removals_one_liner(array, target)
print(f"Minimum removals: {removals}")
Output: Minimum removals: 2
This one-liner defines a lambda function that uses a list comprehension to count the numbers that are less than or equal to the target GCD. However, this method does not guarantee the minimal number
of removals for increasing the GCD, so it should be used with caution.
• Method 1: Brute Force. Strengths: guarantees an optimal solution. Weaknesses: exponential time complexity; not scalable.
• Method 2: Greedy Removal of Non-Multiples. Strengths: fast and easy to implement. Weaknesses: does not guarantee an optimal solution; might be shortsighted in some cases.
• Method 3: Dynamic Programming. Strengths: avoids redundant calculations; more efficient for large arrays. Weaknesses: complex to implement and understand; requires more code.
• Method 4: Mathematical Insight and Optimization. Strengths: potential for great optimization and efficiency. Weaknesses: highly specific and varies per case; can be complicated to derive and
• Method 5: Functional Python Approach. Strengths: concise and elegant. Weaknesses: heuristic, does not ensure minimal removals; may not actually increase GCD. | {"url":"https://blog.finxter.com/5-best-ways-to-minimum-removals-from-array-to-make-gcd-greater-in-python/","timestamp":"2024-11-06T11:07:52Z","content_type":"text/html","content_length":"71507","record_id":"<urn:uuid:3f214315-7ba3-46fe-907b-92e8d3a24b83>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00098.warc.gz"} |
Go to the source code of this file.
subroutine dpbtrf (UPLO, N, KD, AB, LDAB, INFO)
Function/Subroutine Documentation
subroutine dpbtrf ( character UPLO,
integer N,
integer KD,
double precision, dimension( ldab, * ) AB,
integer LDAB,
integer INFO
Download DPBTRF + dependencies
[TGZ] [ZIP] [TXT]
DPBTRF computes the Cholesky factorization of a real symmetric
positive definite band matrix A.
The factorization has the form
A = U**T * U, if UPLO = 'U', or
A = L * L**T, if UPLO = 'L',
where U is an upper triangular matrix and L is lower triangular.
UPLO is CHARACTER*1
[in] UPLO = 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N is INTEGER
[in] N The order of the matrix A. N >= 0.
KD is INTEGER
[in] KD The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB is DOUBLE PRECISION array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
[in,out] AB if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**T*U or A = L*L**T of the band
matrix A, in the same storage format as A.
LDAB is INTEGER
[in] LDAB The leading dimension of the array AB. LDAB >= KD+1.
INFO is INTEGER
= 0: successful exit
[out] INFO < 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading minor of order i is not
positive definite, and the factorization could not be
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Further Details:
The band storage scheme is illustrated by the following example, when
N = 6, KD = 2, and UPLO = 'U':
On entry: On exit:
* * a13 a24 a35 a46 * * u13 u24 u35 u46
* a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
Similarly, if UPLO = 'L' the format of A is as follows:
On entry: On exit:
a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
a31 a42 a53 a64 * * l31 l42 l53 l64 * *
Array elements marked * are not used by the routine.
Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
Definition at line 143 of file dpbtrf.f. | {"url":"https://netlib.org/lapack/explore-html-3.4.2/df/da9/dpbtrf_8f.html","timestamp":"2024-11-11T15:13:01Z","content_type":"application/xhtml+xml","content_length":"12987","record_id":"<urn:uuid:8eb09297-787a-4426-8e16-e4af288cb9bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00384.warc.gz"} |
Distance modulus - Wikiwand
The distance modulus is a way of expressing distances that is often used in astronomy. It describes distances on a logarithmic scale based on the astronomical magnitude system.
The distance modulus ${\displaystyle \mu =m-M}$ is the difference between the apparent magnitude ${\displaystyle m}$ (ideally, corrected from the effects of interstellar absorption) and the absolute
magnitude ${\displaystyle M}$ of an astronomical object. It is related to the luminous distance ${\displaystyle d}$ in parsecs by: {\displaystyle {\begin{aligned}\log _{10}(d)&=1+{\frac {\mu }{5}}\\\
mu &=5\log _{10}(d)-5\end{aligned}}}
This definition is convenient because the observed brightness of a light source is related to its distance by the inverse square law (a source twice as far away appears one quarter as bright) and
because brightnesses are usually expressed not directly, but in magnitudes.
Absolute magnitude ${\displaystyle M}$ is defined as the apparent magnitude of an object when seen at a distance of 10 parsecs. If a light source has flux F(d) when observed from a distance of ${\
displaystyle d}$ parsecs, and flux F(10) when observed from a distance of 10 parsecs, the inverse-square law is then written like: ${\displaystyle F(d)={\frac {F(10)}{\left({\frac {d}{10}}\right)^
The magnitudes and flux are related by: {\displaystyle {\begin{aligned}m&=-2.5\log _{10}F(d)\\[1ex]M&=-2.5\log _{10}F(d=10)\end{aligned}}}
Substituting and rearranging, we get: ${\displaystyle \mu =m-M=5\log _{10}(d)-5=5\log _{10}\left({\frac {d}{10\,\mathrm {pc} }}\right)}$ which means that the apparent magnitude is the absolute
magnitude plus the distance modulus.
Isolating ${\displaystyle d}$ from the equation ${\displaystyle 5\log _{10}(d)-5=\mu }$, finds that the distance (or, the luminosity distance) in parsecs is given by ${\displaystyle d=10^{{\frac {\mu
The uncertainty in the distance in parsecs (δd) can be computed from the uncertainty in the distance modulus (δμ) using ${\displaystyle \delta d=0.2\ln(10)10^{0.2\mu +1}\delta \mu \approx 0.461d\ \
delta \mu }$ which is derived using standard error analysis.^[1]
This section
does not cite any sources
(July 2023)
Distance is not the only quantity relevant in determining the difference between absolute and apparent magnitude. Absorption is another important factor, and it may even be a dominant one in
particular cases (e.g., in the direction of the Galactic Center). Thus a distinction is made between distance moduli uncorrected for interstellar absorption, the values of which would overestimate
distances if used naively, and absorption-corrected moduli.
The first ones are termed visual distance moduli and are denoted by ${\displaystyle {(m-M)}_{v}}$, while the second ones are called true distance moduli and denoted by ${\displaystyle {(m-M)}_{0}}$.
Visual distance moduli are computed by calculating the difference between the observed apparent magnitude and some theoretical estimate of the absolute magnitude. True distance moduli require a
further theoretical step; that is, the estimation of the interstellar absorption coefficient.
Distance moduli are most commonly used when expressing the distance to other galaxies in the relatively nearby universe. For example, the Large Magellanic Cloud (LMC) is at a distance modulus of
18.5,^[2] the Andromeda Galaxy's distance modulus is 24.4,^[3] and the galaxy NGC 4548 in the Virgo Cluster has a DM of 31.0.^[4] In the case of the LMC, this means that Supernova 1987A, with a peak
apparent magnitude of 2.8, had an absolute magnitude of −15.7, which is low by supernova standards.
Using distance moduli makes computing magnitudes easy. As for instance, a solar type star (M= 5) in the Andromeda Galaxy (DM= 24.4) would have an apparent magnitude (m) of 5 + 24.4 = 29.4, so it
would be barely visible for the Hubble Space Telescope which has a limiting magnitude of about 30.^[5] Since it is apparent magnitudes which are actually measured at a telescope, many discussions
about distances in astronomy are really discussions about the putative or derived absolute magnitudes of the distant objects being observed. | {"url":"https://www.wikiwand.com/en/articles/Distance%20modulus","timestamp":"2024-11-13T18:37:37Z","content_type":"text/html","content_length":"262361","record_id":"<urn:uuid:e5f76068-07c7-485d-94b7-3172482fff49>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00798.warc.gz"} |
BFS Tree
To help illustrate the challenge given in this lecture I’ve gone ahead and organised the traversal into a tree diagram. This is a great way to visualised what’s actually going on and should should be
particularly helpful for anyone who is struggling to see how the trace back works when you finally reach the goal.
So, the challenge was to manually work out the shortest path for the following grid, using the directional priority; up, right, down, left:
I won’t go into the specifics of how to actually traverse the grid, since Ben already does a fine job at explaining it in the video.
Now, by following the instructions of our algorithm, you’re building a tree that looks something like this:
Notice that every time a visited node has somewhere to go, those nodes are added as branches on the tree. However, each node can only be added to the tree once.
This gives us a queue in the order A-B-G-C-H-E-D-L-F-M-I-K-Z.
So to find the path back to the start, all you do is work your way back up the tree.
Therefore, the shortest path (according to our algorithm) is A-B-C-E-F-I-Z.
But what about the other paths we could have taken? What happens if we change the directional priority to favour moving down before moving right?
Well, that’s easy! Here’s the tree for the same grid but searched with the down-right directional priority.
This new tree gives us a queue in the order A-G-B-H-C-L-D-E-M-F-K-I-Z.
So now the path shortest path is (A-G-H-L-M-K-Z)
As you can see this, corresponds to moving straight down on the grid and then moving right.
I hope this helps with your understanding of this breadth first search algorithm. | {"url":"https://community.gamedev.tv/t/bfs-tree/64195","timestamp":"2024-11-01T23:35:21Z","content_type":"text/html","content_length":"16644","record_id":"<urn:uuid:d0402fc1-54ac-43f4-8187-f9525d123665>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00376.warc.gz"} |
Wtb: Standard V35 Wheels
If anybody has a set of the standard v35 wheels that they wanted to get rid of, please let me know.
Located in melbourne.
there is a set on ebay, not sure if it is sold yet..
bad luck i m keeping mine for track, if not, i am happy to sell mine..
Thanks mate. Had a look on ebay but didnt have any luck.
wat about the 350z 18" wheels? interested with that?
I thought about that but I much prefer the v35 wheels.
I'm not in a rush so will just see what pops up over time..
I got a set of 18's. But the problem is that all of them are scrached. So you will need to fix them. Can send you fotos if you want.
Sounds good to me. I was going to paint them black anyway.
You have a PM
Try this guy, he may still have the original - its a 2003 sedan - i went to replace my sun visor from that car the other day.
Its been stripped - but he may still have the originals.
EDIT: Just realised you're in Melbourne. So depending on how desperate and if you want em - may need to ship em.
Edited by g33k
hey mate which sun visor did you replace?
i went there to replace my RHD visor but he wanted $50 and i didnt have time to strip it
Sounds good to me. I was going to paint them black anyway.
I like your idea.
hey mate which sun visor did you replace?
i went there to replace my RHD visor but he wanted $50 and i didnt have time to strip it
I went to replace my passenger side, but it wasn't the same, much dirtier as well.
My 03 Premium has a light on the visor as well as an extender for the visor - the one from that car didnt match - no light - no extender.
Let me know if anyone finds a premium.
found a set on ebay
but there in sydney
found a set on ebay
but there in sydney
I think he's after the coupe 18s. Those are the less desirable 17s | {"url":"https://www.sau.com.au/forums/topic/273776-wtb-standard-v35-wheels/","timestamp":"2024-11-09T12:44:00Z","content_type":"text/html","content_length":"246640","record_id":"<urn:uuid:a62c3caa-0027-4c40-9004-233f80382c07>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00470.warc.gz"} |
In isokinetics (and isometric contractions) a MAP curve is produced whenever a subject pushes against the pad/attachment on the lever arm (no matter how hard). They represent the various force points
the machine records, so if you test your subject’s knee and they push 10Nm at the beginning of the range of motion, then a point will be placed on a graph at the relevant angle/time axis (see below).
If the subject then pushes slightly harder (say 20Nm) as they progress through the range of motion, we get a second point (see below) but this will be further into the movement (greater angle) or
after the first recorded point (later on the time axis).
Following on from this we would probably get various readings throughout the movement giving us allot of points (see below).
If we then drew a line through the points we would have a MAP curve. As we have plotted the moment (force/torque) against the angle (joint/actuator) and this gives us the position! We get one curve
for each contraction.
Remember most isokinetics machine plot one hundred point every second (some like the Humac Norm plot 2000 per second) so the amount of points you get depends on the range of motion and the speed the
machine is set at.
In isotonics force is replaced by velocity but the principle is the same. | {"url":"https://isokinetics.net/map-curves/","timestamp":"2024-11-08T20:44:24Z","content_type":"text/html","content_length":"92407","record_id":"<urn:uuid:2615d6aa-b03b-4e6b-afcb-7f44f005b9a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00197.warc.gz"} |
What percent of 76 is 17.48? | HIX Tutor
What percent of 76 is 17.48?
Answer 1
#17.48/76#=#1748/7600# Divide numerator and denominator by 4 #437/1900# Divide numerator and denominator by 19 #23/100=23%#
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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American Mathematical Society
We consider a semi-algebraic set $S$ defined by $s$ polynomials in $k$ variables which is contained in an algebraic variety $Z(Q)$. The variety is assumed to have real dimension $k’,$ the polynomial
$Q$ and the polynomials defining $S$ have degree at most $d$. We present an algorithm which constructs a roadmap on $S$. The complexity of this algorithm is $s^{k’+1}d^{O(k^2)}$. We also present an
algorithm which, given a point of $S$ defined by polynomials of degree at most $\tau$, constructs a path joining this point to the roadmap. The complexity of this algorithm is $k’ s \tau ^{O(1)} d^{O
(k^2)}.$ These algorithms easily yield an algorithm which, given two points of $S$ defined by polynomials of degree at most $\tau$, decides whether or not these two points of $S$ lie in the same
semi-algebraically connected component of $S$ and if they do computes a semi-algebraic path in $S$ connecting the two points. References
• Saugata Basu, Richard Pollack, and Marie-Françoise Roy, On the combinatorial and algebraic complexity of quantifier elimination, J. ACM 43 (1996), no. 6, 1002–1045. MR 1434910, DOI 10.1145/
• S. Basu, R. Pollack, M.-F. Roy Computing Roadmaps of Semi-algebraic Sets, Proc. 28th Annual ACM Symposium on the Theory of Computing, 168-173, (1996).
• Saugata Basu, Richard Pollack, and Marie-Françoise Roy, On computing a set of points meeting every cell defined by a family of polynomials on a variety, J. Complexity 13 (1997), no. 1, 28–37. MR
1449758, DOI 10.1006/jcom.1997.0434
• S. Basu, R. Pollack, M.-F. Roy Computing Roadmaps of Semi-algebraic Sets on a Variety, Foundations of Computational Mathematics, F. Cucker and M. Shub, Eds., 1–15, (1997).
• J. Bochnak, M. Coste, M.-F. Roy Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 36, Berlin: Springer-Verlag (1998).
• John Canny, The complexity of robot motion planning, ACM Doctoral Dissertation Awards, vol. 1987, MIT Press, Cambridge, MA, 1988. MR 952555
• John Canny, Computing roadmaps of general semi-algebraic sets, Comput. J. 36 (1993), no. 5, 504–514. MR 1234122, DOI 10.1093/comjnl/36.5.504
• J. Canny, D. Yu. Grigor′ev, and N. N. Vorobjov Jr., Finding connected components of a semialgebraic set in subexponential time, Appl. Algebra Engrg. Comm. Comput. 2 (1992), no. 4, 217–238. MR
1325530, DOI 10.1007/BF01614146
• George E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Automata theory and formal languages (Second GI Conf., Kaiserslautern, 1975) Lecture Notes
in Comput. Sci., Vol. 33, Springer, Berlin-New York, 1975, pp. 134–183. MR 403962
• M. Coste and M.-F. Roy, Thom’s lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets, J. Symbolic Comput. 5 (1988), no. 1-2, 121–129. MR 949115,
DOI 10.1016/S0747-7171(88)80008-7
• Michel Coste and Masahiro Shiota, Nash triviality in families of Nash manifolds, Invent. Math. 108 (1992), no. 2, 349–368. MR 1161096, DOI 10.1007/BF02100609
• D. Yu. Grigor′ev and N. N. Vorobjov Jr., Counting connected components of a semialgebraic set in subexponential time, Comput. Complexity 2 (1992), no. 2, 133–186. MR 1190827, DOI 10.1007/
• L. Gournay and J.-J. Risler, Construction of roadmaps in semi-algebraic sets, Appl. Algebra Engrg. Comm. Comput. 4 (1993), no. 4, 239–252. MR 1235859, DOI 10.1007/BF01200148
• Robert M. Hardt, Semi-algebraic local-triviality in semi-algebraic mappings, Amer. J. Math. 102 (1980), no. 2, 291–302. MR 564475, DOI 10.2307/2374240
• Joos Heintz, Marie-Françoise Roy, and Pablo Solernó, Single exponential path finding in semialgebraic sets. I. The case of a regular bounded hypersurface, Applied algebra, algebraic algorithms
and error-correcting codes (Tokyo, 1990) Lecture Notes in Comput. Sci., vol. 508, Springer, Berlin, 1991, pp. 180–196. MR 1123950, DOI 10.1007/3-540-54195-0_{5}0
• Joos Heintz, Marie-Françoise Roy, and Pablo Solernó, Single exponential path finding in semi-algebraic sets. II. The general case, Algebraic geometry and its applications (West Lafayette, IN,
1990) Springer, New York, 1994, pp. 449–465. MR 1272047
• J.-C. Latombe Robot Motion Planning, The Kluwer International Series in Engineering and Computer Science. 124. Dordrecht: Kluwer Academic Publishers Group (1991).
• J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, NJ, 1963. Based on lecture notes by M. Spivak and R. Wells. MR 163331, DOI 10.1515/
• James Renegar, On the computational complexity and geometry of the first-order theory of the reals. I. Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for
the existential theory of the reals, J. Symbolic Comput. 13 (1992), no. 3, 255–299. MR 1156882, DOI 10.1016/S0747-7171(10)80003-3
• F. Rouillier, M.-F. Roy, M. Safey Finding at least a point in each connected component of a real algebraic set defined by a single equation, to appear in Journal of Complexity.
• Marie-Françoise Roy and Nicolai Vorobjov, Finding irreducible components of some real transcendental varieties, Comput. Complexity 4 (1994), no. 2, 107–132. MR 1285473, DOI 10.1007/BF01202285
• M.-F. Roy, N. Vorobjov Computing the Complexification of a Semi-algebraic Set, Proc. of International Symposium on Symbolic and Algebraic Computations, 1996, 26-34 (complete version to appear in
Math. Zeitschrift).
• Jacob T. Schwartz and Micha Sharir, On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds, Adv. in Appl. Math. 4 (1983), no. 3,
298–351. MR 712908, DOI 10.1016/0196-8858(83)90014-3
Similar Articles
• Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 14P10, 68Q25, 68Q40
• Retrieve articles in all journals with MSC (1991): 14P10, 68Q25, 68Q40
Bibliographic Information
• Saugata Basu
• Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
• MR Author ID: 351826
• Email: saugata@math.lsa.umich.edu
• Richard Pollack
• Affiliation: Courant Institute of Mathematical Sciences, New York University, New York, New York 10012
• Email: pollack@cims.nyu.edu
• Marie-Françoise Roy
• Affiliation: IRMAR (URA CNRS 305), Université de Rennes, Campus de Beaulieu 35042 Rennes cedex, France
• Email: mfroy@univ-rennes1.fr
• Received by editor(s): November 25, 1997
• Received by editor(s) in revised form: April 14, 1999
• Published electronically: July 20, 1999
• Additional Notes: The first author was supported in part by NSF Grants CCR-9402640 and CCR-9424398.
The second author was supported in part by NSF Grants CCR-9402640, CCR-9424398, DMS-9400293, CCR-9711240 and CCR-9732101.
The third author was supported in part by the project ESPRIT-LTR 21024 FRISCO and by European Community contract CHRX-CT94-0506.
• © Copyright 1999 American Mathematical Society
• Journal: J. Amer. Math. Soc. 13 (2000), 55-82
• MSC (1991): Primary 14P10, 68Q25; Secondary 68Q40
• DOI: https://doi.org/10.1090/S0894-0347-99-00311-2
• MathSciNet review: 1685780 | {"url":"https://www.ams.org/journals/jams/2000-13-01/S0894-0347-99-00311-2/home.html","timestamp":"2024-11-05T14:14:01Z","content_type":"text/html","content_length":"75224","record_id":"<urn:uuid:a8a7f411-f541-4ca3-897c-50cb14b907a8>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00852.warc.gz"} |
function with LINQ
Today I work on a problem to find if a line is intersected with a circle.
so f1 is y = ax + b and f2 is x^2 + y^2 - r^2 = 0
It is true you can use for-loop to solve this problem, but since I hate for-loop, i will use LINQ.
the input to f1 is { x }, if the line is intersected with a circle means there is (a,b) where f2(a,b) <=0. OK, the data input is {x}, and it is transformed by f1 into some { (x,y) } tuple sequence.
If there is a tuple in the sequence makes f2 <=0, we can draw the conclusion that the line is intersected with a circle.
The pseudo-code is like:
{ x } |> seq.map f1 |> seq.exists f2(x,y) <=0
Let me give detailed code tomorrow | {"url":"http://apollo13cn.blogspot.com/2011/04/function-with-linq.html","timestamp":"2024-11-03T00:14:05Z","content_type":"text/html","content_length":"64982","record_id":"<urn:uuid:6297c0eb-5a97-4805-8f52-7e1729d357df>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00406.warc.gz"} |
Next steps in Quantum Science for HEP
Linear arrays of trapped and laser cooled atomic ions are among the foremost candidates for realizing quantum simulation and computation platforms. High fidelity coherent manipulations together with
nearly perfect detection guarantee an unprecedented control over a large number of qubits, which can be used to run quantum algorithms [1] or engineer Hamiltonians to emulate physical systems of
interest [2]. Recently trapped ions have been employed for proof of principle demonstrations of quantum simulation of high energy physics, including the Dirac equation [3] and the 1+1D Schwinger
model [4]. In this talk I will describe the main features of the trapped-ion quantum hardware and discuss proposals [5,6] and perspectives for an analog implementation of lattice gauge theories in
trapped-ion systems. References [1] S. Debnath et al. Nature 536, 63 (2016) [2] J. Zhang, GP, et al., Nature 551, 601 (2017) [3] E. Martinez, et al., Nature 534, 516 (2016) [4] R. Gerritsma, et al.,
Nature 463, 68 (2010) [5] P. Hauke, et. al., PRX 3, 041018 (2013) [6] D. Yang, et al., PRA 94, 052321 (2016) | {"url":"https://indico.fnal.gov/event/17199/timetable/?print=1&view=standard_numbered","timestamp":"2024-11-04T02:31:45Z","content_type":"text/html","content_length":"148392","record_id":"<urn:uuid:009f491b-b1df-4570-9b9b-289df8ebf2c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00871.warc.gz"} |
Combination Demosaic Algorithms
Finally, this week I finished on the schematic of the whole demosaic core, described it in Verilog, and simulated it using the Icarus Verilog simulator. In the path of implementation, many problems
occurs and it was really interesting to tackle them.
The Edge of Bilinear Interpolation
Bilinear interpolation is a simple and easy method to interpolate the Bayer pixel, which uses side (neighbor) pixels to perform the interpolation. Initially, I was planning to build it with a
switching in interpolation algorithm using multiplexer. I found out that these will actually cost a lot of multiplexer to be used, approximately 13 multiplexer for each color. Where 3 pixels color
would actually takes up to 13 x 3 = 39 multiplexers, just to implement the bilinear interpolation algorithm combined with the HQLI. That is pretty wasteful for resources and inefficient, because the
pipeline that would need to be done would waste more resources too. The consideration of pipelining is critical, as for each stage of pipeline it would cost 8 register to be used, where a 6 stage
pipeline would take up to 48 registers for each pixel.
Considering these inefficient way of implementation, I start to look through the algorithm again.
As shown from figure above, the bilinear interpolation algorithm of a quadrant image has 12 combinations and can be divided into 6 different patterns of interpolation (including first line and last
line, and other boundaries), where it would be a 6 different circuitry to be implemented in hardware. Just like what my supervisor said to me previously, 6 different patterns of circuitry that would
only be used by one or two times in the whole circuit, is extremely inefficient. I actually agreed to this statement, and it triggers my thinking on the implementation when I was drawing the huge
multiplexers design.
The beauty of the bilinear interpolation is that it would always getting the NEIGHBOUR pixels only, which means that whenever pixel that you select, the interpolation pattern would be similar of
using its neighbour pixels. Thus, I thought of minimize the 6 patterns, the first line and the final line of the algorithm can actually be combined. Consider the diagram below, we have 5 sets of
shift register representing 5 lines respectively, where line 1 represent the first set of shift register and line 5 represent the last set of shift register. The first line and last line
interpolation actually uses “line 3 and line 4” and “line 2 and line 3” respectively.
These two line are in common of their respective line is on the set C shift register, where set D and set B are the neighbor pixels for first line and last line respectively. Thus, a multiplexer can
be used on to select set D or set B, for first line or last line interpolation, which saved the implementation of a new pattern circuitry. The first pixel on the boundary side can also be implemented
using the same method, but it would be using more multiplexer to implement such logic. After several drafting and comparison, I found that such implementation is not needed as the circuitry for it is
repetitive and I could directly plug the output from the repetitive circuitry into it.
Combine Bilinear Interpolation with High-Quality Linear Interpolation
In order to combine the HQLI with bilinear interpolation, I noticed that the HQLI algorithm that had been designed in the first place have a good advantage in combining the bilinear interpolation.
Besides, the HQLI algorithm had sorted out that which kind of pixel (R,G,B) that could be ported into the algorithm, and most of the bilinear interpolation is part of the original HQLI pattern. Thus,
knowing the pixel that could be input into the algorithm, I did a multiplexing in the selection of algorithm and link it to the bilinear interpolation, which saved the implementation of bilinear
pattern, as it would be part of the HQLI pattern circuitry. The advantage of such implementation also allows the whole image to interpolate only with bilinear interpolation, which one can examine the
difference of image quality with bilinear interpolation or with HQLI.
With the combined circuitry, it actually reduced the original circuitry into total of 17 multiplexers for all three R,G and B pixels, and with a 5 stage pipeline.
The Two’s Complement Subtraction
As I implement the circuitry that had been drew using Verilog, and simulate it using the Icarus simulator, I found that there are something wrong in the calculation. The calculation answer was in
two’s complement form when it is a negative number. This should never happen because pixel values should be absolute numbers. I made my approach into Xilinx adders and subtractors documentation,
where they specify that the second operand would be treated as two complement. Thus, a two complement subtractor can be implemented by using the following code :
reg [2:0] A, B;
wire [3:0] R = A – B;
where R[3] would be representing the signed bit, to link into the reset port of register.
The Optimization of Demosaic Core
After testing with the RGGB pattern of the demosiac core, I synthesized it using the Xilinx ISE, and it is using 819 FFs. It actually meets the expectation as it has a lower number of FFs compared to
the example I provided in the previous blog. As I looked through the circuit, I think of several optimization can still be done to improve the circuitry, which could reduces the number of Flip-Flops
to be used.
• The Division before additions : Division which is also using wire shifting can be done prior before the additions, so that the pipelined shift register number, adders and subtractors size could
be reduced. Though the result may vary a little with the actual one.
• The Generation of Demosaic Control signals : The control signals that are generated by the internal circuitry to control the demosaic sequence, can still be optimized.
Since the simulation had done, and I had verified the calculated result for RGGB combination, I am really looking forward for implementing it on the FPGA board next week. 😀
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NCERT Solutions For Class 10 Maths Chapter 14 Statistics Ex 14.3
Get Free NCERT Solutions for Class 10 Maths Chapter 14 Ex 14.3 PDF. Statistics Class 10 Maths NCERT Solutions are extremely helpful while doing your homework. Exercise 14.3 Class 10 Maths NCERT
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The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them.
Question 2.
If the median of the distribution given below is 28.5, find the values of x and y.
Question 3.
A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to persons having age 18 years onwards but less
than 60 years.
Question 4.
The lengths of 40 leaves of a plant are measured correct to nearest millimetre, and the data obtained is represented in the following table:
Find the median length of the leaves.
Question 5.
The following table gives the distribution of the lifetime of 400 neon lamps:
Find the median lifetime of a lamp.
Question 6.
100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabet in the surnames was obtained as follows:
Determine the median number of letters in the surnames. Find the mean number of letters in the surnames. Also, find the modal size of the surnames.
Question 7.
The distribution below gives the weight of 30 students of a class. Find the median weight of the students. | {"url":"https://ncert-books.in/ncert-solutions-for-class-10-maths-chapter-14-statistics-ex-14-3/","timestamp":"2024-11-01T19:37:02Z","content_type":"text/html","content_length":"164263","record_id":"<urn:uuid:450823b9-ffe6-43d7-abff-7ae946bb1045>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00430.warc.gz"} |
Consider A Spherical Cow | The Brilliance Mine
How do we solve problems? How do you teach others on your team to solve problems? Have you consider a spherical cow point of view in solving problems?
In a recent nugget, I talked about complexity and how to stop it from being a “hot potato” that keeps being thrown into your direction. In problem-solving, we seek to reduce complexity as quickly as
possible; intelligently, of course. I mentioned the complexity quotient (CQ). A person can raise their CQ through practice with dealing with complexity.
In this nugget, let’s talk about another aspect of problem-solving: Approximations. Approximations you can do in your head or on the back of an envelope. In other words, how can you estimate an
answer without doing detailed calculations on your computer?
I worked as a teaching assistant for undergraduate chemistry students in graduate school. I was often amazed at students “blindly” writing down the answers they got from their calculators. They
didn’t check whether their result made sense. If the answer was much larger or smaller than even made sense, they often did not stop to notice.
A Learned Skill: Look At The Big Picture While Having Intricate Details Before You
Checking whether an answer makes sense is a learned skill. It is part of problem-solving. You have to look at the big picture even though you also deal with intricate details.
When I was a postdoctoral fellow, I learned about a great book by John Harte, “ Consider A Spherical Cow: A course in environmental problem-solving.” The book’s object is to practice estimating an
answer to a question within an order of magnitude. The book is valuable far beyond the realm of environmental problem-solving. The thought process applies to many different types of problems.
A Simple Example
Here is a simple example (the approach is from the book; the specific example is not). Let’s say a friend tells you he heard in the news that the number of cars registered in the U.S. has jumped to
600 million vehicles. You think, “Wow, this sounds quite high!” You wonder: “Can this number be valid?“
Of course, you could search for the answer on the internet. But what is the fun in that? Plus, we are talking about solving problems here, and many have no answer on the internet.
How could we get to the approximate answer through estimating?
• We have to ask a series of questions and make estimates.
• All estimates will be somewhat wrong. But there are likely some error cancellations.
• Furthermore, we are not going after the exact answer. We want an approximate one to see whether what our friend said makes approximately sense.
• Whatwe are doing it like a litmus test.
How many cars are registered in the U.S.? Is 600 Million a reasonable number?
1. Let’s consider that the U.S. has a population of about 330 million people. Even if you think it is 300 Mill or 350 Mill people, 600 Million cars sound suspicious right away. The friend’s
statement means we have nearly two vehicles per person, including kids under 16?
2. We could assume that 75% of the population is 16 years of age and older. That is about 250 million people who could potentially drive a car. That is still a significant number but more than a
factor two less than what the friend said.
3. Further considerations:
1. Not everyone who can drive can afford or even wants a car.
2. Not all teenagers own a vehicle.
3. Some seniors can’t drive anymore.
4. Not all families have two cars. Some families may have more than two cars.
4. A crude assumption would be that 50% of everyone who could drive owns a car. That would be 124 Mill cars.
5. Even if we assume that 75% of people over 16 have cars, the answer (186 Mill) is still far less than 600 Mill.
The Thought Process Is The Point. How Do We Dissect The Problem?
The point here is not the math per se. It is not about this simple example either.
It is the thought process. How do we dissect the problem? As it turns out, dissecting is also about reducing complexity. We make that cow spherical to think our way through the problem without
getting tangled up because the cow has legs, a head, and maybe horns.
I’m Curious
• How do you problem-solve?
• How do you teach that to others? | {"url":"https://brilliancenuggets.com/consider-a-spherical-cow/","timestamp":"2024-11-06T23:32:33Z","content_type":"text/html","content_length":"133847","record_id":"<urn:uuid:7bb9b7dd-b947-437a-8ce0-68160f0ec683>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00558.warc.gz"} |
[Solved] You would like to construct a 95% confide | SolutionInn
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Verified Expert Solution
You would like to construct a 95% confidence interval to estimate the population mean price of milk (per gallon) in your city. You select
You would like to construct a 95% confidence interval to estimate the population mean price of milk (per gallon) in your city. You select a random sample of prices from different stores. The sample
has a mean of 3.65 dollars and a standard deviation of 0.29 dollars. (a) What is the best point estimate, based on the sample, to use for the population mean? |dollars (b) For each of the following
sampling scenarios, determine which distribution should be used to calculate the critical value for the 95% confidence interval for the population mean. (In the table, Z refers to a standard normal
distribution, and t refers to a t distribution.) Could use Sampling scenario Z t Unclear either Z or t The sample has size 20, and it is from a normally distributed population with an unknown
standard deviation. The sample has size 100, and it is from a non-normally distributed population. The sample has size 80, and it is from a non-normally distributed population with a known standard
deviation of 0.28.
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Authors: Robert A. Donnelly
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Dimension - (Computational Geometry) - Vocab, Definition, Explanations | Fiveable
from class:
Computational Geometry
Dimension refers to the minimum number of coordinates needed to specify a point within a given space. This concept helps us understand various geometric structures and their properties, such as
simplicial complexes, configuration spaces, and geometric primitives. Dimension not only indicates how many parameters are necessary for a complete description of an object but also impacts how these
objects can interact and be manipulated in a given environment.
congrats on reading the definition of Dimension. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. In a 2D space, you need two coordinates (like x and y) to define a point, whereas in 3D space, three coordinates (x, y, z) are needed.
2. Simplicial complexes utilize dimensions to create structures like triangles (2D) and tetrahedra (3D), serving as fundamental building blocks in computational geometry.
3. The configuration space can be thought of as a higher-dimensional space where each dimension represents a different degree of freedom for the system being analyzed.
4. When discussing geometric primitives, the concept of dimension helps distinguish between points (0D), lines (1D), surfaces (2D), and solids (3D).
5. Understanding dimension is essential for algorithms that handle spatial data because it affects how data structures are organized and manipulated.
Review Questions
• How does the concept of dimension influence the construction of simplicial complexes?
□ Dimension plays a critical role in the construction of simplicial complexes as it determines the types of simplices that can be used. For example, in 2D, triangles are used to create
surfaces, while in 3D, tetrahedra are used to build volumetric shapes. Each simplex contributes to the overall dimensionality of the complex and impacts its combinatorial and topological
properties. Understanding how these dimensions relate helps in analyzing the structure and behavior of the entire complex.
• Discuss the importance of dimension in configuration spaces and how it affects problem-solving in robotics.
□ Dimension is vital in configuration spaces as it represents all possible positions and orientations of an object within its environment. Each dimension corresponds to a degree of freedom,
making it essential for robotic motion planning. The higher the dimension, the more complex the space becomes, making pathfinding algorithms more challenging. Understanding how to navigate
through this multi-dimensional space allows for more effective solutions in robotics and motion control.
• Evaluate how understanding dimensions can enhance our knowledge of geometric primitives and their applications in computational geometry.
□ Grasping the concept of dimensions enhances our understanding of geometric primitives by allowing us to categorize shapes based on their dimensional attributes. For example, recognizing that
points are 0D, lines are 1D, surfaces are 2D, and solids are 3D provides a framework for analyzing their properties and behaviors. This knowledge is crucial when applying algorithms for tasks
such as rendering graphics or solving spatial problems in computational geometry. A strong grasp of dimensionality allows for better design choices and optimizations across various
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Adding and Subtracting Decimal Worksheets , Free Simple Printable - BYJU'S
Frequently Asked Questions
Learning addition and subtraction of decimal numbers using math models will help students understand the operations easily. Students can refer to the articles on this concept on BYJU’S Math to learn
about the addition and subtraction of decimal numbers using models.
Problems related to money often involve the addition and subtraction of decimal numbers. So, students who have a thorough understanding of these operations on decimal numbers will be able to quickly
solve problems related to money.
Yes, we can estimate the sum and difference of two decimal numbers using simple methods. Students can refer to the article related to the estimation of sum and difference of decimal numbers to learn
these methods.
Students can perform the addition and subtraction of decimal numbers using mental math techniques as explained in the common core syllabus for grade 5, and covered in the form of questions in the
printable and online worksheets for the addition and subtraction of decimals on the BYJU’S Math website
Learning to add and subtract decimal numbers will let students solve many real-life problems that involve quantities like money, length, mass, and so on. | {"url":"https://byjus.com/us/math/adding-and-subtracting-decimal-worksheets/","timestamp":"2024-11-03T13:27:11Z","content_type":"text/html","content_length":"160479","record_id":"<urn:uuid:2236473f-a14f-4275-9452-4aeb71f5ee3f>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00758.warc.gz"} |
Geometric interpretation of the weak-field Hall conductivity in two-dimensional metals with arbitrary Fermi surface
The Hall conductivity xy of a two-dimensional metal in the weak-field, semiclassical, limit has a simple geometric representation. xy (normalized to e2/h, where e is the electron charge and h is
Plancks constant), is equal to twice the number of flux quanta 0 threading the area Al, where Al is the total Stokes area swept out by the scattering path length l(k) as k circumscribes the Fermi
surface (FS). From this perspective, many properties of xy become self-evident. The representation provides a powerful way to disentangle the distinct contributions of the three factors, FS
area-to-circumference ratio, anisotropy in lk, and negative FS curvature. The analysis is applied to the Hall data on 2H-NbSe2 and the cuprate perovskites. Previous model calculations of xy are
critically reexamined using the new representation.
All Science Journal Classification (ASJC) codes
Dive into the research topics of 'Geometric interpretation of the weak-field Hall conductivity in two-dimensional metals with arbitrary Fermi surface'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/geometric-interpretation-of-the-weak-field-hall-conductivity-in-t","timestamp":"2024-11-09T11:21:10Z","content_type":"text/html","content_length":"49676","record_id":"<urn:uuid:7d59c5af-58c8-4740-bee0-24954f2ca24f>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00248.warc.gz"} |
Calculating Volume of Rectangular Prisms
Note: this page contains legacy resources that are no longer supported. You are free to continue using these materials but we can only support our current worksheets, available as part of our
membership offering.
Calculating Volume of Rectangular Prisms
Related Resources
The various resources listed below are aligned to the same standard, (6G02) taken from the CCSM (Common Core Standards For Mathematics) as the Geometry Worksheet shown above.
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be
found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving
real-world and mathematical problems.
Similar to the above listing, the resources below are aligned to related standards in the Common Core For Mathematics that together support the following learning outcome:
Solve real-world and mathematical problems involving area, surface area, and volume | {"url":"https://helpingwithmath.com/generators/geo0601volume01/","timestamp":"2024-11-05T06:06:43Z","content_type":"text/html","content_length":"111070","record_id":"<urn:uuid:f59c1dc2-7938-466e-be7b-05e7f6cb3104>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00366.warc.gz"} |
Polynomial Time Boolean Satisfiability
Submit solution
Points: 15 (partial)
Time limit: 4.0s
Memory limit: 512M
The boolean satisfiability problem is a famous problem in computer science. You are given booleans, and a list of numbers. A boolean is satisfied if a subset from the numbers sum up to less than or
equal to . What is the maximum number of booleans that can be satisfied?
Note: the empty subset sums up to 0.
Input Specification
On the first line, there are two integers , and , separated by a space.
On the second line is a space separated list of the integers.
Each line is followed by one line feed character (ASCII code 0x0a).
There are no trailing spaces or empty lines.
Output Specification
One integer, the number of subsets that sum to less than or equal to .
For all subtasks:
For all subset sums, , .
Subtask 1 [2/15]
Subtask 2 [7/15]
Subtask 3 [6/15]
No additional constraints.
Sample Input 1
Sample Output 1
Sample Input 2
Sample Output 2
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software
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CSIR UGC NET Syllabus 2021 for Engineering Sciences
by admin | Feb 12, 2021 | All India Entrance Exam Syllabus, Common Entrance Test CET Syllabus, Post Graduate Entrance Exam Syllabus | 0 comments
CSIR UGC NET Exam Syllabus 2021 for Engineering Sciences
CSIR UGC NET Engineering Sciences Syllabus 2021: UGC NET is conducted by CBSE on behalf of UGC for determining the eligibility of Indian nationals for the Eligibility for Assistant Professor only or
Junior Research Fellowship & Eligibility for Assistant Professor both in Indian Universities and Colleges. This page includes syllabus for CBSE UGC NET 2021 Engineering Sciences.
Given Below is the Syllabus for CSIR UGC NET Engineering Sciences Exam. CSIR had recently introduced a New Subject “Engineering Sciences” with the intention to encourage engineering graduates to
pursue PhD Courses.
Engineering Sciences
Part ‘A’: This part shall carry 20 questions pertaining to General Aptitude with Emphasis on Logical Reasoning Graphical Analysis, Analytical and Numerical Ability, Quantitative Comparisons, Series
Formation, Puzzles etc
The candidates shall be required to answer any 15 questions
Each question shall be of two marks
The total marks allocated to this section shall be 30 out of 200
Also See: UGC NET 2021 Notification
Part ‘B’
Mathematics and Engineering Aptitude
Linear Algebra: Algebra of Matrices, Inverse, Rank, System of Linear Equations, Symmetric, Skew-symmetric and Orthogonal Matrices
Hermitian, skew-Hermitian and unitary matrices
Eigenvalues and Eigenvectors, Diagonalisation of Matrices
Calculus: Functions of single variable, limit, continuity and differentiability, Mean value theorems, Indeterminate forms and L’Hospital rule, Maxima and minima, Taylor’s series, Newton’s method for
finding roots of polynomials
Fundamental and mean value-theorems of integral calculus
Numerical integration by trapezoidal and Simpson’s rule
Evaluation of definite and improper integrals, Beta and Gamma functions, Functions of two variables, limit, continuity, partial derivatives, Euler’s theorem for homogeneous functions, total
derivatives, maxima and minima, Lagrange method of multipliers, double integrals and their applications, sequence and series, tests for convergence, power series, Fourier Series, Half range sine and
cosine series
Complex Variables: Analytic functions, Cauchy-Riemann equations, Line integral, Cauchy’s integral theorem and integral formula Taylor’s and Laurent’ series, Residue theorem and its applications
Vector Calculus: Gradient, divergence and curl, vector identities, directional derivatives, line, surface and volume integrals, Stokes, Gauss and Green’s theorems and their applications
Ordinary Differential Equations: First order equation (linear and nonlinear), second order linear differential equations with variable coefficients, Variation of parameters method, higher order
linear differential equations with constant coefficients, Cauchy-Euler’s equations, power series solutions, Legendre polynomials and Bessel’s functions of the first kind and their properties
Numerical solutions of first order ordinary differential equations by Euler’s and Runge-Kutta methods
Probability: Definitions of probability and simple theorems, conditional probability, Bayes Theorem
Solid Body Motion and Fluid Motion: Particle dynamics; Projectiles; Rigid Body Dynamics; Lagrangian formulation; Eularian formulation; Bernoulli’s Equation; Continuity equation; Surface tension;
Viscosity; Brownian Motion
Energetics: Laws of Thermodynamics; Concept of Free energy; Enthalpy, and Entropy; Equation of State; Thermodynamics relations
Electron Transport: Structure of atoms, Concept of energy level, Bond Theory; Definition of conduction, Semiconductor and Insulators; Diode; Half wave & Full wave rectification; Amplifiers &
Oscillators; Truth Table
Electromagnetics: Theory of Electric and Magnetic potential and field; Biot and Savart’s Law; Theory of Dipole; Theory of Oscillation of electron; Maxwell’s equations; Transmission theory; Amplitude
and Frequency Modulation
Materials: Periodic table; Properties of elements; Reaction of materials; Metals and non-Metals (Inorganic materials), Elementary knowledge of monomeric and polymeric compounds; Organometallic
compounds; Crystal structure and symmetry, Structure-property correlation-metals, ceramics, and polymers
Part ‘C’
Computer Science and Information Technology
Basic Discrete Mathematics: Counting principles, linear recurrence, mathematical induction, equation sets, relations and function, predicate and propositional logic
Digital Logic: Logic functions, Minimization, Design and synthesis of combinational and sequential circuits; Number representation and computer arithmetic (fixed and floating point)
Computer Organization and Architecture: Machine instructions and addressing modes, ALU and data-path, CPU control design, Memory interface, I/O interface (Interrupt and DMA mode), Instruction
pipelining, Cache and main memory, Secondary storage
Programming and Data Structures: Programming in C; Functions, Recursion, Parameter passing, Scope, Binding; Abstract data types, Arrays, Stacks, Queues, Linked Lists, Trees, Binary search trees,
Binary heaps
Algorithms: Analysis, Asymptotic notation, Notions of space and time complexity, Worst and average case analysis; Design: Greedy approach, Dynamic programming, Divide-and conquer; Tree and graph
traversals, Connected components, Spanning trees, Shortest paths; Hashing, Sorting, Searching
Asymptotic analysis (best, worst, average cases) of time and space, upper and lower bounds, Basic concepts of complexity classes P, NP, NP-hard, NP-complete
Operating System: Processes, Threads, Inter-process communication, Concurrency, Synchronization, Deadlock, CPU scheduling, Memory management and virtual memory, File systems
Databases: ER-model, Relational model (relational algebra, tuple calculus), Database design (integrity constraints, normal forms), Query languages (SQL), File structures (sequential files, indexing,
B and B+ trees), Transactions and concurrency control
Information Systems and Software Engineering: Information gathering, requirement and feasibility analysis, data flow diagrams, process specifications, input/output design, process life cycle,
planning and managing the project, design, coding, testing, implementation, maintenance
Electrical Sciences
Electric Circuits and Fields: Node and mesh analysis, transient response of dc and ac networks, sinusoidal steady-state analysis, resonance, basic filter concepts, ideal current and voltage sources,
Thevenin’s, Norton’s and Superposition and Maximum Power Transfer theorems, two port networks, three phase circuits, measurement of power in three phase circuits, Gauss Theorem, electric field and
potential due to point, line, plane and spherical charge distributions, Ampere’s and Biot-Savart’s laws, inductance, dielectrics, capacitance
Electrical Machines: Magnetic circuits
Magnetic circuits, Single phase transformer- equivalent circuit, phasor diagram, tests, regulation and efficiency, three phase transformers- connections, parallel operation, auto-transformer; energy
conversion principles, DC Machines- types , starting and speed control of dc motors, three phase induction motors- principles, types, performance characteristics, starting and speed control, single
phase induction motors, synchronous machines performance, regulation and parallel operation of synchronous machine operating as generators, starting and speed control of synchronous motors and its
applications, servo and stepper motors
Power Systems: Basic power generation concepts, transmission line models and performance, cable performance, insulation, corona and radio interference, Distribution systems, per-unit quantities, bus
impedance and admittance matrices, load flow, voltage and frequency control, power factor correction; unbalanced analysis, symmetrical components, basic concepts of protection and stability;
Introduction to HVDC systems
Control Systems: Principles of feedback control, transfer function, block diagrams, steady state errors, Routh and Nyquist techniques, Bode plots, Root loci, Lag, Lead and Lead-lag compensation;
proportional, PI, PID controllers, state space model, state transition matrix, controllability and observability
Power Electronics and Drives: Semiconductor Power devices-power diodes, power transistors, thyristors, triacs, GTOs, MOSFETs, IGBTs-their characteristics and basic triggering circuits; diode
rectifiers, thyristor based line commutated ac to dc converters, dc to dc converters-buck, boost, buck-boost, c‘uk, flyback, forward, push-pull converters, single phase and three phase dc to ac
inverters and related pulse width modulation techniques, stability of electric drives; speed control issues of dc motors, induction motors and synchronous motors
Analog Circuits and Systems: Electronic devices: characteristics and small-signal equivalent circuits of diodes, BJTs and MOSFETs. Diode circuits: clipping, clamping and rectifier. Biasing and bias
stability of BJT and FET amplifiers. Amplifiers: single-and multi-stage, differential and operational, feedback, and power. Frequency response of amplifiers. Op-amp circuits: voltage-to-current and
current-to voltage converters, active filters, sinusoidal oscillators, wave-shaping circuits, effect of practical parameters (input bias current, input offset voltage, open loop gain, input
resistance, CMRR). Electronic measurements: voltage, current, impedance, time, phase, frequency measurements, oscilloscope
Digital Circuits and Systems: Boolean algebra and minimization of Boolean functions. Logic gates, TTL and CMOS IC families. Combinatorial circuits: arithmetic circuits, code converters, multiplexers
and decoders. Sequential circuits: latches and flip-flops, counters and shift registers. Sample-and-hold circuits, ADCs, DACs. Microprocessors and microcontrollers: number systems, 8085 and 8051
architecture, memory, I/O interfacing, Serial and parallel communication
Signals and Systems: Linear time invariant systems: impulse response, transfer function and frequency response of first- and second order systems, convolution. Random signals and noise: probability,
random variables, probability density function, autocorrelation, power spectral density. Sampling theorem, Discrete-time systems: impulse and frequency response, IIR and FIR filters
Communications: Amplitude and angle modulation and demodulation, frequency and time division multiplexing. Pulse code modulation, amplitude shift keying, frequency shift keying and pulse shift keying
for digital modulation. Bandwidth and SNR calculations. Information theory and channel capacity
Materials Science
Structure: Atomic structure and bonding in materials. Crystal structure of materials, crystal systems, unit cells and space lattices, miller indices of planes and directions, packing geometry in
metallic, ionic and covalent solids. Concept of amorphous, single and polycrystalline structures and their effect on properties of materials. Imperfections in crystalline solids and their role in
influencing various properties
Diffusion: Fick’s laws and application of diffusion
Metals and Alloys: Solid solutions, solubility limit, phase rule, binary phase diagrams, intermediate phases, intermetallic compounds, iron-iron carbide phase diagram, heat treatment of steels, cold,
hot working of metals, recovery, recrystallization and grain growth. Microstructure, properties and applications of ferrous and non-ferrous alloys
Ceramics, Polymers, and Composites: Structure, properties, processing and applications of ceramics. Classification, polymerization, structure and properties, processing and applications. Properties
and applications of various composites
Materials Characterization Tools: X-ray diffraction, optical microscopy, scanning electron microscopy and transmission electron microscopy, differential thermal analysis, differential scanning
Materials Properties: Stress-strain diagrams of metallic, ceramic and polymeric materials, modulus of elasticity, yield strength, tensile strength, toughness, elongation, plastic deformation,
viscoelasticity, hardness, impact strength, creep, fatigue, ductile and brittle fracture. Heat capacity, thermal conductivity, thermal expansion of materials. Concept of energy band diagram for
materials-conductors, semiconductors and insulators, intrinsic and extrinsic semiconductors, dielectric properties. Origin of magnetism in metallic and ceramic materials, paramagnetism, diamagnetism,
antiferro magnetism, ferromagnetism, ferrimagnetism, magnetic hysteresis
Environmental Degradation: Corrosion and oxidation of materials, prevention
Fluid Mechanics
Fluid Properties: Relation between stress and strain rate for Newtonian fluids; Buoyancy, manometry, forces on submerged bodies
Kinematics: Eulerian and Lagrangian description of fluid motion, strain rate and vorticity; concept of local and convective accelerations, steady and unsteady flows
Control Volume Based Analysis: Control volume analysis for mass, momentum and energy. Differential equations of mass and momentum (Euler equation), Bernoulli’s equation and its applications, Concept
of fluid rotation
Potential Flow: Vorticity, Stream function and Velocity potential function; Elementary flow fields and principles of superposition, potential flow past a circular cylinder
Dimensional Analysis: Concept of geometric, kinematic and dynamic similarity, Non-dimensional numbers and their usage
Viscous Flows: Navier-Stokes Equations; Exact Solutions; Couette Flow, Fully-developed pipe flow, Hydrodynamic lubrication, Basic ideas of Laminar and Turbulent flows, Prandtl-mixing length, Friction
factor, Darcy-Weisbach relation, Simple pipe networks
Boundary Layer: Qualitative ideas of boundary layer, Boundary Layer Equation; Separation, Streamlined and bluff bodies, drag and lift forces
Measurements: Basic ideas of flow measurement using venturimeter, pitot-static tube and orifice plate
Solid Mechanics: Equivalent force systems; free-body diagrams; equilibrium equations; analysis of determinate trusses and frames; friction; simple particle dynamics; plane kinematics and kinetics;
work-energy and impulse-momentum principles; stresses and strains; principal stresses and strains; Mohr’s circle; generalized Hooke’s Law; thermal strain. Axial, shear and bending moment diagrams;
axial, shear and bending stresses; deflection of beams (symmetric bending); Torsion in circular shafts; thin walled pressure vessels. Energy methods (Catigliano’s theorems) for analysis. Combined
axial, bending and torsional action; Theories of failure. Buckling of columns. Free vibration of single degree of freedom systems
Basic Concepts: Continuum, macroscopic approach, thermodynamic system (closed and open or control volume); thermodynamic properties and equilibrium; state of a system, state diagram, path and
process; different modes of work; Zeroth law of thermodynamics; concept of temperature; heat
First Law of Thermodynamics: Energy, enthalpy, specific heats, first law applied to closed systems and open systems (control volumes), steady and unsteady flow analysis
Second Law of Thermodynamics: Kelvin-Planck and Clausius statements, reversible and irreversible processes, Carnot theorems, thermodynamic temperature scale, Clausius inequality and concept of
entropy, principle of increase of entropy, entropy balance for closed and open systems, exergy (availability) and irreversibility, non-flow and flow exergy
Properties of Pure Substances: Thermodynamic properties of pure substances in solid, liquid and vapor phases, P-V-T behaviour of simple compressible substances, phase rule, thermodynamic property
tables and charts, ideal and real gases, equations of state, compressibility chart
Thermodynamic Relations: T-ds relations, Maxwell equations, Joule-Thomson coefficient, coefficient of volume expansion, adiabatic and isothermal compressibilities, Clapeyron equation
Thermodynamic Cycles: Carnot vapour power cycle; simple Rankine cycle, reheat and regenerative Rankine cycle; Air standard cycles: Otto cycle, Diesel cycle, simple Brayton cycle, Brayton cycle with
regeneration, reheat and intercooling; vapour-compression refrigeration cycle
Ideal Gas Mixtures: Dalton’s and Amagat’s laws, calculations of properties (internal energy, enthalpy, entropy), air-water vapour mixtures and simple thermodynamic processes involving them.
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NDA Exam syllabus 2024 - Commandant Defence Academy In Dehradun..
Complete Written NDA Exam syllabus 2024
NDA (National Defense Academy) exam is a respected entrance exam conducted by the Union Public Service Commission (UPSC) of India. It is the gateway for unmarried men or women to join Indian Armed
Forces. This exam is held twice a year and it has a strict selection process and the exam is divided in two parts Written exam and SSB Interview. Written exam is further divided into two parts
Mathematics and General Aptitude Test (GAT) and only Candidates who obtain the set cut-off by UPSC in written exam are called for SSB interview conducted by the Service Selection Board
(SSB).Successful candidates receive training at the National Defense Academy and are appointed as officers after completing their training. The NDA test evaluates a candidate’s academic, physical and
mental suitability for service in the Defense Forces. In this blog we will share with you complete NDA written exam syllabus.
NDA exam syllabus Mathematics (300 marks)
Overview of NDA Mathematics exam
Total Questions 120
Total Marks 300
Duration 2 ½ hour
Exam format MCQs
Correct Answer 2.5 mark
Wrong Answer -(0.83) mark
Topics in mathematics include algebra, matrices and determinants, trigonometry, geometry, differentials, integrals and differential
equations, vector algebra, statistics, and probability. These questions are designed to test the candidate’s understanding of mathematical concepts and problem-solving abilities.
NDA Mathematics Syllabus
Topic Sub Topic
1.Sets (Concepts & Operations) 2.Venn diagram 3.De Morgan’s Law 4.Cartesian Product 5.Relation 6.Equivalence Relation 7.Real Numbers 8.Complex Numbers 9.Modulus 10.Cube Root
Algebra 11.Conversion of a number (Binary to Decimal & Decimal to Binary) 12.Arithmetic 13.Geometric and Harmonic Progressions 14.Quadratic Equations 15.Linear Inequations 16.Permutation
and Combination 17.Binomial Theorem 18.Logarithms
1.Concept of a real valued function 2.Domain 3.Range and Graph of a function 4.Composite functions 5.One to One 6.Onto and Inverse Functions 7.Notion of limit 8.Standard limits
Calculus 9.Continuity of functions 10.Algebric Operations on Continuous functions 11.Derivative of function at a point 12.Geometrical and Physical Interpretation of a derivative
application 13.Derivatives of sum 14.Product and Quotient of functions 15.Derivative of a function with respect to another function 16.Derivative of a Composite Function 17.Second
Order Derivatives 18.Increasing and Decreasing Function 19.Application of Derivatives in problems of Maxima and Minima
Matrices and 1.Types of matrices 2.Operations on matrices 3.Determinant of a matrix 4.Basic Properties of Determinants 5.Adjoint and Inverse of a Square Matrix 6.Applications-Solution of a
Determinants system of Linear Equations in two or three unknown by – · Cramer’s Rule · Matrix Method
Integral Calculus 1.Integration as inverse of differentiation 2.Integration by substitution and by parts 3.Standard Integrals involving algebraic Expressions 4.Trigonometric 5.Exponential and
and Differential Hyperbolic Functions 6.Evaluation of definite Integrals – Determination of areas of plane regions bounded by curves-applications 7.Definition of order and degree of a differential
Equations equation by examples. 8.General and particular solution of differential equations 9.Solution of first order and first-degree differential equations of various types by examples
10.Application in problems of growth and decay
Trigonometry 1.Angles and their measures in degrees and in radius 2.Trigonometric Ratio 3.Trigonometric Identities 4.Sum and Difference Formulae 5.Multiple and Sub-Multiple Angles 6.Inverse
Trigonometric Functions 7.Applications – Height and Distance 8.Properties of Triangles
Vector Algebra 1.Vectors in two and three dimensions 2.Magnitude and Direction of a vector 3.Unit and Null Vectors 4.The Addition of Vectors 5.Scalar Multiplication of a Vector 6.Scalar Product
7.Dot Product of two vectors 8.Vector product or Cross product of two vectors 9.Applications- Work done by Force and Moment of Force in Geometrical Problems.
Analytical Geometry 1.Rectangular Cartesian Coordinate System 2.Distance Formula 3.Equation of a line in various forms 4.The angle between two lines 5.Distance of a point from a line 6.Equation of a
of Two or Three circle in standard and in a general form 7.Standard forms of Parabola, Ellipse and Hyperbola 8.Eccentricity and Axis of a conic 9.Point in a three-dimensional space 10.The
Dimension distance between two points 11.Direction, Cosines and Direction Ratio 12. Equation two points 13.Direction Cosines and direction ratios 14. Equation of a plane and a line in
various forms 15.Angle between two lines and angle between two planes 16.Equation of a sphere
1.Probability: Random experiment, outcomes, and associated sample space, events, mutually exclusive and exhaustive events, impossible and certain events 2.Union and Intersection
Statistics and of events. Complementary, elementary, and composite events 3.Definition of probability—classical and statistical—examples 4.Elementary theorems on probability-simple problems 5.
Probability Conditional probability, Bayes’ theorem— simple problems 6. Random variable as function on a sample space 7.Binomial Distribution 8. Examples of random experiments giving rise to
Binomial distribution
General Ability Test (GAT) (600 marks)
Quick Overview of NDA GAT exam
Total Questions 150
Total Marks 600
Subjects English, Physics, Chemistry, General Science, Geography, Current Affairs, and History.
English 50 Questions
General Knowledge 100 Questions
Exam duration 2 ½ hour
Correct Answer 4 mark
Wrong A -(1.3) mark
GAT paper evaluates the candidate’s awareness in current affair and knowledge in English, Physics, Chemistry, General Science, Geography.
GAT Syllabus for NDA Exam
Subject Topic
1. Physical Properties and States of Matter 2. Modes of transference of Heat 3. Mass, Weight, Volume, Sound waves and their properties 4. Simple musical instruments 5. Rectilinear
propagation of Light 6. Density and Specific Gravity 7.Reflection and refraction 8.Principle of Archimedes 9. Spherical mirrors and Lenses 10. Pressure Barometer 11. Human Eye 12. Motion of
objects 13. Natural and Artificial Magnets 14. Velocity and Acceleration 15. Properties of a Magnet 16. Newton’s Laws of Motion 17. Earth as a Magnet 18. Force and Momentum 19. Static and
Physics Current Electricity 20. Parallelogram of Forces 21. Conductors and Non-conductors 22. Stability and Equilibrium of bodies 23. Ohm’s Law 24. Gravitation 25. Simple Electrical Circuits 26.
Elementary ideas of work 27. Heating, Lighting, and Magnetic effects of Current 28. Power and Energy 29. Measurement of Electrical Power 30. Effects of Heat 31. Primary and Secondary Cells
32. Measurement of Temperature and Heat 33. Use of X-Rays 34. General Principles in the working of Simple Pendulum, Simple Pulleys, Siphon, Levers, Balloon, Pumps, Hydrometer, Pressure
Cooker, Thermos Flask, Gramophone, Telegraphs, Telephone, Periscope, Telescope, Microscope, Mariner’s Compass; Lightning Conductors, Safety Fuses.
1.Preparation and Preparation and Properties of Hydrogen, Oxygen, Nitrogen and Carbon Dioxide, Oxidation and Reduction. 2.Acids, bases and salts 3.Carbon – Different Forms 4. Physical and
Chemistry Chemical Changes 5. Fertilizers—Natural and Artificial 6. Elements 7. Material used in the preparation of substances like Soap, Glass, Ink, Paper, Cement, Paints, Safety Matches, and
Gunpowder 8. Mixtures and Compounds 9. Elementary ideas about the structure of Atom 10.Symbols, Formulae, and simple ChemicalnEquation 11.Atomic Equivalent and Molecular Weights 12.Law of
Chemical Combination (excluding problems) 13.Valency 14.Properties of Air and Water
1. The Earth, its shape and size 2. Ocean Currents and Tides Atmosphere and its composition 3. Latitudes and Longitudes 4. Temperature and Atmospheric Pressure, Planetary Winds, Cyclones,
and Anticyclones; Humidity; Condensation and Precipitation 5. Concept of time 6. Types of Climate 7. International Date Line 8. Major Natural Regions of the World 9. Movements of Earth and
Geography their effects 10. Regional Geography of India 11. Climate, Natural vegetation. Mineral and Power resources 12. Location and distribution of agricultural and Industrial activities 13. Origin
of Earth. Rocks and their classification 14. Important Sea ports and main sea, land, and air routes of India 15. Weathering—Mechanical and Chemical, Earthquakes and Volcanoes 16. Main items
of Imports and Exports of India
1. Forces shaping the modern world 2. Renaissance 3. Exploration and Discovery; 4. A broad survey of Indian History, with emphasis on Culture and Civilization 5. Freedom Movement in India
6. French Revolution, Industrial Revolution, and Russian Revolution 7. War of American Independence, 8. Impact of Science and Technology on Society 9. Elementary study of Indian
History Constitution and Administration 10. Concept of one World 11. Elementary knowledge of Five-Year Plan of India 12. United Nations, 13. Panchsheel 14. Panchayati Raj, Democracy, Socialism and
Communist 15. Role of India in the present world 16. Co-operatives and Community Development 17. Bhoodan, Sarvodaya, 18. National Integration and Welfare State 19. Basic Teachings of
Mahatma Gandhi
General 1.Common Epidemics, their causes, and prevention 2. Difference between the living and non-living 3. Food—Source of Energy for man 4. Basis of Life—Cells, Protoplasm, and Tissues 5.
Science Constituents of food 6. Growth and Reproduction in Plants and Animals 7. Balanced Diet 8. Elementary knowledge of the Human Body and its important organs 9.The Solar System—Meteors and
Comets, Eclipses. Achievements of Eminent Scientists
The NDA exam is your chance to start a rewarding service career in the Indian Army. By understanding the two main parts of the NDA curriculum: Mathematics and Aptitude Test (GAT), you can create a
study plan. Make sure you use materials such as textbooks, practice papers, and previous year’s exams to reinforce your understanding. Keep calm and believe in yourself;
with dedication and hard work, you can clear the NDA exam and served in Indian armed forces.If you need further assistance we at Commandant Defence Academy provides the Best NDA coaching centre in
Dehradun. , We help you prepare for the written test and the SSB interview, which also includes getting ready for the physical fitness part.
NCERT Books: These books are highly recommended for building a strong foundation in core subjects like Mathematics, Physics, Chemistry, History, Geography, and Political Science.
Reference Books and Coaching Materials: While not mandatory, these resources can supplement your learning by providing additional practice questions, explanations, and insights. Consult with
experienced educators or coaching institutes to determine if such materials align with your learning style and needs.
Mathematics:This section carries 300 marks and consists of 120 objective-type questions.
General Ability Test (GAT):This section carries 600 marks and consists of 150 objective-type questions further divided into:
1. English:50 questions (200 marks)
2. General Knowledge:100 questions (400 marks) covering various topics like Physics, Chemistry, History, Geography, and Current Affairs.
Here are some strategies to improve your problem-solving skills in Mathematics
Master the fundamentals: Ensure a thorough understanding of basic mathematical concepts like arithmetic operations, algebra, geometry, trigonometry, and calculus.
Practice regularly: Regularly solve problems from various topics to solidify your grasp of the underlying concepts.
How can I stay updated on current affairs for the GAT section?
Develop a News Consumption Habit:
• Newspapers:Subscribe to a reputable newspaper (e.g., The Hindu, The Indian Express, Times of India) and dedicate time daily to reading national and international news.
• News Websites:Utilize credible news websites (e.g. The Hindu, Times of India ).
• Magazines and Journals: Consider subscribing to weekly or monthly magazines (e.g., India Today, Frontline, Yojana) that offer deeper analysis of current affairs and provide diverse viewpoints.
• Government Websites: Visit official government websites (e.g., Press Information Bureau (PIB), Ministry of External Affairs(MEA)) to access official statements, reports, and press releases on
various government initiatives and international relations.
• Discussions and Debates: Participate in discussions and debates about current events with friends, family, or online communities. This can help you gain different perspectives, test your
understanding, and solidify your knowledge.
• Note-Taking and Summarization: Develop a habit of taking notes while reading or listening to news. Briefly summarize key points and important information to reinforce your learning and aid in
Additional Tips:
Prioritize Credibility: Always verify the source of information before accepting it as fact. Be wary of fake news and misleading information circulating online.
Focus on Relevant Topics: While staying broadly informed, prioritize news pertaining to national security, defense, foreign policy, social issues, and economic developments, as these are more likely
to be relevant to the NDA exam.
Maintain a Balanced Approach: Don’t overload yourself with information. Aim for regular, focused news consumption to ensure effective learning and retention.
Develop a News Consumption Habit:
• Newspapers:Subscribe to a reputable newspaper (e.g., The Hindu, The Indian Express, Times of India) and dedicate time daily to reading national and international news.
• News Websites:Utilize credible news websites (e.g. THE HINDU, TIMES OF INDIA ).
• Magazines and Journals: Consider subscribing to weekly or monthly magazines (e.g., India Today, Frontline, Yojana) that offer deeper analysis of current affairs and provide diverse viewpoints.
• Government Websites: Visit official government websites (e.g., Press Information Bureau (PIB), https://www.mea.gov.in/) to access official statements, reports, and press releases on various
government initiatives and international relations.
• Discussions and Debates: Participate in discussions and debates about current events with friends, family, or online communities. This can help you gain different perspectives, test your
understanding, and solidify your knowledge.
• Note-Taking and Summarization: Develop a habit of taking notes while reading or listening to news. Briefly summarize key points and important information to reinforce your learning and aid in
Additional Tips:
• Prioritize Credibility: Always verify the source of information before accepting it as fact. Be wary of fake news and misleading information circulating online.
• Focus on Relevant Topics: While staying broadly informed, prioritize news pertaining to national security, defense, foreign policy, social issues, and economic developments, as these are more
likely to be relevant to the NDA exam.
• Maintain a Balanced Approach: Don’t overload yourself with information. Aim for regular, focused news consumption to ensure effective learning and retention.
Cracking the NDA exam requires dedication, a strategic approach, and a strong foundation in various subjects. While coaching institutes can be a valuable resource, they are not absolutely necessary | {"url":"https://commandantdefenceacademy.com/nda-exam-syllabus-2024/","timestamp":"2024-11-06T02:13:31Z","content_type":"text/html","content_length":"203119","record_id":"<urn:uuid:f02f35b6-0e08-42a9-a4de-b058b4a2f725>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00495.warc.gz"} |
How will Poison's Ratio of a material affect its Strength?How will Poison's Ratio of a material affect its Strength?
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How will Poison’s Ratio of a material Affect its Strength?
1. Poison’s ratio is the ratio of lateral strain to longitudinal strain. It is the property of elasticity of a material.
This means that, if a force is applied in a given direction, say along the axis of the member, then the poison’s ratio is the ratio of the strain in the direction perpendicular to the axis and
the strain along the axis.
The poison’s ratio of concrete is 0.1 to 0.2. Let’s take that it as 0.15. If a force is applied on a concrete specimen along its axis, then, for every 1 unit of deformation in the axis, 0.15 unit
of deformation happens in the perpendicular direction.
Poison’s ratio is a measure of the elastic property of a material. There isn’t any direct relation between the strength of the material and the poison’s ratio.
2. This answer was edited.
Poisson’s Ratio:
1. The metal bar length increases in the direction of applied force when a tensile force is applied to it. The width of the same metal bar decreases in the direction perpendicular to the
applied force.
2. The metal bar length increases in the direction of applied force when a tensile force is applied to it. The width of the same metal bar decreases in the direction perpendicular to the
applied force.
3. The Poisson’s ratio indicated the relationship between change in length and width.
4. Poison’s ratio is to measure the elastic property of a material.
Thank You.
3. Poisson’s ratio is defined as the ratio of the change in the width per unit width of a material, to the change in its length per unit length as a result of strain. Poisson ratio measures the
deformation in the material in a direction perpendicular to the direction of the applied force.
Mathematically, poissons ratio is equal to the negative of the ratio of lateral strain and longitudinal strain. Therefore, if the poisson’s ratio is greater than the strength is greater.
4. Poisson’s ratio is the negative of the ratio of lateral strain to axial strain (Negative because of the decrease in the lateral measurement)
Hence higher the Poisson’s Ratio greater is its ability to withstand the load, hence greater is its strength.
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MAE vs. RMSE: Which Metric Should You Use? | Online Tutorials Library List | Tutoraspire.com
MAE vs. RMSE: Which Metric Should You Use?
by Tutor Aspire
Regression models are used to quantify the relationship between one or more predictor variables and a response variable.
Whenever we fit a regression model, we want to understand how well the model is able to use the values of the predictor variables to predict the value of the response variable.
Two metrics we often use to quantify how well a model fits a dataset are the mean absolute error (MAE) and the root mean squared error (RMSE), which are calculated as follows:
MAE: A metric that tells us the mean absolute difference between the predicted values and the actual values in a dataset. The lower the MAE, the better a model fits a dataset.
MAE = 1/n * Σ|y[i] – ŷ[i]|
• Σ is a symbol that means “sum”
• y[i] is the observed value for the i^th observation
• ŷ[i] is the predicted value for the i^th observation
• n is the sample size
RMSE: A metric that tells us the square root of the average squared difference between the predicted values and the actual values in a dataset. The lower the RMSE, the better a model fits a dataset.
It is calculated as:
RMSE = √Σ(y[i] – ŷ[i])^2 / n
• Σ is a symbol that means “sum”
• ŷ[i] is the predicted value for the i^th observation
• y[i] is the observed value for the i^th observation
• n is the sample size
Example: Calculating RMSE & MAE
Suppose we use a regression model to predict the number of points that 10 players will score in a basketball game.
The following table shows the predicted points from the model vs. the actual points the players scored:
Using the MAE Calculator, we can calculate the MAE to be 3.2
This tells us that the mean absolute difference between the predicted values made by the model and the actual values is 3.2.
Using the RMSE Calculator, we can calculate the RMSE to be 4.
This tells us that the square root of the average squared differences between the predicted points scored and the actual points scored is 4.
Notice that each metric gives us an idea of the typical difference between the predicted value made by the model and the actual value in the dataset, but the interpretation of each metric is slightly
RMSE vs. MAE: Which Metric Should You Use?
If you would like to give more weights to observations that are further from the mean (i.e. if being “off” by 20 is more than twice as bad as being off by 10″) then it’s better to use the RMSE to
measure error because the RMSE is more sensitive to observations that are further from the mean.
However, if being “off” by 20 is twice as bad as being off by 10 then it’s better to use the MAE.
To illustrate this, suppose we have one player who is a clear outlier in their number of points scored:
Using the online calculators mentioned earlier, we can calculate the MAE and RMSE to be:
Notice that the RMSE increases much more than the MAE.
This is because RMSE uses squared differences in its formula and the squared difference between the observed value of 76 and the predicted value of 22 is quite large. This causes the value for RMSE
to increase significantly.
In practice, we typically fit several regression models to a dataset and calculate just one of these metrics for each model.
For example, we might fit three different regression models and calculate the RMSE for each model. We would then select the model with the lowest RMSE value as the “best” model because it is the one
that makes predictions that are closest to the actual values from the dataset.
In either case, just make sure to calculate the same metric for each model. For example, don’t calculate MAE for one model and RMSE for another model and then compare those two metrics.
Additional Resources
The following tutorials explain how to calculate MAE using different statistical software:
How to Calculate Mean Absolute Error in Excel
How to Calculate Mean Absolute Error in R
How to Calculate Mean Absolute Error in Python
The following tutorials explain how to calculate RMSE using different statistical software:
How to Calculate Root Mean Square Error in Excel
How to Calculate Root Mean Square Error in R
How to Calculate Root Mean Square Error in Python
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6.1: Introduction to Ancient Greece
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Introduction to Ancient Greece
Ancient Greek culture spans over a thousand years, from the earliest civilizations to the cultures that became the Ancient Greeks.
Illustrate a timeline of ancient Greece from the Bronze Age to the Hellenistic period
Key Points
• Ancient Greek culture is noted for its government, art, architecture, philosophy, and sports, all of which became foundations for modern western society. It was admired and adopted by others,
including Alexander the Great and the Romans, who helped spread Greek culture around the world. Before Greek culture took root in Greece, early civilizations thrived on the Greek mainland and the
Aegean Islands. The fall of these cultures and the aftermath, known as the Dark Age, is believed to be the time when the Homeric epics were first recited.
• Greek culture began to develop during the Geometric, Orientalizing, and Archaic periods, which lasted from 900 to 480 BCE. During this time the population of city-states began to grow,
Panhellenic traditions were established, and art and architecture began to reflect Greek values .
• The Early, High, and Late Classical periods in Greece occurred from 480 to 323 BCE. During these periods, Greece flourished and the polis of Athens saw its Golden Age under the leadership of
Pericles . However, city-state rivalries lead to wars, and Greece was never truly stable until conquered.
• The Hellenistic period in Greece is the last period before Greek culture becomes a subset of Roman hegemony . This period occurs from the death of Alexander the Great in 323 BCE, to the Greek
defeat at the Battle of Actium in 30 BCE. It marks the spread of Greek culture across the Mediterranean.
Key Terms
• polis: A city, or a city-state. Its plural is poleis.
Ancient Greek Culture
Ancient Greek culture covers over a thousand years of history, from the earliest civilizations in the area to the cultures that became the Ancient Greeks. Following a Greek Dark Age, Greece once more
flourished and developed into the ancient culture that we recognize today .
Classical Greece: Map of Ancient Greece.
Greek culture is based on a series of shared values that connected independent city-states throughout the region, and expanded as far north as Mount Olympus. Greek society was insular, and loyalties
were focused around one’s polis (city-state). Greeks considered themselves civilized and considered outsiders to be barbaric.
While Greek daily life and loyalty was centered on one’s polis, the Greeks did create leagues, which vied for control of the peninsula, and were able to unite together against a common threat (such
as the Persians).
Greek culture is focused on their government, art, architecture, philosophy, and sport. Athens was intensely proud of its creation of democracy, and citizens from all poleis (city-states) took part
in civic duties. Cities commissioned artists and architects to honor their gods and beautify their cities.
Greek philosophers, mathematicians, and thinkers are still honored in society today. As a religious people, the Greeks worshipped a number of gods through sacrifices, rituals, and festivals.
Bronze Age and Proto-Greek Civilizations
Cycladic Civilization
During the Bronze Age, several distinct cultures developed around the Aegean. The Cycladic civilization, around the Cyclades Islands, thrived from 3,000 to 2,000 BCE. Little is known about the
Cycladic civilization because they left no written records. Their material culture is mainly excavated from grave sites, which reveal that the people produced unique, geometric marble figures.
Minoan Civilization
The Minoan civilization stretches from 3700 BCE until 1200 BCE, and thrived during their Neopalatial period (from 1700 to 1400 BCE), with the large-scale building of communal palaces. Numerous
archives have been discovered at Minoan sites; however their language, Linear A , has yet to be deciphered. The culture was centered on trade and production, and the Minoans were great seafarers on
the Mediterranean Sea.
Mycenaean Civilization
A proto-Greek culture known as the Mycenaeans developed and flourished on the mainland, eventually conquering the Aegean Islands and Crete, where the Minoan civilization was centered. The Mycenaeans
developed a fractious, war-like culture that was centered on the authority of a single ruler. Their culture eventually collapsed, but many of their citadel sites were occupied through the Greek Dark
Age and rebuilt into Greek city-states.
The Dark Age
From around 1200 BCE, the palace centers and outlying settlements of the Mycenaeans’ culture began to be abandoned or destroyed. By 1050 BCE, the recognizable features of Mycenaean culture had
Many explanations attribute the fall of the Mycenaean civilization and the collapse of the Bronze Age to climatic or environmental catastrophe, combined with an invasion by the Dorians or by the Sea
Peoples, or to the widespread availability of edged weapons of iron, but no single explanation fits the available archaeological evidence.
This two- to three-century span of history is also known as the Homeric Age. It is believed that the Homeric epics The Iliad and The Odyssey were first recited around this time.
The Geometric and Orientalizing Periods
The Geometric period (c. 900–700 BCE), which derives its name from the proliferation of geometric designs and rendering of figures in art, witnessed the emergence of a new culture on the Greek
mainland. The culture’s change in language, its adaptation of the Phoenician alphabet, and its new funerary practices and material culture suggest the ethnic population changed from the mainland’s
previous inhabitants, the Mycenaeans.
During this time, the new culture was centered on the people and independent poleis, which divided the land into regional populations. This period witnessed a growth in population and the revival of
The Orientalizing period (c. 700–600 BCE) is named for the cultural exchanges the Greeks had with Eastern, or Oriental civilizations. During this time, international trade began to flourish. Art from
this period reflects contact with locations such as Egypt, Syria, Assyria, Phoenicia, and Israel.
Archaic Greece
Greece’s Archaic period lasted from 600 to 480 BCE, in which the Greek culture expanded. The population in Greece began to rise and the Greeks began to colonize along the coasts of the Mediterranean
and the Black Sea. The poleis at this time were typically ruled by a single ruler who commanded the city by force.
For the city of Athens, this led to the creation of democracy. Several city-states emerged as major powers, including Athens, Sparta, Corinth, and Thebes. These poleis were often warring with each
other, and formed coalitions to gain power and allies. The Persian invasion of Greece in 480 BCE marked the end of the Archaic period.
Classical Greece
The era of Classical Greece began in 480 BCE with the sacking of Athens by the Persians. The Persian invasion of Greece, first lead by Darius I and then by his son Xerxes, united Greece against a
common enemy.
With the defeat of the Persian threat, Athens became the most powerful polis until the start of the Peloponnesian War in 431 BCE. These wars continued on and off until 400 BCE. While marred by war,
the Classical period saw the height of Greek culture and the creation of some of Greece’s most famous art and architecture.
However, peace and stability in Greece was not achieved until it was conquered and united by Macedonia under the leadership of Philip II and Alexander the Great in the mid-third century BCE.
Hellenistic Greece
The Hellenistic period began with the death of Alexander the Great in 323 BCE, and ended with the Roman victory at the Battle of Actium in 30 BCE. Greece poleis spent this time under the hegemony of
foreign rulers, first the Macedons and then the Romans, starting in 146 BCE.
New centers of Hellenic culture flourished through Greece and on foreign soil, including the cities of Pergamon, Antioch, and Alexandria—the capitals of the Attalids, Seleucids, and Ptolemies.
The Ancient Greek Gods and Their Temples
Greek religion played a central and daily role in the life of ancient Greeks, and group worship was centered on the temple and cult sites.
Describe the ways in which Greek life and art was influenced by the gods
Key Points
• The history of the Greek pantheon begins with the primordial deities Gaia and Uranus and their children, the Titans. The pantheon of Greek gods consisted of twelve Olympian gods plus a variety of
additional principal and minor gods and goddesses. The gods had human characteristics and personalities, and their lives were detailed by the mythologies told about them.
• The gods played a central role in Greek daily life. They were consulted, blamed, and honored for a variety of reasons, including natural occurrences (from earthquakes to rain), as well as for the
public and private affairs of the polis and its people.
• The mythologies and cult worship of heroes also played an important role in Greek religion and ritual . Heroes—especially Perseus, Hercules , Theseus, and those involved in the Trojan War—were
often depicted in art, and the location of their feats became cult sites.
• The temple was considered the home of the god and was often an expensive and lavishly decorated building. The temple included a naos , the main room that held the cult statue. Offerings and
dedications were left for the gods, and sacrifices took place outdoors.
Key Terms
• primordial: Existing at or before the beginning of time.
• demigod: A half-god or hero; the offspring of a deity and a mortal.
• libation: The act of pouring a liquid or liquor, usually wine, either on the ground or on a victim in sacrifice, in honor of some deity.
• naos: The central room in the god’s temple, where a cult statue of the god is erected.
• polytheistic: A religious system whose members worship many deities.
• votive: A small religious offering deposited at a temple without the purpose of display or retrieval.
Greek religious traditions encompassed a large pantheon of gods, complex mythologies, rituals, and cult practices. Greece was a polytheistic society, and looked to its gods and mythology to explain
natural mysteries as well as current events. Religious festivals and ceremonies were held throughout the year, and animal sacrifice and votive offerings were popular ways to appease and worship the
gods. Religious life, rituals, and practices were one of the unifying aspects of Greece across regions and poleis (cities, or city-states , such as Athens and Sparta).
The principal religious sanctuaries of the Greek Aegean: This map lists the major Greek gods and shows where their principal religious sanctuaries are located throughout the Greek Aegean region.
Greek Gods
Greek gods were immortal beings who possessed human-like qualities and were represented as completely human in visual art. They were moral and immoral, petty and just, and often vain. The gods were
invoked to intervene and assist in matters large, small, private and public.
City-states claimed individual gods and goddess as their patrons . Temples and sanctuaries to the gods were built in every city. Many cities became cult sites due to their connection with a god or
goddess and specific myths. For instance, the city of Delphi was known for its oracle and sanctuary of Apollo, because Apollo was believed to have killed a dragon that inhabited Delphi.
The history of the Greek pantheon begins with the primordial deities Gaia (Mother Earth) and Uranus (Father Sky), who were the parents of the first of twelve giants known as Titans. Among these
Titans were six males and six females.
• The males were named Oceanus, Hyperion, Coeus, Crius, Iapetus, and Kronos.
• The females were named Themis, Mnemosyne, Tethys, Theia, Phoebe, and Rhea.
Kronos eventually overthrew Uranus and ruled during a mythological Golden Age. Over time, he and Rhea had twelve children who would become the Olympian gods. However, Kronos heard a prophecy that his
son would overthrow him, as he did to Uranus. In an effort to avert fate, he ordered Rhea to allow him to devour each of the children upon their birth.
The Olympian Gods
Best known among the pantheon are the twelve Olympian gods and goddesses who resided on Mt. Olympus in northern Greece. Zeus, the youngest son of Rhea and Kronos, was hidden from his father, instead
of being swallowed. Once he became a man, he challenged his father’s rule, forcing Kronos to regurgitate the rest of his swallowed children. These children were Zeus’s siblings, and together they
overthrew Kronos, making Zeus the father of gods and men.
Violence and power struggles were common in Greek mythology, and the Greeks used their mythologies to explain their lives around them, from the change in seasons to why the Persians were able to sack
The traditional pantheon of Greek gods includes
• Zeus, the king of gods and the ruler of the sky,
• Zeus’ two brothers, Poseidon (who ruled over the sea) and Hades (who ruled the underworld).
• Zeus’s sister and wife, Hera, the goddess of marriage, who is frequently jealous and vindictive of Zeus’s other lovers.
• Their sisters Hestia, the goddess of the hearth, and Demeter, the goddess of grain and culture .
• Zeus’s children:
• Athena (goddess of warfare and wisdom).
• Hermes (a messenger god and god of commerce).
• the twins Apollo (god of the sun, music, and prophecy) and Artemis (goddess of the hunt and of wild animals).
• Dionysos (god of wine and theatre).
• Aphrodite (goddess of beauty and love), who was married to Hephaestus (deformed god of the forge).
• Ares (god of war and lover of Aphrodite) are also part of the traditional pantheon.
• Hephaestus was in some mythologies the son of Zeus while in others the fatherless son of Hera.
Heroes, who were often demigods , were also important characters in Greek mythology. The two most important heroes are Perseus and Hercules.
Perseus is known for defeating the Gorgon, Medusa. He slew her with help from the gods: Athena gave him armor and a reflective shield, and Hermes provided Perseus with winged sandals so he could fly.
Hercules was a strong but unkind man, a drunkard who conducted huge misdeeds and social faux pas. Hercules was sent on twelve labors to atone for his sins as punishment for his misdeeds. These deeds,
and several other stories, were often depicted in art, on ceramic pots, or on temple metopes . The most famous of his deeds include slaying both the Nemean Lion and the Hydra, capturing Cerberus (the
dog of the underworld), and obtaining the apples of the Hesperides.
A third hero, Theseus, was an Athenian hero known for slaying King Minos’s Minotaur . Other major heros in Greek mythology include the warriors and participants of the Trojan War, such as Achilles,
Ajax, Odysseus, Agamemnon, Paris, Hector, and Helen.
Hero cults were another popular form of Greek worship that involved the honoring of the dead, specifically the dead heroes of the Trojan War. The sites of hero worship were usually old Bronze Age
sites or tombs that the ancient Greeks recognized as important or sacred, which they then connected to their own legends and stories.
Hercules and Cerberus : Hercules bringing Cerberus back to King Eurystheus. Black figure hydra. c. 525 BCE.
Sacred Spaces
Greek worship was centered on the temple. The temple was considered the home of the god, and a cult statue of the god would be erected in the central room, or the naos. Temples generally followed the
same basic rectangular plan, although a round temple, known as a tholos , were used at some sites in starting in the Classical period.
Temples were oriented east to face the rising sun. Patrons would leave offerings for the gods, such as small votives, large statues, libations or costly goods. Due to the wealth dedicated to the
gods, the temples often became treasuries that held and preserved the wealth of the city. Greek temples would be extensively decorated, and their construction was a long and costly endeavor.
Rituals and animal sacrifices in honor of the god or goddess would take place outside, in front of the temple. Rituals often included a large number of people, and sacrifice was a messy business that
was best done outdoors. The development and decoration of temples is a primary focus in the study of Greek art and culture.
Sacrificial scene: Scene of a sacrifice. Attic red-figure bell krater. Circa 430–420 BCE. Athens, Greece.
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• polis. Provided by: Wiktionary. Located at: en.wiktionary.org/wiki/polis. License: CC BY-SA: Attribution-ShareAlike
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Inflation-Indexed Rate in context of percentage rate
31 Aug 2024
Title: Understanding Inflation-Indexed Rates: A Conceptual Framework for Percentage-Based Interest Rates
Abstract: Inflation-indexed rates have gained prominence as a means to combat inflation and stabilize economic growth. This article provides an in-depth analysis of inflation-indexed rates, focusing
on their conceptual framework within the context of percentage-based interest rates. We explore the mathematical formulation of these rates and discuss their implications for financial
Introduction: Inflation-indexed rates are designed to account for the effects of inflation on the purchasing power of money. These rates are typically expressed as a percentage of the principal
amount, adjusted for inflation. The primary objective of this article is to provide a comprehensive understanding of inflation-indexed rates within the context of percentage-based interest rates.
Mathematical Formulation: The formula for calculating an inflation-indexed rate (IIR) can be represented in ASCII format as:
IIR = (1 + r) ^ n - 1
• r is the nominal interest rate
• n is the number of periods
This formula indicates that the IIR is a function of the nominal interest rate and the number of periods.
Conceptual Framework: Inflation-indexed rates can be viewed as a percentage-based interest rate that takes into account the effects of inflation on the principal amount. The conceptual framework for
IIRs involves adjusting the nominal interest rate to reflect the expected rate of inflation. This adjustment ensures that the purchasing power of the principal amount remains constant over time.
Implications: The implications of inflation-indexed rates are far-reaching, affecting financial decision-making and economic growth. By accounting for inflation, IIRs provide a more accurate
representation of the true cost of borrowing or investing. This, in turn, can lead to more informed financial decisions and improved economic stability.
Conclusion: Inflation-indexed rates offer a valuable tool for managing inflation and stabilizing economic growth. By understanding the conceptual framework and mathematical formulation of these
rates, we can better appreciate their implications for financial decision-making. As the global economy continues to evolve, the importance of IIRs is likely to increase, making them an essential
component of modern finance.
• [Insert relevant references here]
Note: The article does not provide numerical examples as per your request.
Related articles for ‘percentage rate’ :
• Reading: Inflation-Indexed Rate in context of percentage rate
Calculators for ‘percentage rate’ | {"url":"https://blog.truegeometry.com/tutorials/education/8b0e5358de3a109f02621d5ae8e6832d/JSON_TO_ARTCL_Inflation_Indexed_Rate_in_context_of_percentage_rate.html","timestamp":"2024-11-08T15:07:27Z","content_type":"text/html","content_length":"16057","record_id":"<urn:uuid:1d1c9de5-30ee-4a25-aeee-7234c036a8ed>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00277.warc.gz"} |
Monte Carlo simulation of homopolymer chains. I. Second virial coefficient Microscopic theory of the dynamics of polymeric liquids: General formulation of a mode–mode-coupling approach Polymer reference interaction site model theory: New molecular closures for phase separating fluids and alloys Monte Carlo simulation of homopolymer chains. I. Second virial coefficient
Monte Carlo simulation of homopolymer chains. I. Second virial coefficient Microscopic theory of the dynamics of polymeric liquids: General formulation of a mode–mode-coupling approach Polymer
reference interaction site model theory: New molecular closures for phase separating fluids and alloys Monte Carlo simulation of homopolymer chains. I. Second virial coefficient
Publication , Journal Article
Withers, IM; Dobrynin, AV; Berkowitz, ML; Rubinstein, M
Published in: The Journal of Chemical Physics Journal of Chemical Physics The Journal of Chemical Physics II The Journal of Chemical Physics The Journal of Chemical Physics The Journal of Chemical
Physics Journal of Chemical Physics
The second virial coefficient, A 2 , is evaluated between pairs of short chain molecules by direct simulations using a parallel tempering Monte Carlo method where the centers of mass of the two
molecules are coupled by a harmonic spring. Three off-lattice polymer models are considered, one with rigid bonds and two with flexible bonds, represented by the finitely extensible nonlinear elastic
potential with different stiffness. All the models considered account for excluded volume interactions via the Lennard-Jones potential. In order to obtain the second virial coefficient we calculate
the effective intermolecular interaction between the two polymer chains. As expected this intermolecular interaction is found to be strongly dependent upon chain length and temperature. For all three
models the temperature ( n), defined as the temperature at which the second virial coefficient vanishes for chains of finite length, varies as n Ϫ ϱ ϰn Ϫ1/2 , where n is the number of bonds in
the polymer chains and ϱ is the point for an infinitely long chain. Introducing flexibility into the model has two effects upon n ; the temperature is reduced with increasing flexibility, and
the n dependence of n is suppressed. For a particular choice of spring constant an n-independent temperature is found. We also compare our results with those obtained from experimental studies of
polystyrene in decalin and cyclohexane, and for poly͑methyl methacrylate͒ in a water and tert-butyl alcohol mixture, and show that all the data can be collapsed onto a single universal curve without
any adjustable parameters. We are thus able to relate both A 2 and the excluded volume parameter v, to the chain interaction parameter z, in a way relating not only the data for different molecular
weights and temperatures, but also for different polymers in different solvents.
Duke Scholars
Published In
The Journal of Chemical Physics Journal of Chemical Physics The Journal of Chemical Physics II The Journal of Chemical Physics The Journal of Chemical Physics The Journal of Chemical Physics Journal
of Chemical Physics
Publication Date
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Related Subject Headings
• Chemical Physics
• 51 Physical sciences
• 40 Engineering
• 34 Chemical sciences
• 09 Engineering
• 03 Chemical Sciences
• 02 Physical Sciences
Published In
The Journal of Chemical Physics Journal of Chemical Physics The Journal of Chemical Physics II The Journal of Chemical Physics The Journal of Chemical Physics The Journal of Chemical Physics Journal
of Chemical Physics
Publication Date
Start / End Page
Related Subject Headings
• Chemical Physics
• 51 Physical sciences
• 40 Engineering
• 34 Chemical sciences
• 09 Engineering
• 03 Chemical Sciences
• 02 Physical Sciences | {"url":"https://scholars.duke.edu/publication/1323857","timestamp":"2024-11-13T13:23:47Z","content_type":"text/html","content_length":"65739","record_id":"<urn:uuid:d34c0d4a-c315-4569-8e5d-32ef06d56868>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00229.warc.gz"} |
Cellfunktionen. Funktionerna Cells and Range kan du berätta för ditt VBA-skript 4 Fel du kan undvika när du programmerar Excel Makroer med VBA 4-fel som
Applications (VBA) som används i programmet Excel. Verktyget är uppbyggt av Excel Formulas and Functions For Dummies. 3. uppl. Indiana:
When I start typing =MyFunction in the cell, Excel lists it as an option, but doesn't show the parameters and then produces this result. This is in a 2007 .xlsm file. When Excel opens an unkown
workbook containing VBA-Code, it usually asks for macros to be enabled by the user (depending on the application settings). If the user then enables the macros, all event-driven procedures will be
started, such as auto_open or others. Custom VBA Functions however require for a full recalculation of the workbook.
It can be used as a VBA function (VBA) in Excel. As a VBA function, you can use this function in macro code that is entered through the Microsoft Visual Basic Editor. 2018-09-14 2020-08-28 How to
make a function using VBA. In this lesson, we make 2 functions from scratch, =Lastrow() and =LastrowC().Lastrow function, when typed into any cell, gi Se hela listan på docs.microsoft.com In Visual
Basic, the Excel worksheet functions are available through the WorksheetFunction object. The following Sub procedure uses the Min worksheet function to determine the smallest value in a range of
cells. First, the variable myRange is declared as a Range object, and then it is set to range A1:C10 on Sheet1. VBA Text Functions.
This is required if you want to return a value from a function. For example, you can pass two numbers in a function and then you can expect from the function to return their multiplication in
your calling program. 2015-11-04 If you want Excel VBA to perform a task that returns a result, you can use a function.
Within Excel, there is a function called Convert and it already In this quick tutorial, I will show you how to
And if there is any change in the syntax, we will be able to see that also when we are about to use VBA function. To declare a function in VBA, first, open a Private-End procedure and select the data
type as per functional need. 2020-01-21 This post provides an in-depth look at the VBA array which is a very important part of the Excel VBA programming language. It covers everything you need to
know about the VBA array.
How to make a function using VBA. In this lesson, we make 2 functions from scratch, =Lastrow() and =LastrowC().Lastrow function, when typed into any cell, gi
If you would like a list of these functions sorted by category, click on the following button: Home VBA Tutorial Top VBA Functions. Once you get started with VBA, the next important thing is to learn
how to use in-built VBA functions while writing a VBA code. So here I have listed the Top 100 VBA Functions (Category Wise) in detail with examples and sample codes, and these functions are listed
under specific categories (10) to make you understand the purpose of each function easily. Most Excel worksheet functions are available for use with Visual Basic for Applications. However, there are
a few cases where you can't use an Excel worksheet function in VBA because it isn't a method of the WorksheetFunction object. In those cases you'll generally find an equivalent within Visual Basic
for Applications. It is also possible to use Excel functions in VBA code, as we will see on this page.
Om du vill lära dig hantera stora databaser som ett proffs då är detta kursen för dig. . The Excel Functions Guide - onlineThis online course will both guide See more ideas about excel tutorials,
microsoft excel formulas, excel shortcuts. Excel Dashboard Examples and Template Files — Excel Dashboards VBA. Från heltäckande referensverk till böcker som är specialiserade på VBA, pivottabeller,
presentation och Formulas and Functions: Microsoft Excel 2013. Que. Från kursen: Excel VBA: Managing Files and Data (2014) with filters; manage workbooks and worksheets; access built-in functions;
create charts; and build However, Excel supports User-Defined Functions (UDFs) using the Microsoft Visual Basic for Applications (VBA) operations on cells based on background or There are two
classes of functions in Excel and Google sheets: Spreadsheet user-defined function that calculates buyer discounts written VBA code. Collection Vba Wait.
Value call option formula
Intermediate knowledge of Microsoft Excel. Last Updated 2/2021.
5m 3s Jämför och hitta det billigaste priset på Excel VBA innan du gör ditt köp. you wanted to automate different processes and functions that you perform on Excel? VBA-Excel: Date-Time Functions –
CDate() – Excel-Macro. Appendix C - CONOPS Function Template | Guidance for C Date And Time Functions png images | VBA and Macros for Microsoft Office Excel 2007 Develop your Excel macro programming
skills using VBA instantly with Build User-Defined Functions.
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Det skulle kräva att Excel kör VBA parallellt, dvs i två trådar. https://www.exceltrick.com/formulas_macros/vba-wait-and-sleep-functions/?
I’ve included several of my favorite VBA UDFs in my Excel VBA Examples page. Defining Functions . Subroutines VS Functions This line simply tells VBA (Visual Basic for Applications, which is the
language used by macros in Excel) that you are defining a function, that it's name is "NumXs", that the function will work with a range of cells you want to refer to as "rng", and that the result of
the function will be an Integer value (a whole number). In my VBA Module Module1: Public Function MyFunction(anything As String) MyFunction = anything End Function in the cell: =MyFunction("a")
Result: #NAME?
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VBA has no ROUND function, but Excel does. Therefore, to use ROUND in this statement, you tell VBA to look for the Round method (function) in the Application object (Excel). You do that by adding the
word Application before the word Round. Use this syntax whenever you need to access an Excel function from a VBA module.
Returns a substring from the start of a supplied string. In Visual Basic, the Excel worksheet functions are available through the WorksheetFunction object. The following Sub procedure uses the Min
worksheet function to determine the smallest value in a range of cells. First, the variable myRange is declared as a Range … Excel VBA Functions. We have seen that we can use the worksheet functions
in VBA, i.e.
In Excel, there are more than 450 functions and some of them highly useful in your daily work. But, Excel gives you the ability to create a custom function using VBA. Yes, you get it right. USER
DEFINED Function, in short UDF, or you can also call it a Custom VBA function.
2018-09-14 · When you create your own function in Excel using VBA, it is called a user defined function, or UDF. VBA user defined functions are extremely powerful and they can make your Excel
spreadsheets far more userful.
error, CVerror constants, CVerr function, GoTo statement, and On Error statement. Excel VBA : How to debug a user defined function (UDF). | {"url":"https://affarereiny.web.app/25490/85890.html","timestamp":"2024-11-03T19:07:02Z","content_type":"text/html","content_length":"13246","record_id":"<urn:uuid:9e18bdd3-da25-4a5e-b75d-8c9b8606351f>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00742.warc.gz"} |
Positive-Negative Charge Model For Integers
Williams, Barbara J. N. Thorp School
To represent operations on integers with positive and negative
Overhead projector, drawing of an empty beaker on an acetate, bingo
chips (two colors needed).
The "positive-negative" model was used to represent addition
and subtraction, however, this model can also be extended to represent
division and multiplication.
To use this model the blue chips represent negative charges. The
red chips represent positive charges. The beaker represented on the
acetate will be used to combine the charges. We are not concerned
with individual charges, but with collection of charges in the beaker.
Therefore, an empty jar would have a collective charge of zero. If the
jar contains an equal number of blue and red chips, the charge is also
zero. Three positive (red) chips and three negative (blue) chips form
a 1:1 correspondence and they therefore cancel each other. The
collective charge of the beaker is zero.
Ex. 1. +3+(4)=+7
Place three red chips in the beaker, add four red chips. The collective
charge is now a positive seven.
Ex. 2. -5+(2)=-3
Place five blue chips in the beaker then add two red chips. Match a
blue and and red chip 1:1 until two sets of zeroes are matched.
Remove the matched chips from the beaker. There are now three blue
chips remaining in the beaker. The collective charge is now
represented by as negative three. The answer -3 represented by the
three blue chips.
Ex. 3. -2-(+7)
Place two blue chips in the beaker. We now must create +7. Add
zeroes (seven red chips and seven blue chips), to the beaker. The
collective charge is -2. Now, remove the seven positive (red) charges.
Only blue (negative) chips remain. Your answer is represented by the
nine blue chips that remain.
Multiplication and division can be similarly represented.
A COMPLETE MODEL FOR OPERATIONS ON INTEGERS by Michael Battista
Arithmetic Teacher, May 1983.
Return to Mathematics Index | {"url":"https://smileprogram.info/ma8718.html","timestamp":"2024-11-07T18:33:29Z","content_type":"text/html","content_length":"2711","record_id":"<urn:uuid:00fbce9f-4e7f-47c9-a1dc-88a8599c4780>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00012.warc.gz"} |
Understanding Mathematical Functions: What Are Zeros Of A Function
When it comes to mathematics, understanding mathematical functions is crucial for grasping the concepts and principles of the subject. One important aspect of functions is their zeros, or the values
of the independent variable that make the function equal to zero. In this blog post, we will delve into the significance of understanding zeros of a function and how they play a vital role in solving
equations and analyzing the behavior of functions.
Key Takeaways
• Understanding zeros of a function is crucial for grasping the concepts and principles of mathematics.
• Zeros of a function are the values of the independent variable that make the function equal to zero.
• Zeros play a vital role in solving equations and analyzing the behavior of functions.
• There are different types of zeros, including real zeros, complex zeros, and multiple zeros.
• Zeros of a function have practical applications in engineering, finance, and real-world problem-solving.
Understanding Mathematical Functions: What are Zeros of a Function
In mathematics, a function is a relation between a set of inputs and a set of possible outputs, where each input is related to exactly one output. Functions are a fundamental concept in mathematics
and are used to describe various real-world phenomena and relationships.
A. Definition of a Function
A mathematical function is a rule that assigns each input exactly one output. It can be represented by an equation, a graph, or a table of values.
B. Examples of Common Mathematical Functions
1. Linear Function: A linear function is a function that can be graphically represented as a straight line. It has the form f(x) = ax + b, where a and b are constants.
2. Quadratic Function: A quadratic function is a function that can be graphically represented as a parabola. It has the form f(x) = ax^2 + bx + c, where a, b, and c are constants.
3. Exponential Function: An exponential function is a function in which the variable appears in the exponent. It has the form f(x) = a^x, where a is a constant.
4. Trigonometric Function: Trigonometric functions such as sine, cosine, and tangent are used to model periodic phenomena and oscillatory behavior.
What are Zeros of a Function
The zeros of a function, also known as roots, are the values of the input that make the output equal to zero. In other words, the zeros of a function are the solutions to the equation f(x) = 0.
• Example: Consider the function f(x) = x^2 - 4. The zeros of this function can be found by setting f(x) equal to zero and solving for x. In this case, the solutions are x = 2 and x = -2.
Understanding the zeros of a function is important in many areas of mathematics and science, as they provide valuable information about the behavior and properties of the function.
Understanding zeros of a function
Mathematical functions play a crucial role in various fields such as physics, engineering, and economics. Understanding the concept of zeros of a function is essential for solving equations and
analyzing the behavior of functions.
A. Definition of zeros of a function
Zeros of a function refer to the values of the independent variable that make the function equal to zero. In other words, the zeros of a function are the points where the graph of the function
intersects the x-axis.
B. How to find zeros of a function
• One way to find the zeros of a function is by setting the function equal to zero and solving for the independent variable. For example, if the function is f(x) = x^2 - 4, the zeros can be found
by setting x^2 - 4 = 0 and solving for x.
• Another method to find zeros is using graphical methods such as plotting the graph of the function and identifying the points where the graph crosses the x-axis.
• In some cases, zeros can also be found using numerical methods such as the bisection method or Newton's method.
C. Importance of zeros in mathematical analysis
Zeros of a function hold significant importance in mathematical analysis for several reasons. Firstly, they provide insights into the behavior and characteristics of the function. The number and
nature of zeros can indicate the function's properties such as roots, extrema, and the behavior of the function at different intervals.
Moreover, the zeros of a function play a crucial role in solving equations and systems of equations. By finding the zeros of a function, one can determine the solutions to equations and establish
relationships between different variables.
Furthermore, zeros of a function are fundamental in calculus for finding the integration and differentiation of functions. They are also essential in the study of complex analysis and the behavior of
complex functions.
Different types of zeros
Mathematical functions can have different types of zeros, which are the values of the variable that make the function equal to zero. These zeros can be categorized into different types based on their
nature and characteristics.
• Real zeros
Real zeros of a function are the values of the variable for which the function equals zero and are real numbers. These zeros are often found on the x-axis of the graph of the function and
represent the points where the graph intersects the x-axis.
• Complex zeros
Complex zeros of a function are the values of the variable for which the function equals zero and are complex numbers. Complex zeros often come in conjugate pairs, and they are not found on the
real number line. Instead, they exist in the complex plane.
• Multiple zeros
Multiple zeros of a function occur when a particular value of the variable causes the function to equal zero more than once. This means that the graph of the function touches or crosses the
x-axis at the same point multiple times. These multiple zeros can have different behaviors and implications for the function, depending on their multiplicities.
Application of zeros in real-world problems
Mathematical functions play a crucial role in various real-world problems, and understanding the zeros of a function is essential for solving these problems.
Examples of how zeros of a function are used in engineering
• Structural engineering: Engineers use the zeros of a function to analyze and design complex structures such as bridges and buildings. By finding the zeros of a function representing the forces
acting on the structure, engineers can determine the points where the forces are balanced, which is crucial for ensuring the stability and safety of the structure.
• Electrical engineering: Zeros of a function are used to analyze and design electrical circuits. Engineers use the zeros to determine the points where the voltage or current is zero, which helps
in optimizing the performance and efficiency of the circuits.
Examples of how zeros of a function are used in finance
• Financial modeling: In finance, the zeros of a function are used to analyze and predict the behavior of financial assets such as stocks and bonds. By finding the zeros of a function representing
the price or value of an asset, financial analysts can identify the points where the asset's value is zero, which is essential for making investment decisions.
• Risk management: Zeros of a function are also used in finance to assess and manage risk. Financial institutions use the zeros to identify the points where the risk is minimized or mitigated,
which is crucial for maintaining financial stability and minimizing potential losses.
Common misconceptions about zeros of a function
Understanding the concept of zeros of a function is crucial in mathematics, but it is also an area that is prone to misconceptions. Here are some common misconceptions about zeros of a function:
A. Misunderstanding the concept of zero
One of the most common misconceptions about zeros of a function is a misunderstanding of the concept of zero itself. Some individuals may confuse the concept of zero with the absence of value, rather
than understanding it as the value where the function equals zero.
B. Confusing zeros with critical points
Another misconception is the confusion between zeros and critical points of a function. While critical points are where the derivative of the function is zero, zeros are the values where the function
itself equals zero. It is important to differentiate between these two concepts to have a clear understanding of the behavior of a function.
C. Not recognizing the significance of zeros in a function
Some individuals may not fully grasp the significance of zeros in a function. Zeros play a crucial role in determining the roots of a function, which in turn helps in solving equations and
understanding the behavior of the function. Failing to recognize the importance of zeros can lead to a limited understanding of the function as a whole.
Recap: Understanding the zeros of a function is crucial in mathematics as it helps us find the roots of equations and understand the behavior of the function. It allows us to solve real-world
problems and make predictions based on the data.
Exploration: I encourage you to further explore the topic of mathematical functions and zeros for a better understanding of how they are used in various fields such as engineering, economics, and
science. Delving deeper into this topic will not only enhance your mathematical skills but also open up new opportunities for problem-solving and critical thinking.
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Mountain 64314 - math word problem (64314)
Mountain 64314
The hiker set out on a hike at 5 km/h. After 30 minutes, a cyclist on a mountain bike set off on the same route at 20 km/h. How many minutes will the cyclist overtake the tourist?
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
Tips for related online calculators
Do you have a linear equation or system of equations and are looking for its
? Or do you have
a quadratic equation
Do you want to
convert velocity (speed) units
Do you want to
convert time units
like minutes to seconds?
You need to know the following knowledge to solve this word math problem:
Units of physical quantities:
Themes, topics:
We encourage you to watch this tutorial video on this math problem:
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MRS. FORTUNATO - AT Cycle 9 (10/24 - 10/29)
🔴 2: Th 10/24, 🟡 4: Th 10/24, 🔵 8: Th 10/24 - center of mass problems (1)
Daily Check-in: center of mass summation (a similar example if you missed class)
Today, we'll collaboratively solve center of mass problems from Chapter 9:
Support: 2, 4
Required: Sample prob 9.02, 5, 7, 8, 15, 16, 17, 114, G65
Enrichment: 14, G66, G67
Some of these problems will require you to use the calculus and what you learned about different kinds of densities. If you finish the ones that don't require calculus, watch the homework videos
Homework: Quiz on center of mass, rotational kinematics, and tipping - Wednesday, October 30th. Watch the following two videos about how to find the center of mass of an extended object with
calculus. Also, make sure that you're comfortable with Example 9-14 on page 223 of the Giancoli textbook.
🟥 2: M 10/28 lab, 🟨 4: F 10/25 lab, 🟦 8: F 10/25 lab - CoM (2) & rotational kinematics
Daily Check-in (hour 1): center of mass integral
Today, we'll collaboratively finish solving center of mass problems from Chapter 9:
Support: 2, 4
Required: Sample prob 9.02, 5, 7, 8, 15, 16, 17, 114, G65
Enrichment: 14, G66, G67
And if you need a review of the center of mass integral that we did in class, watch my video:
Then during the second hour, watch the video below on rotational kinematic variables. Take notes while you watch! Understanding these concepts are EXTREMELY IMPORTANT in being successful in the
rest of this unit, so take the time to rewind and rewatch as needed. (When Mr. Fullerton derives centripetal acceleration in minute 13, he talks about unit vectors. "I-hat" is a unit vector
magnitude 1 in the x direction. "J-hat" is a unit vector magnitude 1 in the y direction. Unit vectors are really just multipliers which turn scalar magnitudes into vectors with direction.)
Fill out the handout first which will relate Translational & Rotational Kinematics Variables using what you learned in the video. Then, we'll do problems from Chapter 10 which will allow you to
practice applying the rotational kinematics formulas.
Support: 3, 4, 10
Required: 6, 7, 14, 16, 22, 26, 28, 32
Enrichment: 8, 17, 31
Homework: Quiz on center of mass, rotational kinematics, and tipping - Wednesday, October 30th. Complete the required center of mass problems above. Try recreate the solution the problem from the
3rd and 4th video from last night independently. That means to write out your own solution without watching the video. The problem is to find the center of mass of a non-uniform rod length L where
λ = kx3. Then, answer the following questions in a ✏️ Google Classroom assignment by Monday, October 28th at 10pm.
• How do you know that you need an integral to solve this problem?
• What general formula do you need to use?
• Why do you need to change the formula to dx?
• How do you change the formula to dx?
• How do you know which variable determines the limits?
• How do you know where x = 0? (kind of a trick question)
• How do I know the units of my solution are correct?
If you get stuck, rewatch the videos:
❤️ 2: T 10/29, 💛 4: M 10/28, 💙 8: M 10/28 - moment of inertia
Daily Check-in: translational vs. rotational variables
Today, we'll start by learning first about what moment of inertia is and then trying to figure out how to calculate the moment of inertia of different objects. We'll learn formulas for moment of
inertia using a summation for discrete objects and then an integral for continuous bodies. With any time remaining, start problems from the next post. Demos: two different rods with the same mass
but different moments of inertia; ring and solid disk rolling down ramp.
Homework: Quiz on center of mass, rotational kinematics, and tipping next class - Wednesday, October 30th. If you have any questions about the lecture today, you may watch my video lecture, and
make sure you watch the part about parallel axis theorem starting at minute 33. | {"url":"https://www.pirate-physics.net/at-physics/previous-cycles/ATc09","timestamp":"2024-11-10T09:14:22Z","content_type":"text/html","content_length":"189089","record_id":"<urn:uuid:9844580e-ac10-494b-bb8f-2a0c7d9db925>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00381.warc.gz"} |
'Metcalf, R. Benton'
Searching for codes credited to 'Metcalf, R. Benton'
➥ Tip! Refine or expand your search. Authors are sometimes listed as 'Smith, J. K.' instead of 'Smith, John' so it is useful to search for last names only. Note this is currently a simple phrase
[ascl:1505.026] Lensed: Forward parametric modelling of strong lenses
Lensed performs forward parametric modelling of strong lenses. Using a provided model, Lensed renders the expected image of the lensing event for a large number of parameter settings, thereby
exploring the space of possible realizations of the observation. It compares the expectation to the observed image by calculating the likelihood that the observation was indeed produced by the
assumed model, thus reconstructing the probability distribution over the parameter space of the model. Written in C, the code uses a massively parallel ray-tracing kernel to perform the necessary
calculations on a graphics processing unit (GPU), making the precise rendering of the background lensed sources fast and allowing the simultaneous optimization of tens of parameters for the selected | {"url":"https://ascl.net/code/cs/Metcalf%2C%20R.%20Benton","timestamp":"2024-11-11T08:16:13Z","content_type":"text/html","content_length":"5170","record_id":"<urn:uuid:584d2f3d-d514-47ea-8f2e-f30d27833948>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00368.warc.gz"} |
Force and Acceleration in context of initial acceleration
30 Aug 2024
Journal of Physics and Engineering
Volume 12, Issue 3, 2023
Force and Acceleration: An Exploration of Initial Acceleration
The relationship between force and acceleration is a fundamental concept in physics, with far-reaching implications for our understanding of the natural world. In this article, we delve into the
specifics of initial acceleration, examining the mathematical framework that underlies this phenomenon.
When an object is subjected to a constant force, it accelerates at a rate proportional to the magnitude of the force and inversely proportional to its mass. This fundamental principle is encapsulated
in Newton’s second law of motion:
F = ma
where F represents the net force acting on the object, m is its mass, and a is the resulting acceleration.
Initial Acceleration
In the context of initial acceleration, we consider a scenario where an object is suddenly subjected to a constant force. The key question here is: what is the instantaneous rate at which the object
accelerates when the force is first applied?
To address this query, we can differentiate Newton’s second law with respect to time:
dF/dt = d(ma)/dt
Since m is assumed to be constant, the equation simplifies to:
dF/dt = a \* dm/dt + m \* da/dt
However, since the force is applied instantaneously, we can assume that the mass remains unchanged (dm/dt = 0). Therefore, the equation reduces to:
a = F / m
This result indicates that the initial acceleration of an object subjected to a constant force is directly proportional to the magnitude of the force and inversely proportional to its mass.
In conclusion, our analysis has provided insight into the relationship between force and acceleration in the context of initial acceleration. The mathematical framework developed here highlights the
fundamental principles governing this phenomenon, underscoring the importance of Newton’s second law of motion in understanding the behavior of physical systems.
• Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica.
• Feynman, R. P., Leighton, R. B., & Sands, M. L. (1963). The Feynman Lectures on Physics.
Related articles for ‘initial acceleration’ :
• Reading: Force and Acceleration in context of initial acceleration
Calculators for ‘initial acceleration’ | {"url":"https://blog.truegeometry.com/tutorials/education/ca96391a37f5adeb3220a7b9cc92639a/JSON_TO_ARTCL_Force_and_Acceleration_in_context_of_initial_acceleration.html","timestamp":"2024-11-05T15:37:59Z","content_type":"text/html","content_length":"15936","record_id":"<urn:uuid:b011a6b5-35ce-4c21-b4f6-61d99a765e46>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00595.warc.gz"} |
За лятото и връзката между UVA и UVB | PolinaSofiaЗа лятото и връзката между UVA и UVB - PolinaSofiaЗа лятото и връзката между UVA и UVB
July 17, 2020
За лятото и връзката между UVA и UVB
Кой е любимият ви сезон? Моят определено е лятото, колкото и взискателно откъм грижи да е то! Да, да, именно взискателно, момичета. Това, което може би не знаете е, че SPF (sun protection factor) има
отношение само към UVB лъчите. Това означава, че при избора на слънцезащитен продукт трябва търсим и UVA защита, изрично спомената от производителя върху опаковката на продукта. Съотношението между
стойностите на SPF и UVA трябва да бъде поне 1:3, а в най-добрия случай 1:1 и трябва да е придружено с логото (буквите UVA), което е Официален печат на Европейската комисия/COLIPA. Това е от
изключително значение, тъй като UVA минават дори чрез стъклата на прозорците (камо ли в ‘убежещето’ ни под чадъра на плажа). Общо взето, в горещите дни няма къде да избягаме на 100% от UVA, така че
онова, което трябва да направим за себе си е просто да изберем правилен продукт и да го подновяваме/повтаряме през 2 часа (това с подновяването не е маркетингов трик, наистина филтрите си губят
ефекта след 1,5-2 часа). Отделно, нека не забравяме и това, че кожата лятото изсъхва драстично бързо и изпозването на т. нар. ‘продукти за след слънце’ e задължително. Абсолютно всеки тип кожа, във
всяка възраст се нуждае от добрата хидратация. Това е по отношение на козметиката при мен.
По отношение на мода, за постигане на максимално разнообразни и смели летни визии, това лято ще залагам на ефектни аксесоари. Любимите ми находки са от новата колекция аксесоари и бижута на
А що се отнася разкрасяването отвътре, на всички нас, дамите, препоръчвам да не пропускаме вечерите по женски с приятелки (колкото и да сме заети в забързаното ежедневие) и поне един морски
уикенд2020г., отново по женски. Има нещо същински магическо в това как често и ние, и човешките отношения разцъфтяват именно лятото!
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Hubble's Law Disproves Young Earth Creationism
I take the position that the Universe is about 13.8 billion years old and the Earth is about 4.5 billion years old. In my opinion, one of the larger pieces of evidence against a young Earth, is
Hubble's Law.
Hubble's law states that galaxies outside of the Local Group are moving away from earth, and the speed at which they are moving away is proportional to the distance they are from Earth.
In a formula Hubble's Law is the following:
Meaning the speed at which a distant galaxy is moving in km/s is equivalent to its distance from earth in megaparsecs (mpc) multiplied by Hubble's Constant, which is about 71.
With this formula we can calculate the age of the earth.
Distance divided by velocity is time. So we now know the age of the universe is equal to the inverse of Hubble's Constant.
Hubble's Constant is in [km/s]/[mpc] so we can plug that in.
There are 3.0857e19 kilometers in a megaparsec.
We can cancel out the kilometers and get the following:
So the universe is 4.3460563e17 seconds old. Which is 13.78 billion years.
So my question to young earth creationists is:
How do you reconcile a belief that the earth is less than 10,000 years old with Hubble's Law?
Edited by Calvin, : No reason given. | {"url":"https://evcforum.net/dm.php?control=newerthread&t=19674","timestamp":"2024-11-02T08:23:52Z","content_type":"text/html","content_length":"39707","record_id":"<urn:uuid:dfb5939d-debe-4848-9b12-f0ed40cdf2da>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00852.warc.gz"} |
Book Value per Share Formula | Calculator (Excel template)
Updated November 23, 2023
Book Value Per Share Formula (Table of Contents)
Book Value Per Share Formula
Total liabilities are the total debt and financial obligations payable by the company to organizations or individuals at any defined period of time. Total liabilities are stated on the balance sheet
by the company.
Total Assets are the total assets owned by an entity or an individual. Assets are items of monetary value used over time to produce a benefit for the asset’s holder. If a company owns assets, it
includes them in the balance sheet to maintain accurate accounting records.
Examples of Book Value Per Share Formula
Let’s take an example to find out the Book Value Per Share for a company: –
Example #1
Let’s assume Company Anand Pvt Ltd has $25,000,000 of stockholders’ equity, $5,000,000 preferred stock, and total outstanding shares of $10,000,000 shares outstanding. We must calculate the book
value per share for the Anand Group of companies.
Now, we can calculate the Book Value Per Share for Anand Pvt Ltd by using the Book Value Per Share Formula:
• Stockholder’s Equity = $25,000,000
• Preferred Equity = $5,000,000
• Total Outstanding Common Shares = $10,000,000
By using the Book Value per Share Formula
• Book Value per Share = (Shareholders’ Equity – Preferred Equity) / Total Outstanding Common Shares
• Book Value per Share = $(25,000,000- $5,000,000) / $10,000,000
• Book Value per Share = $2
This shows Anand Group has a book value per share of $2.
Example #2
Jagriti Group of Companies has the following details as per its financials for the year ended 2017-18:
• Total assets = $200,000
• Total liabilities = $50,000
• Preferred shares = $25,000
• Number of outstanding common shares = 5000 shares
We must calculate the Book Value Per Share of Jagriti Group of Companies.
As we can see in the above case, the Shareholder’s Equity is not provided, then we have to calculate the Shareholder’s Equity by using the below formula:
• Total assets = $200,000
• Total liabilities = $50,000
Shareholder’s Equity Formula
• Shareholder’s Equity =Total assets – Total Liabilities
• Shareholder’s equity = $200,000 – $50,000
• Shareholder’s Equity = $1,50,000
Now, we have to calculate how much common shareholders will be getting from the shareholders’ equity. So, we must deduct the Preferred stocks from the Shareholders’ equity.
• Common shareholders’ equity = Shareholder’s equity – Preferred Share
• Common shareholder’s equity = $1,50,000- $25,000
• Common shareholder’s equity = $1,25,000
Now, by using the formula below, we can calculate Book Value Per Share:
• Book Value per Share = (Shareholders’ Equity – Preferred Equity) / Total Outstanding Common Shares
• Book Value per share = $1,50,000- $25,000/ 5,000
• Book Value per share = $1,25,000/ 5,000
• Book Value per share = $25
The book Value per share of Jagriti Group of Companies is $25
Example #3
Calculate the Book Value per share for Anand Group of Companies using the following extracts available:
• Current Assets = $70,000
• Non-current Assets = $230,000
• Current Liabilities = $60,000
• Non-Current Liabilities = $30,000
• Preferred shares = $45,000
• Number of outstanding common shares = 3500 shares
For calculating Book Value Per Share, we need Shareholders’ Equity, which can be calculated as below:
• Total assets = Current Assets + Non-current Assets
• Total Liabilities = Current Liabilities + Non-Current Liabilities
Shareholder’s Equity =Total assets – Total Liabilities
• Shareholder’s Equity = (Current Assets + Non-current Assets) – (Current Liabilities + Non-Liabilities)
• Shareholder’s Equity = ($70,000 + $230,000) – ($60,000 + $30,000)
• Shareholder’s Equity = $3,00,000 – $90,000
• Shareholder’s Equity = $2,10,000
Now, by using the formula below, we can calculate Book Value Per Share:
Book Value per Share = (Shareholder’s Equity – Preferred Equity) / Total Outstanding Common Shares
• Book Value per share = ($2,10,000- $45,000)/3500
• Book Value per share = $47.14
The book Value per share of Jagriti Group of Companies is $47.14.
You can calculate the book value per share to determine the value of a company per share. The calculation is based on the equity available to common shareholders after paying off the debts and
preferred shareholders for which the company is legally obliged. One must subtract preferred shares from the shareholders’ equity when calculating book value per share.
A company’s “Book Value”, also referred to as Shareholder’s Equity or Owner’s Equity, can be calculated by subtracting Total Liabilities from Total Assets.
Therefore, Shareholder’s Equity =Total assets – Total Liabilities
And, Book Value per Share = (Shareholders’ Equity – Preferred Equity) / Total Outstanding Common Shares. The data mentioned above can be found on the company’s balance sheet.
Significance and Use of Book Value Per Share
The investors can use book value per share to determine the equity in a company compared to the company’s current market value, that is, the current price of the stock. For example, Let’s assume
Anand Ltd is currently trading for $30. But it has a book value of $15. This shows the stock of Anand Ltd is selling at double, I.e., two times its equity. The above example is used in valuation
methodology, i.e., Multiple Valuation (price to book value or P/B) or relative valuation; in this formula, book value per share is used in the denominator.
The stock’s current market price reflects its growth potential in contrast to its Book Value. One can look at their book value per share to compare the value of different companies. The price-to-book
value ratio, also known as P/B, is the value of a company’s common stock, which can be determined by using its book value per share or by Company B’s price-to-book value ratio or the industry ratio.
To compute the return on equity formula, investors can use the book value per share, abbreviated as ROE. In this scenario, one calculates ROE on a per-share basis.
Simply divide the stockholder’s equity by the net income to calculate the ROE. ROE per share = (Net Income Per share or EPS)/Book Value per share. Per per-share basis of Net income is referred to as
Earnings per share or EPS. As the article demonstrates, the book value per share represents the stockholder’s equity per share.
Book Value Per Share Calculator
You can use the following Book Value per Share Calculator
Shareholder's Equity
Preferred Equity
Total Outstanding Common Share's
Book Value per Share Formula =
Shareholder's Equity − Preferred Equity
Book Value per Share Formula = =
Total Outstanding Common Share's
Book Value Per Share Formula in Excel (With Excel Template)
Here, we will do the same example of the Book Value per Share in Excel. It is very easy and simple. You need to provide the two inputs, i.e., Shareholders’ Equity and Preferred Equity
You can easily calculate the Book Value per Share using the formula in the template provided.
In this template, we have to solve the Book Value per Share Formula
Hence first, we are calculating the Shareholder’s Equity by using the Shareholder’s Equity Formula.
Now, we will calculate the Book Value per Share by using the formula.
Recommended Articles
This has guided the Book Value per Share Formula; here, we discuss its uses and practical examples. We also provide a Book Value per Share calculator and a downloadable Excel template. | {"url":"https://www.educba.com/book-value-per-share-formula/","timestamp":"2024-11-13T21:30:36Z","content_type":"text/html","content_length":"347102","record_id":"<urn:uuid:f0def0c5-a18a-4662-b308-0a499080cde6>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00638.warc.gz"} |
What is a linear regression t-test used for?
A T-test is used to compare the means of two different sets of observed data and to find to what extent such difference is ‘by chance’. Linear Regression is used to find the relationship between one
dependent or outcome variable and one or more independent or predictor variables.
What is the T value in regression?
The t statistic is the coefficient divided by its standard error. The standard error is an estimate of the standard deviation of the coefficient, the amount it varies across cases. It can be thought
of as a measure of the precision with which the regression coefficient is measured.
What is omnibus in regression?
Omnibus Tests in Multiple Regression. In Multiple Regression the omnibus test is an ANOVA F test on all the coefficients, that is equivalent to the multiple correlations R Square F test.
What is a multivariate regression test?
Multivariate Regression is a method used to measure the degree at which more than one independent variable (predictors) and more than one dependent variable (responses), are linearly related. A
mathematical model, based on multivariate regression analysis will address this and other more complicated questions.
How do you test multiple regression?
Test for Significance of Regression. The test for significance of regression in the case of multiple linear regression analysis is carried out using the analysis of variance. The test is used to
check if a linear statistical relationship exists between the response variable and at least one of the predictor variables.
What is a strong t-value?
A t-value between 1.5 to 2.0 indicates some evidence of learning. c. A t-value between 2 to 3 indicates strong evidence of learning. d. A t-value above 3 indicates very strong strong evidence of
Why is my t-value so high?
Higher values of the t-value, also called t-score, indicate that a large difference exists between the two sample sets. The smaller the t-value, the more similarity exists between the two sample
sets. A large t-score indicates that the groups are different.
Which is the best practice to deal with Heteroskedasticity?
The solution. The two most common strategies for dealing with the possibility of heteroskedasticity is heteroskedasticity-consistent standard errors (or robust errors) developed by White and Weighted
Least Squares.
When would you use a multivariate regression?
Multivariate regression comes into the picture when we have more than one independent variable, and simple linear regression does not work. Real-world data involves multiple variables or features and
when these are present in data, we would require Multivariate regression for better analysis.
What is multivariate regression used for?
Multivariable regression models are used to establish the relationship between a dependent variable (i.e. an outcome of interest) and more than 1 independent variable.
How is the t statistic used in linear regression?
In linear regression, the t -statistic is useful for making inferences about the regression coefficients. The hypothesis test on coefficient i tests the null hypothesis that it is equal to zero –
meaning the corresponding term is not significant – versus the alternate hypothesis that the coefficient is different from zero.
How to test for significance of regression coefficients?
This example shows how to test for the significance of the regression coefficients using t-statistic. Load the sample data and fit the linear regression model.
Where do I find TSTAT for the Hypotheses test?
You can see that for each coefficient, tStat = Estimate/SE. The -values for the hypotheses tests are in the pValue column. Each -statistic tests for the significance of each term given other terms in
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6. Differences between percentages and paired alternatives | The BMJ6. Differences between percentages and paired alternatives
Standard error of difference between percentages or proportions
The surgical registrar who investigated appendicitis cases, referred to in
Chapter 3
, wonders whether the percentages of men and women in the sample differ from the percentages of all the other men and women aged 65 and over admitted to the surgical wards during the same period.
After excluding his sample of appendicitis cases, so that they are not counted twice, he makes a rough estimate of the number of patients admitted in those 10 years and finds it to be about 12-13
000. He selects a systematic random sample of 640 patients, of whom 363 (56.7%) were women and 277 (43.3%) men.
The percentage of women in the appendicitis sample was 60.8% and differs from the percentage of women in the general surgical sample by 60.8 – 56.7 = 4.1%. Is this difference of any significance? In
other words, could this have arisen by chance?
There are two ways of calculating the standard error of the difference between two percentages: one is based on the null hypothesis that the two groups come from the same population; the other on the
alternative hypothesis that they are different. For Normally distributed variables these two are the same if the standard deviations are assumed to be the same, but in the binomial case the standard
deviations depend on the estimates of the proportions, and so if these are different so are the standard deviations. Usually both methods give almost the same result.
Confidence interval for a difference in proportions or percentages
The calculation of the standard error of a difference in proportions p1 – p2 follows the same logic as the calculation of the standard error of two means; sum the squares of the individual standard
errors and then take the square root. It is based on the alternative hypothesis that there is a real difference in proportions (further discussion on this point is given in Common questions at the
end of this chapter).
Note that this is an approximate formula; the exact one would use the population proportions rather than the sample estimates. With our appendicitis data we have:
Thus a 95% confidence interval for the difference in percentages is
4.1 – 1.96 x 4.87 to 4.1 + 1.96 x 4.87 = -5.4 to 13.6%.
Significance test for a difference in two proportions
For a significance test we have to use a slightly different formula, based on the null hypothesis that both samples have a common population proportion, estimated by p.
To obtain p we must amalgamate the two samples and calculate the percentage of women in the two combined; 100 – p is then the percentage of men in the two combined. The numbers in each sample are
Number of women in the samples: 73 + 363 = 436
Number of people in the samples: 120 + 640 = 760
Percentage of women: (436 x 100)/760 = 57.4
Percentage of men: (324 x 100)/760 = 42.6
Putting these numbers in the formula, we find the standard error of the difference between the percentages is 4.1-1.96 x 4.87 to 4.1 + 1.96 x 4.87 = -5.4 to 13.6%
This is very close to the standard error estimated under the alternative hypothesis.
The difference between the percentage of women (and men) in the two samples was 4.1%. To find the probability attached to this difference we divide it by its standard error: z = 4.1/4.92 = 0.83. From
Table A (
Appendix table A.pdf
) we find that P is about 0.4 and so the difference between the percentages in the two samples could have been due to chance alone, as might have been expected from the confidence interval. Note that
this test gives results identical to those obtained by the
Standard error of a total
The total number of deaths in a town from a particular disease varies from year to year. If the population of the town or area where they occur is fairly large, say, some thousands, and provided that
the deaths are independent of one another, the standard error of the number of deaths from a specified cause is given approximately by its square root,
This can be used to estimate the significance of a difference between two totals by dividing the difference by its standard error:
It is important to note that the deaths must be independently caused; for example, they must not be the result of an epidemic such as influenza. The reports of the deaths must likewise be
independent; for example, the criteria for diagnosis must be consistent from year to year and not suddenly change in accordance with a new fashion or test, and the population at risk must be the same
size over the period of study.
In spite of its limitations this method has its uses. For instance, in Carlisle the number of deaths from ischaemic heart disease in 1973 was 276. Is this significantly higher than the total for
1972, which was 246? The difference is 30. The standard error of the difference is
table A shows that P = 0.2. The difference could therefore easily be a chance fluctuation.
This method should be regarded as giving no more than approximate but useful guidance, and is unlikely to be valid over a period of more than very few years owing to changes in diagnostic techniques.
An extension of it to the study of paired alternatives follows.
Paired alternatives
Sometimes it is possible to record the results of treatment or some sort of test or investigation as one of two alternatives. For instance, two treatments or tests might be carried out on pairs
obtained by matching individuals chosen by random sampling, or the pairs might consist of successive treatments of the same individual (see
Chapter 7
for a comparison of pairs by the tt test). The result might then be recorded as “responded or did not respond”, “improved or did not improve”, “positive or negative”, and so on. This type of study
yields results that can be set out as shown in table 6.1.
Table 6.1
Member of pair receiving treatment A Member of pair receiving treatment B
Responded Responded (1)
Responded Did not respond (2)
Did not respond Responded (3)
Did not respond Did not respond (2)
The significance of the results can then be simply tested by McNemar’s test in the following way. Ignore rows (1) and (4), and examine rows (2) and (3). Let the larger number of pairs in either of
rows (2) or (3) be called n1 and the smaller number of pairs in either of those two rows be n2. We may then use formula ( 6.1 ) to obtain the result, z. This is approximately Normally distributed
under the null hypothesis, and its probability can be read from
table A.
However, in practice, the fairly small numbers that form the subject of this type of investigation make a correction advisable. We therefore diminish the difference between n1and n2 by using the
following formula:
where the vertical lines mean “take the absolute value”.
Again, the result is Normally distributed, and its probability can be read from. As for the unpaired case, there is a slightly different formula for the standard error used to calculate the
confidence interval.(1) Suppose N is the total number of pairs, then
For example, a registrar in the gastroenterological unit of a large hospital in an industrial city sees a considerable number of patients with severe recurrent aphthous ulcer of the mouth. Claims
have been made that a recently introduced preparation stops the pain of these ulcers and promotes quicker healing than existing preparations.
Over a period of 6 months the registrar selected every patient with this disorder and paired them off as far as possible by reference to age, sex, and frequency of ulceration. Finally she had 108
patients in 54 pairs. To one member of each pair, chosen by the toss of a coin, she gave treatment A, which she and her colleagues in the unit had hitherto regarded as the best; to the other member
she gave the new treatment, B. Both forms of treatment are local applications, and they cannot be made to look alike. Consequently to avoid bias in the assessment of the results a colleague recorded
the results of treatment without knowing which patient in each pair had which treatment. The results are shown in Table 6.2
Member of pair receiving treatment A Member of pair receiving treatment B Pairs of patients
Responded Responded 16
Responded Did not respond 23
Did not respond Responded 10
Did not respond Did not respond 5
Total 54
Here n[1]=23, n[2]=10. Entering these values in formula (6.1) we obtain
The probability value associated with 2.089 is about 0.04
Table A
. Therefore we may conclude that treatment A gave significantly better results than treatment B. The standard error for the confidence interval is
23/54 – 10/54 = 0.241
The 95% confidence interval for the difference in proportions is
0.241 – 1.96 x 0.101 to 0.241 + 1.96 x 0.10 that is, 0.043 to 0.439.
Although this does not include zero, the confidence interval is quite wide, reflecting uncertainty as to the true difference because the sample size is small. An exact method is also available.
Common questions
Why is the standard error used for calculating a confidence interval for the difference in two proportions different from the standard error used for calculating the significance?
For nominal variables the standard deviation is not independent of the mean. If we suppose that a nominal variable simply takes the value 0 or 1, then the mean is simply the proportion of is and the
standard deviation is directly dependent on the mean, being largest when the mean is 0.5. The null and alternative hypotheses are hypotheses about means, either that they are the same (null) or
different (alternative). Thus for nominal variables the standard deviations (and thus the standard errors) will also be different for the null and alternative hypotheses. For a confidence interval,
the alternative hypothesis is assumed to be true, whereas for a significance test the null hypothesis is assumed to be true. In general the difference in the values of the two methods of calculating
the standard errors is likely to be small, and use of either would lead to the same inferences. The reason this is mentioned here is that there is a close connection between the test of significance
described in this chapter and the Chi square test described in Chapter 8. The difference in the arithmetic for the significance test, and that for calculating the confidence interval, could lead some
readers to believe that they are unrelated, whereas in fact they are complementary. The problem does not arise with continuous variables, where the standard deviation is usually assumed independent
of the mean, and is also assumed to be the same value under both the null and alternative hypotheses.
It is worth pointing out that the formula for calculating the standard error of an estimate is not necessarily unique: it depends on underlying assumptions, and so different assumptions or study
designs will lead to different estimates for standard errors for data sets that might be numerically identical.
1. Gardner MJ, Altman DG, editors. Statistics with Confidence. London: BMJ Publishing, 1989:31.
Exercise 6.1
In an obstetric hospital l7.8% of 320 women were delivered by forceps in 1975. What is the standard error of this percentage? In another hospital in the same region 21.2% of 185 women were delivered
by forceps. What is the standard error of the difference between the percentages at this hospital and the first? What is the difference between these percentages of forceps delivery with a 95%
confidence interval and what is its significance?
Exercise 6.2
A dermatologist tested a new topical application for the treatment of psoriasis on 47 patients. He applied it to the lesions on one part of the patient’s body and what he considered to be the best
traditional remedy to the lesions on another but comparable part of the body, the choice of area being made by the toss of a coin. In three patients both areas of psoriasis responded; in 28 patients
the disease responded to the traditional remedy but hardly or not at all to the new one; in 13 it responded to the new one but hardly or not at all to the traditional remedy; and in four cases
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- T
Lesson 7
Comparing Numbers and Distance from Zero
7.1: Opposites (10 minutes)
The purpose of this warm-up is to use opposites to get students to think about distance from 0. Problem 3 also reminds students that the opposite of a negative number is positive.
Notice students who choose 0 or a negative number for \(a\) and how they reason about \(\text-a\).
Arrange students in groups of 2. Give students 5 minutes of quiet think time, then 2 minutes of partner discussion. Follow with whole-class discussion.
Student Facing
1. \(a\) is a rational number. Choose a value for \(a\) and plot it on the number line.
1. Based on where you plotted \(a\), plot \(\text- a\) on the same number line.
2. What is the value of \(\text- a\) that you plotted?
3. Noah said, “If \(a\) is a rational number, \(\text- a\) will always be a negative number.” Do you agree with Noah? Explain your reasoning.
Anticipated Misconceptions
For problem 3, students might assume that \(\text-a\) is always a negative number. Ask these students to start with a negative number and find its opposite. For example, starting with \(a = \text-3\)
, we can find its opposite, \(\text-(\text-3)\), to be equal to 3.
Activity Synthesis
The main idea of discussion is that opposites have the same distance to 0 (i.e., same absolute value) and that the opposite of a negative number is positive. Ask students to discuss their reasoning
with a partner. In a whole-class discussion, ask a student who chose \(a\) to be positive to share their reasoning about how to plot \(\text-a\) and whether they agreed with Noah in problem 3.
Then, select previously identified students who chose \(a\) to be negative to share their thinking. If not mentioned by students, emphasize both symbolic and geometric statements of the fact that the
opposite of a negative number is positive. For example, if \(a=\text-3\), write \(\text-(\text-3) = 3\) and show that 3 is the opposite of -3 on the number line because they are the same distance to
0. If time allows, select a student who chose \(a\) to be 0 and compare to cases where \(a\) is negative or positive. The number 0 is its own opposite because no other number is 0 units away from 0.
Sequencing the discussion to look at positive, negative, and 0 values of \(a\) helps students to visualize and generalize the concept of opposites for rational numbers.
7.2: Submarine (15 minutes)
Students distinguish between absolute value and order in the context of elevation. Students express their ideas carefully using symbols, verbally, and using a vertical number line. Placing possible
elevations on the number line serves as a transition to thinking about solutions to inequalities. Look for students who choose positive and negative elevations for Han and Lin to compare in the
Arrange students in groups of 4. Distribute one set of sticky notes to each group, where each note contains one name: Clare, Andre, Han, Lin, and Priya. Display the image for all to see throughout
the activity.
Ask students to read the instructions for the task and the description of each person's elevation. Give them a few minutes to use their sticky notes, as a group, to decide where each person (except
Priya) could be located.
Place Clare’s sticky note on the number line according to the completed first row of the table. Explain the completed first row of the table to students as it pertains to Clare’s description. Use
precise language when explaining the symbols in the table:
• One possible elevation for Clare is 150 feet because 150 is greater than -100, and it is also farther from sea level.
• 150 is greater than -100.
• The absolute value of 150 is greater than the absolute value of -100.
Ask groups to complete the rest of the table for the other people (except Priya), and then answer the question about Priya. Note that it is possible to come up with different, correct responses that
fit the descriptions. Give students 10 minutes to work followed by whole-class discussion.
Representation: Access for Perception. Activate or supply background knowledge. Give students 1–2 minutes to review the first row of the table that shows a possible elevation for Clare. Invite 1–2
students to think aloud and share connections they make between the display with the sticky notes, and the values in the table. Record their thinking on a display of the table and keep the work
visible as students work.
Supports accessibility for: Organization; Attention
Student Facing
A submarine is at an elevation of -100 feet (100 feet below sea level). Let’s compare the elevations of these four people to that of the submarine:
• Clare’s elevation is greater than the elevation of the submarine. Clare is farther from sea level than the submarine.
• Andre’s elevation is less than the elevation of the submarine. Andre is farther away from sea level than the submarine.
• Han’s elevation is greater than the elevation of the submarine. Han is closer to sea level than is the submarine.
• Lin’s elevation is the same distance away from sea level as the submarine’s.
1. Complete the table as follows.
1. Write a possible elevation for each person.
2. Use \(<\), \(>\), or \(=\) to compare the elevation of that person to that of the submarine.
3. Use absolute value to tell how far away the person is from sea level (elevation 0).
As an example, the first row has been filled with a possible elevation for Clare.
│ │possible │ compare to │ distance from │
│ │elevation│ submarine │ sea level │
│Clare│150 feet │\(150 > \text-100\)│\(|150|\) or 150 feet │
│Andre│ │ │ │
│ Han │ │ │ │
│ Lin │ │ │ │
2. Priya says her elevation is less than the submarine’s and she is closer to sea level. Is this possible? Explain your reasoning.
Activity Synthesis
The purpose of the discussion is to let students practice using proper vocabulary to express ideas that distinguish order from absolute value with positive and negative numbers. Select previously
identified students to share different elevations for Han and for Lin that show both positive and negative possibilities. Encourage students to explain why the elevation they chose satisfies the
description in the problem. As students speak, record their statements using \(<,>,=\) and \(|\boldcdot |\). Allow students to rearrange sticky notes on the vertical number line display. If time
allows, use the sticky notes to show the range of possible solutions for each character; this will help to further prepare students for the concept of graphing solutions of an inequality on the
number line.
Speaking: MLR8 Discussion Supports. To support students’ use of vocabulary related to absolute value and positive and negative numbers, provide sentence frames related to each column heading. Some
examples include: “_____ could have an elevation of _____ because _____,” “Comparing _____’s elevation to the submarine’s, I notice _____,” or “_____’s distance from sea level is _____ because
Design Principle(s): Cultivate conversation
7.3: Info Gap: Points on the Number Line (15 minutes)
Optional activity
In this info gap activity, students use comparisons of order and absolute value of rational numbers to determine the location of unknown points on the number line. In doing so students reinforce
their understanding that a number and its absolute value are different properties. Students will also begin to understand that the distance between two numbers, while being positive, could be in
either direction between the numbers. This concept is expanded on further when students study arithmetic with rational numbers in grade 7.
The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of
discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the
information they need (MP6).
Here is the text of the cards for reference and planning:
Arrange students in groups of 2. In each group, distribute the first problem card to one student and a data card to the other student. After debriefing on the first problem, distribute the cards for
the second problem, in which students switch roles.
Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their
own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization
Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to determine the location of unknown points on the number line. Display questions
or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation
Student Facing
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the problem card:
1. Silently read your card and think about what information you need to be able to answer the question.
2. Ask your partner for the specific information that you need.
3. Explain how you are using the information to solve the problem.
Continue to ask questions until you have enough information to solve the problem.
4. Share the problem card and solve the problem independently.
5. Read the data card and discuss your reasoning.
If your teacher gives you the data card:
1. Silently read your card.
2. Ask your partner “What specific information do you need?” and wait for them to ask for information.
If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.
3. Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.
4. Read the problem card and solve the problem independently.
5. Share the data card and discuss your reasoning.
Anticipated Misconceptions
Students may struggle to make sense of the abstract information they are given if they don't choose to draw a number line. Rather than specifically instructing them to use this strategy, consider
asking them a question like “How could you keep track of the information you've learned about the points so far?”
Activity Synthesis
Select students with different strategies to share their approaches. Invite them to share which of the clues they thought were more helpful and which were least helpful. Ask students to explain how
drawing a number line helped them and how they decided on the appropriate order for the unknown numbers.
7.4: Inequality Mix and Match (15 minutes)
Optional activity
The goal of this activity is for students to practice comparing rational numbers.
Notice students who compare fractions to decimals, fractions to integers, or who compare absolute values to negative numbers.
Arrange students in groups of 2. Give students 10 minutes to work before whole-class discussion.
Action and Expression: Provide Access for Physical Action. Create alternatives for physically interacting with materials. Consider creating a set of cards for each of the numbers and inequality
symbols that students can select from and sequence to create true comparison statements. Invite students to talk about their statements before writing them down.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Speaking: MLR5 Co-Craft Questions. To create space for students to produce the language of mathematical questions themselves, display only the array of numbers that the students will be using in this
activity. Ask students to think about the values of the numbers and write a mathematical question using two or more numbers from the array. Students may generate questions such as “How many values
are greater than zero?” or “Which numbers are opposites?” Notice students that have questions about comparing and ordering the numbers and ask them to share their questions. This will help students
use conversation skills to generate, choose, and improve their questions as well as develop meta-awareness of the language used in mathematical questions.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
Here are some numbers and inequality symbols. Work with your partner to write true comparison statements.
\(|\text{-}\frac {5}{2}|\)
One partner should select two numbers and one comparison symbol and use them to write a true statement using symbols. The other partner should write a sentence in words with the same meaning, using
the following phrases:
• is equal to
• is the absolute value of
• is greater than
• is less than
For example, one partner could write \(4 < 8\) and the other would write, “4 is less than 8.” Switch roles until each partner has three true mathematical statements and three sentences written down.
Student Facing
Are you ready for more?
For each question, choose a value for each variable to make the whole statement true. (When the word and is used in math, both parts have to be true for the whole statement to be true.) Can you do it
if one variable is negative and one is positive? Can you do it if both values are negative?
1. \(x < y\) and \(|x| < y\).
2. \(a < b\) and \(|a| < |b|\).
3. \(c < d\) and \(|c| > d\).
4. \(t < u\) and \(|t| > |u|\).
Activity Synthesis
The goal of discussion is to allow students to use precise language when comparing rational numbers and absolute values verbally. Select previously identified students to share their responses
that compare fractions to decimals, fractions to integers, or absolute values to negative numbers. Display their responses using absolute value and \(>, <, =\) symbols for all to see. Ask students to
indicate whether they agree that each response is true, and ask students to share their reasoning about whether they agree or disagree.
Lesson Synthesis
During this lesson, students have used precise language to distinguish absolute value from order of rational numbers. Display \(|\text-8|\) and 3 questions for all to see:
• “How do you say this?” (The absolute value of -8.)
• “What does it mean in an elevation situation?” (It’s the distance from 8 feet below sea level to sea level.)
• “What does it mean on a number line?” (It’s the distance from -8 to 0 on the number line.)
• “What is its value?” (8.)
Next, display \(|\text-8| < 5\) and two questions for all to see:
• “How do you say this?” (The absolute value of -8 is less than 5.)
• “What does it mean on a number line?” (-8 is less than 5 units away from 0.)
• “Is it true?” (No, -8 is more than 5 units away from 0.)
7.5: Cool-down - True or False? (5 minutes)
Student Facing
We can use elevation to help us compare two rational numbers or two absolute values.
• Suppose an anchor has an elevation of -10 meters and a house has an elevation of 12 meters. To describe the anchor having a lower elevation than the house, we can write \(\text-10<12\) and say
“-10 is less than 12.”
• The anchor is closer to sea level than the house is to sea level (or elevation of 0). To describe this, we can write \(|\text-10|<|12|\) and say “the distance between -10 and 0 is less than the
distance between 12 and 0.”
We can use similar descriptions to compare rational numbers and their absolute values outside of the context of elevation.
• To compare the distance of -47.5 and 5.2 from 0, we can say: \(|\text-47.5|\) is 47.5 units away from 0, and \(|5.2|\) is 5.2 units away from 0, so \(|\text-47.5|>|5.2|\).
• \(|\text-18|>4\) means that the absolute value of -18 is greater than 4. This is true because 18 is greater than 4. | {"url":"https://curriculum.illustrativemathematics.org/MS/teachers/1/7/7/index.html","timestamp":"2024-11-07T23:58:17Z","content_type":"text/html","content_length":"114566","record_id":"<urn:uuid:b527db78-c294-4343-aed7-eca4e92db3ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00448.warc.gz"} |
Items where Division is "College of Engineering & Physical Sciences > Systems analytics research institute (SARI)" and Year is 2006
Number of items: 27.
Alamino, Roberto C. and Caticha, Nestor (2006). Online learning in discrete hidden Markov models. IN: Bayesian inference and maximum entropy methods In science and engineering. Mohammad-Djafari, Ali
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Nerukh, A., Ruzhytska, N. and Nerukh, D. (2006). Quasi-intermittency of waves and their complexity in modulated dielectric medium. IN: Conference Proceedings - 11th International Conference on
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Yurkevich, I.V., Lerner, I.V., Stepanenko, A.S. and Constantinou, C.C. (2006). Random walks in local dynamics of network losses. Physical Review E, 74 (4), | {"url":"https://publications.aston.ac.uk/view/divisions/NRG2016/2006.default.html","timestamp":"2024-11-09T15:39:12Z","content_type":"application/xhtml+xml","content_length":"33352","record_id":"<urn:uuid:7f1a90c6-3523-4231-a911-aff836e5d89a>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00259.warc.gz"} |
numpy.polynomial.chebyshev.chebder(c, m=1, scl=1, axis=0)[source]¶
Differentiate a Chebyshev series.
Returns the Chebyshev series coefficients c differentiated m times along axis. At each iteration the result is multiplied by scl (the scaling factor is for use in a linear change of variable).
The argument c is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series 1*T_0 + 2*T_1 + 3*T_2 while [[1,2],[1,2]] represents 1*T_0(x)*T_0(y) +
1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y) if axis=0 is x and axis=1 is y.
Array of Chebyshev series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index.
mint, optional
Number of derivatives taken, must be non-negative. (Default: 1)
sclscalar, optional
Each differentiation is multiplied by scl. The end result is multiplication by scl**m. This is for use in a linear change of variable. (Default: 1)
axisint, optional
Axis over which the derivative is taken. (Default: 0).
Chebyshev series of the derivative.
In general, the result of differentiating a C-series needs to be “reprojected” onto the C-series basis set. Thus, typically, the result of this function is “unintuitive,” albeit correct; see
Examples section below.
>>> from numpy.polynomial import chebyshev as C
>>> c = (1,2,3,4)
>>> C.chebder(c)
array([14., 12., 24.])
>>> C.chebder(c,3)
>>> C.chebder(c,scl=-1)
array([-14., -12., -24.])
>>> C.chebder(c,2,-1)
array([12., 96.]) | {"url":"https://numpy.org/doc/1.18/reference/generated/numpy.polynomial.chebyshev.chebder.html","timestamp":"2024-11-05T17:15:45Z","content_type":"text/html","content_length":"12707","record_id":"<urn:uuid:91154f9e-49a3-4d9e-86f0-d4a36bff0349>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00732.warc.gz"} |
NCERT Solutions
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students eyes as per child psychology.
As a parent, now you need not insist on your child for studies. Children enjoy studying for hours together on computer, T.V & Tablet. Help in increasing overall % of the students.
Home Revise Features for Maths Class 1 NCERT Solutions
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It's a unique feature of Home Revise. We have added the meaning of difficult words from each paragraph for better understanding of students.
Benefits to students:
• No need to search for the meaning.
• Parents also can understand the meaning of difficult words then & there and can explain the concepts to their kids if necessary.
• Teachers can also use it as a teaching aid or reference to teach the lesson.
The Best Sound Quality
International Voice quality which is crisp & clear to understand.
Interactive Objective Test
We have provided interactive objective tests after every paragraph, which help students to check their grasping level of the lesson & to revise it thoroughly.
Result Oriented Approach
Every Education System is based on Exam system, where students have to answer the questions based on NCERT Class 1 Maths syllabus.
• We help students to understand the lessons as well as to learn the questions & answers thoroughly with the help of Maths Class 1 NCERT Solutions .
• To make the revision easy, we have explained questions & answers after every paragraph.
Exercise with Sound
• NCERT Maths Class 1 We have explained all the questions & answers (Swadhyay) at the end of lessons.
• Students can learn, by-heart the question answers then and there itself.
Three Times Revision
Students can revise the questions & answers three times with Home Revise.
• Through Interactive Test after every paragraph.
• Through the given question answers at the end of the paragraph.
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It is simple and clearly structured, which helps learners to easily and their way. | {"url":"https://homerevise.co.in/cbse/ncert-solutions-class-1-maths.html","timestamp":"2024-11-13T08:18:50Z","content_type":"text/html","content_length":"45934","record_id":"<urn:uuid:3e3d0c9f-0d5b-4831-a4cd-db9bf71facf5>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00056.warc.gz"} |
Replace an expression containing '/' with div function call in Python
Replacing Division (/) with div in Python: A Clear Explanation
Python's div function might not be as familiar as the standard division operator (/), but it can be useful in specific scenarios. This article will guide you through understanding the div function,
how it differs from traditional division, and how to replace expressions containing / with div function calls.
Understanding the Problem:
Let's say you have a Python code snippet like this:
result = 10 / 3
print(result) # Output: 3.3333333333333335
In this example, we're using the / operator for division, which results in a floating-point number. However, you might want to obtain the integer quotient of the division, discarding the remainder.
This is where the div function comes in.
Diving into the div Function
The div function, part of the math module, provides the integer quotient of a division operation. It's important to note that div is not a built-in function in Python, so you need to import the math
module first:
import math
result = math.divmod(10, 3)
print(result) # Output: (3, 1)
The divmod() function returns a tuple containing both the quotient (3) and the remainder (1) of the division.
Replacing '/' with div
To directly replace / with div, we need to extract only the quotient from the divmod() result:
import math
result = math.divmod(10, 3)[0] # Extracts the quotient
print(result) # Output: 3
This code snippet will give you the integer quotient of the division (3), discarding the remainder.
Why Use div?
While you can achieve the same result with the // operator (floor division), the div function provides a more explicit and understandable way to represent integer division in your code, especially
when working with complex mathematical calculations.
Here are some scenarios where div could be helpful:
• Clarity: When dealing with complex mathematical expressions, div can make your code easier to understand, particularly for other programmers.
• Specific Applications: In situations requiring explicit integer division for specific algorithms, div helps maintain code clarity and intention.
Replacing division expressions with div function calls in Python can improve code readability and maintain a clear intention of performing integer division. Remember to import the math module and
extract the quotient from the divmod() result when utilizing the div function.
By understanding the nuances of division in Python and leveraging the div function effectively, you can write more robust and understandable code for your mathematical operations. | {"url":"https://laganvalleydup.co.uk/post/replace-an-expression-containing-with-div-function-call-in","timestamp":"2024-11-10T20:56:59Z","content_type":"text/html","content_length":"82040","record_id":"<urn:uuid:50845808-dbe3-427e-a3c0-cda1a338a5c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00240.warc.gz"} |
Practicing exercise of multiplying and division of rational expressions
Author Message
Tafed Panmir Posted: Saturday 23rd of Dec 17:42
I think God would have been in a really bad mood that he came up with something called math to trouble us! I’ve spent hours working on this algebra problem which relates to
practicing exercise of multiplying and division of rational expressions and I still can’t solve it. I’m particularly stuck on linear equations, proportions and graphing. Can
anyone throw some light on how to go about finding a solution to such problems? I’ve tried various means that I could think of, but none helped. I need some urgent help now.
Registered: 23.11.2004
oc_rana Posted: Sunday 24th of Dec 10:00
Dear Friend , don't get strained . Check out https://algebra-test.com/features.html, https://algebra-test.com/comparison.html and https://algebra-test.com/features.html.
There is a utility by name Algebra Master available at all the three sites. This instrument would give all the details that you would require on the title College Algebra.
But, ensure that you go through all the lessons carefully.
Registered: 08.03.2007
From: egypt,alexandria
Svizes Posted: Sunday 24th of Dec 21:35
It is good to know that you wish to improve your math and are taking efforts to do so. I think you should try Algebra Master. This is not exactly a tutoring device but it
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Registered: 10.03.2003
From: Slovenia
Dolknankey Posted: Tuesday 26th of Dec 10:25
Algebra Master is the program that I have used through several algebra classes - Remedial Algebra, Pre Algebra and College Algebra. It is a truly a great piece of math
software. I remember of going through problems with adding fractions, least common denominator and difference of squares. I would simply type in a problem from the workbook ,
click on Solve – and step by step solution to my algebra homework. I highly recommend the program.
Registered: 24.10.2003
From: Where the trout
streams flow and the air
is nice
Ljrekol_G Posted: Wednesday 27th of Dec 17:18
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Registered: 25.11.2004
From: London
TC Posted: Thursday 28th of Dec 20:10
There you go https://algebra-test.com/comparison.html.
Registered: 25.09.2001
From: Kµlt °ƒ Ø, working
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If we have an estimate pˆ = 0.25 and if we want the margin of error to be ±3% (with 99% confidence), how many individuals should we sample? -
If we have an estimate pˆ = 0.25 and if we want the margin of error to be ±3% (with 99% confidence), how many individuals should we sample?
1. 361 students recorded the number of hours of television they watch per week. The sample mean is 6.5 hours with a sample standard deviation 2.5 hours. Construct a 99% confidence interval for the
population mean number of hours of television students watch per week. 2. In a survey of 1000 adults in the U.S., 20% say that they never exercise. Find a 90% confidence interval for the proportion
of all adults in the U.S. who never exercise. 3. We have a sample of 2752 adults; 832 of them have children in the house and 1920 have no children in the house. They were asked if they visited a
library in the last 12 months. 421 people who have children in the house visited a library and 810 people who do not have children in the house visited a library. Find a 92% confidence interval for
pC − pN , the difference in proportions visiting the library between those with children in the house and those that have no children in the house. 4. We want to study the difference in means when a
group of people pays individually for a dinner versus splitting the bill equally (as a group). We have a group of 40 people who split the bill equally – their sample mean cost was $50.92 with a
sample standard deviation $14.33. We also have another group of 40 people who paid for dinners individually – their sample mean cost was $38.78 with a sample standard deviation of $12.54. Find a 95%
confidence interval for μS − μI , the difference in population means costs between people who split their bills equally and people who pay their bills individually. 5. We would like to know what
proportion of people would support taxes on junk food. If we have an estimate pˆ = 0.25 and if we want the margin of error to be ±3% (with 99% confidence), how many individuals should we sample? | {"url":"https://topassignmenthelp.com/if-we-have-an-estimate-p%CB%86-0-25-and-if-we-want-the-margin-of-error-to-be-%C2%B13-with-99-confidence-how-many-individuals-should-we-sample/","timestamp":"2024-11-10T07:34:13Z","content_type":"text/html","content_length":"90075","record_id":"<urn:uuid:cc9b9839-7686-43a1-be71-196de025d4f7>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00044.warc.gz"} |
Analytic Modelling – Solved - Ideal coders
• The homework will be peer-graded. In analytics modeling, there are often lots of different approaches that work well, and I want you to see not just your own, but also others.
• The homework grading scale reflects the fact that the primary purpose of homework is learning:
Rating Meaning Point value (out of 100)
4 All correct (perhaps except a few details) with a deeper solution than expected 100
3 Most or all correct 90
2 Not correct, but a reasonable attempt 75
1 Not correct, insufficient effort 50
0 Not submitted 0
Question 2.1
Describe a situation or problem from your job, everyday life, current events, etc., for which a classification model would be appropriate. List some (up to 5) predictors that you might use.
Question 2.2
The files credit_card_data.txt (without headers) and credit_card_data-headers.txt
(with headers) contain a dataset with 654 data points, 6 continuous and 4 binary predictor variables. It has anonymized credit card applications with a binary response variable (last column)
indicating if the application was positive or negative. The dataset is the “Credit Approval Data Set” from the UCI Machine Learning Repository (https://archive.ics.uci.edu/ml/datasets/
Credit+Approval) without the categorical variables and without data points that have missing values.
1. Using the support vector machine function ksvm contained in the R package kernlab, find a good classifier for this data. Show the equation of your classifier, and how well it classifies the data
points in the full data set. (Don’t worry about test/validation data yet; we’ll cover that topic soon.)
Notes on ksvm
• You can use scaled=TRUE to get ksvm to scale the data as part of calculating a classifier.
• The term λ we used in the SVM lesson to trade off the two components of correctness and margin is called C in ksvm. One of the challenges of this homework is to find a
value of C that works well; for many values of C, almost all predictions will be “yes” or almost all predictions will be “no”.
• ksvm does not directly return the coefficients a0 and a1…am. Instead, you need to do the last step of the calculation yourself. Here’s an example of the steps to take (assuming your data is stored
in a matrix called data):
# call ksvm. Vanilladot is a simple linear kernel.
model <- ksvm(data[,1:10],data[,11],type=”Csvc”,kernel=”vanilladot”,C=100,scaled=TRUE)
# calculate a1…am
a <- colSums(model@xmatrix[[1]] * model@coef[[1]]) a
# calculate a0 a0 <- –model@b a0
# see what the model predicts pred <- predict(model,data[,1:10]) pred
# see what fraction of the model’s predictions match the actual classification
sum(pred == data[,11]) / nrow(data)
Hint: You might want to view the predictions your model makes; if C is too large or too small, they’ll almost all be the same (all zero or all one) and the predictive value of the model will be poor.
Even finding the right order of magnitude for C might take a little trial-and-error.
Note: If you get the error “Error in vanilladot(length = 4, lambda = 0.5) : unused arguments (length = 4, lambda = 0.5)”, it means you need to convert data into matrix format:
model <-
2. You are welcome, but not required, to try other (nonlinear) kernels as well; we’re not covering them in this course, but they can sometimes be useful and might provide better predictions than
3. Using the k-nearest-neighbors classification function kknn contained in the R kknn package, suggest a good value of k, and show how well it classifies that data points in the full data set. Don’t
forget to scale the data (scale=TRUE in kknn).
Notes on kknn
• You need to be a little careful. If you give it the whole data set to find the closest points to i, it’ll use i itself (which is in the data set) as one of the nearest neighbors. A helpful feature
of R is the index –i, which means “all indices except i”. For example, data[i,] is all the data except for the ith data point. For our data file where the first 10 columns are predictors and the 11th
column is the response, data[-i,11] is the response for all but the ith data point, and data[-i,1:10] are the predictors for all but the ith data point.
(There are other, easier ways to get around this problem, but I want you to get practice doing some basic data manipulation and extraction, and maybe some looping too.)
• Note that kknn will read the responses as continuous, and return the fraction of the k closest responses that are 1 (rather than the most common response, 1 or 0).
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Evaluation Metrics: Sharpe Ratio
The Sharpe Ratio is a measure of the risk-adjusted return of an investment. It was developed by economist William F. Sharpe and is widely used as a trading evaluation metric. It is calculated as the
average return of an investment minus the risk-free rate, divided by the standard deviation of the returns. The risk-free rate is the return on an investment with no risk, such as a Treasury bill,
and the standard deviation is a measure of the volatility of the returns.
The higher the Sharpe Ratio, the better the risk-adjusted return of the investment. A Sharpe Ratio of 1.0 is considered good, while a ratio of 2.0 or higher is considered excellent. A ratio below 1.0
indicates that the investment has underperformed a risk-free investment.
The Sharpe Ratio is often used to compare the performance of different investments or trading strategies. It can be especially useful for evaluating the performance of investments with high
volatility, since it takes into account the risk of the investment.
Here’s the formula for the Sharpe ratio:
Sharpe ratio = (portfolio return – risk-free rate of return) / portfolio standard deviation
For example, let’s say a portfolio has a return of 8% and a standard deviation of 4%, and the risk-free rate of return is 2%. The Sharpe ratio would be:
(8 – 2) / 4 = 1
A Sharpe ratio of 1 indicates that the portfolio is generating a return that is equal to the level of risk it is taking on. A Sharpe ratio greater than 1 indicates that the portfolio is generating a
return that is higher than the level of risk it is taking on, while a Sharpe ratio less than 1 indicates that the portfolio is generating a return that is lower than the level of risk it is taking
The Sharpe ratio is a commonly used metric for evaluating the risk-adjusted performance of a portfolio and it is a good way to compare the performance of different portfolios on a level playing
field. However, it’s important to note that the Sharpe ratio does not account for the skewness or the kurtosis of the return distribution, which are other important aspects of the risk.
Additionally, it is also important to consider the Sharpe ratio in the context of the investor’s risk tolerance. For example, if an investor is willing to take on more risk for the chance of higher
returns, a portfolio with a higher Sharpe ratio may be more attractive than one with a lower Sharpe ratio. | {"url":"https://quant.fish/wiki/evaluation-metrics-sharpe-ratio/","timestamp":"2024-11-14T00:50:43Z","content_type":"text/html","content_length":"122910","record_id":"<urn:uuid:ca715967-01e6-43c6-93e8-ba4603796ecc>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00574.warc.gz"} |
∃ - (Mathematical Logic) - Vocab, Definition, Explanations | Fiveable
from class:
Mathematical Logic
The symbol ∃ represents the existential quantifier in logic, indicating that there exists at least one element in a domain that satisfies a given property. It is used to assert the existence of such
an element, and its application can influence the structure of statements and proofs significantly.
congrats on reading the definition of ∃. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The existential quantifier is often used in conjunction with predicates to formulate statements such as 'There exists an x such that P(x) is true.'
2. When negating a statement with an existential quantifier, it transforms into a statement with a universal quantifier, such as 'It is not the case that there exists an x such that P(x) is true'
becomes 'For all x, P(x) is false.'
3. The notation ∃! signifies that there exists exactly one element in the domain satisfying the property, which can be crucial in distinguishing between existence and uniqueness.
4. In first-order logic, existential statements play a key role in forming valid proofs and deriving conclusions from premises.
5. Understanding how to manipulate statements involving ∃ is essential for developing proof strategies and applying inference rules effectively.
Review Questions
• How does the existential quantifier ∃ interact with predicates to form meaningful logical statements?
□ The existential quantifier ∃ is used alongside predicates to assert that there is at least one element in a specific domain that satisfies the condition described by the predicate. For
example, if we have a predicate P(x), stating '∃x P(x)' means that there is at least one value of x for which P(x) is true. This interaction is fundamental in constructing logical expressions
and proofs.
• What happens when you negate a statement containing the existential quantifier ∃, and why is this important for understanding logical implications?
□ When negating a statement that contains the existential quantifier ∃, it transitions into a statement with the universal quantifier ∀. For instance, negating '∃x P(x)' results in '∀x ¬P(x)',
meaning for every x, P(x) is not true. This transformation is crucial because it helps understand how existential claims relate to universal claims and assists in proving or disproving
statements within logical arguments.
• Evaluate how existential quantification influences proof strategies and logical reasoning in mathematical contexts.
□ Existential quantification plays a pivotal role in proof strategies by allowing mathematicians to assert the existence of elements satisfying certain conditions without specifying them. In
proofs, such assertions can lead to deriving conclusions based on known properties or previously established results. Furthermore, using ∃ enables techniques like proof by contradiction and
constructing counterexamples, making it essential for effective reasoning and establishing mathematical truths.
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Method for efficient computation of the density of states in water-explicit biopolymer simulations on a lattice
We present a method for fast computation of the density of states of binary systems. The contributions of each of the components to the density of states can be separated based on the conditional
independence of the individual components' degrees of freedom. The conditions establishing independence are the degrees of freedom of the interfacial region between the two components. The separate
contributions of the components to the density of states can then be calculated using the Wang-Landau algorithm [Wang, F.; Landau, D. P. Phys. Rev. Lett. 2001, 86, 2050]. We apply this method to a 2D
lattice model of a hydrophobic homopolytmer in water that exhibits protein-like cold, pressure, and thermal unfolding. The separate computation of the protein and water density of states
contributions is faster and more accurate than the combined simulation of both components and allows for the investigation of larger systems.
All Science Journal Classification (ASJC) codes
• Physical and Theoretical Chemistry
Dive into the research topics of 'Method for efficient computation of the density of states in water-explicit biopolymer simulations on a lattice'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/method-for-efficient-computation-of-the-density-of-states-in-wate","timestamp":"2024-11-10T01:52:03Z","content_type":"text/html","content_length":"50235","record_id":"<urn:uuid:0477e6f1-4e8c-4fd9-96a6-0804fba13ae7>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00158.warc.gz"} |
Uses old Rust 2015
0.1.3 Apr 23, 2016
0.1.2 Apr 26, 2015
0.1.1 [DEL:Apr 25, 2015:DEL]
0.1.0 Apr 24, 2015
Calculate order statistics.
This crates allows one to compute the kth smallest element in (expected) linear time, and estimate a median element via the median-of-medians algorithm.
Ensure your Cargo.toml contains:
order-stat = "0.1"
The kth function allows computing order statistics of slices of Ord types.
let mut v = [4, 1, 3, 2, 0];
println!("the 2nd smallest element is {}", // 1
order_stat::kth(&mut v, 1));
The kth_by function takes an arbitrary closure, designed for order statistics of slices of floating point and more general comparisons.
let mut v = [4.0, 1.0, 3.0, 2.0, 0.0];
println!("the 3rd smallest element is {}", // 2
order_stat::kth_by(&mut v, 2, |x, y| x.partial_cmp(y).unwrap()));
struct Foo(i32);
let mut v = [Foo(4), Foo(1), Foo(3), Foo(2), Foo(0)];
println!("the element with the 4th smallest field is {:?}", // Foo(3)
order_stat::kth_by(&mut v, 3, |x, y| x.0.cmp(&y.0)));
The median_of_medians function gives an approximation to the median of a slice of an Ord type.
let mut v = [4, 1, 3, 2, 0];
println!("{} is close to the median",
order_stat::median_of_medians(&mut v).1);
It also has a median_of_medians_by variant to work with non-Ord types and more general comparisons.
let mut v = [4.0, 1.0, 3.0, 2.0, 0.0];
println!("{} is close to the median",
order_stat::median_of_medians_by(&mut v, |x, y| x.partial_cmp(y).unwrap()).1);
struct Foo(i32);
let mut v = [Foo(4), Foo(1), Foo(3), Foo(2), Foo(0)];
println!("{:?}'s field is close to the median of the fields",
order_stat::median_of_medians_by(&mut v, |x, y| x.0.cmp(&y.0)).1); | {"url":"https://lib.rs/crates/order-stat","timestamp":"2024-11-14T13:58:10Z","content_type":"text/html","content_length":"30970","record_id":"<urn:uuid:964662b1-313d-4fb3-9603-43676fa25508>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00593.warc.gz"} |
Lesson Topic: One-Step Equations (Addition, Subtraction, Multiplication, and Division) Lesson Objective: I can… Solve one-step equations by relating. - ppt download
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Factor Variables in R | BridgeText - Online Dissertation Writing Service and Help
Learning how to create factor variables is indispensable for carrying out many statistical tests. In this blog, you’ll learn how to use R to create factor variables, change the names of factors, and
use factors to conduct an ANOVA.
Create a Single Factor
Try the following R code:
height <- factor(c("short","medium","tall"))
Factors have levels, as you can see. These levels are part of statistical analyses such as ANOVAs. Let’s see that in action.
Create a Factor in a Data Frame
Try the following R code:
status <- factor(c(rep("single", 10), rep("married",10), rep("divorced",10)))
sat <- rnorm(30, mean=100, sd=15)
satisfaction <- round(sat)
subj <- 1:30
df <- data.frame(subj, status, satisfaction)
Let’s say these are the satisfaction levels of 30 people, 10 of whom are single, 10 married, and 10 divorced. You used the factor command combined with rep to create 10 each of these factors. Now,
before you conduct an ANOVA using these factors, let’s say we want to change the factor names to sentence case. Begin by installing the dplyr package if you have not done so already, and load it as a
Next, try the following R code:
df <- df %>% mutate(status=recode(status,
'single' = 'Single',
'married' = 'Married',
'divorced' = 'Divorced'))
That changed the factor names into:
Now that you have a factor with more than 2 levels, you can run an ANOVA followed by a Tukey’s HSD. Try the following R code:
mod.aov <- aov(satisfaction~status, data=df)
The ANOVA isn’t significant. The Tukey’s pairwise comparisons also show no statistically significant differences between factor level comparisons:
This analysis was only possible because of the existence of a factor variable, status. Now you know how to create your own factor variables in R.
BridgeText can help you with all of your statistical analysis needs.
Position, Velocity, and Acceleration: A Worked Calculus Problem
3 August
Navigating Time-Series Forecasting with ARIMA in R
25 February
Chi Square in Stata: The Basics
18 December
Cloning Variables in Stata
24 December | {"url":"https://www.bridgetext.com/factor-variables-in-r","timestamp":"2024-11-02T07:57:59Z","content_type":"application/xhtml+xml","content_length":"35642","record_id":"<urn:uuid:10d95535-414f-4318-a73d-71a88d97d73c>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00815.warc.gz"} |
What is the Vertex of a Quadratic Function?
Each quadratic equation has either a maximum or minimum, but did you that this point has a special name? In a quadratic equation, this point is called the vertex! Take a look at the vertex of a
quadratic equation by watching this tutorial. | {"url":"https://virtualnerd.com/algebra-1/quadratic-equations-functions/graphing/graph-basics/vertex-definition","timestamp":"2024-11-13T12:16:02Z","content_type":"text/html","content_length":"21427","record_id":"<urn:uuid:c829a9a7-03ab-4c49-a689-29f6a2cce5db>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00074.warc.gz"} |
Multi-Step Word Problems
Madison is reading a book that has 232 pages in it. She read 42 pages over the weekend. Then she read 30 more pages on Monday night. How many pages does Madison have left to read?
What is 160 pages?
Tricia's mother let her play outside for 15 minutes. When she went outside, she played with her dog for 5 minutes. Then she rode her bike for 4 minutes. She spent the rest of the time catching bugs.
How many minutes did Tricia spend catching bugs?
What is 6 minutes?
Christopher and his brother played hockey in their living room. Each goal was worth 2 points. Christopher scored 6 points. His brother scored 8 points. How many GOALS did they score altogether?
What is 7 goals?
The bakers at the Beverly Hills Bakery baked 200 loaves of bread on Monday morning. They sold 93 loaves in the morning and 39 loaves in the afternoon. How many loaves of bread did they have left?
What is 68 loaves.
Allini and Dennis make cookies for the school bake sale. Allini baked 72 cookies. Dennis baked twice as many as Allini. How many cookies did they bake altogether?
What is 216 cookies?
Luke had (2) ten dollar bills. His younger sister Nichelle had a five dollar bill. They combined their money to buy a gift for their father that cost $22. How much change did they receive?
What is $3?
Jimmy and Kelly ran a snack stand in their front yard. They sold cookies for 5¢, apples for 6¢ and cupcakes for 10¢. They sold 3 cookies, 2 cupcakes, and 1 apple. How much money did they make
What is 41 cents?
Peter has four horses. Each one eats 4 pounds of oats, twice a day. How many pounds of oats does he need to feed his horses for 3 days?
What is 96 pounds of oats
Troymiah has 34 stuffed animals. Ty'Johnna has twice as many as Troymiah. How many stuffed animals do the two girls have in all?
What is 102 stuffed animals.
Michael and David went to Fun Park. Michael won 152 tickets. David won 84 tickets. They want to put their tickets together to get a large toy monkey that costs 300 tickets. How many more tickets do
they need?
What is 64 more tickets?
Mia bought a sketch book with 125 blank pages in it. She tears out 7 blank pages for her friend to draw on. Mia draws pictures on 64 pages. How many blank pages does she have left?
What is 54 blank pages?
Ashton had two boxes of pencils with 14 pencils in each box. He gave six pencils to his brother. How many pencils did Ashton have left?
What is 22 pencils?
Mr. Parker has 982 pounds of grain. He feeds 240 pounds to his pigs and 460 to his cows. How much grain does he have left?
What is 282 pounds of grain?
Dr. Banks had 330 toothbrushes to give away to his patients. He gave away 53 toothbrushes in January. He gave away 67 toothbrushes in February. In March he gave away 46 toothbrushes. How many
toothbrushes does he have left?
What is 164 toothbrushes.
Lamar Reese Magnet School of the Arts has 124 first graders and 130 second graders. On Friday, 12 first graders and 9 second graders were absent. How many first and second graders were in school on
What is 233 first and second graders?
In football, a player scores 6 points for a touchdown and 3 points for a field goal. Kingston's team scored 2 touchdowns in this week's game. Last week they scored 1 touchdown and 1 field goal. How
many points did Kingston's team score in both games combined?
What is 21 points?
At the Tasty Bakery, cupcakes cost fifty-cents each. Bagels cost a dollar twenty-five. How much more do two bagels cost than two cupcakes?
What is $1.50.
Aunt Mary milks her cows twice a day. This morning, she got 365 gallons of milk. This evening she got 380 gallons. She sold 612 gallons to the local ice cream factory. How many gallons of milk does
she have left?
What is 133 gallons?
Carlton and Will went fishing together. Carlton caught 21 bass and 7 walleye. Will caught 13 bass and 8 walleye. How many more fish did Carlton catch than Will?
What is 7 more fish.
Sam has tomato plants in his backyard. This year the plants grew 127 tomatoes. Birds had eaten 19 of the tomatoes. 23 tomatoes had been ruined by bugs. He picked the rest. How many tomatoes did Sam
What is 85 tomatoes?
Steven bought two boxes of erasers. One box had 24 erasers in it. The other box had 36 erasers in it. He then gave 18 of his erasers to his friend. How many erasers did Steven have left?
What is 42 erasers?
Joe is learning to play the trumpet. On Monday he practiced from 6:30 until 7:05. On Tuesday he practiced from 3:55 until 4:15. How many minutes did he practice in all over the two days?
What is 55 minutes? ** DOUBLE POINTS **
Uncle Ben has 440 chickens on his farm. 39 are roosters and the rest are hens. 15 of his hens do not lay eggs. The rest lay eggs. How many egg-laying hens does Uncle Ben have on his farm?
What is 386 hens?
Hillary's teacher assigned 60 minutes of reading during the weekend. On Friday night, Hillary read for 16 minutes. On Saturday she read for 28 minutes. How many minutes does Hillary have to read on
Sunday to complete the assignment?
What is 16 minutes
Noah brought 60 cupcakes to school on his birthday. He gave a cupcake to each of the 25 students in Mrs. Smith's class. He also gave a cupcake to each of the 25 students in Mrs. Middlebrook's class.
He also gave a cupcake to Mrs. Smith, Mrs. Middlebrooks, Ms. Burks, Ms. Whitmire, Mrs. Bradford, and Dr. Shumate. How many cupcakes did he have left over?
What is 4 cupcakes? | {"url":"https://jeopardylabs.com/play/multi-step-word-problems-328","timestamp":"2024-11-02T11:02:24Z","content_type":"application/xhtml+xml","content_length":"61977","record_id":"<urn:uuid:c4be59e2-fc3d-4cea-b2a6-d0f3446ea20e>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00777.warc.gz"} |
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슬롯 용가리
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On improvements of the dissipation integral method for the calculation of turbulent boundary layers
The prediction of incompressible two dimensional turbulent boundary layers is discussed. An integral method based on the momentum and the mean kinetic energy equations was developed. A third integral
equation was added to calculate the dissipation integral. This equation was derived from the turbulent energy equation and takes into account the flow.
Contributions on Transport Phenomena in Fluid Mechanics and Related Topics (ESA-TT-498)
Pub Date:
March 1979
□ Boundary Layer Flow;
□ Integral Equations;
□ Turbulent Boundary Layer;
□ Differential Equations;
□ Flow Equations;
□ Incompressible Flow;
□ Kinetic Energy;
□ Navier-Stokes Equation;
□ Two Dimensional Boundary Layer;
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□ Fluid Mechanics and Heat Transfer | {"url":"https://ui.adsabs.harvard.edu/abs/1979ctpf.rept..197J/abstract","timestamp":"2024-11-08T14:51:20Z","content_type":"text/html","content_length":"34024","record_id":"<urn:uuid:bf215130-33f1-4c06-9ad0-2f01469cb247>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00350.warc.gz"} |
Multiplication 0 9 Worksheets
Math, especially multiplication, creates the keystone of many academic self-controls and real-world applications. Yet, for many students, understanding multiplication can posture an obstacle. To
resolve this hurdle, teachers and moms and dads have welcomed a powerful tool: Multiplication 0 9 Worksheets.
Intro to Multiplication 0 9 Worksheets
Multiplication 0 9 Worksheets
Multiplication 0 9 Worksheets -
Multiplication facts with 9 s Students multiply 9 times numbers between 1 and 12 The first worksheet is a table of all multiplication facts 1 12 with nine as a factor 9 times table Worksheet 1 49
questions Worksheet 2 Worksheet 3 100 questions Worksheet 4 Worksheet 5 3 More Similar Multiplying by 10 Multiplying by 11 What is K5
Our multiplication worksheets start with the basic multiplication facts and progress to multiplying large numbers in columns We emphasize mental multiplication exercises to improve numeracy skills
Choose your grade topic Grade 2 multiplication worksheets Grade 3 multiplication worksheets Grade 4 mental multiplication worksheets
Relevance of Multiplication Practice Recognizing multiplication is crucial, laying a solid structure for sophisticated mathematical principles. Multiplication 0 9 Worksheets use structured and
targeted method, cultivating a deeper understanding of this basic arithmetic procedure.
Advancement of Multiplication 0 9 Worksheets
Pin On Multiplication worksheets Ideas For Kids
Pin On Multiplication worksheets Ideas For Kids
Welcome to The Multiplying 1 to 10 by 1 to 9 100 Questions A Math Worksheet from the Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 22 and
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Have your third graders demonstrate their grasp of multiplication facts 0 9 with this math assessment This worksheet asks students to multiply two factors to find the product for 20 different
multiplication problems
From typical pen-and-paper workouts to digitized interactive layouts, Multiplication 0 9 Worksheets have advanced, catering to diverse discovering styles and choices.
Types of Multiplication 0 9 Worksheets
Basic Multiplication Sheets Basic exercises focusing on multiplication tables, assisting learners build a strong arithmetic base.
Word Trouble Worksheets
Real-life scenarios integrated into issues, boosting critical thinking and application skills.
Timed Multiplication Drills Examinations developed to improve speed and accuracy, assisting in rapid psychological mathematics.
Advantages of Using Multiplication 0 9 Worksheets
Multiplication Across Zeros Worksheets Free Printable
Multiplication Across Zeros Worksheets Free Printable
These multiplication worksheets are appropriate for Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade and 5th Grade You may vary the numbers of problems on each worksheet from 12 to 30 You can
click Clear to reset or All to select all of the numbers Choose the numbers for the first factor Choose the numbers for the second factor
Multiplying by 9 Worksheets Basic Facts with Factors of 9 Multiplying by 9s Use these printables to help your students master basic multiplication facts with 9 as a factor Also find sheets with all
facts up to 9s Multiplication by 9s Only Multiplication Nines Trick
Enhanced Mathematical Skills
Constant method hones multiplication proficiency, improving general math capacities.
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Worksheets fit private knowing rates, cultivating a comfortable and versatile discovering environment.
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Integrating Visuals and Shades Dynamic visuals and shades record attention, making worksheets aesthetically appealing and engaging.
Including Real-Life Circumstances
Connecting multiplication to day-to-day circumstances includes relevance and usefulness to exercises.
Customizing Worksheets to Various Skill Levels Personalizing worksheets based upon differing effectiveness degrees makes sure inclusive learning. Interactive and Online Multiplication Resources
Digital Multiplication Tools and Gamings Technology-based resources provide interactive understanding experiences, making multiplication appealing and satisfying. Interactive Internet Sites and Apps
On the internet systems give diverse and accessible multiplication practice, supplementing traditional worksheets. Personalizing Worksheets for Different Discovering Styles Aesthetic Students
Aesthetic help and diagrams aid comprehension for students inclined toward visual understanding. Auditory Learners Verbal multiplication issues or mnemonics deal with students that understand
principles through auditory ways. Kinesthetic Students Hands-on tasks and manipulatives sustain kinesthetic students in understanding multiplication. Tips for Effective Implementation in
Understanding Uniformity in Practice Routine practice reinforces multiplication skills, advertising retention and fluency. Balancing Rep and Variety A mix of repetitive exercises and diverse problem
formats maintains rate of interest and comprehension. Supplying Useful Responses Feedback aids in determining locations of improvement, urging ongoing development. Challenges in Multiplication
Technique and Solutions Inspiration and Interaction Difficulties Boring drills can result in uninterest; innovative methods can reignite motivation. Overcoming Anxiety of Math Negative understandings
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Addition Table Patterns Worksheets Worksheet Hero
multiplication Worksheet For Kids Archives EduMonitor
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Printable Times Tables Worksheets Activity Shelter
Multiplication Facts Worksheets 0 9 Free Printable
Single Digit Multiplication 0 9 Worksheet For 2nd 3rd Grade Lesson Planet
FREE PRINTABLE MULTIPLICATION WORKSHEETS WonkyWonderful
100 Vertical Questions Multiplication Facts 1 31 10 A Free Printable Multiplication
Multiplication Strategies 3rd Grade Worksheets Times Tables Worksheets
Multiplication Worksheets K5 Learning
Our multiplication worksheets start with the basic multiplication facts and progress to multiplying large numbers in columns We emphasize mental multiplication exercises to improve numeracy skills
Choose your grade topic Grade 2 multiplication worksheets Grade 3 multiplication worksheets Grade 4 mental multiplication worksheets
Printable Multiplication Worksheets Super Teacher Worksheets
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Our multiplication worksheets start with the basic multiplication facts and progress to multiplying large numbers in columns We emphasize mental multiplication exercises to improve numeracy skills
Choose your grade topic Grade 2 multiplication worksheets Grade 3 multiplication worksheets Grade 4 mental multiplication worksheets
Multiplication by 9s When you re teaching students to multiply only by the number nine use these printable worksheets
FREE PRINTABLE MULTIPLICATION WORKSHEETS WonkyWonderful
Multiplication Facts Worksheets 0 9 Free Printable
100 Vertical Questions Multiplication Facts 1 31 10 A Free Printable Multiplication
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Printable Timed Multiplication Quiz PrintableMultiplication
Multiplication Times Table Worksheets Numeracy Warm Up Multiplication worksheets Math
Multiplication Times Table Worksheets Numeracy Warm Up Multiplication worksheets Math
Free Printable Maths Worksheets Ks2 Multiplication Free Printable Maths Worksheets Ks2 Multi
Frequently Asked Questions (Frequently Asked Questions).
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How To Solve Linear Programming Problems
Linear programming is the field of mathematics concerned with maximizing or minimizing linear functions under constraints. A linear programming problem includes an objective function and constraints.
To solve the linear programming problem, you must meet the requirements of the constraints in a way that maximizes or minimizes the objective function. The ability to solve linear programming
problems is important and useful in many fields, including operations research, business and economics.
Step 1
Graph the feasible region of your problem. The feasible region is the region in space defined by the linear constraints of the problem. For example, if your problem contains the inequalities x + 2y >
4, 3x – 4y < 12, x > 1 and y > 0, you graph the intersection of these regions as your feasible region.
Step 2
Find the corner points of the region. If your problem is solvable, there will be visible sharp points, or corners, in your region. Mark these points on your graph.
Step 3
Calculate the coordinates of these points. If you graphed the feasible region well, you often will be able to know immediately the coordinates of the corner points. If not, you can calculate them by
hand by substituting your inequalities into each other and solving for x and y. In the given example, you will find (4,0) is a corner point, as well as (1,1.5).
Step 4
Substitute these corner points into the objective function of the linear programming problem. You will have as many answers as you do corner points. For example, assume your objective function is to
maximize the function x + y. In this example, you will have two answers: one for the point (4,0) and one for the point (1,1.5). The answers these points yield are 4 and 2.5, respectively.
Step 5
Compare all your answers. If your objective function is one of maximization, you inspect your answers to find the largest one. Likewise, if your objective function is one of minimization, you inspect
your answers, looking for the smallest one. In our example, since the objective function is for the purpose of maximization, the point (4,0) solves the linear programming problem, yielding an answer
of 4.
• "An Introduction to Linear Programming and Game Theory"; Thie and Keough; 2008
Cite This Article
Verial, Damon. "How To Solve Linear Programming Problems" sciencing.com, https://www.sciencing.com/solve-linear-programming-problems-7797465/. 24 April 2017.
Verial, Damon. (2017, April 24). How To Solve Linear Programming Problems. sciencing.com. Retrieved from https://www.sciencing.com/solve-linear-programming-problems-7797465/
Verial, Damon. How To Solve Linear Programming Problems last modified March 24, 2022. https://www.sciencing.com/solve-linear-programming-problems-7797465/ | {"url":"https://www.sciencing.com:443/solve-linear-programming-problems-7797465/","timestamp":"2024-11-09T13:31:21Z","content_type":"application/xhtml+xml","content_length":"71270","record_id":"<urn:uuid:3173d079-1ad4-4bd4-88c5-fa2f6cb4df03>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00306.warc.gz"} |
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Advanced Vedic Division Techniques
[First Half: Anurupyena Vargs Approach]
4.1: Introduction to Anurupyena Vargs
The Anurupyena Vargs approach is a powerful Vedic division technique that allows you to solve complex division problems efficiently. This method is based on the principle of proportionality, where
the dividend and divisor are related in a specific way.
In Anurupyena Vargs, the key is to identify the relationship between the dividend and divisor, and then leverage this relationship to simplify the division process. By understanding the underlying
mathematical principles, you'll be able to apply this technique to a wide range of division problems, from simple to more advanced.
The step-by-step process of the Anurupyena Vargs approach involves:
1. Identifying the relationship between the dividend and divisor: Analyze the numbers to determine the proportional relationship between them.
2. Expressing the divisor in terms of the dividend: Rewrite the divisor as a function of the dividend, using the identified relationship.
3. Performing the division: Apply the division process, leveraging the simplified divisor expression to streamline the calculations.
4. Verifying the result: Check the accuracy of the obtained solution by performing a reverse calculation or confirming the remainder.
By mastering the Anurupyena Vargs approach, you'll be able to solve division problems with greater speed, accuracy, and a deeper understanding of the underlying mathematical principles.
Key Takeaways:
• Anurupyena Vargs is a Vedic division technique based on the principle of proportionality.
• It involves identifying the relationship between the dividend and divisor, and then leveraging this relationship to simplify the division process.
• The step-by-step approach includes analyzing the numbers, expressing the divisor in terms of the dividend, performing the division, and verifying the result.
4.2: Practical Applications of Anurupyena Vargs
In this sub-chapter, we'll explore the practical applications of the Anurupyena Vargs approach by solving various types of division problems.
Example 1: Dividing by a One-Digit Divisor Let's consider the division problem: 1234 ÷ 7
Using the Anurupyena Vargs approach:
1. Identify the relationship between the dividend and divisor:
□ The divisor, 7, is a one-digit number.
□ The dividend, 1234, is a four-digit number.
2. Express the divisor in terms of the dividend:
□ Since the divisor is a one-digit number, we can express it as a fraction of the dividend.
□ In this case, the divisor 7 is 1/176 of the dividend 1234.
3. Perform the division:
□ 1234 ÷ 7 = 1234 × (1/176) = 176
The answer is 176, with no remainder.
Example 2: Dividing by a Two-Digit Divisor Let's consider the division problem: 12345 ÷ 67
Using the Anurupyena Vargs approach:
1. Identify the relationship between the dividend and divisor:
□ The divisor, 67, is a two-digit number.
□ The dividend, 12345, is a five-digit number.
2. Express the divisor in terms of the dividend:
□ Since the divisor is a two-digit number, we can express it as a fraction of the dividend.
□ In this case, the divisor 67 is 1/184 of the dividend 12345.
3. Perform the division:
□ 12345 ÷ 67 = 12345 × (1/184) = 184
The answer is 184, with no remainder.
Example 3: Dividing by a Recurring Digit Divisor Let's consider the division problem: 12345 ÷ 111
Using the Anurupyena Vargs approach:
1. Identify the relationship between the dividend and divisor:
□ The divisor, 111, has a recurring digit pattern of 1.
□ The dividend, 12345, is a five-digit number.
2. Express the divisor in terms of the dividend:
□ Since the divisor has a recurring digit pattern, we can express it as a fraction of the dividend.
□ In this case, the divisor 111 is 1/111 of the dividend 12345.
3. Perform the division:
□ 12345 ÷ 111 = 12345 × (1/111) = 111
The answer is 111, with no remainder.
These examples demonstrate the versatility of the Anurupyena Vargs approach in solving various types of division problems. By identifying the relationship between the dividend and divisor, you can
streamline the division process and arrive at the solution efficiently.
Key Takeaways:
• The Anurupyena Vargs approach can be applied to division problems with one-digit, two-digit, and recurring digit divisors.
• The key is to express the divisor as a fraction of the dividend based on the identified relationship.
• This simplifies the division process and allows for efficient problem-solving.
4.3: Anurupyena Vargs Shortcuts and Techniques
As you become more proficient with the Anurupyena Vargs approach, you can explore various shortcuts and techniques to further enhance your division skills.
Shortcut 1: Using Convenient Multiples of the Divisor In some cases, you can simplify the division process by using convenient multiples of the divisor. For example, if the divisor is 25, you can
express it as 1/4 of the dividend, as 25 is a multiple of 4.
Shortcut 2: Identifying Patterns in Recurring Digit Divisors When dealing with divisors with recurring digit patterns, you can identify specific patterns that allow for quicker calculations. For
instance, if the divisor is 111, you can recognize that it is 1/111 of the dividend, as the recurring digit pattern is 1.
Technique 1: Decomposing the Dividend In some cases, you can decompose the dividend into more manageable parts to simplify the division process. This technique is particularly useful when the
dividend is a large number.
Technique 2: Identifying Complementary Relationships Sometimes, the divisor and dividend may have a complementary relationship, where the divisor can be expressed as a function of the dividend. This
can be leveraged to simplify the division.
Technique 3: Utilizing Divisibility Rules Applying divisibility rules can help you quickly identify if a number is divisible by a given divisor, without having to perform the full division process.
By mastering these shortcuts and techniques, you'll be able to solve Anurupyena Vargs division problems with greater speed, efficiency, and accuracy.
Key Takeaways:
• Shortcuts, such as using convenient multiples of the divisor and identifying patterns in recurring digit divisors, can streamline the division process.
• Techniques like decomposing the dividend, identifying complementary relationships, and utilizing divisibility rules can further simplify the division process.
• Mastering these shortcuts and techniques will improve your problem-solving skills and deepen your understanding of the Anurupyena Vargs approach.
4.4: Mastering Anurupyena Vargs: Troubleshooting and Error Handling
As you progress in your journey of mastering the Anurupyena Vargs approach, it's important to be aware of common pitfalls and develop strategies for identifying and correcting errors.
Common Pitfalls and Errors:
1. Misidentifying the relationship between the dividend and divisor: Accurately recognizing the proportional relationship is crucial for the success of the Anurupyena Vargs approach. Carefully
analyzing the numbers is essential to avoid mistakes.
2. Errors in expressing the divisor in terms of the dividend: Mistakes in the mathematical manipulation required to rewrite the divisor can lead to incorrect solutions.
3. Computational errors during the division process: Even after simplifying the division, inaccuracies in the calculations can still occur, especially when dealing with larger numbers.
4. Overlooking the verification step: Failing to double-check the obtained solution by performing a reverse calculation or confirming the remainder can result in undetected errors.
Troubleshooting Strategies:
1. Develop a systematic approach: Establish a consistent step-by-step process for applying the Anurupyena Vargs technique. This will help you identify the stage at which an error might have
2. Practice regularly: Consistent practice with a variety of division problems will help you become more adept at recognizing patterns and relationships between the dividend and divisor.
3. Utilize visual aids: Diagrams or illustrations can assist in visualizing the proportional relationships and simplifying the division process.
4. Engage in self-reflection: Carefully analyze your work, identify areas of weakness, and develop strategies to address them. This will help you improve your problem-solving skills over time.
5. Seek feedback and guidance: Consult with peers, instructors, or expert resources to get feedback on your approach and identify potential areas for improvement.
By being mindful of common pitfalls, developing effective troubleshooting strategies, and continuously refining your skills, you'll be able to master the Anurupyena Vargs approach and solve complex
division problems with confidence and accuracy.
Key Takeaways:
• Common pitfalls in the Anurupyena Vargs approach include misidentifying relationships, errors in expressing the divisor, computational errors, and overlooking the verification step.
• Effective troubleshooting strategies involve developing a systematic approach, practicing regularly, utilizing visual aids, engaging in self-reflection, and seeking feedback.
• Mastering the Anurupyena Vargs approach requires a combination of conceptual understanding, problem-solving skills, and a commitment to continuous improvement.
[Second Half: Adyamam Vargs Approach]
4.5: Introduction to Adyamam Vargs
The Adyamam Vargs approach is another powerful Vedic division technique that complements the Anurupyena Vargs method. While Anurupyena Vargs focuses on the proportional relationship between the
dividend and divisor, Adyamam Vargs leverages the concept of the "first factor" to simplify the division process.
The key principle behind Adyamam Vargs is the identification of the "first factor" of the dividend and divisor. The "first factor" refers to the largest common factor between the two numbers. By
expressing the divisor in terms of this first factor, you can streamline the division calculations.
The step-by-step process of the Adyamam Vargs approach involves:
1. Identifying the first factor: Determine the largest common factor between the dividend and divisor.
2. Expressing the divisor in terms of the first factor: Rewrite the divisor as a function of the first factor.
3. Performing the division: Apply the division process, leveraging the simplified divisor expression to streamline the calculations.
4. Verifying the result: Check the accuracy of the obtained solution by performing a reverse calculation or confirming the remainder.
By mastering the Adyamam Vargs approach, you'll be able to solve a wide range of division problems efficiently, complementing the skills you've developed through the Anurupyena Vargs technique.
Key Takeaways:
• Adyamam Vargs is a Vedic division technique that focuses on the concept of the "first factor" between the dividend and divisor.
• The step-by-step process involves identifying the first factor, expressing the divisor in terms of the first factor, performing the division, and verifying the result.
• Adyamam Vargs complements the Anurupyena Vargs approach, providing an additional tool for solving complex division problems.
4.6: Practical Applications of Adyamam Vargs
In this sub-chapter, we'll explore the practical applications of the Adyamam Vargs approach by solving various types of division problems.
Example 1: Dividing by a One-Digit Divisor Let's consider the division problem: 1234 ÷ 7
Using the Adyamam Vargs approach:
1. Identify the first factor:
□ The largest common factor between the dividend 1234 and the divisor 7 is 7.
2. Express the divisor in terms of the first factor:
□ The divisor 7 can be expressed as 7/7 = 1.
3. Perform the division:
□ 1234 ÷ 7 = 1234 ÷ 1 = 1234
The answer is 176, with no remainder.
Example 2: Dividing by a Two-Digit Divisor Let's consider the division problem: 12345 ÷ 67
Using the Adyamam Vargs approach:
1. Identify the first factor:
□ The largest common factor between the dividend 12345 and the divisor 67 is 1.
2. Express the divisor in terms of the first factor:
□ The divisor 67 can be expressed as 67/1 = 67.
3. Perform the division:
□ 12345 ÷ 67 = 12345 ÷ 67 = 184
The answer is 184, with no remainder.
Example 3: Dividing by a Recurring Digit Divisor Let's consider the division problem: 12345 ÷ 111
Using the Adyamam Vargs approach:
1. Identify the first factor:
□ The largest common factor between the dividend 12345 and the divisor 111 is 3.
2. Express the divisor in terms of the first factor:
□ The divisor 111 can be expressed as 111/3 = 37.
3. Perform the division:
□ 12345 ÷ 111 = 12345 ÷ 37 = 333
The answer is 333, with no remainder.
These examples demonstrate the versatility of the Adyamam Vargs approach in solving various types of division problems. By identifying the first factor and expressing the divisor in terms of this
factor, you can streamline the division process and arrive at the solution efficiently.
Key Takeaways:
• The Adyamam Vargs approach can be applied to division problems with one-digit, two-digit, and recurring digit divisors.
• The key is to identify the first factor (the largest common factor) between the dividend and divisor, and then express the divisor in terms of this factor.
• This simplifies the division process and allows for efficient problem-solving.
4.7: Adyamam Vargs Shortcuts and Techniques
As you become more proficient with the Adyamam Vargs approach, you can explore various shortcuts and techniques to further enhance your division skills.
Shortcut 1: Recognizing Common First Factors Certain divisors may have common first factors that you can readily identify. For example, if the divisor is a multiple of 5 or 10, the first factor is
likely to be 5 or 10, respectively.
Shortcut 2: Leveraging Divisibility Rules Applying divisibility rules can help you quickly determine the first factor, without having to perform extensive factorization.
Technique 1: Combining Anurupyena and Adyamam Vargs In some cases, you can combine the Anurupyena Vargs and Adyamam Vargs approaches to achieve even greater efficiency. By using the strengths of both
techniques, you can tackle a wider range of division problems.
Technique 2: Decomposing the Dividend and Divisor Similar to the Anurupyena Vargs approach, you can decompose the dividend and divisor into more manageable parts to simplify the division process.
Technique 3: Identifying Patterns in Recurring Digit Divisors When dealing with divisors with recurring digit patterns, you can identify specific patterns that allow for quicker calculations.
By mastering these shortcuts and techniques, you'll be able to solve Adyamam Vargs division problems with greater speed, efficiency, and accuracy.
Key Takeaways:
• Shortcuts, such as recognizing common first factors and leveraging divisibility rules, can streamline the division process.
• Techniques like combining Anurupyena and Adyamam Vargs, decomposing the dividend and divisor, and identifying patterns in recurring digit divisors can further simplify the division process.
• Mastering these shortcuts and techniques will improve your problem-solving skills and deepen your understanding of the Adyamam Vargs approach.
4.8: Mastering Adyamam Vargs: Troubleshooting and Error Handling
As you progress in your journey of mastering the Adyamam Vargs approach, it's important to be aware of common pitf | {"url":"https://wisemen.ai/app/courses/66152e844519edd487aca545/4","timestamp":"2024-11-10T19:11:25Z","content_type":"text/html","content_length":"70812","record_id":"<urn:uuid:0ddf8df2-52a4-4661-b206-06fc7ba94d33>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00762.warc.gz"} |
Learning Trajectories for Grades 6-8 Rational Number Reasoning
An interview with Jere Confrey about her Gates Foundation and NSF-funded project to build and validate learning trajectories (LTs) for grades 6-8 mathematics around rational number reasoning.
What is the big idea of your project work?
Our team is working on an interactive learning map in for middle school math. This work is part of our Scaling Up Digital Design Studies (SUDDS) project work, with funding from NSF and the Bill &
Melinda Gates Foundation.
A learning map, in contrast to a straight disciplinary map, is a map that is based on research on learning and tries to delineate the landmarks/or obstacles that are essential to understanding a big
idea. Our learning map helps teachers create personalized learning resources for students in middle grades math in a coherent way. It supports flexible grouping based on students’ choices, paths, and
What does the interactive learning map cover and how does it work?
Our middle school map covers 4 fields: Geometry & Measurement, Algebra, Number, and Statistics & Probability (see Figure 1). Within each field you see boxes, which we call regions or “big ideas” of
learning math. There are 9 big ideas across the fields. If you click on a region label, you can see what it stands for. So in Numbers, the two big ideas around middle grades are one-dimensional
number (how do you locate things on the number line, what’s bigger/smaller, what’s the order of fractions and decimals); and the second is ratio, rate and percent. In Statistics & Probability, there
is displaying data using statistics around centered variation, and probability and inference.
Figure 2. Middle school map covering 4 fields. Each field contains regions or big ideas, each of which contain clusters of constructs.
If you double click on a big idea, it zooms in and you can see circles that we call “clusters”. The circles are constructs that go together to make up an idea. So if you look at key ratio relations,
there’s identifying ratio equivalents, finding base ratios, and finding unit ratios. The structure reflects that they first need to learn ratio equivalents, and then they can learn these other two in
any order. The idea is that instead of big unidimensional learning trajectories, they are broken down into clusters. A learning trajectory at the level of a construct can still cross grades but they
are more compact than in my previous work.
If you click on a construct within a cluster, like Identifying Ratio Equivalence, it will expand and show a window that we call a stack (see Figure 2). The stack is the delineation of the learning
trajectory and shows that when you’re learning ratio equivalence, the first thing you have to learn is that you can make more of something and maintain the same ratio, like in a recipe for lemonade
that you double. When the numbers are even, it’s easier for kids to do it than when the numbers are odd. As you go up the stack, you’ll see that at 6th grade, students should learn all of these
proficiency levels in that stack. If you click on CCSS-M at the bottom, it puts up the standards that are associated with this particular construct. And you can go the other way, too: when you click
on a standard, it shows you what constructs it corresponds to on the map. Another major change from my prior work (turnonccmath.net) is that a standard can be associated with multiple locations on
the map.
Figure 2. The Identifying Ratio Equivalence construct and corresponding stack and link to CCSS-M standard.
How do teachers use the map?
We think the map is useful to organize the mathematical space. We’re designing it for teachers to use with classes, and home schoolers could also use it. One concern we have is that there’s a drift
on the web toward individualized learning and instruction. Our cyberlearning and science of learning community knows how important discourse and interaction are. So we are designing this to try to
support teachers who want to do interactive classes but still respond to the diversity of needs among their students.
We also have a way that teachers can build a scope and sequence for the year based on which regions are taught or not taught, and in what kind of order, and then they can go in and order the clusters
(see Figure 3). Below the clusters we don’t let them order them because we want those things taught in a certain order as related learning concepts. From the concepts, we also link out to resources
on the web. We are populating the map this summer.
Figure 3. An example scope and sequence for a year that reflects what big ideas and constructs a teacher chooses to teach, and in what order.
So that’s the notion of the learning map – being able to use web resources and getting teachers deep information about how kids think. In our NSF project, we are building diagnostics for each of the
clusters so that when students finish a cluster, they can take an assessment and the assessment will give them feedback on how well they’ve learned the topics. Teachers can use this information to
group students or students can use it to find other students to work with.
What do you sees as unique about your work?
NSF Project Information
Title: Completing, Validating, and Linking Learning Trajectories for K-8 Rational Number Reasoning Tied to the Common Core Standards
Award Details
Investigators: Jere Confrey, Alan Maloney
I think of this whole thing as a digital learning system, and this is where I think the field’s going to go. It’s not complete now; this is really the bookends: The navigation and the information on
the underlying concepts that kids need to learn and what it looks like when they’ve learned it. It also leverages visualization to make navigation of Common Core easy and intuitive. But you still
need the middle part: how do you create a workspace where kids can use mathematical tools as they solve problems. All of that, and the backend analytics, would have to be built. But you’re seeing two
critical components.
Learn more about this work on the Scaling Up Digital Design Studies (SUDDS) project web site. | {"url":"https://circlcenter.org/learning-trajectories-for-6-8-rational-number-reasoning/","timestamp":"2024-11-10T15:36:05Z","content_type":"text/html","content_length":"76376","record_id":"<urn:uuid:6b06dacf-2103-4f8c-81af-4ff43203e83d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00190.warc.gz"} |
A Separation Logic for Negative Dependence (POPL 2022 - POPL Research Papers) - POPL 2022
POPL 2022 (series) / POPL Research Papers /
A Separation Logic for Negative Dependence
Hashing plays a fundamental role in many probabilistic data structures, from basic hash tables to more sophisticated schemes such as Bloom filters. In these applications, a \emph{hash function} $h$
maps a \emph{universe} of possible values, typically large, to a set of \emph{buckets}, typically much smaller. Hash-based data structures satisfy a variety of probabilistic guarantees. For instance,
we may be interested in the \emph{false positive rate}: what is the probability that a data structure mistakenly identifies an element as being stored in a collection, when it was never stored
before? We may also be interested in \emph{load properties}, such as: what is the chance that some bucket in the data structure overflows? A typical way to prove these properties is to treat random
hash functions as a \emph{balls-into-bins} process: randomly throw $N$ balls into $B$ bins one at a time, where each bin is chosen from some known distribution, and report the number of balls in each
bin at the end of the process. For example, hashing $N$ unique elements into $B$ bins can be modeled as throwing $N$ balls into $B$ bins, where each bin is drawn uniformly at random.
While this modeling is convenient, one complication is that the counts of the elements in the different buckets are not probabilistically independent: one bin containing many elements makes it more
likely that other bins contain few elements. The lack of independence makes it difficult to reason about multiple bins, for instance bounding the number of empty bins. Moreover, many common tools for
analyzing probabilistic processes, like concentration bounds, usually require independence. This subtlety has also been a source of problems in pen-and-paper analyses of probabilistic data
structures. For instance, the standard analysis of the Bloom filter bounds the number of occupied bins in order to bound the false positive rate. The original version of this analysis~\citep
{DBLP:journals/cacm/Bloom70}, and also repeated in many papers, assumes that the bin counts are independent. However, \citet{DBLP:journals/ipl/BoseGKMMMST08} pointed out that this assumption is
incorrect, and in fact the claimed upper-bound on the false-positive rate is actually a \emph{lower} bound; proving a correct bound on the false-positive rate required a substantially more
complicated argument. Recently, \citet{DBLP:conf/cav/GopinathanS20} mechanized a correct, but complex proof in Coq.
We aim to develop a simpler method to formally reason about hash-based data structures and balls-into-bins processes, drawing on a key concept in probability theory: \emph{negative dependence}.
Wed 19 Jan
Displayed time zone:
Eastern Time (US & Canada) change
16:40 - 17:30
16:40 A Separation Logic for Negative DependenceRemote
25m POPL
Research paper DOI Media Attached
17:05 Reasoning about “Reasoning about Reasoning”: Semantics and Contextual Equivalence for Probabilistic Programs with Nested Queries and RecursionRemote
25m POPL
Research paper DOI Media Attached | {"url":"https://popl22.sigplan.org/details/POPL-2022-popl-research-papers/57/A-Separation-Logic-for-Negative-Dependence","timestamp":"2024-11-03T21:53:16Z","content_type":"text/html","content_length":"79195","record_id":"<urn:uuid:ca73315f-d26d-4fd4-a357-087ae27f6cb5>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00758.warc.gz"} |
Introduction to fido::Pibble
An introduction to fido
fido (Justin D. Silverman et al. 2019) is a loose acronym for “(Bayesian) Multinomial Logistic-Normal Models”. In particular the development of fido stems from the need for fast inference for
time-invariant MALLARD models(Justin D. Silverman et al. 2018). fido is very fast! It uses closed form solutions for model gradients and Hessians written in C++ to preform MAP estimation in
combination with parameter uncertainty estimation using a Laplace Approximation. One of the main models in fido is the function pibble which fits a Multinomial Logistic-Normal Linear Regression
So what is a fido model exactly? First let me give the broad description from 10,000ft up: Basically its a model for multinomial count data (e.g., each sample contains the counts of \(D\) “types of
things”). Importantly, unlike the more common Poisson count models, the multinomial models a “competition to be counted” (i.e., cases in which counting more of one type of thing means that I have
less resources available to count other types of things).
This may seem vague so let me give an example. Pretend there is a ball pit with red, green, and blue balls. Pretend that the ball pit is very large and I don’t know the total number of balls in the
ball pit, yet I want to say something about the relative number of red, blue, and green balls in the pit. One way I may choose to measure the ball pit is by grabbing an armful of balls and counting
the number of balls of each color (e.g., in one armful I may collect 5 red, 3 blue, and 6 green). My arms can only contain so many balls (in this example about 14) and so if I were to have (randomly)
gotten another green ball in my armful (making 7 total) I would likely not have been able to measure one of the red or blue balls; hence the “competition to be counted”. It turns out that this type
of sampling occurs all the time in many situations (Wikipedia has an example with political polling). Perhaps one of the most notable examples of this type of count data occurs with modern
high-throughput sequencing studies such as 16S rRNA studies to profile microbial communities or bulk/single-cell RNA-seq studies to study expression profiles of cells. In all cases, transcripts are
sequenced and the number of different types of transcripts are counted. The important part is that sequencing only samples a small portion of the total genetic material available and leads to similar
competition to be counted.
The pibble model
Pibble is one type of fido model. In particular its a fido model for multivariate linear regression.
Let \(Y\) denote an \(D\times N\) matrix of counts. Let us denote the \(j\)-th column of \(Y\) as \(Y_j\). Thus each “sample” in the dataset is a measurement of the relative amount of \(D\) “types of
things”. Suppose we also have have covariate information in the form of a \(Q\times N\) matrix \(X\).
The following is the pibble model including likelihood and priors: \[ Y_j & \sim \text{Multinomial}\left(\pi_j \right) \\ \pi_j & = \phi^{-1}(\eta_j) \\ \eta_j &\sim N(\Lambda X_j, \Sigma) \\ \Lambda
&\sim MN_{(D-1) \times Q}(\Theta, \Sigma, \Gamma) \\ \Sigma &\sim W^{-1}(\Xi, \upsilon) \] Here \(MN_{(D-1) \times Q}\) denotes a Matrix Normal distribution for a matrix \(\Lambda\) of regression
coefficients of dimension \((D-1)\times Q\). Essentially you can think of the Matrix normal as having two covariance matrices one describing the covariation between the rows of \(\Lambda\) (\(\Sigma
\)) and another describing the covariation of the columns of \(\Lambda\) (\(\Gamma\)). and \(W^{-1}\) refers to the Inverse Wishart distribution (which is a common distribution over covariance
matrices). The line \(\pi_j = \phi^{-1}(\eta_j)\) represents a transformation between the parameters \(\pi_j\) which exist on a simplex (e.g., \(\pi_j\) must sum to 1) and the transformed parameters
\(\eta_j\) that exist in real space. In particular we define \(\phi^{-1}\) to be the inverse additive log ratio transform (which conversely implies that \(\eta_j = ALR(\pi_j)\)) also known as the
identified softmax transform (as it is more commonly known in the Machine Learning community). While I will say more on this later in this tutorial, one thing to know is that I have the model
implemented using the ALR transform as it is computationally simple and fast; the results of the model can be viewed as if any number of transforms had been used (instead of the ALR) including the
isometric log-ratio transform, or the centered log-ratio transform.
Before moving on, I would like to give a more intuitive description of pibble. Essentially the main modeling component of pibble is the third equation above (\(\eta_j \sim N(\Lambda X_j, \Sigma)\))
which is just a multivariate linear model. That is, \(X\) are your covariates (which can be continuous, discrete, binary, etc…), and \(\Sigma\) is the covariance matrix for the regression residuals.
Example analysis of microbiome data
This analysis is the same as that presented in the fido manuscript (Justin D. Silverman et al. 2019). I will reanalyze a previously published study comparing microbial composition in the terminal
ileum of subjects with Crohn’s Disease (CD) to healthy controls (Gevers et al. 2014). To do this I will fit a pibble model using CD status, inflammation status and age as covariates (plus a constant
intercept term).
For convienece, we have added a copy of the data set to fido. The data was obtained from the MicrobeDS repository on GitHub.
# making into a phyloseq object
CCFA_phylo <- phyloseq(otu_table(as.matrix(RISK_CCFA_otu), taxa_are_rows = TRUE), sample_data(RISK_CCFA_sam), tax_table(as.matrix(RISK_CCFA_tax)))
# drop low abundant taxa and samples
dat <- CCFA_phylo %>%
immunosup!="missing") %>%
subset_samples(diagnosis %in% c("no", "CD")) %>%
subset_samples(steroids=="false") %>%
subset_samples(antibiotics=="false") %>%
subset_samples(biologics=="false") %>%
subset_samples(biopsy_location=="Terminal ileum") %>%
tax_glom("Family") %>%
prune_samples(sample_sums(.) >= 5000,.) %>%
filter_taxa(function(x) sum(x > 3) > 0.10*length(x), TRUE)
Create Design Matrix and OTU Table
sample_dat <- as.data.frame(as(sample_data(dat),"matrix")) %>%
mutate(age = as.numeric(as.character(age)),
diagnosis = relevel(factor(diagnosis, ordered = FALSE), ref="no"),
disease_stat = relevel(factor(disease_stat, ordered = FALSE), ref="non-inflamed"))
X <- t(model.matrix(~diagnosis + disease_stat+age, data=sample_dat))
Y <- otu_table(dat)
# Investigate X and Y look like
#> 1939.SKBTI.0175 1939.SKBTI047 1939.SKBTI051 1939.SKBTI063
#> (Intercept) 1.00000 1.00000 1.00 1.00000
#> diagnosisCD 1.00000 1.00000 1.00 1.00000
#> disease_statinflamed 0.00000 1.00000 1.00 1.00000
#> age 15.16667 14.33333 15.75 13.58333
#> 1939.SKBTI072
#> (Intercept) 1.00
#> diagnosisCD 1.00
#> disease_statinflamed 1.00
#> age 15.75
#> OTU Table: [5 taxa and 5 samples]
#> taxa are rows
#> 1939.SKBTI.0175 1939.SKBTI047 1939.SKBTI051 1939.SKBTI063 1939.SKBTI072
#> 4442127 0 9 0 14 2
#> 74305 1 2 35 1 0
#> 663573 36 1 0 2 1
#> 2685602 10 264 211 276 83
#> 4339819 0 37 42 70 22
Next specify priors. We are going to start by specifying a prior on the covariance between log-ratios \(\Sigma\). I like to do this by thinking about a prior on the covariance between taxa on the
log-scale (i.e., between the log of their absolute abundances not the log-ratios). I will refer to this covariance on log-absolute abundances \(\Omega\). For example, here I will build a prior that
states that the mean of \(\Omega\) is the identity matrix \(I_D\). From From Aitchison (1986), we know that if we assume that the taxa have a covariance \(\Omega\) in terms of log-absolute abundance
then their correlation in the \(\text{ALR}_D\) is given by \[ \Sigma = G \Omega G^T \] where \(G\) is a \(D-1 \times D\) matrix given by \(G = [I_{D-1}; -1_{D-1}]\) (i.e., \(G\) is the \(\text{ALR}_D
\) contrast matrix). Additionally, we know that the Inverse Wishart mode is given by \(\frac{\Xi}{\upsilon + D}\). Finally, note that \(\upsilon\) essentially controls our uncertainty in \(\Sigma\)
about this prior mean. Here I will take \(\upsilon = D+3\). This then gives us \(\Xi = (\upsilon - D) GIG^T\). We scale \(\Xi\) by a factor of 1/2 to make \(Tr(\Xi)=D-1\).
upsilon <- ntaxa(dat)+3
Omega <- diag(ntaxa(dat))
G <- cbind(diag(ntaxa(dat)-1), -1)
Xi <- (upsilon-ntaxa(dat))*G%*%Omega%*%t(G)
Finally I specify my priors for \(\Theta\) (mean of \(\Lambda\)) and \(\Gamma\) (covariance between columns of \(\Lambda\); i.e., covariance between the covariates). I will center my prior for \(\
Lambda\) about zero, and assume that the covariates are independent.
I strongly recommend users perform prior predictive checks to make sure their priors make sense to them. fido makes this easy, all the main fitting functions (e.g., pibble) will automatically sample
from the prior predictive distribution if Y is left as NULL (e.g., without data your posterior is just your prior).
priors <- pibble(NULL, X, upsilon, Theta, Gamma, Xi)
#> pibblefit Object (Priors Only):
#> Number of Samples: 250
#> Number of Categories: 49
#> Number of Covariates: 4
#> Number of Posterior Samples: 2000
#> Contains Samples of Parameters:Eta Lambda Sigma
#> Coordinate System: alr, reference category: 49
The main fitting functions in the fido package output special fit objects (e.g., pibble outputs an object of class pibblefit). These fit objects are just lists with some extra metadata that allows
special method dispatch. For example, if you call print on a pibblefit object you will get a nice summary of what is in the object.
Note: Currently, the function pibble takes expects inputs and outputs in the “default” coordinate system; this is simply the ALR coordinate system where the last category (49 above) is taken as
reference (this will be generalized in future versions). More specifically for a vector \(x\) representing the proportions of categories \(\{1, \dots, D\}\) we can write \[x^* = \left( \log \frac
{x_1}{x_D}, \dots, \log \frac{x_{D-1}}{x_D}\right).\] As mentioned above however, I have designed fido to work with many different coordinate systems including the ALR (with respect to any category),
CLR, ILR, or proportions. To help transform things between these coordinate systems I have written a series of transformation functions that transform any pibblefit object into a desired coordinate
system. Importantly, pibblefit objects keep track of what coordinate system they are currently in so as a user you only need to specify the coordinate system that you want to change into. Keep in
mind that covariance matrices cannot be represented in proportions and so visualizations or summaries based on covariance matrices will be suppressed when pibblefit objects are in the proportions
coordinate system. As an example, lets look at viewing a summary of the prior for \(\Lambda\) with respect to the CLR coordinate system.
priors <- to_clr(priors)
summary(priors, pars="Lambda", gather_prob=TRUE, as_factor=TRUE, use_names=TRUE)
#> $Lambda
#> # A tibble: 784 × 9
#> Parameter coord covariate val .lower .upper .width .point .interval
#> <chr> <int> <int> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
#> 1 Lambda 1 1 0.0509 -0.528 0.596 0.5 mean qi
#> 2 Lambda 1 2 0.00199 -0.567 0.575 0.5 mean qi
#> 3 Lambda 1 3 -0.0373 -0.620 0.532 0.5 mean qi
#> 4 Lambda 1 4 -0.00205 -0.554 0.549 0.5 mean qi
#> 5 Lambda 2 1 0.00991 -0.564 0.535 0.5 mean qi
#> 6 Lambda 2 2 -0.00000782 -0.572 0.580 0.5 mean qi
#> 7 Lambda 2 3 -0.0247 -0.570 0.518 0.5 mean qi
#> 8 Lambda 2 4 0.0165 -0.536 0.589 0.5 mean qi
#> 9 Lambda 3 1 0.0138 -0.584 0.620 0.5 mean qi
#> 10 Lambda 3 2 -0.00502 -0.542 0.581 0.5 mean qi
#> # ℹ 774 more rows
By default the summary function returns a list (with possible elements Lambda, Sigma, and Eta) summarizing each posterior parameter based on quantiles and mean (e.g., p2.5 is the 0.025 percentile of
the posterior distribution). As this type of table may be hard to take in due to how large it is, pibblefit objects also come with a default plotting option for each of the parameters. Also the
returned plot objects are ggplot objects so normal ggplot2 commands work on them. Before doing that though we are going to use one of the names functions for pibblefit objects to provide some more
specific names for the covariates (helpful when we then plot).
names_covariates(priors) <- rownames(X)
p <- plot(priors, par="Lambda")
#> Scale for colour is already present.
#> Adding another scale for colour, which will replace the existing scale.
p + ggplot2::xlim(c(-10, 10))
This looks fairly reasonable to me. So I am going to go ahead and fit the model with data. fido provides a helper method called refit that we will use to avoid passing prior parameters again.
priors$Y <- Y # remember pibblefit objects are just lists
posterior <- refit(priors, optim_method="lbfgs")
Unlike the main pibble function, the refit method can be called on objects in any coordinate system and all transformations to and from the default coordinate system are handled internally. This is
one nice thing about using the refit method. That said, new objects added to the pibblefit object need to be added in the proper coordinates For example, if we wanted to replace our prior for \(\Xi\)
for an object in CLR coordinates, we would had to transform our prior for Xi to CLR coordinates before adding it to the priors object.
Now I are also going to add in the taxa names to make it easier to interpret the results.
tax <- tax_table(dat)[,c("Class", "Family")]
tax <- apply(tax, 1, paste, collapse="_")
names_categories(posterior) <- tax
Before doing anything else lets look at the posterior predictive distribution to assess model fit. This can be accessed through the method ppc.
There are a few things to note about this plot. First, when zoomed out like this it looks it is hard to make much of it. This is a fairly large dataset we are analyzing and its hard to view an
uncertainty interval; in this case its plotting the median and 95% confidence interval in grey and black and the observed counts in green. fido also has a simpler function that summarizes the
posterior predictive check.
Here we see that the model appears to be fitting well (at least based on the posterior predictive check) and that only about 1.5% of observations fall outside of the 95% posterior predictive density
(this is good).
Some readers will look at the above ppc plots and think “looks like over-fitting”. However, note that there are two ways of using ppc. One is to predict the counts based on the samples of \(\eta\)
(Eta; as we did above); the other is to predict “from scratch” that is to predict starting form the posterior samples of \(\Lambda\) (Lambda) then sampling \(\eta\) and only then sampling \(Y\). This
later functionality can be accessed by also passing the parameters from_scratch=TRUE to the ppc function. Note: these two posterior predictive checks have different meanings, one is not better than
the other.
ppc_summary(posterior, from_scratch=TRUE)
#> Proportions of Observations within 95% Credible Interval: 0.9725714
Now we are going to finally look at the posterior distribution of our regression parameters, but because there are so many we will focus on just those that have a 95% credible interval not including
zero (i.e., those that the model is fairly certain are non-zero). We are also going to ignore the intercept term and just look at parameters associated with age and disease status.
posterior_summary <- summary(posterior, pars="Lambda")$Lambda
focus <- posterior_summary[sign(posterior_summary$p2.5) == sign(posterior_summary$p97.5),]
focus <- unique(focus$coord)
plot(posterior, par="Lambda", focus.coord = focus, focus.cov = rownames(X)[2:4])
#> Scale for colour is already present.
#> Adding another scale for colour, which will replace the existing scale.
The first, and most obvious ting to notice is that the covariate age has pretty much no effect at all, whatever effect it may have is incredibly weak. So we are going to remove age from the plot and
just look at those coordinates with non-zero effect for diagnosis CD
posterior_summary <- filter(posterior_summary, covariate=="diagnosisCD")
focus <- posterior_summary[sign(posterior_summary$p2.5) == sign(posterior_summary$p97.5),]
focus <- unique(focus$coord)
tax_table(dat)[taxa_names(dat)[which(names_coords(posterior) %in% focus)]]
#> Taxonomy Table: [13 taxa by 7 taxonomic ranks]:
#> Kingdom Phylum Class Order
#> 74305 "Bacteria" "Proteobacteria" "Epsilonproteobacteria" "Campylobacterales"
#> 4449236 "Bacteria" "Proteobacteria" "Betaproteobacteria" "Burkholderiales"
#> 1105919 "Bacteria" "Proteobacteria" "Betaproteobacteria" "Burkholderiales"
#> 4477696 "Bacteria" "Proteobacteria" "Gammaproteobacteria" "Pasteurellales"
#> 4448331 "Bacteria" "Proteobacteria" "Gammaproteobacteria" "Enterobacteriales"
#> 4154872 "Bacteria" "Bacteroidetes" "Flavobacteriia" "Flavobacteriales"
#> 4452538 "Bacteria" "Fusobacteria" "Fusobacteriia" "Fusobacteriales"
#> 341322 "Bacteria" "Firmicutes" "Bacilli" "Turicibacterales"
#> 1015143 "Bacteria" "Firmicutes" "Bacilli" "Gemellales"
#> 176318 "Bacteria" "Firmicutes" "Clostridia" "Clostridiales"
#> 1788466 "Bacteria" "Firmicutes" "Clostridia" "Clostridiales"
#> 1896700 "Bacteria" "Firmicutes" "Clostridia" "Clostridiales"
#> 191718 "Bacteria" "Firmicutes" "Erysipelotrichi" "Erysipelotrichales"
#> Family Genus Species
#> 74305 "Helicobacteraceae" NA NA
#> 4449236 "Alcaligenaceae" NA NA
#> 1105919 "Oxalobacteraceae" NA NA
#> 4477696 "Pasteurellaceae" NA NA
#> 4448331 "Enterobacteriaceae" NA NA
#> 4154872 "[Weeksellaceae]" NA NA
#> 4452538 "Fusobacteriaceae" NA NA
#> 341322 "Turicibacteraceae" NA NA
#> 1015143 "Gemellaceae" NA NA
#> 176318 "Christensenellaceae" NA NA
#> 1788466 "Lachnospiraceae" NA NA
#> 1896700 "Peptostreptococcaceae" NA NA
#> 191718 "Erysipelotrichaceae" NA NA
plot(posterior, par="Lambda", focus.coord = focus, focus.cov = rownames(X)[2])
#> Scale for colour is already present.
#> Adding another scale for colour, which will replace the existing scale. | {"url":"https://cran.mirror.garr.it/CRAN/web/packages/fido/vignettes/introduction-to-fido.html","timestamp":"2024-11-03T00:47:15Z","content_type":"text/html","content_length":"116710","record_id":"<urn:uuid:3c1a6c6b-0217-424c-b0f4-a77b8bc6a99c>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00586.warc.gz"} |
Kovariančný derivát youtube
We derived the equation of motion by differentiating the 4-velocity. Rewrite „ua „t =„x b „t ∑ua ∑xb =ub ∑u a b and insert to get ub ∑u a ∑xb +Ga bgu g =0. Contraction is a tensor operation.
In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the form of physical laws under arbitrary differentiable
coordinate transformations. Covariance is the measure of changes between two random variables in statistics. Learn about its types and how it differs from correlation along with formulas and the
solved example here at BYJU'S. In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis.The transformation that
describes the new basis vectors as a linear combination of the old basis vectors is defined as a covariant transformation. One can motivate the covariant differentiation using only vector calculus.
It works for an oversimplified case though (but since the OP doesn't accept either the definition via Ehresmann connection nor the vector bundle definition, may be it's justified.) Many text books on
differential geometry motivate covariant derivative more or less by saying that if you have a vector field along a curve on a manifold (that is a curve $\gamma(t)$ and an assignm Metric compatible.
In the coordinate-specific section of this article, it is stated "By the way, this particular expression is equal to zero, because the covariant derivative of a function solely of the metric is
always zero.".
Oct 10, 2019 · We can calculate the covariance between two asset returns given the joint probability distribution. Consider the following example: Example. Suppose we wish to find the variance of
each asset and the covariance between the returns of ABC and XYZ, given that the amount invested in each company is $1,000. Jan 01, 1990 · JGP - Vol. 7, n.
See full list on corporatefinanceinstitute.com
Suppose we wish to find the variance of each asset and the covariance between the returns of ABC and XYZ, given that the amount invested in each company is $1,000. Jan 01, 1990 · JGP - Vol. 7, n.
We derived the equation of motion by differentiating the 4-velocity. Rewrite „ua „t =„x b „t ∑ua ∑xb =ub ∑u a b and insert to get ub ∑u a ∑xb +Ga bgu g =0. Contraction is a tensor operation.
Applied to find the equation of heat diffusion on a curved surface. The covariant derivative is used to derive In mathematics, the covariant derivative is a way of specifying a derivative along
tangent vectors of a manifold.
Covariation definition is - correlated variation of two or more variables. Jun 21, 2009 · Homework Statement The problem concerns how to transform a covariant differentiation. Using this formula for
covariant differentiation and demanding that it is a (1,1) tensor: \ abla_cX^a=\\partial_cX^a+\\Gamma^a_{bc}X^b it should be proven that \\Gamma'^a_{bc}= Stack Exchange network consists of 176 Q&A
communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Stack Exchange network consists of 176 Q&A
communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Mar 04, 2020 · Covariance is a measure of the
degree to which returns on two risky assets move in tandem. A positive covariance means that asset returns move together, while a negative covariance means returns ti, kteří s kovem dělají, práci s
kovem milují nebo se o ni alespoň zajímají. Feb 21, 2008 · Homework Statement Help!
Kovarianz erklärenHier bist du genau richtig, wenn für dich Mathe in der Schule wie chinesisch ist, wenn du dich sehr schnell und produktiv verbessern möchte Lecture # 8 General Relativity &
Cosmology Lecture Series In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.Alternatively, the covariant derivative is a way of
introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine
connection. The covariant derivative is the derivative that under a general coordinate transformation transforms covariantly, i.e., linearly via the Jacobian matrix of the coordinate transformation.
Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. The connection is chosen so that the covariant derivative of the metric
is zero. The vanishing covariant metric derivative is not a consequence of using "any" connection, it's a condition that allows us to choose a specific connection $\Gamma^{\sigma}_{\mu \beta}$. After
the company’s founding in 2005, YouTube rose quickly through the ranks of online video websites to become an industry leader that streams more than a billion hours of video a day.
Rewrite „ua „t =„x b „t ∑ua ∑xb =ub ∑u a b and insert to get ub ∑u a ∑xb +Ga bgu g =0. Contraction is a tensor operation. The connection is chosen so that the covariant derivative of the metric is
zero. The vanishing covariant metric derivative is not a consequence of using "any" connection, it's a condition that allows us to choose a specific connection $\Gamma^{\sigma}_{\mu \beta}$. YouTube
kanály Gameball.
Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6 Danielson, Donald A. (2003 Sep 25, 2012 · The covariant derivative
of a 1-form [itex]\omega[/itex] is a 1-form [itex] abla_X\omega[/itex]. And a 1-form (i.e. a field of covectors) eating a vector field Y does not depend on the partial derivatives of the components
of Y: This video looks at the process of how to derive an expression for the covariant derivative from first principles that involves changes in basis vectors on s Differential Geometry Covariant
Derivatives. Applied to find the equation of heat diffusion on a curved surface.
Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Mar 04, 2020 · Covariance is a measure of the degree to which returns on two risky assets move in tandem. A positive covariance means that asset returns move together, while a negative covariance
means returns ti, kteří s kovem dělají, práci s kovem milují nebo se o ni alespoň zajímají. Feb 21, 2008 · Homework Statement Help!
ako paypal na bankový účetčo je eos coinbasedres kupovať a predávať motorky2,99 dolárov v pásochprevádzať libry na rupie v minciiphone 11 nedokáže poslať smsaktualizácie môjho telefónu
Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Dne 16. | {"url":"https://enklapengarqkir.web.app/40328/65242.html","timestamp":"2024-11-04T16:52:39Z","content_type":"text/html","content_length":"19468","record_id":"<urn:uuid:b0c8cdd5-bc91-4958-b276-ce0aaa1a64ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00471.warc.gz"} |
who participated in rodeos in the outsiders
Answer :
Dallas participates in Rodeos!
Answer Link
Dallas (Dally) Winston and I believe Sherri (Cherry) and her friend, Martia also participated in rodeos. I also think that the guy who opened the door for Pony and Johnny (after the murder) rode in
rodeos as well.
Answer Link | {"url":"https://mis.kyeop.go.ke/shelf/49394","timestamp":"2024-11-08T17:44:28Z","content_type":"text/html","content_length":"153889","record_id":"<urn:uuid:bdc2c6e8-afe0-4975-9a9b-f489e3a8507a>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00124.warc.gz"} |
The Beginner’s Guide to Excel’s Formulas and Functions - Techregister
Formulas and functions, the nuts and bolts of Excel, both enable you to perform calculations, but they differ in how they’re created, what they do, and how they work. In this article, we’ll strip the
two right back to their bare bones, so you can use Excel like a pro, both at home and at work.
What Are Functions and Formulas?
The key difference between formulas and functions is that anyone can create a formula, whereas functions are predefined by Microsoft’s programmers.
Excel Formulas
Excel’s formulas enable you to perform basic mathematical calculations. To create a formula, start by typing the equals (=) sign, and then create the parameters for the calculation.
For example, typing
into cell B2 and pressing Enter would result in Excel adding 20 and 40, producing the outcome (60) in the cell where the formula is typed.
Excel can also calculate numerical values that are already in the spreadsheet. Typing
into cell B3 and pressing Enter would multiply the value in cell B2 by ten.
Similarly, typing
into cell B4 and pressing Enter would multiply the values in cells B2 (60) and B3 (600) together, resulting in 36,000.
That said, you’re not limited to two arguments when creating an Excel formula. For example, typing
into cell B5 would sum two and eight, then multiply the value in B3 by five, before subtracting the former from the latter, with an outcome of 2,990.
Excel follows the
standard mathematical order of operations—PEMDAS
. In other words, it first performs calculations in parentheses, then evaluates exponents (or indices), then deals with multiplication and division, and finishes with addition and subtraction.
Excel Functions
Excel’s functions work in a similar way. Indeed, they start with the = sign, and enable you to perform calculations. However, where Excel’s formulas are limited to the basic mathematical operations,
its functions let you do a lot more.
For example, the AVERAGE function takes a set of numbers and finds the mean, and MAX tells you the largest number in a range.
Excel’s functions follow a very specific syntax:
where a is the name of the function (such as AVERAGE or MAX), and b are the arguments used to enable that function to perform calculations.
For example, typing
into cell A1 and pressing Enter would calculate the average of 20 and 30, returning 25.
We could also type
in cell B2 to make Excel calculate the mean of all the values in cells A1 to A5 (the colon tells Excel to include the cells mentioned and all those in between).
There are hundreds of Excel functions, ranging from the most basic functions to the more complex ones. Remembering them all is pretty much impossible, especially given that Microsoft’s developers are
always adding new ones to the list. Instead, Excel is ready to help you choose which function best suits the job you need to carry out, and helps you through the process.
To launch this assistant, click the “fx” icon above the first row of your spreadsheet, or press Shift+F3.
You can then type some words in the Search For A Function field to find the function you need. The Select A Category drop-down shows you the different function groups, including the financial,
statistical, and logical categories. When you select a function in the list under the categories, you will see a brief description that tells you what the function does.
When you’ve found the function you want to use, click “OK.” Then, you will see a new dialog box that walks you through the process.
Formulas and Functions Together
Formulas and functions don’t have to be used independently. For example, typing
would add the values in cells A1 to A10, before dividing the total by two.
Cells vs. The Formula Bar
Once you’ve typed a formula or used a function in a given cell in Excel, it is replaced by the result. For example, when we type
into cell A3 and press Enter, we no longer see what we typed in that cell. Instead, we see the result.
If you realize you made an error and need to amend the formula you typed, this is where you can use the Formula Bar, which runs along the top of your Excel worksheet. You can also see the Name Box in
the top-left corner, which indicates the active cell.
In other words, in the example below, the Name Box tells us that A3 is the active cell, the Formula Bar tells us what we typed into cell A3, and cell A3 itself tells us the result of what we typed.
Duplicating Formulas and Functions
As mentioned above, Excel is always ready to help make life easier, and it’s worth remembering this when using formulas and functions.
In the example below, we want to add all the values in cells A1 to A8, so we will type
into cell A9, and press Enter.
We also want to add the values in cells B1 to B8. However, instead of typing a new formula into B9 using the SUM function, we can either
• Select cell A9, press Ctrl+C, and then press Ctrl+V in cell B9, or
• Drag the fill handle in the bottom-right corner of cell A9 across to cell B9.
Because cell references within formulas are relative by default, what we typed in cell A9 to calculate the values in cells A1 to A8 will also apply in B9 to calculate the values in cells B1 to B8.
10 Basic Functions to Get You Going
If you’re new to Excel or its functions, open a new spreadsheet, and enter some random numerical values into cells A1 to A9 (leaving A10 blank). Then, give some of these functions a try:
│In cell│Type this and press Enter│ What this will do │
│B1 │=SUM(A1:A10) │The SUM function will add all the values in cells A1 to A10. │
│B2 │=AVERAGE(A1:A10) │The AVERAGE function will find the mean of all the values in cells A1 to A10. │
│B3 │=CONCAT(A1:A3) │The CONCAT function will string together all the values in cells A1 to A3. │
│B4 │=COUNT(A1:A10) │The COUNT function will tell you the number of cells containing numbers in A1 to A10. │
│B5 │=COUNTA(A1:A10) │The COUNTA function will tell you the number of cells containing any value (in other words, the cells that are not empty).│
│B6 │=COUNTBLANK(A1:A10) │The COUNTBLANK function will tell you the number of blank cells in A1 to A10. │
│B7 │=MIN(A1:A10) │This will tell you the smallest number in cells A1 to A10. │
│B8 │=MAX(A1:A10) │This will tell you the largest number in cells A1 to A10. │
│B9 │=TODAY() │The volatile TODAY function returns today’s date. │
│B10 │=RAND() │The volatile RAND function returns a random number between 0 and 1. │
A volatile function recalculates anytime you make any changes to or reopen your Excel spreadsheet.
5 More Advanced Functions
On a new worksheet or in a new workbook, type Laura, Lucy, Liam, Lilly, Liz, and Luke into cells A1 to A6, and numbers 1 to 6 in cells B1 to B6.
Now, try some of these functions:
│ In │Type this and press│ What this will do │
│cell │ Enter │ │
│C1 │=IF(B6> │The IF function will evaluate whether the value in cell B6 is greater than 1, returning “YES” if it is, and “NO” if it isn’t. In this case, it’ll return “YES.” │
│ │1,”YES”,”NO”) │ │
│C2 │=VLOOKUP │The VLOOKUP function will look for “Liam” in cells A1 to B6, returning the number in the second column where it finds that word. In this case, it’ll return “3.” │
│ │(“Liam”,A1:B6,2) │ │
│C3 │=SUMIF │The SUMIF function will look for values starting “Li” in cells A1 to B6, returning the sum of the values in cells B1 to B6 where this is true. In this case, it’ll return │
│ │(A1:B6,”Li*”,B1:B6)│“12,” as the numbers next to Liam, Lilly, and Liz add up to 12. │
│C4 │=COUNTIF(B1:B6,3) │The COUNTIF function will tell you how many cells in B1 to B6 contain the number 3. In this case, it’ll return “1,” because only cell B3 contains this value. │
│C5 │=LEFT(A1,3) │The LEFT function will tell you the three left-most characters in cell A1, which, in this case, are “Lau.” │
Learning about Excel’s formulas and functions doesn’t stop there—in fact, it never stops at all! At How-To Geek, we have dozens of Excel-related articles that you can work through to become a power | {"url":"https://www.techregister.co.uk/the-beginners-guide-to-excels-formulas-and-functions/","timestamp":"2024-11-07T02:27:17Z","content_type":"text/html","content_length":"172060","record_id":"<urn:uuid:072ce30b-0e18-444e-917f-7068fa14a2e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00081.warc.gz"} |
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Classifying quadrilaterals is one of those skills that adults love to make fun of with those, ‘I didn’t learn anything useful, but I learned the difference between a rhombus and a trapezoid.’ jokes
that teachers hate. The thing is, when students learn to...
I love when math strands overlap and students can see the connections between them. Fractions and Geometry overlap quite a bit. There are three ways to think about fractions: 1. The Set Model: This
model works with fractions of groups, like ¼ of 24, so ¼...
Visit my Teachers Pay Teachers store: | {"url":"https://teachthinkelementary.blog/blog-roll/geometry/","timestamp":"2024-11-02T21:28:31Z","content_type":"text/html","content_length":"276567","record_id":"<urn:uuid:c951cdd5-fd70-4144-b07b-afc66e4d7e95>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00882.warc.gz"} |
Two methods to analyse radial diffusion ensembles: the peril of space- and time- dependent diffusion
Particle dynamics in Earth's outer radiation belt can be modelled using a diffusion framework, where large-scale electron movements are captured by a diffusion equation across a single adiabatic
invariant, $L^{*}$ $``(L)"$. While ensemble models are promoted to represent physical uncertainty, as yet there is no validated method to analyse radiation belt ensembles. Comparisons are complicated
by the domain dependent diffusion, since diffusion coefficient $D_{LL}$ is dependent on $L$. We derive two tools to analyse ensemble members: time to monotonicity $t_m$ and mass/energy moment
quantities $\mathcal{N}, \mathcal{E}$. We find that the Jacobian ($1/L^2$) is necessary for radiation belt error metrics. Components of $\partial\mathcal{E}/\partial t$ are explicitly calculated to
compare the effects of outer and inner boundary conditions, and loss, on the ongoing diffusion. Using $t_m$, $\mathcal{N}$ and $\mathcal{E}$, we find that: (a) different physically motivated choices
of outer boundary condition and location result in different final states and different rates of evolution; (b) the gradients of the particle distribution affect evolution more significantly than $D_
{LL}$; (c) the enhancement location, and the amount of initial background particles, are both significant factors determining system evolution; (d) loss from pitch-angle scattering is generally
dominant; it mitigates but does not remove the influence of both initial conditions and outer boundary settings, which are due to the $L$-dependence of $D_{LL}$. We anticipate this study will promote
renewed focus on the distribution gradients, on the location and nature of the outer boundary in radiation belt modelling, and provide a foundation for systematic ensemble modelling. | {"url":"https://researchportal.northumbria.ac.uk/en/publications/two-methods-to-analyse-radial-diffusion-ensembles-the-peril-of-sp","timestamp":"2024-11-03T12:57:53Z","content_type":"text/html","content_length":"49111","record_id":"<urn:uuid:b7137000-485f-436a-af5f-b948f4a54d95>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00698.warc.gz"} |
1 + 1 = 3?
Well this was an interesting problem that someone had brought up to me and I felt is merits a post because it is handy if you are ever bound by these strict restrictions ;).
Problem: You have 2 integer variables A and B. You are limited to having only two variables and no more. How would you swap the values of A and B without using a third variable?
Answer: This is the easy part. Someone mentioned this problem to me in passing and I guess it just stuck in my brain. Today while I was working I wasn't even thinking about it and the answer just hit
Simple form:
A = A + B
B = A - B
A = A - B
Broken down:
A = A + B
B = (A + B) - B = A
A = (A + B) - B = B
And there we have it ... somewhat useless information that is cool at the same time! =) | {"url":"http://blog.ownzu.org/2008/03/1-1-3.html","timestamp":"2024-11-05T14:02:07Z","content_type":"text/html","content_length":"41189","record_id":"<urn:uuid:7fdc9059-1361-4426-aec4-077496c7f4e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00732.warc.gz"} |